Further progress in analysis: Proceedings of the 6th int. ISAAC Congress, Turkey,2007 9812837329, 9789812837325

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urther Progress in Analysis

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urther Progress in Analysis Proceedings of the 6th International ISAAC Congress Middle East Technical University,

13 - 18 August 2007

Ankara, Turkey

Editors

H. G. W. Begehr Freie Universitat Berlin, Germany

A. O. gelebi Yeditepe University, Turkey

R. P. Gilbert University of Delaware, USA

Editorial Assistant

H. T. Kaptanoglu Bilkent University, Turkey

World Scientific N E W J E R S E Y • L O N D O N • S I N G A P O R E • BEIJING • S H A N G H A I • H O N G K O N G • T A I P E I • C H E N N A I

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

FURTHER PROGRESS IN ANALYSIS Proceedings of the 6th International ISAAC Congress Copyright © 2009 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-283-732-5 ISBN-10 981-283-732-9

Printed in Singapore.

v

PREFACE The 6th International ISAAC Congress took place on the beautiful campus of Middle East Technical University (METU), Ankara, Turkey, during the period 13–18 August 2007. The Congress was carried out by the Ankara Branch of the Turkish Mathematical Society with the generous support of METU and particularly of the Department of Mathematics. There were 303 registered participants from 49 countries of whom more than 250 were from abroad. These numbers made the Congress the largest mathematical scientific meeting ever held in Turkey. The governmental support obtained enabled the organizers to introduce 25 well-known mathematicians from Central Asia to the western world. The ISAAC Board acted as the Scientific Committee of the Congress. ¨ The Local Organizing Committee consisted of Oner C ¸ akar (Ankara University), A. Okay C ¸ elebi (then METU, now Yeditepe University), H. Turgay Kaptano˘ glu (Bilkent University) and A˘ gacık Zafer (METU). They had a long and exhausting preparation period. The illness and hospitalization of Prof. C ¸ elebi for an extended time scared but did not slow them. Luckily, he was back to his normal health and activity well before the Congress. The program of the Congress was a combination of plenary talks and special sessions. The 7 plenary speakers, who are international experts in their fields of interest, were selected by the ISAAC Board. The plenary speakers and the titles of their presentations were the following: G. A. Barsegian (Armenia) : Some Interrelated Results in Different Branches of Geometry and Analysis A. Bourgeat (France) : Spatial and Random Scaling Up of the Source Terms in a Diffusion Convection Model P. L. Butzer (Germany) : Prediction in Terms of Samples from the Past: Error Estimates by a Modulus Covering Discontinuous Signals W. D. Evans (UK) : Recent Results on Hardy and Rellich Inequalities B. Gramsch (Germany) : Spectral Invariance for Pseudodifferential Operators and Fredholm Theory

vi

J. Sj¨ ostrand (France) : Some Results on Nonselfadjoint Operators: A Survey C. Y. Yıldırım (Turkey) : The Distribution of Primes: Conjectures vs. Hitherto Provables In addition, there were 281 contributed talks in 15 special sessions. The final form of the Special Sessions and their organizers were as follows: Analytic Function Spaces and Their Operators R. Aulaskari (Finland), H. T. Kaptano˘ glu (Turkey) Clifford and Quaternion Analysis I. M. Sabadini (Italy), M. Shapiro (Mexico), F. Sommen (Belgium) Complex Analysis and Potential Theory T. A. Aliyev (Turkey), M. Lanza de Cristoforis (Italy), P. M. Tamrazov (Ukraine) Complex Analytic Methods for Applied Sciences S. V. Rogosin (Belarus), V. V. Mityushev (Poland) Complex and Functional Analytic Methods in Partial Differential Equations H. G. W. Begehr (Germany), D.-Q. Dai (China), J. Y. Du (China) Dispersive Equations V. Georgiev (Italy), M. Reissig (Germany) Inverse and Ill-Posed Problems A. Hasanov (Turkey), M. Yamamoto (Japan), S. I. Kabanikhin (Russia) Modern Aspects of the Theory of Integral Transforms and Their Applications A. Kilbas (Belarus), S. Saitoh (Japan), J. J. Trujillo (Spain) Oscillation of Functional-Differential and Difference Equations L. Berezansky (Israel), A. Zafer (Turkey) Pseudodifferential Operators L. Rodino (Italy), M. W. Wong (Canada) Reproducing Kernels and Related Topics D. Alpay (Israel), A. Berlinet (France), S. Saitoh (Japan), D.-X. Zhou (China) Spaces of Differentiable Functions of Several Real Variables and Applications V. I. Burenkov (UK), S. G. Samko (Portugal) Numerical Functional Analysis P. E. Sobolevski (Brazil), A. Ashyralyev (Turkey)

vii

Integrable Systems M. G¨ urses (Turkey), Ismagil T. Habibullin (Turkey) General Session (topics not suitable for any of the sessions listed above) A. O. C ¸ elebi (Turkey) The contributions to the Proceedings of the Congress are organized so as to follow the special sessions. Thanks are due to the organizers for accepting to be the editors of their sessions and for their extraordinary efforts to keep the scientific quality of the Proceedings at a high level. The texts of the plenary talks were edited by the editorial board of this volume. The organizers of two sessions, of Inverse and Ill-Posed Problems and of Pseudodifferential Operators, decided to publish their contributed papers elsewhere. Thus this volume contains 75 articles from 13 sessions along with the texts of 4 plenary talks, all refereed. A few of the manuscripts were not in the style of the publisher or even TEX. Dr. U˘ gur Y¨ uksel converted them into expected forms. But the main job of preparing the Proceedings for publication was done by Prof. H. Turgay Kaptano˘ glu who acted as editorial assistant. He did major work in compiling the individual papers into book form and correcting them in every respect, from mathematics to language to TEX. His efficient and meticulous work is evident in this volume. He was also the editor of the book of abstracts and the schedule of the Congress. Two important events occurred during the Congress. The first was the presentation of the ISAAC award for researchers under the age of 40 to Dr. Michael Ruzhansky during the opening ceremony. The second was the meeting of the ISAAC Board. The Board reelected Prof. M. W. Wong as the president of ISAAC, and decided that the next Congress would take place in the United Kingdom at the host institution of the ISAAC awardee. So the 7th International ISAAC Congress will be held at Imperial College London, July 13–18, 2009. The high scientific quality of the Congress can be seen from the contributions in this Proceedings volume. The presentations covered a wide range topics varying from analytic number theory to computational aspects of applied analysis. There were plenary talks directed to a more general audience as well as to specialists. The sessions of the Congress were well-attended despite the lack of air conditioning and the heat wave. The closing ceremony extraordinarily attracted a majority of the participants. The support of the institutions listed below made the Congress possible. More than 25% of the participants were supported one way or another.

viii

Republic of Turkey Prime Ministry Promotion Fund ˙ Turkish International Cooperation and Development Agency (TIKA) ¨ ˙ Scientific and Technological Research Council of Turkey (TUBITAK) European Mathematical Society Middle East Technical University Ankara University Bilkent University Gama Trade and Tourism Y¨ uksel Construction Kavaklıdere Anatolian Wines Ertem Printing House Birkh¨ auser Also, volunteer work of colleagues and friends are gratefully acknowledged. The last, but not the least, the students in orange T-shirts should be thanked for their ever presence to help and joviality. Congress services were successfully and efficiently handled by Asterya Congress and Event Management. They and the local organizers prepared an excellent social program and excursions with wining and dining, and music and dancing, sometimes in surprise times and locations. These all made the whole event a memorable one. The Editors 15 October 2008

ix

SCIENTIFIC COMMITTEE of the Congress (The ISAAC Board of 2007 ) H. G. W. Begehr A. Berlinet E. Br¨ uning V. I. Burenkov A. O. C ¸ elebi R. P. Gilbert A. Kilbas M. Lanza de Cristoforis W. Lin M. Reissig L. Rodino J. Ryan S. Saitoh B.-W. Schulze J. Toft M. W. Wong M. Yamamoto S. Zhang

Freie Universit¨ at, Berlin, Germany Universit´e des Sciences et Techniques de Languedoc, Montpellier, France University of KwaZulu-Natal, Durban, South Africa Cardiff University, Cardiff, Wales, UK ˙ Yeditepe University, Istanbul, Turkey University of Delaware, Newark, DE, USA Belarusian State University, Minsk, Belarus Universit` a di Padova, Padova, Italy Zhongshan University, Guangzhou, China Technische Universit¨ at, Freiberg, Germany Universit` a di Torino, Torino, Italy University of Arkansas, Fayetteville, AR, USA Gunma University, Kiryu, Japan Universit¨ at Potsdam, Potsdam, Germany V¨ axj¨ o University, V¨ axj¨ o, Sweden York University, Toronto, Canada University of Tokyo, Tokyo, Japan University of Delaware, Newark, DE, USA

LOCAL ORGANIZING COMMITTEE of the Congress ¨ O. A. H. A.

C ¸ akar O. C ¸ elebi T. Kaptano˘ glu Zafer

Ankara University, Ankara, Turkey ˙ Yeditepe University, Istanbul, Turkey Bilkent University, Ankara, Turkey Middle East Technical University, Ankara, Turkey

x

xi

CONTENTS Preface

v

Committees

ix

Plenary Talks

1

Some Interrelated Results in Different Branches of Geometry and Analysis G. A. Barsegian

3

Recent Results on Hardy and Rellich Inequalities W. D. Evans

33

Some Results on Nonselfadjoint Operators: A Survey J. Sj¨ ostrand

45

The Distribution of Primes: Conjectures vs. Hitherto Provables C. Y. Yıldırım

75

Session 01 Analytic Function Spaces and Their Operators A C ∗ -Algebra of Functional Operators with Shifts Having a Nonempty Set of Periodic Points M. A. Bastos, C. A. Fernandes & Y. I. Karlovich On Bloch and Normal Functions on Complex Banach Manifolds P. V. Dovbush

109 111

122

xii

Strong-Type Estimates and Carleson Measures for Weighted Besov-Lipschitz Spaces V. S. Guliyev & Z. Wu

132

Composition Theorems in Besov Spaces, the Vector-Valued Case M. Moussai

142

Pointwise Approximation for Certain Mixed Sz´ asz-Beta Operators Q. Qi & Y. Zhang

152

Session 02

Clifford and Quaternion Analysis

Criteria for Monogenicity of Clifford-Algebra-Valued Functions R. Abreu Blaya, J. Bory Reyes, D. Pe˜ na Pe˜ na & F. Sommen

165 167

Hypermonogenic Functions and Bihypermonogenic Functions in Real Clifford Analysis Y. Qiao, H. Yang & X. Bian

175

Initial Value Problems for Regular Quaternion-Valued Initial Functions L. H. Son, N. C. Luong & N. Q. Hung

185

Session 03 Complex Analysis and Potential Theory The Geometry of Blaschke Product Mappings I. Barza & D. Ghisa The Layer Potentials of Some Partial Differential Operators: Real Analytic Dependence upon Perturbations M. Dalla Riva

195 197

208

Differential Properties of (α, Q)-Homeomorphisms A. Golberg

218

On Characterization of the Extension Property A. P. Goncharov

229

xiii

On Some Boundary Properties of Conformal Mapping O. W. Karupu The Jump Problem on Nonrectifiable Curves and Metric Dimension B. A. Kats A Functional Analytic Aproach to the Analysis of the Asymptotic Behavior of the Energy Integral of a Nonlinear Transmission Problem M. Lanza de Cristoforis

233

241

249

Analogue of Jackson-Bernstein Theorem in Lp on Closed Curves in the Complex Plane II J. I. Mamedkhanov & I. B. Dadashova

260

An Infinite-Dimensional Commutative Banach Algebra and Spatial Potential Fields S. A. Plaksa

268

Session 04 Complex Analytic Methods for Applied Sciences

279

On the Curvature of an Inner Curve in a Schwarz-Christoffel Mapping A. Andersson

281

An R-Linear Problem with Derivatives for Doubly Periodic Functions and its Application P. Dryga´s

291

Session 05 Complex and Functional Analytic Methods in Partial Differential Equations Some Problems for Elliptic Systems on the Plane G. Akhalaia, G. Makatsaria & N. Manjavidze

301 303

xiv

Neumann Problem for Generalized Poisson and Bi-Poisson Equations ¨ Aksoy & A. O. C U. ¸ elebi

311

A New Solution of Some Weighted Problems for Riemann-Liouville and Weyl Operators M. Avcı, S. O˘gra¸s & R. A. Mashiev

321

A Polyharmonic Dirichlet Problem of Arbitrary Order for Complex Plane Domains H. Begehr & T. S. Vaitekhovich

327

The Fourier Transform Method in Controllability Problems for the Finite String Equation with a Boundary Control Bounded by a Hard Constant L. V. Fardigola Density Problem of Monodromy Representation of Fuchsian Systems G. K. Giorgadze About One Class of Second-Order Linear Hyperbolic Equations for Which All of the Boundary Consist of Singular Lines N. Rajabov Asymptotic Study of an Anisotropic Periodic Rotating MHD System R. Selmi Hilbert-Type Boundary Value Problem for Polyanalytic Functions Y. Wang & J. Wang

337

347

356

368

379

Existence and Uniqueness of the Solution of a Free Boundary Problem for a Parabolic Complex Equation Y. Xu

387

First-Order Differential Operators Associated to the Cauchy-Riemann Operator of Clifford Analysis U. Y¨ uksel

395

xv

Singular Integral Equation on the Real Axis with Solution Having Singularity of Higher Order at Infinite Point S. Zhong

Session 06

Dispersive Equations

Generalized Energy Conservation C. B¨ ohme & M. Reissig The Cauchy Problem for a Hyperbolic Operator with Log-Zygmund Coefficients D. Del Santo An Optimal Transportation Metric for Two Nonlinear PDEs M. Fonte

404

413 415

425

434

Generalized Energy Conservation for Wave Equations with Time-Depending Coefficients under Stabilization Properties F. Hirosawa

444

A Unified Treatment of Models of Thermoelasticity, Decay and Diffusion Phenomena K. Jachmann & M. Reissig

454

Some Results on Spectral Analysis of Nonselfadjoint Perturbations for Schr¨ odinger and Wave Equations M. Kadowaki, H. Nakazawa & K. Watanabe

465

On Scattering for Evolution Equations with Time-Dependent Small Perturbations K. Mochizuki

476

Comparison of Estimates for Dispersive Equations M. Ruzhansky & M. Sugimoto

486

Dispersive Estimates in Anisotropic Thermoelasticity J. Wirth

495

xvi

Session 11 Oscillation of Functional Differential and Difference Equations

505

Oscillation Criteria for a Class of Third-Order Nonlinear Delay Differential Equations M. F. Akta¸s & A. Tiryaki

507

A Criterion for Existence of Positive Solutions of the Equation y(t) ˙ = −p(t)y α (t − r) J. Dibl´ık, M. R˚ uˇziˇckov´ a & Z. Svoboda

515

Discreteness of the Spectrum of a Nonselfadjoint Second-Order Difference Operator E. Erg¨ un & G. S. Guseinov

525

Eventually Positive Solutions of Second-Order Superlinear Dynamic Equations R. Mert & A. Zafer

535

Oscillation Criteria for Second-Order Nonlinear Impulsive Differential Equations ¨ A. Ozbekler

545

Positive Solutions of a Nonlinear Three-Point Boundary Value Problem on Time Scales S. G. Topal & A. Yantır

555

Session 13 Reproducing Kernels and Related Topics Reproducing Kernels for Harmonic Functions on Some Balls K. Fujita Numerical Real Inversion of the Laplace Transform by Using a High-Accuracy Numerical Method H. Fujiwara, T. Matsuura, S. Saitoh & Y. Sawano Practical Inversion Formulas for Linear Physical Systems M. Yamada & S. Saitoh

565 567

574

584

xvii

Session 14 Spaces of Differentiable Functions of Several Real Variables and Applications Weighted H¨ older Estimates of Singular Integrals Generated by a Generalized Shift Operator S. K. Abdullayev & A. A. Akperov

591 593

On Interpolation Properties of Generalized Besov Spaces A. Almeida

601

Partial Hypoellipticity of Differential Operators T. G. Ayele & W. T. Bitew

611

Some Embeddings and Equivalent Norms of the Lλ,s p,q Spaces D. Drihem

622

Session 15

Numerical Functional Analysis

Global Exponential Periodicity for Discrete-Time Hopfield Neural Networks with Finite Distributed Delays and Impulses H. Ak¸ca, V. Covachev, Z. Covacheva & S. Mohamad A Note on the Modified Pad´e Difference Schemes A. Ashyralyev

633 635

645

Numerical Solution of a One-Dimensional Parabolic Inverse Problem A. Ashyralyev, A. S. Erdo˘gan & E. Demirci

654

Numerical Solution of Nonlocal Boundary Value Problems for Elliptic-Parabolic Equations A. Ashyralyev & O. Ger¸cek

663

On One Difference Scheme of Second Order of Accuracy for Hyperbolic Equations A. Ashyralyev & M. E. K¨ oksal

670

On Well-Posedness of Abstract Hyperbolic Problems in Function Spaces A. Ashyralyev, M. Martinez, J. Pastor & S. Piskarev

679

xviii

A Note on Difference Schemes of Second Order of Accuracy for Hyperbolic-Parabolic Equations ¨ A. Ashyralyev & Y. Ozdemir

689

Numerical Solutions of Bitsadze-Samarskii Problem for Elliptic Equations ¨ urk A. Ashyralyev & E. Ozt¨

698

A Note on Parabolic Difference Equations A. Ashyralyev & Y. S¨ ozen

707

Semiexplicit Multisymplectic Integration of Nonlinear Schr¨ odinger Equation A. Aydın & B. Karas¨ ozen

717

Nonlocal Problems with Special Gluing for a Parabolic-Hyperbolic Equation A. S. Berdyshev & N. A. Rakhmatullaeva

727

Exact embedding theorems for traces and extensions M. D. Ramazanov

Session 16

Integrable Systems

The Characteristic Chern-Type Classes and Integrability of Muiltidimensional Differential Systems on Riemannian Manifolds A. K. Prykarpaysky & N. N. Bogolubov Introductory Background to Modern Quantum Mathematics with Application to Nonlinear Dynamical Systems A. K. Prykarpatsky, J. Golenia, N. N. Bogolubov & U. Taneri

Session 17

General Session

Cohen p-Nuclear Multilinear Mappings D. Achour

734

741

743

760

781 783

xix

Numerical Modeling of a System of Mutual Reaction-Diffusion Type M. Aripov & A. Khydarov l2 On Complemented Subspaces of E0l2 (a) × E∞ (b) E. Karapınar

Asymptotic Formulas for Eigenvalues and Eigenfunctions of a Nonselfadjoint Sturm-Liouville Operator K. R. Mamedov & H. Menken

790

794

798

On the Multisublinear Operators L. Mezrag & K. Saadi

806

Probabilistic Modular Spaces K. Nourouzi

814

An Application of Extremal Points in Dual Spaces S. Rezapour

819

On a Problem for a Parabolic-Hyperbolic Equation with a Nonsmooth Line of Type Changing A. K. Urinov & I. U. Khaydarov

824

Addenda to Session 06 Dispersive Equations

833

Asymptotic Behavior and Regularity for Nonlinear Dissipative Wave Equations in Rn G. Todorova, D. U˘gurlu & B. Yordanov

835

The Blow-up Boundary for a System of Semilinear Wave Equations H. Uesaka

845

Author Index

855

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Plenary Talks

EDITORS H. G. W. Begehr A. O. C ¸ elebi R. P. Gilbert

Freie Universit¨ at, Berlin, Germany Middle East Technical University, Ankara, Turkey University of Delaware, Newark, DE, USA

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3

SOME INTERRELATED RESULTS IN DIFFERENT BRANCHES OF GEOMETRY AND ANALYSIS G. A. BARSEGIAN Institute of Mathematics of NAS, Armenia E-mail: [email protected] This paper presents some new identities and inequalities in integral and differential geometry, in real and complex analysis in ODE. Ideas, methods and problems come from some preceding studies related to Gamma-lines. There are also certain bridges between the presented results and Nevanlinna theory of meromorphic functions. Keywords: Differential Geometry, Integral Geometry, Real Analysis, Complex Analysis, ODE, Nevanlinna Theory, Gamma Lines.

Introduction In the sections 1-4 of this paper we give some new interrelated inequalities for the broken lines, plane curves, real and complex functions of one variable. Derivation of these inequalities is based on a new principle of angles and lengths for curves (Section 1). Then we obtain some modifications that refer to the broken lines (Section 2) and to the real functions of one variable (Section 3). In the Section 4 we apply the principle of angles and lengths to the complex functions and obtain, particularly we obtain an inequality, principle of derivatives, valid for arbitrary analytic function in a given domain. In the Section 5 we apply the principle of angles and lengths to describe the windings of the solutions of some broad classes of ODE. Some of these results resemble the second fundamental theorem of Nevanlinna theory and its deficiency relation. Thus we see that Nevanlinna type results (Nevanlinna theory [16], Ahlfors theory [1], Gamma-lines [7,8]) are valid far beyond complex analysis since we have now their corresponding analogues in differential geometry, real analysis and ODE. Section 6 presents some generalized identities and inequalities in integral geometry. The last results are applied to the real functions of two variables in Section 7. We obtain an inequality, principle of zeros, which deals with

4

G. A. Barsegian

the zeros of these functions and their derivatives. The titles of the sections. 1. Some Nevanlinna type inequalities for the plane curves. 2. Consequences for the broken lines. 3. Consequences for the real functions of one variable. 4. Applications in complex analysis, particularly principle of derivatives of analytic functions. 5. Applications in ODE: the windings of solutions. 6. Some identities and inequalities in integral geometry. 7. Principles of zeros for the real functions of two variables. The author thanks Professors R. Ambartzumian, H. Begehr, A. Fernandez, and D. T. Le for valuable discussions of results. Also, I should like to thank the ICTP at Trieste and the Departments of Mathematics of UNED at Madrid for their hospitality as well as INTAS (for two grants) and ANSEF (for two grants). 1. Some Nevanlinna type inequalities for the plane curves 1.1. The principle of angles and lengths and consequent deficiency relation for curves Let γ be a curve in the plane (x, y). We can consider it as a curve in the complex plane x+iy respectively γ := f (t) := f1 (t)+if2 (t), t ∈ [0, 1], where (j) f (t) is a complex function of the real argument t. Denoting by f1 (t) and (j) f2 (t) the derivatives we say γ := f (t) ∈ F (k), k is an integer ≥ 1, if (j) (j) γ (j) := f (j) (t) := f1 (t) + if2 (t) is continuous in [0, 1] for any j, 1 ≤ j ≤ k + 1, and if for any j, 0 ≤ j ≤ k + 1 and any t ∈ [0, 1] we have f (j) (t) 6= 0. Notice that arg f (t0 ) means the angle between the x-axis and the vector connecting 0 and f (t0 ), while arg f ′ (t) means the angle between tangent to γ at the point f (t0 ) and the x-axis. Thus, if a is a point on the plane (x, y) then Z 1 (arg(f (t) − a))′ dt R(a, γ) := 0

is the total rotation of γ around this point a. Then we consider the curve (k) (k) γ (k) := f1 (t) + if2 (t) and denote by Z 1  ′ (k) (k) (k) T (γ ) := R(0, γ ) := arg f (t) dt 0

the total integral curvature of γ around a = 0).

(k−1)

(or the total rotation of the curve γ (k)

Some Interrelated Results in Different Branches of Geometry and Analysis

5

Denote by l(γ) the length of γ. Theorem 1.1 (principle of angles and lengths). For any γ := f (t) ∈ F (k), any integer k ≥ 1, any point a   (1.1) R (a, γ) ≤ T γ (k) + kπ. For any collection of pairwise different points aν , ν = 1, 2, ..., q q  2kπq  X R (aν , γ) ≤ T γ (k) + l(γ) + kπ, ρ ν=1

(1.2)

where ρ is the minimal distance between the points aν .

The inequalities have simple geometric meaning: the total rotation of γ around a does not exceed total integral curvature of γ plus π. For k > 1, T (γ (k) ) equals total rotation of the curve γ (k) so that both (1.1) and (1.2) admit corresponding interpretations. The reader familiar with Nevanlinna’s value distribution theory and (or) with Ahlfors theory of covering surfaces will see an analogy between (1.2) and the second fundamental theorem in Nevanlinna’s theory; we will discuss this below. Sharpness. We consider the case where k = 1. Let γ := f (t), t ∈ [0, 1] be the segment connecting the points (−1, ε) and (1, ε) in the plane. Then R(0, γ) is as close to π as we please when we take ε sufficiently small, meantime T (γ (1) ) is equal to zero. Thus, (1.1) can not be improved. Assume that our curve γ approaches to a circumference by a spiral. Then the part of γ having N “coils” contributes to both the first and the second integrals asymptotically as N when N tends to infinity. Thus the ratio of the left and the right magnitudes in (1.1) tends to 1 when N tends to infinity. The inequality (1.2) is sharp as well. Let γ be the graph of the function √ 1 , t ∈ [0, 1], where 0 < ε < 21 and take aν = 2(ν − 1), fε := ε sin t+ε ν = 1, 2, ..., q. When ε tends to zero then both the left and the right sides of (1.2) tend to infinity but their ratio tends to 1. Let γi ∈ F (1) be a sequence of curves,
each satisfying the conditions of Theorem 1.1, γi ⊂ γi+1 , for which T γ (k) → ∞ when i → ∞ and l(γi )
→ 0, T γ (k)

i → ∞.

(1.3)

This is a bee sequence (frequent excursions on small distances in different directions) respectively T is comparatively large and l is comparatively

6

G. A. Barsegian

small. The rotations of γi around aν determines the following magnitude ∆(aν ) := lim inf i→∞

R (aν , γi )
T γ (k)

which we refer as deficiency. Inequality (1.2) implies the following Deficiency relation (for the curves). For any bee sequence of curves and an arbitrary collection of pairwise different points aν , ν = 1, 2, ..., q, q X

ν=1

∆(aν ) ≤ 1.

(1.4)

1.2. Bridges between inequalities (1.2), (1.4) and Nevanlinna theory We would like to stress the similarity between (1.2) and the second fundamental theorems in Nevanlinna theory [16], Ahlfors theory [1]. In Nevanlinna version it looks as follows: for any meromorphic in the complex plane function w(z) and any collection of pairwise different points aν , ν = 1, 2, ..., q we have q X

ν=1

m(r, aν , w) ≤ 2T (r, w) + C(q)λ(r)T (r, w),

r = rn → ∞,

(1.5)

where m(r, aν , w) is the approximation function, T (r, w) is Nevanlinna characteristic function, C(q) is a constant depending on q and λ(r) → 0 when r → ∞, see [16]. The fact that the second term on the right side of (1.5) is essentially less than the first term plays a crucial role. Particularly it leads to the well known Nevanlinna deficiency relation: q X

ν=1

δ(aν ) ≤ 2,

(1.6)

where δ(aν ) := lim inf r→∞ {m(r, aν , w)/T (r, w)}. Thus we see that inequality (1.2) is similar to the second fundamental theorem (1.5) and (1.4) is similar to the Nevanlinna deficiency relation (1.6). 1.3. The proof Theorem 1.1. and some comments An inequality similar to (1.2) was proved in 1978 (see Lemma 2 in [3]). In this paper we studied meromorphic functions w in the disks D(r) := {z| |z| < r} and in Lemma 2 we obtained an inequality of type (1.2) but for

Some Interrelated Results in Different Branches of Geometry and Analysis

7

the particular class of curves γ which are the images w(∂D(r)) and with the spherical length instead of ordinary length l(γ) in (1.2). However, the proof was valid in fact for arbitrary smooth curves. Much later, passing to study ordinary differential equations [6] we again used this proof, this time in full generality (for arbitrary smooth curves). In an excellent book by Sheill-Small [15] (2002) the same magnitudes (R and T ) were studied but for closed curves only and for only one value a. He proved (pp. 387-389) that in this case R (a, γ) ≤ T (γ ′ ) .

(1.7)

Unfortunately in [15] we did not find any hint as regards the origin of this inequality. Seemingly it should be Sheill-Small. Below we prove this inequality making use in fact just a part of the proof of Lemma 2 in [3]. Observe that arg f (t0 ) means the angle between x-axis and vector connecting 0 and f (t0 ) respectively arg f ′ (t) means the angle between tangent to γ at the point f (t0 ) and x-axis. on the plane (x, y) R 1 Thus, if a is a ′point then the magnitude R(a, γ) := 0 (arg(f (t) − a)) dt means total rotation of γ around this point a. First we assume that on the curve γ do not involve some sub curves ′ consisting only of points, where (arg(f (t) − a)) = 0. This assumption does not restrict generality since we can consider the result for a new a∗ (very close to the point a) such that the magnitudes occurring in (1.7) for a∗ and a are as close as we please. Then proving the inequality for a∗ we obtain the inequality for a. Now we assume that our closed curve do not have points, where (arg(f (t) − a))′ change its sign. This means that the curve rotate around clockwise (or anti Rclockwise). Since the curve is closed we have Ra 1only (arg(f (t) − a))′ dt = 1 (arg(f (t) − a))′ dt = 2πk, where k is an in0 0 teger. But then the tangential angle arg f ′ (t) also rotates at least k time so that inequality (1.7) is true in this simplest case. Now we consider the case when on the curve we have some points, ′ where(arg(f (t) − a)) change its sign. Denote corresponding points by t1 < t2 < ... < tI . Since the curve is closed we have f (0) = f (1) so that we can assume that t1 = 0 take as tI+1 the point 1. Consider three type of intervals: (type 1) those intervals (ti , t+1 ) at whose both endpoints function |f (t)| increases (or decreases); (type 2) those intervals (ti , t+1 ) for which |f (t)| increases at ti and

8

G. A. Barsegian

decreases at ti+1 ; (type 3) those intervals (ti , t+1 ) for which |f (t)| decreases at ti and increases ti+1 . ′ Since (arg(f (t) − a)) > 0 and |arg(f (ti+1 ) − a) − arg(f (ti ) − a)| = |arg f ′ (ti+1 ) − arg f ′ (ti )| we have for the intervals (ti , ti+1 ) of type 1 R t+1 (arg(f (t) − a))′ dt ti = |arg(f (ti+1 ) − a) − arg(f (ti ) − a)| = |arg f ′ (ti+1 ) − arg f ′ (ti )| Rt ′ ≤ ti+1 (arg f ′ (t)) dt.

For a given interval (ti , t+1 ) of type 2 we observe that there is a point t∗ ∈ (ti , t+1 ) where |f (t)| is maximal; if we have more than one similar points we take arbitrary one. Denote by α, α1 and α2 the angles R ′ (arg(f (t) − a)) dt taken for interval (ti , t+1 ), (ti , t∗ ) and (t∗ , t+1 ) respectively and by η1 , η ∗ and η2 the straight lines passing trough zero and the points having f (ti ), f (t∗ ) and f (ti+1 ) respectively. Since the tangent of our curve at the point ti coincides with η1 and direction of arg f ′ (t∗ ) is perpendicular to the direction of η ∗ and since |f (t)| increases at ti we have R t∗ ′ α1 + π/2 ≤ |arg f ′ (t∗ ) − arg f ′ (ti )| ≤ ti (arg f ′ (t)) dt. Quite similarly we Rt ′ have α2 + π/2 ≤ |arg f ′ (ti+1 ) − arg f ′ (t∗ )| ≤ ∗i+1 (arg f ′ (t)) dt so that t

we obtain for any interval of type 2 Z Z t+1 (arg(f (t) − a))′ dt ≤ ti

t+1

ti

′ (arg f ′ (t)) dt − π.

Consider now intervals of type 3 and denote by t∗ ∈ (ti , t+1 ) the point where |f (t)| is minimal. Making use the above notations for angles and the straight lines and taking into account that the tangent of our curve at the point ti coincides with η1 , that the direction of arg f ′ (t∗ ) is perpendicular to the direction of η ∗ and |f (t)| decreases at the point we observe that for R t∗ ′ α1 > π/2 we have α1 − π/2 ≤ |arg f ′ (t∗ ) − arg f ′ (ti )| ≤ ti (arg f ′ (t)) dt. The same inequality is true for any α1 since for α1 ≤ π/2 this inequality is obvious. Quite similarly we have α2 − π/2 ≤ |arg f ′ (ti+1 ) − arg f ′ (t∗ )| ≤ R ti+1 (arg f ′ (t))′ dt so that we obtain for any interval of type 3 t∗ Z t+1 Z t+1 ′ (arg(f (t) − a))′ dt ≤ (arg f ′ (t)) dt + π. ti

ti

Now we observe that when our curve is closed then the intervals of type 2 and 3 occur in pair since if the module |f (t)| increases (decreases) at a given point ti then at another point ti+k this module |f (t)| should decrease (increase) due to this closeness. From here and the above three inequalities

Some Interrelated Results in Different Branches of Geometry and Analysis

9

for the intervals of type 1-3 we obtain inequality (1.1) for any closed curve γ which do with pass trough zero. Now we consider a given non closed curves γ1 . Denote the coordinate of the plane by (x, y) and make use complex coordinates z = x + iy, so that |z| = r. Assume that r∗ > |f1 (0)| , |f1 (1)|. Let T (0) (T (1)) be the tangent of γ1 at the terminal points f1 (0) (f1 (1)). Let us continue the curve γ1 by adding to γ1 the segment γ2 lying on T (1) and connecting f1 (1) with a point t(1) lying on intersection of T (1) with the circumference {z| |z| = r∗ }. Similarly we define the segment γ4 lying on T (0) and connecting f1 (0) with a point t(0) lying on intersection of T (0) with the circumference {z| |z| = r ∗ }. The points t(0) and t(1) divide the circumference {z| |z| = r∗ } onto two parts. Denote by γ3 that part of {z| |z| = r∗ } which corresponds to the smaller central angle. Notice that the curve γ = γ1 ∪ γ2 ∪ γ3 ∪ γ4 is a closed curve. Also notice that if r∗ tends to infinity then the angle formed by γ2 and γ3 at the point t(1) tends to π/2 as well as the angle formed γ3 and γ4 at the point t(0). In other words, for a given “small” ε > 0 we can chose such a big r∗ that the differences between these angles and π/2 are less than ε/4. After choosing similar r∗ we cut from γ3 two ends of γ3 with the central angles less than ε/4; the rest part we denote by γ˜3 . Now we construct a new curve γ2,3 which connects the point f1 (1) with the initial point of γ˜3 in such a way that the union γ1 ∪ γ2,3 ∪ γ˜3 is a smooth curve. Similarly we will construct a new curve γ3,4 connecting the terminal point of γ˜3 with the point f1 (0) and again such that γ˜3 ∪ γ3,4 ∪ γ4 is a smooth curve. We clearly can construct γ2,3 and γ3,4 such that: (a) the curve γ˜ = γ1 ∪ γ2 ∪ γ2,3 ∪ γ˜3 ∪ γ3,4 ∪ γ4

(1.8)

is a closed, smooth curve; (b) π ε T (γ2,3 ) − ≤ 2 2

and

π ε T (γ3,4 ) − ≤ . 2 2

Taking into account (a), (b), the inequality R(0, γ˜3 ) = T (˜ γ3′ ) (since γ˜3 is ′ the part of the circumference centered at zero) and |T (γ2 )| = |T (γ4′ )| = 0

10

G. A. Barsegian

(since γ2 and γ4 are straightforward) we obtain from (1.7) and (1.8) R(0, γ1 ) ≤ R(0, γ˜) ≤ T (˜ γ ′)

′ ′ := T (γ1′ ) + T (γ2′ ) + T γ2,3 + T (γ4′ ) + T (γ3′ ) + T γ3,4 ′ ≤ T (γ1 ) + π + ε. and since ε is arbitrary we obtain (1.1) when k = 1. Taking into account that T (γ1′ ) := R (0, γ1′ ) we can repeat the obtained (k) inequality for γ1′ , γ1′′ , ..., γ1 and come to (1.1) for arbitrary integer k. 1.4. The proof of Theorem 1.2 Denote D(r, h a) := {z : i|z − a| < r}, γ(D(ρ, a)) := γ ∩ D(ρ, a), |a −a | ρ = min mini6=,j i 2 j ; 1 . For any aν we can consider our curve γ

as a collection of the following type of curves. Type 1: curves γ1 (i, aν ), i = 1, 2, ..., I(ν), which lie in D(ρ, aν ) and have the length ≥ ρ/2. Type 2: curves γ2 (j, aν ), j = 1, 2, ..., J(ν), which lie in D(ρ, a)\D( ρ2 , a) and have the length < ρ/2. Type 3: curves γ3 (s, aν ), s = 1, 2, ..., S(ν), which lie out of ∪ν D(ρ, aν ). Type 4: curve γ4 , which have common points with D( ρ2 , a) and have the length < ρ/2; obviously this is the case when our curve γ coincides with γ4 . We will proceed now assuming that γ is not of view γ4 . Obviously we have Pq ν=1 R (aν , γ) Pq PI(ν) Pq PJ(ν) (1.9) = ν=1 i=1 R (aν , γ1 (i, aν )) + ν=1 j=1 R (aν , γ2 (j, aν )) Pq PS(ν) + ν=1 s=1 R (aν , γ3 (s, aν )) Denoting by l(X) Euclidean length of a curve X and taking into account the definition of curves of type 1 we have q I(ν) X X

ν=1 i=1

kπ ≤

q I(ν) 2kπ X X l (γ1 (i, aν )) ρ ν=1 i=1

so that (1.1) yields Pq PI(ν) ν=1 i=1 R (aν , γ1 (i, aν )) d Pq PI(ν) Pq PI(ν) R ≤ ν=1 i=1 z∈γ1 (i,aν ) arg f (k) dt + 2kπ ν=1 i=1 l (γ1 (i, aν )) ρ dt
Pq PI(ν) ≤ R 0, γ (k) + 2kπ ν=1 i=1 l (γ1 (i, aν )) . ρ (1.10)

Some Interrelated Results in Different Branches of Geometry and Analysis

11

Observe that since γ2 (j, aν ) lie in D(ρ, a)\D( ρ2 , a) we have that the increment of arg(z − aν ) on γ2 (j, aν ) is comparable with the length l (γ2 (j, aν )) so that follows R (aν , γ2 (j, aν )) ≤

2 l (γ2 (j, aν )) ρ

consequently q J(ν) X X ν=1 j=1

q

R (aν , γ2 (j, aν )) ≤

J(ν)

2XX l (γ2 (j, aν )) . ρ ν=1 j=1

(1.11)

Further we have d arg (z − a) dt = d arctan y − Ima dt dt dt x − Rea  ′ y − Ima  y ′ (x − Rea) − x′ (y − Ima) x − Rea   t  = dt 2 dt = y − Ima (x − Rea)2 + (y − Ima)2   1+ x − Rea

|y ′ | |x − Rea| q ≤ q dt 2 2 2 2 (x − Rea) + (y − Ima) (x − Rea) + (y − Ima) |y − Ima| |x′ | q dt +q 2 2 2 2 (x − Rea) + (y − Ima) (x − Rea) + (y − Ima) q |y ′ |2 + |x′ |2 ≤ q (x − Rea)2 + (y − Ima)2

so that taking into account that for any z ∈ γ3 (s, aν ) is valid q (x − Reaν )2 + (y − Imaν )2 ≥ ρ

we obtain

q S(ν) X X

ν=1 s=1

q

R (aν , γ3 (s, aν )) ≤

q

≤

S(ν) Z

1XX ρ ν=1 s=1

z∈γ3 (s,aν )

q 2 2 |y ′ | + |x′ | dt

S(ν)

1XX l (γ3 (s, aν )) . ρ ν=1 s=1

(1.12)

12

G. A. Barsegian

Summing up (1.9)-(1.12) we have
Pq (k) ν=1 R (aν , γ) ≤ R 0, γ Pq PJ(ν) 2kπ hPq PI(ν) + ν=1 i=1 l (γ1 (i, aν )) + ν=1 j=1 l (γ2 (j, aν )) ρ i Pq PJ(ν) + ν=1 j=1 l (γ2 (j, aν )) and since the magnitude in the brackets ≤ ql(γ) we obtain  2kπq  l(γ). R (aν , γ) ≤ R 0, γ (k) + ρ ν=1 q X

(1.13)

Remember that we proved (1.13) provided that γ is not of view γ4 . Notice that any curve γ is of view γ4 should lie in one of the disks D(ρ, aν ), say in D(ρ, a1 ), and then inequality (1.1) applied for a1 yields   R (a1 , γ) ≤ R 0, γ (k) + kπ meantime repeating for aν 6= a1 the above proofs of (1.11) and (1.12) we obtain 2 R (aν , γ) ≤ l(γ) ρ so that summing up we have  2 {q − 1}  R (aν , γ) ≤ R 0, γ (k) + l(γ) + kπ. ρ ν=1 q X

(1.14)

Inequality (1.2) of Theorem 1.1 follow now from (1.13) and (1.14). 2. Consequences for the broken lines Let Γ be an arbitrary broken line (closed or open) in the plane (x, y) composed of n successive segments γi , i = 1, 2, ..., n. Denote by α(a, γi ) the angle under which γi is seen from the point a in the plane and by L(X) the length of X. We state first a very simple inequality: for an arbitrary broken line Γ (closed or open) and an arbitrary collection of pairwise different points aν , ν = 1, 2, ..., q, in the plane

where A(aν , Γ) =

Pn i

q X

ν=1

A(aν , Γ) ≤ πn +

α(aν , γi ).

2πq L(Γ), ρ

(2.1)

Some Interrelated Results in Different Branches of Geometry and Analysis

13

The inequality is a very rough corollary of our Theorem 2.2 below. Meantime, surprisingly, (2.1) appears to be rather sharp as shows the first example given after Theorem 1.1. We may prescribe a direction to the segment γi (that is consider them as some successive vectors) and denote by β(γi ) the absolute value∗ of the Pn angle between vectors γi and γi+1 . Further we write B(Γ) := i β(γi ), where n∗ = n if Γ is closed and n∗ = n − 1 if Γ is an open broken line: for an open Γ clearly β(γn ) is not defined. Theorem 2.1. For an arbitrary broken line Γ (closed or open) in the plane and an arbitrary point a in the plane A(a, Γ) ≤ B(Γ) + π.

(2.2)

For any collection of pairwise different points aν , ν = 1, 2, ..., q, q X

ν=1

A(aν , Γ) ≤ B(Γ) +

2kπq L(Γ) + π, ρ

(2.3)

where ρ is the minimal distance between aν . Sharpness, inequalities (2.2) and (2.3) can be verified by considering broken lines “very close” to the curves that demonstrated sharpness in the previous section. Let now Γ be a broken line consisting of infinitely many successive segments γi and Γn := ∪nν=1 γi . We say that Γ is a bee broken line if L(Γn ) → 0, B(Γn )

n → ∞.

(2.4)

Defining the deficiency of Γ as ∆(aν ) := lim inf n→∞

A(aν , Γn ) B(Γn )

from the inequality (2.3) we obtain Deficiency relation for arbitrary bee broken line. For an arbitrary bee broken line in the plane and an arbitrary collection of pairwise different points aν , ν = 1, 2, ..., q in the plane q X

ν=1

∆(aν ) ≤ 1.

(2.5)

Clearly the inequalities (2.3) and (2.5) also can be considered as analogues of the second fundamental theorem in Nevanlinna-Ahlfors’ theories.

14

G. A. Barsegian

Theorem 2.1 is an immediate consequence of Theorem 1.1. Indeed, rounding up Γ at the vertices (the ends of γi and the initial points of γi+1 ) we can obtain a curve γ ∈ F (1) as close to Γ as we please. Then
A(aν , Γ) and B(Γ) will be as close as we please to R(aν , γi ) and T γ (k) Γ correspondingly. Thus Theorem 1.1 implies Theorem 2.1. 3. Consequences for the real functions of one variable Inequalities (1.1) and (1.2) for the curves imply corresponding corollaries for the real smooth functions of one variable. The graph of a real function of one variable ϕ(x) ∈ C 1 [0, 1] is the curve γ := f (t)
:= x + iϕ(x) ∈ F (1), x ∈ [0, 1]. With the notation S(u) := u′ / 1 + u2 (the spherical derivative of u) and aν = (xν , yν ) we have Theorem 3.1. For any ϕ(x) ∈ C 1 [0, 1],  Z 1 Z 1  ϕ(x) dx ≤ S (ϕ′ (x)) dx + π. (3.1) S x 0 0

For any collection of pairwise different points aν = (xν , yν ), ν = 1, 2, ..., q,  Z 1 Z q q Z 1  X 2πq 1 ϕ(x) − yν S (ϕ′ (x)) dx + dx ≤ 1 + (ϕ′ (x))2 dx + π, S x − x ρ ν 0 0 0 ν=1 (3.2) where ρ is the minimal distance between aν . Thus we again obtain the Nevanlinna type inequality for real functions. Proof. To derive these inequalities from Theorem 1.1 we need just to observe that   ϕ(x) − yν ϕ(x) − yν ′ , =S (arg(f (t) − aν )) = arctan x − xν x − xν ′

(arg f ′ (t)) = S (ϕ′ (x)) , so that the inequality (1.1) of Theorem 1.1 applied to the curve γ and a1 = 0 immediately yields (3.1) and inequality (1.2) yields (3.2). Sharpness of Theorem 3.1. To show that (3.1) is sharp we consider the function ϕ(x) = h(2x − 1), x ∈ [0, 1], for which  Z 1  ϕ(x) dx → π, as h → ∞. S x 0

Clearly, the integral means the angle under which we see the line connecting the points (−h, 0) and (1, h). On the other hand Rsegment 1 ′ S (ϕ (x)) dx = 0 so that the difference between the left and the right 0

Some Interrelated Results in Different Branches of Geometry and Analysis

15

sides of (3.1) will be as small as we please provided we take h sufficiently large. √ 1 , Sharpness of (3.2) can be checked using the functions fε := ε sin x+ε x ∈ [0, 1] by the same arguments as in Section 1. 4. Applications in complex analysis, particularly principle of derivatives of analytic functions We feel pertinent to mention that the last principle is a “pure” type assertion in the sense that we deal only with the functions w and its derivatives like Cauchy formula and its consequences, Carleman formula for analytic functions and Cauchy-Pompeiu formula for smooth functions. One can remember that the majority of results valid for arbitrary analytic functions w in arbitrary domains are not of pure type since they deal either with a-points of w or their generalizations such as Ahlfors’ islands (NevanlinnaAhlfors theories [1,16]) or they deal with Gamma-lines ([7,8]). In what follows we denote by D a bounded domain with piecewise smooth boundary whose intersection with any line consists of finite number of intervals. Let l(∂D) be the length of the boundary ∂D. Theorem 4.1 (principle of derivatives). For any meromorphic function f in the closure of a given domain D and any integer k ≥ 1, Z Z (k+1) Z Z ′ f f (z) (z) kπ dσ ≤ (4.1) f (k) (z) dσ + 2 l(∂D). f (z) D D

For any collection of pairwise different points aν , ν = 1, 2, ..., q, q ZZ X f′ f − aν dydx D ν=1 ZZ Z Z (k+1) f kπ kπ 2 q l(∂D), |f ′ | dydx + ≤ f (k) dydx + ρ 2 D D

(4.2)

where ρ is the minimal distance between the aν ’s.

Sharpness. For function f (z) = exp z in the disk |z| < r we have R R f (k+1) (z) R R f ′ (z) 2 dσ = 2πr , dσ = 2πr2 and l(∂D) = 2πr so that D f (k) (z) D f (z) the ratio of the left and the right sides in (4.1) tends to 1 when r → ∞. This means that (4.1) is asymptotically sharp.

Clearly (4.2) is an analogue of Nevanlinna theorem. The case k = 1 was considered in [4].

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G. A. Barsegian

Now we pass to analogous results for “reasonably smooth” complex functions. The results can be proved for more general classes of functions but for simplicity we put some rather common restrictions. We write f (z) := u(x, y) + iv(x, y) ∈ C(D, k) if for any integer j, 1 ≤ j ≤ k + 1, u and v ¯ of D and if have continuous j-th derivatives in x and y in the closure D (j) ¯ for any j there is at most finite number of points in D, where ux (x, y) = (j) (j) (j) vx (x, y) = 0 or uy (x, y) = vy (x, y) = 0. Also we need the following Definition (projections with multiplicity). Let Γ be a given smooth plane curve Γ and γ be a given straight line in the plane P . Consider Γ as a union of subsets {ωm } ∪ {¯ ωp } (ωm := ωm (γ), ω ¯=ω ¯ p (γ)) such that for any m the ortogonal projection π (ωm ) of ωm onto γ is a homeomorphic map and for any p the length l(π(¯ ωp )) of the ortogonal projections π(¯ ωp ) onto γ is equal to zero. Denote by Γ ⊥ γ the union ∪m π(ωm ) and by l(Γ ⊥ γ) P the sum m l(π(ωm ). Thus the mentioned union counts the projections of Γ onto γ with multiplicities. We call l(Γ ⊥ γ) the projection length of Γ on γ. Denote by l(∂D ⊥ x) the projection length of ∂D on the axis x. Similarly we define l(∂D ⊥ x) substituting x by y. Theorem 4.2. For any f (z) ∈ C(D, k) and any integer k ≥ 1,  ′ (k) dydx + kπ l(∂D ⊥ x). arg fy...y 2 y D

ZZ ZZ ′ f ) dydx ≤ (arg y D

(4.3)

For any collection of pairwise different points aν , ν = 1, 2, ..., q, ′ (arg(f − aν )y dydx ≤  ′ R ′ (k) f dydx + intD arg fy...y dydx + 2kπq y ρ D Pq

ν=1

y

(4.4) kπ 2 l(∂D

⊥ x).

Thus we have again an Nevanlinna type inequality, this time for broad classes of (non analytic) functions belonging to C(D, k). Clearly these inequalities remain true if we substitute y by x.√Summing up and taking into account that l(∂D ⊥ x) + l(∂D ⊥ y) ≤ 2l(D) we obtain i R h (arg f )′ + (arg f )′ dydx ≤ x y D √     ′  ′ R (k) (k) arg fx...x + arg fy...y dydx + 2kπ l(D) D 2 y y

Some Interrelated Results in Different Branches of Geometry and Analysis

and

17

i R h (arg(f − aν )′ + (arg(f − aν )′ dydx ≤ y x  ′  ′  R  (k) (k) dydx+ arg fx...x + arg fy...y D Pq

ν=1 D

y

y

2kπq R D ρ

√ ′
2kπ ′ |fx | + fy dydx + l(D). 2

mx

mx

Proof of Theorem 4.2. Denote by Ix the straight line passing trough point (x, 0) and perpendicular to the axes x. Similarly we define Iy as the straight line passing trough point (0, y) and perpendicular to the axes y. Since the domain D is proper we have only finite number of intervals (p) on D ∩ Ix ; denote these intervals by mx , where p is the counting index. Consider the following function g(y) := f (x, y) := u(x, y) + iv(x, y) ∈ (p) (p) C(k, D) (of one variable) on mx . Function g(y) on mx determines a curve. (p) Denote by (x, y ∗ (p)) and (x, y ∗∗ (p)) the endpoints of mx . The determined curve we can consider on [0, 1] after linear transformation [y ∗ (p), y ∗∗ (p)] to [0, 1]. Since f ∈ C(D, k) we conclude that this curve on [0, 1] belongs to the class F (k). So that we can apply inequality (1.1) to this curve. This yields  Z Z ′ ′ (k) (4.5) arg gy...y dy + kπ (arg g)y dy ≤ (p) (p) y (p)

so that summing up by all mx we have Z  Z ′ X ′ (k) dy + kπ. arg g (arg g)y dy ≤ y...y y Ix

Ix

(k)

{mx }

Now we integrate this inequality by x and obtain R R ′ g) (arg dydx ≤ y D|x Ix  P ′  R R R (k) arg gy...y dydx + kπ dx, (k) 1 D|x D|x Ix {m } y

(4.6)

x

where D|x is the interval on x-axis which is the ortogonal projection of D on x-axis. Suppose that the part D(Ix′ , Ix′′ ) of the domain D contained in the strip between Ix′ and Ix′′ satisfies the following conditions: (A) the part is decomposed into n connected components Dj (Ix′ , Ix′′ ), j = 1, 2, ..., n, each with boundaries having common points both with Ix′ and Ix′′ ;

18

G. A. Barsegian

(B) intersection of each of these components Dj (Ix′ , Ix′′ ) with any Ix , x′ ≤ x ≤ x′′ consists of only one interval on Ix . Then for every x ∈ (x′ , x′′ ) we have X

1 = n,

(k) {mx }

where the constant n is independent of x and consequently   Z x′′ X  1 dx = n(x′′ − x′ ). x′

(k)

{mx }

But the quantity n(x′′ − x′ ) is half of the sum of the lengths of total projections on axis x of all boundary components of Dj (Ix′ , Ix′′ ) occurring in the mentioned strip; here the multiplier 2 (half) arise since each Dj (Ix′ , Ix′′ ) has two boundary components projected on x. Hence, since the domain D is assumed to have a piecewise smooth boundary, we can split the interval (x1 , x2 ) onto appropriate parts and obtain   Z X l(∂D ⊥ x)  . (4.7) 1 dx = 2 D|x (k) {mx }

Now (4.6) and (4.7) yield the inequality Z Z Z Z  ′ ′ (k) dydx + kπ l(∂D ⊥ x) arg g g) dydx ≤ (arg y y...y 2 y D|x Ix D|x Ix (4.8) which implies inequality (4.3) of Theorem 4.2. To prove inequality (4.4) we need to apply inequality (1.2) similarly as we have applied above inequality (1.1). This yields R Pq R ′ − a ) (arg(f ν y dydx ≤ ν=1 D|x Ix  ′ R R (k) arg gy...y dydx+ D|x Ix y

2kπq ρ

R

D|x

P  ′ R fy dydx + kπ dx (k) 1 D|x Ix {m }

R

x

so that taking into account (4.7) we obtain (4.4).

Proof of Theorem 4.1. Now we deal with the meromorphic functions f ¯ First we assume that f , f ′ , f (2) ,...,f (k+1) 6= 0, ∞ in D. ¯ From the in D.

Some Interrelated Results in Different Branches of Geometry and Analysis

19

definition of g(y) we have  vy′ u − vu′y v ′ ′ (arg g)y = arctan = u y u2 + v 2 q   (u′y )2 + (vy′ )2 vy′ u′y v u q  √ √ q −√ = 2 + v2 2 + v2 ′ 2 ′ 2 ′ 2 ′ 2 u2 + v 2 u u (uy ) + (vy ) (uy ) + (vy ) q
2
2 ′ u′y + vy′ f

√ sin arg f − arg fy′ = − sin arg f − arg fy′ =− f u2 + v 2

and similarly we have

f (k+1) ′    y...y (k) (k) (k+1) arg gy...y = − (k) sin arg fy...y − arg fy...y fy...y y

so that (4.3) yields
R R fy′ sin arg f − arg fy′ dσ ≤ D f (k+1)   R R fy...y (k+1) (k) arg f − arg f sin dσ + y...y y...y D (k) fy...y

kπ 2 l(∂D

⊥ x),

where dσ is the area element. ¯ Let X(θ) be the straight line {(x, y)| 0 ≤ θ := arctan(y/x) < 2π} on ¯ the plane (x, y), Y¯ (θ) is the straight line perpendicular to X(θ). ′ Denoting by fη the partial derivative of f in direction η perpendicular ¯ ¯ to X(θ) we rewrite the last inequality in the new coordinates (X(θ), Y¯ (θ)) as follows:
R R fη′ ′ D f sin arg f − arg fη dσ ≤ (4.9) (k+1)   R R fη...η (k+1) (k) kπ ¯ dσ + 2 l(∂D ⊥ X(θ)). D (k) sin arg fη...η − arg fη...η fη...η Now we observe that due to conformity of f , f ′ , f (2) ,...,f (k+1) we have ′ ′ fη f (z) = f f (z)

and

f (k+1) f (k+1) (z) η...η . (k) = (k) fη...η f (z)

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G. A. Barsegian

′ (z) Substituting the above two equalities in (4.9), observing that ff (z) and (k+1) f (z) f (k) (z) do not depend on θ and integrating the obtained inequality in θ we have  
R 2π R R f ′ sin arg f − arg fη′ dσ dθ 0 D f R R f ′ hR 2π
i sin arg f − arg fη′ dθ dσ = D f 0 (k+1)     R 2π R R f (z) (k) (k+1) ≤ 0 (4.10) sin arg fη...η − arg fη...η dσ dθ D (k) R 2π kπ f (z) ¯ + 0 2 l(∂D ⊥ X(θ))dθ (k+1) h  i R R f (z) R 2π  (k) (k+1) arg f − arg f = sin dθ dσ η...η η...η 0 D f (k) (z) R 2π kπ ¯ + 0 2 l(∂D ⊥ X(θ))dθ.

Since arg fη′ = arg fx′ + θ + π2 = arg f ′ + θ + π2 (conformity of f ) and the magnitudes arg f , arg f ′ do not depend on θ we have Z 2π  Z 2π   π  ′ sin arg f − arg fX(θ) dθ = sin arg f − arg f ′ − θ − dθ 2 0 0 Z 2π = |sin t| dt = 4. 0

On the other hand we have fζ′ = f ′ eiθ , where ζ is the direction of the ¯ straight line X(θ) composing angle θ with x-axis. Observing that eiθ is a (k) ¯ constant when we move along X(θ) we obtain fζ...ζ = f (k) eikθ . Applied to (k)

π

the direction η this yields fη...η = f (k) eik(θ+ 2 ) . Therefore Z 2π   (k) (k+1) − arg fη...η sin arg fη...η dθ 0

=

=

Z

Z

2π

0 2π 0

  π π sin arg f (k) + arg eik(θ+ 2 ) − arg f (k+1) − arg ei(k+1)(θ+ 2 ) dθ Z 2π  π  (k) (k+1) arg f − arg f − θ − |sin t| dt = 4. dθ = sin 2 0

The above two equalities and (4.10) yield Z Z ′ Z Z (k+1) Z 2π f (z) f (z) kπ ¯ 4 l(∂D ⊥ X(θ))dθ. f (z) dσ ≤ 4 f (k) (z) dσ + 2 D D 0 (4.11)

Some Interrelated Results in Different Branches of Geometry and Analysis

21

Due to the main identity of integral geometry ([9], [14], see also Section 6 of the present paper) we have Z 2π ¯ l(∂D ⊥ X(θ))dθ = 4l(D). (4.12) 0

The last two assertions imply inequality (4.1) in Theorem 4.1 in the case ¯ The general case can be proved when f , f ′ , f (2) ,...,f (k+1) 6= 0, ∞ in D. ¯ some making use quite standard arguments. Namely, we exclude from D small neighborhoods sc of all the zeros and poles of all these functions. Then we apply in the remained domain the proved inequality and then shrink the neighborhoods to the corresponding points (zeros or poles). When sc tends to the corresponding point we have Z Z (k+1) Z Z ′ f f (z) (z) dσ → 0, f (k) (z) dσ → 0, l(∂sc ) → 0 f (z) sc sc what completes the proof of (4.1).

In the proof of (4.1) we have utilized inequality (4.3) to obtain (4.11). Arguing similarly and making use (4.4) we obtain instead of (4.11) the following inequality Pq RR f ′ 4 ν=1 D f −a dydx ≤ ν R 2π RR ′ R 2π kπ RR (k+1) ¯ 4 D f f (k) dydx + 2kπq ρ 2 l(∂D ⊥ X(θ))dθ. 0 D fη dydxdθ + 0 Inequality (4.2) follows now from the last inequality, (4.12) and identity ′ fη = |f ′ |. 5. Applications in ODE: the windings of the solutions

5.1. The problem, its connections with Poincar´ e theory and applicability Many phenomena in physics, technics, biology, economics have a cyclic nature. Often these phenomena are described by the differential equations y ′ = F1 (t, x, y), x′ = F2 (t, x, y). In similar cases the solutions are the curves γ := (x (t) , y (t)), t ∈ (t1 , t2 ), that rotate (wind up) around a given center a on the plane (x, y). There are numerous studies (in pure mathematics and in all the mentioned sciences) which utilize Poincar´e theory to describe the windings of the solutions (x (t) , y (t)) of these equations. The differential geometric principle of the angles and lengths in Section 1 leads to another approach for studying these windings.

22

G. A. Barsegian

Below we discuss some parallels and differences between the Poincar´e theory and the new approach. Poincar´e theory deals with the concepts (the spiral and periodic solutions, the limit cycles) that determine the asymptotic shape or asymptotic windings of the solutions. Respectively the theory considers the solutions on infinite intervals t ∈ (−∞, +∞), (−∞, 0), (0, +∞). This is an essential restriction in applications since in the practice we very often need to consider namely finite time intervals t ∈ (t1 , t2 ). Also the Poincar´e theory deals in fact with only one center. The new approach permits to study the solutions on finite time intervals. In this case the windings are determined by ordinary total rotation of γ defined in Section 1. Also we able to study an interplay between the windings around different centers what leads to some Nevanlinna type consequences for the solutions. The approach leads to some new problems related to the interplay between the solutions on finite and infinite intervals. We state qualitatively one problem connected with Hilbert Problem 16 (part b) [10] which asks about the number of limit cycles for the solutions with polynomial F1 and F2 of the degree n. Following this problem we can consider (instead of the limit cycles) the solutions with rather strong rotations and we may ask whether the number of all possible similar solutions also depend on n. The approach first arose in the study [6] by K. Barseghyan and the author, where we dealt with some particular classes of equations in biomathematics (Lotka-Volterra’s [11], [17] and Kolmogorov’s [12] equations). Below we study the windings of the solutions of much larger classes of autonomous equations. 5.2. The windings of solutions of autonomous equations Consider the following autonomous system of equations  ′ y = F1 (x, y) x′ = F2 (x, y)

(5.1)

¯ of a given with continuously differentiable F1 and F2 in the closure D domain D. Assume that the coefficient satisfy the usual restrictions (F1 (x, y))′ , (F1 (x, y))′ , (F2 (x, y))′ , (F2 (x, y))′ ≤ C(D) = const. x y x y

(5.2) Let γ := (x (t) , y (t)), t ∈ (t1 , t2 ), be a part of an integral curve of (5.1) ¯ We will refer simply γ as a is a solution of (5.1), [13]. lying in D.

Some Interrelated Results in Different Branches of Geometry and Analysis

23

Theorem 5.1. For any smooth solution γ := (x (t) , y (t)), t ∈ (t1 , t2 ), of equation (5.1) satisfying (5.2) R (a, γ) ≤ T (γ ′ ) + π ≤ 3C(D) |t2 − t1 | + π.

(5.3)

Notice that (5.3) gives upper bounds both for the rotations R of the solutions around a and for the integral curvature T of γ. Another advantage is that the upper bounds can be given simply by making use |t2 − t1 | and the derivatives of F1 and F2 . As in section 1 we consider again a collection of pairwise different points i q, q ≥ 2, in the plane (x, y) and denote ρ = h aν , ν = 1, 2, ..., |ai −aj | min mini6=,j ; 1 . Then we consider a collection of the bounded non 2 intersecting domains Dν ∋ aν , ν = 1, 2, ..., q, in the plane (x, y). Denote d := ∪qν=1 Dν and assume that for (x, y) ∈ d¯ functions F1 (x, y) and F2 (x, y) in (5.1) satisfy |(F1 (x, y))| , |(F2 (x, y))| ≤ K(d) = const.

(5.4)

The set γ ∩ D consists of some (one or more) curves γi (Dν ), i = 1, 2, ..., Iν . Denote by A(aν , Dν , γ) the total rotations of all similar curves PIν R (aν , γi (Dν )). around a, that is A(aν , Dν , γ) := i=1 For the similar total rotations we prove the following Nevanlinna type result. Theorem 5.2. Assume that for a given above type set d the coefficients of the equation (5.1) satisfy (5.2) and (5.4) in d. Then for any smooth solution (x (t) , y (t)), t ∈ (t1 , t2 ), we have q X

ν=1

A(aν , Dν , γ) ≤

(

) √ 4 2π K(d) |t2 − t1 | + π. 3C(d) + ρ

(5.6)

Remark 5.1. It is interesting that ρ is the only magnitude depending on the geometry of points aν , ν = 1, 2, ..., q, and that the geometry of domains Dν does not affect to inequality (5.6). Proof of Theorem 5.1. Consider a solution γ := (x(t), y(t)), t ∈ (t1 , t2 ), of (5.1) lying in D. Upper bounds for T (γ ′ ) can be given easily. Indeed, according to the definitions we have

′

T (γ ) :=

Z

t2

t1

Z d arg (x′ (t) + iy ′ (t)) dt = dt

t2

t1

′ d arctan y (t) dt. dt x′ (t)

24

G. A. Barsegian

Further

′′ ′ ′ ′′ ′ d arctan y (t) := y x − x y . dt (x′ )2 + (y ′ )2 x′ (t)

Since

′

′

y ′′ (t) = (F1 (x, y))x x′ + (F1 (x, y))y y ′ = ′

′

(F1 (x, y))x F2 (x, y) + (F1 (x, y))y F1 (x, y) and ′

′

x′′ (t) = (F2 (x, y))x x′ + (F2 (x, y))y y ′ = ′

′

(F2 (x, y))x F2 (x, y) + (F2 (x, y))y F1 (x, y) we have y ′′ x′ − x′′ y ′

2 2 = (x′ ) + (y ′ ) h i ′ ′ (F1 (x, y))x F2 (x, y) + (F1 (x, y))y F1 (x, y) F2 (x, y)

F22 (x, y) + F12 (x, y)

h i ′ ′ (F2 (x, y))x F2 (x, y) + (F2 (x, y))y F1 (x, y) F1 (x, y) F22 (x, y) + F12 (x, y)

′

− =

′

(F1 (x, y))x F22 (x, y) − (F2 (x, y))y F12 (x, y)

+ F22 (x, y) + F12 (x, y) h i ′ ′ (F1 (x, y))y − (F2 (x, y))x F1 (x, y) F2 (x, y) F22 (x, y) + F12 (x, y)

so that

y ′′ x′ − x′′ y ′ ′ ≤ (F1 (x, y))x + (x′ )2 + (y ′ )2 ′ ′ (F1 (x, y))y + (F2 (x, y))x + (F2 (x, y))′y 2

and applying (5.2) we obtain the inequality

T (γ ′ ) ≤ 3C(D) |t2 − t1 |

(5.7)

which being applied to inequality (1.1) of Theorem 1.1 yields Theorem 4.1.

Some Interrelated Results in Different Branches of Geometry and Analysis

25

Proof of Theorem 5.2. Arguing similarly as in the proof of inequality (1.2) we can obtain the following quite similar inequality q X

A(aν , Dν , γ) :=

ν=1

≤

q X Iν X

R (aν , γi (Dν ))

ν=1 i=1 q X Iν X

T (γi′ (Dν )) +

ν=1 i=1

q Iν 4π X X l(γi (Dν )) + π. ρ ν=1 i=1

(5.8)

Thus to prove Theorem 5.2 we need to obtain upper bounds for the sums in the right side of (5.8). Assume that γi (Dν ) is the image of (t′i,ν , t′′i,ν ) ⊂ (t1 , t2 ). Applying (5.7) to this curve and obtain T (γi′ (Dν )) ≤ 3C(Dν ) t′′i,ν − t′i,ν so that q X Iν X

ν=1 i=1

T (γi′ (Dν )) ≤ 3

q X Iν X ν=1 i=1

C(Dν ) t′′i,ν − t′i,ν ≤ 3C(d) |t2 − t1 | .

p R For the length we have l(γi (Dν )) := γi (Dν ) (x′ (t))2 + (y ′ (t))2 dt and from the equation we have x′ (t) = F1 (x, y), y ′ (t) = F2 (x, y) so that (5.4) yields q X Iν X

ν=1 i=1

l(γi (Dν )) ≤

q X Iν X √ ′′ √ t − t′ ≤ 2K(d) |t2 − t1 | . 2K(d) i,ν i,ν ν=1 i=1

Substituting the last two inequalities in (5.8) we obtain Theorem 5.2. 6. Some identities and inequalities in integral geometry 6.1. Introduction

In this section we present some new identities and inequalities concerning integrals of function along the curves, among them those generalizing some classical formulae of integral geometry. The study has arisen as follows. The theory of Gamma-lines [7] (complex analysis) studies particularly the level sets of harmonic functions. The level sets occur in numerous branches of pure and applied mathematics (isoterm, isobar, potential line, stream line etc.). Naturally we tried to find some ways to apply and extend Gamma-lines technic for studying level sets of much larger classes of functions u(x, y) that can be of interest in different applied situations. This have led us to an extensive list of open problems [8] in nearly thirty fields. Our attempts to develop these ideas and solve some of the problems posed in [8] have led to some new inequalities [5] for curves, real and complex functions. Then we have applied the Gamma-lines’ methods to obtain

26

G. A. Barsegian

some other identities and inequalities which we present in this paper. It is interesting that the obtained results turned out to be rather close to those obtained earlier in Integral Geometry [9], [14] and Combinatorial integral Geometry [2]. I got idea about the mentioned interrelation with Integral geometry thanks to R. Ambartzumyan and V. Oganian who kindly presented me the key ideas and results in [2], [9], [14]. I express my deep thanks to them for that as well as for their valuable comments. In this section we present some of the obtained results in this direction. In full generality they will be presented elsewhere. 6.2. The main identity in integral geometry We say that a given oriented plane curve Γ := (x(t)) , y(t)), t ∈ [0, 1], is proper if Γ is a smooth curve with bounded length l(Γ) and uniformly continuous on Γ curvature k. Let f be aRfunction given on Γ. In this paper we give an identity for f ds. In the particular case when f ≡ 1 this identity coincides with the classical Identity 1 (see below ) in integral geometry. R To present the identity for f ds we need to modify the classical concepts and to give an interpretation. Definition (projections with multiplicity). Let Γ be a given smooth plane curve Γ and γ be a given straight line in the plane P . Consider Γ as a union of subsets {ωm } ∪ {¯ ωp } (ωm := ωm (γ), ω ¯=ω ¯ p (γ)) such that for any m the orthogonal projection π (ωm ) of ωm onto γ is a homeomorphic map and for any p the length l(π(¯ ωp )) of the orthogonal projections π(¯ ωp ) onto γ is equal to zero. Denote by Γ ⊥ γ the union ∪m π(ωm ) and by l(Γ ⊥ γ) P the sum m l(π(ωm ). Thus the mentioned union counts the projections of Γ onto γ with multiplicities. We call l(Γ ⊥ γ) the projection length of Γ on γ. ¯ Let X(θ) be the straight line {(x, y)| 0 ≤ θ := arctan(y/x) < 2π} on the ¯ plane (x, y). We will use notation X(θ) for the coordinate on X(θ). Denote ¯ ¯ by JX(θ) the straight line perpendicular to X(θ) and intersecting X(θ) at the point X(θ). Notice that Γ ∩ JX(θ) may consist of some points and some intervals on the straight lines JX(θ) : these intervals clearly should be the intervals ¯ of type ω ¯ p (X(θ)) belonging to Γ. Ignoring similar intervals we denote by N (Γ, JX(θ) ) the total number of similar points. ¯ We make use notations lθ (Γ) for l(Γ ⊥ X(θ)) when γ is the straight line ¯ ¯ stands for the set of all those points X(θ) on X(θ) X(θ).Notations Γ|X(θ) ¯ for which JX(θ) intersects Γ in at least one point.

Some Interrelated Results in Different Branches of Geometry and Analysis

27

With the above notations and definitions we able now to present Identity 6.1. For any proper curve Γ we have Z Z Z 1 2π 1 2π l(Γ) = N (Γ, JX(θ) ) |dX(θ)| dθ = lθ (Γ)dθ, (6.1) 4 0 4 0 Γ|X(θ) ¯ (here we make use Lebesgue integration). Comment 6.1. Notice that lθ (Γ) = lθ+π (Γ) and N (Γ, JX(θ) ) = N (Γ, JX(θ+π) ) for any θ so that the above inequality we may rewrite as Z Z Z 1 π 1 π N (Γ, JX(θ) ) |dX(θ)| dθ = lθ (Γ)dθ. (6.1′ ) l(Γ) = 2 0 Γ|X(θ) 2 0 ¯ Similar comments are true for other identities and inequalities below. Comment 6.2. Identity (6.1) is a bit simplified and complemented version of the classical identity whose modifications and generalizations constitute in fact an essential part of integral geometry: here contributed Barbier, Poincar´e, Blaschke, Santalo (see [14]). The difference between the first identity in (6.1) and its classical counterparts is that we make use more simple pointwise intersections N (Γ, JX(θ) ) (instead of forms and densities). Also we have the second part in (6.1) which despite its triviality we did not meet elsewhere. Notice that in (6.1) we deal with all length projections lθ (Γ) for all θ, 0 ≤ θ ≤ 2π. Now we give another identity for the length of Γ which make use only two projections of Γ on perpendicular axes JX(θ) and JX(η) , where η := θ + π/2. Since our curve Γ is oriented we can fix starting point of Γ and define the angle α(s) between the tangent to Γ at the point s ∈ Γ and ¯ X(0). Notice that the last straight line is x-axis. Let us consider identity (6.1) from a bit different point of view. Moving along the curve Γ we fix the elements si(θ) of Γ ∩ JX(θ) , si(θ) = 1, 2, ..., N (Γ, JX(θ)): here i is, clearly, the index of these elements (points or curves). Thus we can rewrite (6.1′ ) as   N (Γ,JX(θ) )  Z π Z πZ  X 1 1 lθ (Γ)dθ = 1 |dX(θ)| dθ. l(Γ) =   2 0 2 0 Γ|X(θ) ¯ i(θ)=1

6.3. Identities for the integrals along curves

In this section we give some formulas of the above types for the integrals Z f (s)ds, Γ

28

G. A. Barsegian

where s is the natural parameter of Γ. Particularly, when f (s) ≡ 1 this formulae coincide with identities (6.1) and (6.2). ¯ Instead of the projection length of Γ on X(θ), we consider now the ¯ following weighted projection length of Γ on X(θ) Z XZ f (s) |cos (α(s) − θ)| ds = f (s) |cos (α(s) − θ)| ds Lθ (Γ, f ) := Γ

m

¯ ωm (X(θ)

and, instead of the length Dθ (Γ) of double R projection, we consider the following weighted double projection length π(ωm (X(θ)) ¯ Z Z X f cos2 (α(s) − θ) ds. f (s) cos2 (α(s) − θ) ds = Iθ (Γ, f ) := Γ

m

¯ ωm (X(θ)

Theorem 6.1. For any proper curve Γ and any continuous function f (s) > 0 on Γ we have Z Z 1 2π Lθ (Γ, f )dθ f (s)ds = 4 0 Γ N (Γ,JX(θ) ) Z Z X 1 2π = f (si(θ) ) |dX(θ)| dθ. (6.2) 4 0 Γ|X(θ) i(θ)=1

6.4. Inequalities for the lengths of curves and inequalities for the integrals along curves From the above identities we derive some inequalities giving upper bounds R f (s)ds in terms of the integral curvature C(Γ) := for the integrals Γ R k(s)ds, where k(s) is the ordinary curvature of Γ at the point s ∈ Γ. Γ Observe that for the proper curves C(Γ) = VarΓ α(s) so that we can consider also C(Γ) as the variation of the tangential angle. We give first some simple inequalities related to the particular case when R f (s) ≡ 1; respectively Γ f (s)ds become ordinary length l(Γ) of Γ. Denote by ∆θ (Γ) the length of Γ|X(θ) . Inequality 6.1. For any proper curve Γ and any θ we have   Z 1 |k(s) sin α(s)| ds + 1 ∆0 (Γ) l(Γ) ≤ 2 Γ  Z  1 + |k(s) cos α(s)| ds + 1 ∆π/2 (Γ). (6.3) 2 Γ If Γ is closed we have  Z   Z  1 1 l(Γ) ≤ |k(s) sin α(s)| ds ∆0 (Γ) + |k(s) cos α(s)| ds ∆π/2 (Γ). 2 Γ 2 Γ

Some Interrelated Results in Different Branches of Geometry and Analysis

29

Notice that (6.3) implies the following more simple (and more rough) inequality  Z 
1 l(Γ) ≤ |k(s)| ds + 1 ∆0 (Γ) + ∆π/2 (Γ) . (6.3′ ) 2 Γ

Notice also that when Γ is a segment of a straight line then k(s) ≡ 0 and inequality (6.3′ ) become a usual Pythagorean (or triangular) inequality in the form: l(Γ) ≤ ∆0 (Γ) + ∆π/2 (Γ). Sharpness. We have equality in (6.3) and (6.3′ ) and for any segment lying ¯ ¯ on X(0) and X(π/2). For the proper closed curves Γ Santalo proved the following interesting inequality, see [14], p. 37: Z 1 |k(s)| ds∆max (Γ). l(Γ) ≤ 2 Γ

We establish similar result for arbitrary (non closed) proper curves. Inequality 6.2. For any proper curve Γ we have Z Z 1 π 1 l(Γ) ≤ |k(s)| ds∆max (Γ) + ∆θ (Γ)dθ 2 Γ 2 0 Z 1 π ≤ (6.4) |k(s)| ds∆max (Γ) + ∆max (Γ). 2 Γ 2

Sharpness The ratio of the left and the right sides in (6.4) tends to 1 for 1 )eiω(r) , rn ∈ (0, 1), where ϕ(r) and ω(r) are any sequence of spirals (1 − ϕ(r) some monotone differentiable functions tending to infinity when rn → 1. Thus (6.4) is asymptotically sharp. 6.5. Inequalities for the integrals along curves

Theorem 6.2. For any proper curve Γ and any continuously differentiable function f (s) > 0 on Γ and any θ we have R Γ f (s)ds R R ≤ 12 supΓ |f | Γ |k(s) sin α(s)| ds∆0 (Γ) + Γ |k(s) cos α(s)| ds∆π/2 (Γ)
 + 12 supΓ |fλ′ | l(Γ) + supΓ |f | ∆0 (Γ) + ∆π/2 (Γ)  R
≤ 12 supΓ |f | Γ k(s)ds + 21 supΓ |fλ′ | l(Γ) + supΓ |f | ∆0 (Γ) + ∆π/2 (Γ) (6.5) and  Z 2π  Z Z 1 ′ sup |f | |k(s)| ds + sup |fλ | l(Γ) + 2 sup |f | ∆θ (Γ)dθ. f (s)ds ≤ 2π Γ Γ Γ Γ 0 Γ (6.6)

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G. A. Barsegian

Notice that (6.5) implies (6.3) when f (s) ≡ 1. In turn the sharpness of (6.5) follows from the sharpness of (6.3). 7. Principle of zeros for real functions of two variables In this section we present an inequality related to the zeros of real functions of two variables and the zeros of derivatives of this function. We show that these inequalities can be considered as some analogues of Rolle’s theorem for functions of two variables. In what follows we denote by D domains with piecewise smooth boundary ∂D of length l(∂D). ¯ D ¯ = The results are valid for arbitrary functions u(x, y) ∈ C 2 (D), ¯ D ∪ ∂D, but below we present only the case of the proper functions in D, 2 ¯ that is functions u ∈ C (D) which admit only finite number of points in ¯ where gradu = 0 or gradu′ = 0. Observe that the zeros of u(x, y) (that D, is the solutions of u(x, y) = 0 or level sets of u(x, y)) are piecewise smooth curves γi (u) for the proper functions. The same is true for zeros u′θ (x, y) = 0 (for any θ, 0 ≤ θ < π). Corresponding curves we denote by γj (u′θ ). Denote by L(D, 0, u) (by L(D, 0, u′θ ) the total length of the curves γi (u) ¯ (γj (u′θ )) in D. Theorem 1 (Principle of zeros for proper functions). For any proper ¯ function u(x, y) in D Z 1 1 π L(D, 0, u′θ )dθ + l(∂D) L(D, 0, u) ≤ 2 0 2 1 π sup L(D, 0, u′θ ) + l(D). (7.1) ≤ 2 0≤θ 0, x ∈ Rn . Cases of (16) were earlier established in [8], [9] and [10], the proofs being based on wavelet decompositions together with weakℓ1 type estimates and interpolation theory. Ledoux’s approach relies on the use of pseudo-Poincar´e inequalities and the argument easily extends to more general frameworks, including manifolds or graphs which are out of reach of wavelet methods. The inequality (16) is easily seen to include the classical Sobolev inequality (3) on using the readily established estimate kPt f kL∞ (Rn ) ≤ Ct−n/2r kf kLr (Rn )

for any r ≥ 1. For then with r = q = p∗ = np/(n − p), we have θ/(θ − 1) + n/r = 0 and so kf kB θ/(θ−1) ≤ Ckf kLq (Rn ) . ∞,∞

Ledoux’s technique requires specific information on the heat semi-group et∆ in L2 (Rn ). The operator semigroup associated with the inequality in [2] is ∗ e−tL L , where L = x · ∇. In [2], Theorem 2, it is proved that Z 2 ∗ e−tn /4 −n/2 ∞ − (ln r−ln s)2 −n/2 4t s ψ(s, ω) sn−1 ds. (17) r e (e−tL L ψ)(x) = √ 4πt 0 A significant fact is that after a co-ordinate change Φ, L∗ L is related to the Laplacian on R. To be specific, Φ is an isometry of L2 (Rn ) onto L2 (R×Sn−1 ) defined by (Φψ)(s, ω) := esn/2 ψ(es , ω),

ω ∈ Sn−1 ,

s ∈ R.

It is readily seen that L = iA−n/2, where A is the non-negative self-adjoint generator of dilations in L2 (Rn ) which in fact is the momentum operator on R. It follows that ΦAΦ−1 = −i∂r ⊗ ISn−1

(18)

where ISn−1 denotes the identity on the unit sphere S and, consistent with (11), L∗ L = A2 + n2 /4 ≥ n2 /4. This leads to the representation Z 2 1 1 (19) exp{− (r − s)2 }g(s, ω)ds Φe−tA Φ−1 g(r, ω) = √ 4t 4πt R n−1

38

W. D. Evans ∗

2

2

2

and e−tL L = e−tA e−tn /4 . The fact that Φe−tA Φ−1 in (19) is essentially radial means that the analogue of (16) derived by Ledoux’s technique is a consequence of the one-dimensional case of (16). Defining B α to be the space of all tempered distributions g on R × Sn−1 for which the norm 2

kgkB α := sup{t−α/2 kΦe−tA Φ−1 g|kL∞ (R×Sn−1 ) } < ∞,

(20)

t>0

one obtains from the n = 1 case of (16), that for any ω ∈ Sn−1 , Z Z ∂g(r, ω) p q q |g(r, ω)| dr ≤ C ∂r dr R R q(1−θ)  Z 1 2 × sup tθ/2(1−θ) √ e−(r−s) /4t g(s, ω)ds 4πt R t>0,r∈R p  Z q(1−θ) ∂g(r, ω) θ/2(1−θ) −tA2 −1 dr = Cq sup t Φ g(r, ω) Φe ∂r t>0,r∈R R q(1−θ)  Z

∂g(r, ω) p −tA2 −1 dr sup tθ/2(1−θ) Φ g ≤ Cq

Φe

∂r L∞ (R×Sn−1 ) t>0 R p Z ∂g(r, ω) q(1−θ) ≤ Cq ∂r drkgkB θ/(θ−1) . R

On integrating with respect to ω over Sn−1 we obtain the following result. Theorem 2.1. Let 1 ≤ p < q < ∞ and suppose that g is such that ΦAΦ−1 g ≡ −i(∂/∂r)g ∈ Lp (R × Sn−1 ) and g ∈ B θ/(θ−1) , θ = p/q. Then there exists a positive constant C, depending on p and q, such that 1−θ kgkLq (R×Sn−1 ) ≤ Ck(∂/∂r)gkθLp (R×Sn−1 ) kgkB θ/(θ−1) .

(21)

Various corollaries of Hardy-Sobolev type featuring the inequality (2.1) are derived in [2]. Corollary 2.1. Let p∗ := np/(n−p), 1 ≤ p ≤ n−1, and suppose (∂/∂r)g ∈ Lp (R × Sn−1 ) and supω∈Sn−1 kg(·, ω)kLp(R) < ∞. Then (n−1)/n

1/n

kgkLp∗ (R×Sn−1 ) ≤ Ck(∂/∂r)gkLp (R×Sn−1 ) sup kg(·, ω)kLp(R)

.

(22)

ω∈Sn−1

If G = M(g) denotes the integral mean of g, namely, Z 1 G(r) = M(g)(r) := n−1 g(r, ω)dω, |S | Sn−1

then if g, (∂/∂r)g ∈ Lp (R × Sn−1 ),

1/n

(n−1)/n

kGkLp∗ (R) ≤ Ck(∂/∂r)gkLp (R×Sn−1 ) kgkLp(R×Sn−1 ) .

(23)

Recent Results on Hardy and Rellich Inequalities

39

If g is supported in [−Λ, Λ] × Sn−1 , then 2

(n−1)/n

kgkLp∗ (R) ≤ CΛ(n−1)/n k(∂/∂r)gkLp(R×Sn−1 ) sup kg(·, ω)kLp∗ (R) ; (24) ω∈Sn−1

also kGkLp∗ (R) ≤ CΛ(n−1)/n k(∂/∂r)gkLp(R×Sn−1 ) .

(25)

The case p = 2 is of special interest; for instance one gets from (22) and (23) the following. Corollary 2.2. Let f be such that Lf ∈ L2 (Rn ) supω∈Sn−1 kf kL2(R+ ;dµ) < ∞, where L = x · ∇. Then for n ≥ 3,

and

1/n  n2 kf k2L2(Rn ) krf (r, ω)k2L2∗ (Rn ) ≤ C kLf k2L2(Rn ) − 4 2(1−1/n)

× sup kf kL2(R+ ;dµ)) ,

(26)

ω∈Sn−1

where 2∗ = 2n/(n − 2) and dµ = rn−1 dr. If f, Lf ∈ L2 (Rn ) then with F = M(f ), 1/n  n2 kf k2L2(Rn ) krF (r)k2L2∗ (R+ ;dµ) ≤ C kLf k2L2(Rn ) − 4 2(1−1/n)

× kf kL2 (Rn ) .

(27)

On setting f = h/| · |, n ≥ 3, in (26) and (27) it follows that there exist positive constants C depending only on n such that  1/n n − 2
2 khk2L2∗ (Rn ) ≤ C k∇hk2L2 (Rn ) − kh/| · |k2L2 (Rn ) 2  1−1/n 2 × sup kh(·, ω)/| · |kL2 (R+ ;dµ)

(28)

ω∈Sn−1

and

1/n  n − 2
2 kM(h)k2L2∗ (R+ ;dµ) ≤ C k∇hk2L2 (Rn ) − kh/| · |k2L2 (Rn ) 2 1−1/n  . (29) × kh/| · |k2L2 (Rn )

From (29) and Young’s inequality, we have that, for any ε > 0, ε1−1/n kM(h)k2L2∗ (R+ ;dµ) ≤ C{k∇hk2L2 (Rn ) n − 2
2 −[ − ε]kh/| · |k2L2 (Rn ) }, 2

40

W. D. Evans

kM(h)k2L2∗ (R+ ;dµ)


n−2 2 2

− ε > 0, yields   (n−1) h 2 (n − 2)2 − δ]− n kL2 (Rn ) . ≤ C[ k∇hk2L2 (Rn ) − δk 4 |·| (30)

which, on taking δ =

Since ∗

kM(h)k2L2∗ (R+ ;dµ) ≤

∗ 1 khk2L2∗ (Rn ) |Sn−1 |

by H¨ older’s inequality, it follows that (30) is a consequence of Stubbe’s inequality in [29], Theorem 1, namely, for δ < (n − 2)2 /4, khk2L2∗ (Rn ) ≤ C 2 [1 −

(n−1) h 2 4δ ]− n {k∇hk2L2 (Rn ) − δk k 2 n }, (31) 2 (n − 2) | · | L (R )

where the optimal constant C is that for (3) in the case p = 2, that is, 1/n

C = [πn(n − 2)]−1/2 (Γ(n)/Γ(n/2))

.

(32)

The fact that the integral mean, Mh, rather than h appears on the lefthand-side of (30) is due to the fact that (30) is derived from (23), where 2 the presence of the integral mean is a consequence of the fact that e−tA does not map any Lp (R × Sn−1 ), 1 < p < ∞, into L∞ (R × Sn−1 ). This is in contrast to et∆ which maps Lp (Rn ) into L∞ (Rn ). If in (25) we put g = Φf, where f is supported in the annulus A(1/R, R) := {x ∈ Rn : 1/R ≤ |x| ≤ R}, then G is supported in the interval [− ln R, ln R] and we obtain the following local form of a HardySobolev inequality. Corollary 2.3. Let f ∈ C0∞ (A(1/R, R)). Then   2(n−1) (n − 2)2 f 2 k∇f k2L2 (Rn ) − k kL2 (Rn ) . kM(f )k2L2∗ (R+ ;dµ) ≤ C(ln R) n 4 |·| (33) The following related result is proved in [7] and [19] (as the case s = 1 of (2.6) whose proof is given in section 6.4): if f ∈ C0∞ (Ω) and 2 ≤ q < 2∗ ,  

∗ (n − 2)2

f 2 , (34) kf k2Lq (Rn ) ≤ C|Ω|2(1/q−1/2 ) k∇f k2L2 (Rn ) −

4 | · | L2 (Rn )

where |Ω| denotes the volume of Ω. It is noted in [19], Remark 2.4, that, in contrast to (33), the q in (34) must be strictly less than the critical Sobolev exponent 2∗ = 2n/(n − 2) if Ω includes the origin. A similar inequality to (34) in Lp (Ω), 1 < p < n, when Ω is convex and f (x)/|x| is replaced by f (x)/dist(x, ∂Ω), was established in [18]; see the discussion after (41) in the next section.

Recent Results on Hardy and Rellich Inequalities

41

3. Hardy’s inequality on bounded domains It is known (see [24] and [25]) that if Ω is a convex domain in Rn , the best constant in (6) is C = cp := [(p − 1)/p]p , but there are smooth domains for which C < cp . In [6], Br´ezis and Markus proved that for any convex Ω, the largest possible constant λ(Ω) such that Z Z Z |f (x)|2 |f (x)|2 dx, f ∈ C0∞ (Ω), dx + λ(Ω) |∇f (x)|2 dx ≥ (1/4) 2 δ(x) Ω Ω Ω (35) satisfies λ(Ω) ≥ (4 diam(Ω)2 )−1 .

(36)

This was improved by M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and A. Laptev, to λ(Ω) ≥ µn |Ω|−2/n ,

µn = n(n−2)/n |Sn−1 |2/n /4

(37)

in [20] and subsequently by Evans and Lewis in [16] to λ(Ω) ≥ 6µn |Ω|−2/n .

(38)

Filippas, Maz’ya and Tertikas [18] obtained the following bound as a special case of Lp Hardy inequalities: λ(Ω) ≥ 3Dint (Ω)−2

(39)

where Dint (Ω) := 2 supx∈Ω δ(x), the interior diameter of Ω. Since we have 3Dint (Ω)−2 ≥ (3/n)µn |Ω|−2/n , this is an improvement of (37) when n = 2, 3, but not of (38); for n > 3, the results are not comparable. The analogue of (35) in Lp for p > 1 has been studied in [17], [18] and [31]; in [17] weighted versions of the inequality and also arbitrary domains were considered. As illustrations, we quote the following results for a convex domain Ω from [31] and [18] respectively: Tidblom proves an extension of (35) and (37) to the following in Lp (Ω), 1 < p < ∞ : Z Z Z |f (x)|p p dx + λp (Ω) |f (x)|p dx, f ∈ C0∞ (Ω), (40) |∇f | dx ≥ cp p Ω Ω Ω δ(x) where λp (Ω) ≥ µn,p |Ω|−p/n and µn,p =

(p − 1)p+1 pp



|Sn−1 | n

p/n √ πΓ( n+p 2 )

n Γ( p+1 2 )Γ( 2 )

.

42

W. D. Evans

In [18] the result is a corollary of the following Hardy-Sobolev type inequality: for 1 < p < n and p ≤ q < p∗ := np/(n − p), we have Z p/q Z Z |f (x)|p p q |∇f | d x ≥ cp d x + λp,q (Ω) |f (x)| d x , (41) p Ω Ω δ(x) Ω for f ∈ C0∞ (Ω), where, for some positive constants ki = ki (p, q, n), i = 1, 2, independent of Ω, k1 (p, q, n)Dint (Ω)n−p−

np q

≥ λp,q (Ω) ≥ k2 (p, q, n)Dint (Ω)n−p−

np q

.

As we have already observed, Dint (Ω) ≤ (n/µn )1/2 |Ω|1/n . Hence (41) yields the following analogue of (34) in Lp (Ω) when Ω is convex with finite internal diameter Dint (Ω) : for 1 < p < n, p ≤ q < p∗ , and f ∈ C0∞ (Ω), o n ∗ kf kpLq (Ω) ≤ C(n, p, q)|Ω|p(1/q−1/p ) k∇f kpLp (Ω) − cp kf /δkpLp(Ω) . (42) A comprehensive and unified treatment of improvements to Hardy’s inequality is given in [3] in which distance functions other than δ are also considered and optimal weights and best constants established. The technique used in [17], [20] and [31] is to first derive an appropriate one-dimensional Hardy-type inequality on an interval and then use this to give an inequality of type (35) in which δ is replaced by the mean distance δM of (7). This leads to (35) when Ω is convex, on using the following estimate given in [13] for p = 2 and [31] for general p: Z 1 dω(ν) ≥ B(n, p)/δM (x)p , p δ (x) n−1 M S

where, with e a fixed unit vector in Rn , Z Γ( p+1 ) · Γ( n2 ) B(n, p) := |e · ν|p dω(ν) = √ 2 . π · Γ( n+p Sn−1 2 ) 4. Rellich’s inequality

The same technique as that described at the end of the last section has been used in [4] and [17] to prove extensions of the Rellich inequality(10) of the form Z Z Z |f (x)|2 2 |f (x)|2 d x d x + λ(Ω) |∆f (x)| ≥ µ(Ω) 4 Ω Ω δ(x) Ω for a convex Ω : weighted versions are also given in [17]. To incorporate the weights, the following is established in [17], Proposition 1, in which

Recent Results on Hardy and Rellich Inequalities

∂ν := ν · ∇ and Ω is any domain in Rn : for all u ∈ C 2 (Rn ) Z n 2 i h X 1 ∂ u(x) 2 |∆u(x)|2 + 2 |∂ν2 u(x)|2 dω(ν) = . n(n + 2) ∂xi ∂xj Sn−1 i,j=1

A consequence of this is that for all u ∈ C02 (Ω) Z Z Z 3 2 2 |∆u(x)|2 dx, |∂ν u(x)| dω(ν)dx = n(n + 2) Ω Ω Sn−1

43

(43)

(44)

an identity first proved by Owen in [26]. When Ω is convex, the best constant in (10) is 9/16 (see [26]), and in [4] it is proved that Z Z h Sn−1 i4/nZ |u(x)|2 9 2 dx + cn(n+2) |u(x)|2 dx, (45) |∆u(x)| dx ≥ 16 Ω δ(x)4 n|Ω| Ω Ω with c = 11/48; the corresponding result in [17]) has the improvement c ≈ 1.25. References 1. A. Ancona, On strong barriers and an inequality of Hardy for domains in Rn , J. London Math. Soc. 34 (1986), 274-290. 2. A. Balinsky, W. D. Evans, D. Hundertmark and R. T. Lewis, On inequalities of Hardy-Sobolev type, preprint. 3. G. Barbatis, S. Filippas and A. Tertikas, A unified approach to improved Lp Hardy inequalities with best constants, Trans. Amer. Math. Soc. 356 (6) (2004), 2169-2196. 4. G. Barbatis, Improved Rellich inequalities for the polyharmonic operator, Indiana Univ. Math. J. 55(4) (2006), 1401-1422. 5. D. M. Bennett, An extension of Rellich’s inequality, Proc. Amer. Math. Soc. 106 (1989), 987-993. 6. H. Brezis and M. Marcus, Hardy’s inequalities revisited, Dedicated to Ennio De Giorgi, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25 (1997, 1998), 217-237. 7. H. Brezis and J.-L. V´ azquez, Blow-up solutions of some non-linear elliptic problems, Rev. Mat. Univ. Comp. Madrid 10 (1997), 443-469. 8. A. Cohen, W. Dahmen, I. Daubechies and R. DeVore, Harmonic analysis of the space BV, Rev. Mat. Iberoamericana 19 (2003), 235-263. 9. A. Cohen, R. DeVore, P. Petrushev and H. Xu, Non-linear approximation and the space BV (R2 ), Amer. J. Math. 121 (1999), 587-628. 10. A. Cohen, Y. Meyer and F. Oru, Improved Sobolev inequalities, S´eminaires ´ X-EDP, Ecole Polytechnique (1998). 11. E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Univ. Press, Cambridge, 1989. 12. E. B. Davies, The Hardy constant, Quart. J. Math. Oxford (2), 46 (1995), 417-431.

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13. E. B. Davies, Spectral Theory and Differential Operators, Cambridge Studies in Advanced Mathematics, Vol. 42, Cambridge Univ. Press, Cambridge, 1995. 14. E. B. Davies and A. Hinz, Explicit constants for Rellich inequalities in Lp (Ω), Math. Z. 227 (1998), 511-523. 15. D. E. Edmunds and W.D. Evans, Hardy Operators, Function Spaces and Embeddings, Springer, Berlin, Heidelberg, New York, 2004. 16. W. D. Evans and R. T. Lewis, On the Rellich inequality with magnetic potentials, Math. Z. 251 (2005), 267-284. 17. W. D. Evans and R. T. Lewis, Hardy and Rellich inequalities with remainders, to appear in J. Math. Inequal. 18. S. Filippas, V. Maz’ya, and A. Tertikas, On a question of Brezis and Marcus, Calc. Var. Partial Differential Equations 25(4) (2006), 491-501. 19. R. L. Frank, E. H. Lieb and R. Seiringer, Hardy-Lieb-Thirring inequalities for fractional Schr¨ odinger operators, preprint (2006). http://xxx.lanl.gov/math.SP/0610593 20. M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and A. Laptev, A geometrical version of Hardy’s inequality, J. Funct. Anal. 189 (2002), 539-548. 21. A. Laptev and T. Weidl, Hardy inequalities for magnetic Dirichlet forms, in Mathematical results in quantum mechanics (Prague, 1998), 299-305, Oper. Theory Adv. Appl., 108, Birkh¨ auser, Basel, 1999. 22. R. T. Lewis, Singular elliptic operators of second-order with purely discrete spectrum, Trans. Amer. Math. Soc. 271 (1982), 653-666. 23. M. Ledoux, On improved Sobolev embedding theorems, Math. Res. Lett. 10 (2003), 659-669. 24. M. Marcus, V. J. Mizel, and Y. Pinchover, On the best constant for Hardy’s inequality in Rn , Trans. Amer. Math. Soc. 350 (1998), 3237-3255. 25. T. Matskewich and P. E. Sobolevskii, The best possible constant in a generalized Hardy’s inequality for convex domains in Rn , Nonlinear Analysis TMA, 28 (1997), 1601-1610. 26. M. P. Owen, The Hardy-Rellich inequality for polyharmonic operators, Proc. Royal Society of Edinburgh A 129 (1999), 825-839. 27. F. Rellich, Halbbeschr¨ ankte Differentialoperatoren h¨ oherer Ordnung, in Proceedings of the International Congress of Mathematicians 1954 (J. C. H. Gerretsen, J. de Groot, eds.), Vol. III (pp. 243-250), Groningen, Noordhoff, 1956. 28. F. Rellich and J. Berkowitz, Perturbation Theory of Eigenvalue Problems, Gordon and Breach, New York, 1969. 29. J. Stubbe, Bounds on the number of bound states for potentials with critical decay at infinity, J. Math. Phys. 31, no. 5, (1990), 1177-1180. 30. J. C. Thomas, Some problems associated with sum and integral inequalities, Ph.D. thesis, Cardiff University, 2007. 31. J. Tidblom, A geometrical version of Hardy’s inequality for W01,p (Ω), Proc. Amer. Math. Soc. 132(8) (2004), 2265-2271.

45

SOME RESULTS ON NONSELFADJOINT OPERATORS: A SURVEY ¨ JOHANNES SJOSTRAND Centre de Math´ ematiques Laurent Schwartz, Ecole Polytechnique, FR-91128 Palaiseau C´ edex, France and UMR 7640, CNRS E-mail: [email protected] This text is a survey of recent results obtained by the author and collaborators on different problems for non-self-adjoint operators. The topics are: Kramers-Fokker-Planck type operators, spectral asymptotics in two dimensions and Weyl asymptotics for the eigenvalues of non-self-adjoint operators with small random perturbations. In the introduction we also review the notion of pseudo-spectrum and its relation to non-self-adjoint spectral problems. Keywords: Non-self-adjoint, spectrum, resolvent, random, Fokker-Planck.

1. Introduction 1.1. Some background For self-adjoint and more generally normal operators on some complex Hilbert space H we have a nice theory, including the spectral theorem and a nice estimate on the norm of the resolvent: k(z − P )−1 k ≤ (dist (z, σ(P )))−1 ,

σ(P ) = the spectrum of P.

(1)

This has a consequence for the corresponding evolution problem: If σ(P ) ⊂ {z ∈ C; ℜz ≥ λ0 }, then ke−tP k ≤ e−λ0 t , t ≥ 0.

(2)

However, non-normal operators appear frequently: Scattering poles, convection-diffusion problems, Kramers-Fokker-Planck equation, damped wave equations, linearized operators in fluid dynamics. Then typically, k(z − P )−1 k may be very large even when z is far from the spectrum. This implies mathematical difficulties: – When studying the distribution of eigenvalues,

46

J. Sj¨ ostrand

– When studying functions of the operator, like e−tP and its norm. It also implies numerical difficulties like: – Eigenvalue instability. There are (in the author’s opinion) two ways out: – Change the Hilbert space norm to make the operators look more normal. (Complex scaling methods.) – Recognize that the region of the z-plane where k(z − P )−1 k is large, has its own interest. (Pseudospectrum.) The option to choose depends on the problem. • In some problems, like those related to scattering poles, there is no obvious choice of Hilbert space and we are free to make the most natural one. This option is particularly natural when considering a differential operator with analytic coefficients. • In other problems the canonical Hilbert space is L2 and we are at most allowed to change the norm into an equivalent one. Here the notion of pseudospectrum is likely to be important. Let P : H → H be closed, densely defined, H a complex Hilbert space and let ρ(P ) = C\σ(P ) denote the resolvent set. The notion of pseudospectrum is important in numerical analysis and we refer to L.N. Trefethen [52], Trefethen–M. Embree [54] and further references given there. Thanks to works of E.B. Davies [5], [7], M. Zworski [57] and others it has become popular in the non-self-adjoint spectral theory of differential operators. Definition 1.1. Let ǫ > 0. The ǫ-pseudospectrum of P is σǫ (P ) := σ(P ) ∪ {z ∈ ρ(P ); k(z − P )−1 k > 1/ǫ}. Unlike the spectrum, the pseudospectrum will in general change when we change the norm on H. Moreover, it can be characterized as a set of spectral instability as follows from the following version of a theorem of Roch-Silberman [43]: Theorem 1.1. σǫ (P ) =

S { σ(P + Q) : Q ∈ L(H, H), kQk < ǫ }.

In his survey [52] L.N. Trefethen discusses some linearized operators from fluid dynamics: • Orr-Sommerfeld equation (Orzag, Reddy, Schmid, Hennigson). • Plane Poiseuille flow (L.N and A.N Trefethen, Schmid). • Pipe Poiseuille flow (L.N and A.N Trefethen, Reddy, Driscoll),

Some Results on Nonselfadjoint Operators: A Survey

47

and to what extent stability can be predicted from the study of the spectrum of these non-self-adjoint operators: Eigenvalue analysis alone leads in some cases to the prediction of stability for Reynolds numbers R < 5772. Experimentally however, we have stability only for R < 1000. The rough explanation of this is that the ǫ-pseudospectrum (for a suitable ǫ) crosses the imaginary axis before the spectrum does, when R increases. Then ke−tP k will grow fast for a limited time even though the growth for very large times is determined by the spectrum. However, since P appears as a linearization of a non-linear problem, that suffices to cause instability. In the case of differential operators the pseudospectral phenomenon is very general and related to classical works in PDE on local solvability and non-hypoellipticity. E.B. Davies [5] studied the non-self-adjoint semiclasscial Schr¨ odinger operator with a smooth (complex-valued) potential in dimension 1 and showed under “generic” assumptions that one can construct quasimodes with the spectral parameter varying in an open complex set, containing points that are possibly very far from the spectrum (as can be verified in the case of the complex harmonic oscillator). M. Zworski [57] observed that this is essentially a rediscovery of an old result of H¨ ormander [32,33], and was able to generalize considerably Davies’ result by adapting the one of H¨ ormander to the semi-classical case. With N. Dencker and M.Zworski [9] we also gave a direct proof and a corresponding adaptation of old results of Sato-Kawai-Kashiwara [44] to the analytic case: Theorem 1.2. ([9,57]) Let P (x, hDx ) =

X

|α|≤m

aα (x)(hDx )α , Dx =

∂ ∂x

(3)

have smooth coefficients in the open set Ω ⊂ Rn . Put p(x, ξ) = P α = p(x0 , ξ0 ) with the Poisson bracket |α|≤m aα (x)ξ . Assume z 1 ∞ i {p, p}(x0 , ξ0 ) > 0. Then there exists a u = uh ∈ C0 (Ω) with kuk = 1 ∞ such that k(P − z)uk = O(h ) when h → 0. Analytic case: We can replace “h∞ ” by “e−1/Ch ”. Here, we have used standard multi-index notation: ξ α = ξ1α1 · ... · ξnαn , |α| = α1 + ... + αn , and the norms k · k are the ones of L2 or ℓ2 if nothing else is indicated. This result was subsequently generalized by K. Pravda-Starov [42]. Notice that this implies that when the theorem applies and if the resolvent

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J. Sj¨ ostrand

(P − z)−1 exists then its norm is greater than any negative power of h when h → 0. Example 1.1. Let P = −h2 ∆+V (x), p(x, ξ) = ξ 2 +V (x), ξ 2 = ξ12 +...+ξn2 . Then 1i {p, p} = −4ξ · ℑV ′ (x). “Generically”, if z = p(x, ξ), then {p, p}(x, ξ) 6= 0 and one can show quite generally that if this happens then there is also another point (y, η) with p(y, η) = z such that {p, p}(x, ξ) takes the opposite sign. This justifies the following simplified terminology in the semi-classical limit: The semi-classical pseudospectrum of P is the range R(p) of p. In [9] we also showed under suitable assumptions (inspired from scattering theory and from the theory of sub-elliptic operators), that there is no spectrum near the boundary of the semi-classical pseudospectrum and that we may have quite a good control of the norm of the resolvent there. Generalizations to the case of systems were given by Dencker [8]. 1.2. The topics of this survey We will discuss three subjects involving non-self-adjoint differential and pseudodifferential operators, we will always work in the semi-classical limit, which means that our operators are of the form P (x, hDx ; h), where P is a suitable symbol and 0 < h ≪ 1. It is quite clear however that some of our results will also apply to non-semiclassical situations in the limit of large eigenvalues. The subjects are: • The Kramers-Fokker-Planck operator, • Bohr-Sommerfeld rules in dimension 2, • Weyl asymptotics for non-self-adjoint operators with small random perturbations. and most of the works discussed are the results of collaborations with A. Melin, M. Hitrik, F. H´erau, C. Stolk, S. V˜ u Ngo.c and M. Hager. In the first two topics we exploit the possibility of changing the Hilbert space norm by introducing exponential weights on phase space. In the case of the Kramers-Fokker-Planck operator, we make no analyticity assumptions and the phase-space weights are correspondingly quite weak. In this case however our operator is a differential one, so we are allowed to apply strong exponential weights depending only on the base variables, and this is important when studying small exponential corrections of the eigenvalues via the so called tunnel effect.

Some Results on Nonselfadjoint Operators: A Survey

49

For the Bohr-Sommerfeld rules, we make analyticity assumptions that allow stronger phase space weights. In both cases the effect of the exponential weights is to make the operator under consideration more normal. In the third topic, we do not use any deformations of the given Hilbert space, but exponential weights play an important role at another level, namely to count zeros of holomorphic functions with exponential growth. The pseudospectrum will not be discussed explicitly below. In the Kramers-Fokker-Planck case, the problems are located near the boundary of the semi-classical pseudospectrum, and it turns out that we have a very nice control of the resolvent there. In the 2 dimensional Bohr-Sommerfeld rules, we have stronger exponential weights, reflecting stronger pseudospectral phenomena. Finally in the subject of Weyl asymptotics, we often have strong pseudospectral behaviour for the unperturbed operator. From the proofs it appears that the random perturbations will weaken the pseudospectral behaviour and this might have very interesting consequences for the associated evolution problems. This is still very much an open problem. 2. Kramers-Fokker-Planck type operators, spectrum and return to equilibrium 2.1. Introduction There has been a renewed interest in the problem of “return to equilibrium” for various 2nd order operators. One example is the Kramers-Fokker-Planck operator: γ (4) P = y · h∂x − V ′ (x) · h∂y + (−h∂y + y) · (h∂y + y), 2 where x, y ∈ Rn correspond respectively to position and speed of the particles and h > 0 corresponds to temperature. The constant γ > 0 is the friction. (Since we will only discuss L2 aspects we here present right away an adapted version of the operator, obtained after conjugation by a Maxwellian factor.) The associated evolution equation is: (h∂t + P )u(t, x, y) = 0. Problem of return to equilibrium: Study the rate of convergence of u(t, x, y) 2 to a multiple of the “ground state” u0 (x, y) = e−(y /2+V (x))/h when t → +∞, assuming that V (x) → +∞ sufficiently fast when x → ∞ so that u0 ∈ L2 (R2n ). Notice here that P (u0 ) = 0 and that the vector field part of P is h times the Hamilton field of y 2 /2 + V (x), when we identify R2n x,y with the cotangent space of Rnx .

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J. Sj¨ ostrand

A closely related problem is to study the difference between the first eigenvalue (0) and the next one, µ(h). (Since our operator is non-selfadjoint, this is only a very approximate formulation however.) Some contributions: L. Desvillettes–C. Villani [10], J.P. Eckmann– M. Hairer [12], F. H´erau–F. Nier [27], B. Helffer–F. Nier [22], Villani [55]. In the work [27] precise estimates on the exponential rates of return to equilibrium were obtained with methods close to those used in hypoellipticity studies and this work was our starting point. With H´erau and C. Stolk [28] we made a study in the semi-classical limit and studied small eigenvalues modulo O(h∞ ). More recently with H´erau and M. Hitrik [25] we have made a precise study of the exponential decay of µ(h) when V has two local minima (and in that case µ(h) turns out to be real). This involves tunneling, i.e. the study of the exponential decay of eigenfunctions. As an application we have a precise result on the return to equilibrium [26]. This has many similarities with older work on the tunnel effect for Schr¨ odinger operators in the semi-classical limit by B. Helffer–Sj¨ ostrand [23,24] and B. Simon [45] but for the Kramers-Fokker-Planck operator the problem is richer and more difficult since P is neither elliptic nor self-adjoint. We have used a supersymmetry observation of J.M. Bismut [1] and J. Tailleur– S. Tanase-Nicola–J. Kurchan [51] that allow arguments similar to those for the standard Witten complex [24]. 2.2. Statement of the main results Let P be given by (4) where V ∈ C ∞ (Rn ; R), and ∂ α V (x) = O(1), |α| ≥ 2,

(5)

|∇V (x)| ≥ 1/C, |x| ≥ C,

(6)

V is a Morse function.

(7)

We also let P denote the graph closure of P from S(R2n ) which coincides with the maximal extension of P in L2 (see [22,26,27]). We have ℜP ≥ 0 and the spectrum of P is contained in the right half plane. In [28] the spectrum in any strip 0 ≤ ℜz ≤ Ch (and actually in a larger parabolic neighborhood of the imaginary axis, in the spirit of [27]) was determined asymptotically mod (O(h∞ )). It is discrete and contained in a sector |ℑz| ≤ Cℜz + O(h∞ ): Theorem 2.1. The eigenvalues in the strip 0 ≤ ℜz ≤ Ch are of the form λj,k (h) ∼ h(µj,k + h1/Nj,k µj,k,1 + h2/Nj,k µj,k,2 + ..)

(8)

Some Results on Nonselfadjoint Operators: A Survey

51

where µj,k are the eigenvalues of the quadratic approximation (“nonselfadjoint oscillator”) γ y · ∂x − V ′′ (xj )x · ∂y + (−∂y + y) · (∂y + y), 2 at the points (xj , 0), where xj are the critical points of V . The µj,k are known explicitly and it follows that when xj is not a local minimum, then ℜλj,k ≥ h/C for some C > 0. When xj is a local minimum, then precisely one of the λj,k is O(h∞ ) while the others have real part ≥ h/C. Furthermore, when V → +∞ as x → ∞, then 0 is a simple eigenvalue. In particular, if V has only one local minimum, then inf ℜ(σ(P ) \ {0}) ∼ h(µ1 + hµ2 + . . . ),

µ1 > 0

(or possibly an expansion in fractional powers) and we obtained a corresponding result for the problem of return to equilibrium. It should be added that when µj,k is a simple eigenvalue of the quadratic approximation then Nj,k = 1 so there are no fractional powers of h in (8). The following is the main new result that we obtained with F. H´erau and M. Hitrik in [25]: Theorem 2.2. Assume that V has precisely 3 critical points; 2 local minima, x±1 and one “saddle point”, x0 of index 1. Then for C > 0 sufficiently large and h sufficiently small, P has precisely 2 eigenvalues in the strip 0 ≤ ℜz ≤ h/C, namely 0 and µ(h), where µ(h) is real and of the form µ(h) = h(a1 (h)e−2S1 /h + a−1 (h)e−2S−1 /h ),

(9)

where aj are real, aj (h) ∼ aj,0 + haj,1 + ..., h → 0,

aj,0 > 0,

Sj = V (x0 ) − V (xj ). As for the problem of return to equilibrium, we obtained the following result with F. H´erau and M. Hitrik in [26]: Theorem 2.3. We make the same assumptions as in Theorem 2.2 and let Πj be the spectral projection associated with the eigenvalue µj , j = 0, 1, where µ0 = 0, µ1 = µ(h). Then we have Πj = O(1) : L2 → L2 ,

h → 0.

(10)

We have furthermore, uniformly as t ≥ 0 and h → 0,

e−tP/h = Π0 + e−tµ1 /h Π1 + O(1)e−t/C , in L(L2 , L2 ),

(11)

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J. Sj¨ ostrand

where C > 0 is a constant. Actually, as we shall see in the outline of the proofs, these results (as well as (8)) hold for more general classes of supersymmetric operators. 2.3. A partial generalization of [28] Consider on Rn (2n is now replaced by n): X 1X (cj (x)h∂xj + h∂xj ◦ cj (x)) + p0 (x) P = hDxj bj,k (x)hDxk + 2 j j,k

= P2 + iP1 + P0 ,

where bj,k , cj , p0 are real and smooth. The associated symbols are: p(x, ξ) = p2 (x, ξ) + ip1 (x, ξ) + p0 (x), X X p2 = bj,k ξj ξk , p1 = cj ξj .

Assume p2 ≥ 0, p0 ≥ 0,

∂xα bj,k = O(1), |α| ≥ 0, ∂xα cj = O(1), |α| ≥ 1,

∂xα p0 = O(1), |α| ≥ 2. Assume that

{x; p0 (x) = c1 (x) = .. = cn (x) = 0} is finite = {x1 , ..., xN } and put C = {ρ1 , ..., ρn }, ρj = (xj , 0). Put p pe(x, ξ) = hξi−2 p2 (x, ξ) + p0 (x), hξi = 1 + |ξ|2 he piT0 =

1 T0

Z

T0 /2

−T0 /2

pe ◦ exp(tHp1 )dt, T0 > 0 fixed.

∂ ∂ − a′x · ∂ξ denote the Hamilton field of Here in general we let Ha = a′ξ · ∂x 1 the C -function a = a(x, ξ). Dynamical assumptions: Near each ρj we have he piT0 ∼ |ρ − ρj |2 and in any compact set disjoint from C we have he piT0 ≥ 1/C. (Near infinity this last assumption has to be modified slightly and we refer to [25] for the details.) The following result from [25] is very close to the main result of [28] and generalizes Theorem 2.1:

Theorem 2.4. Under the above assumptions, the spectrum of P is discrete in any band 0 ≤ ℜz ≤ Ch and the eigenvalues have asymptotic expansions as in (8).

Some Results on Nonselfadjoint Operators: A Survey

53

Put q(x, ξ) = −p(x, iξ) = p2 (x, ξ) + p1 (x, ξ) − p0 (x). The linearization of the Hamilton field Hq at ρj (for any fixed j) has eigenvalues ±αk , k = 1, .., n with real part 6= 0. Let Λ+ = Λ+,j be the unstable manifold through ρj for the Hq -flow. Then Λ+ is Lagrangian and of the form ξ = φ′+ (x) near xj (φ+ = φ+,j ), where φ+ (xj ) = 0, φ′+ (xj ) = 0, φ′′+ (xj ) > 0. The next result is from [25]: Theorem 2.5. Let λj,k (h) be a simple eigenvalue as in (8) and assume there is no other eigenvalue in a disc D(λj,k , h/C) for some C > 0. Then, in the L2 sense, the corresponding eigenfunction is of the form e−φ+ (x)/h (a(x; h) + O(h∞ )) near xj , where a(x; h) is smooth in x with an asymptotic expansion in powers of h. Away from a small neighborhood of xj it is exponentially decreasing. The proof of the first theorem uses microlocal weak exponential estimates, while the one of the last theorem also uses local exponential estimates. 2.4. Averaging and exponential weights. The basic idea of the proof of Theorem 2.4 is taken from [28], but we reworked it in order to allow for non-hypoelliptic operators. We will introduce a weight on T ∗ Rn of the form Z t (12) pǫ ◦ exp(tHp1 )dt, ψǫ = − J( )e T0

for 0 < ǫ ≪ 1. Here J(t) is the odd function given by  0, |t| ≥ 21 , J(t) = 1 1 2 − t, 0 < t ≤ 2 ,

(13)

and we choose peǫ (ρ) to be equal to pe(ρ) when dist (ρ, C) ≤ ǫ, and flatten out to ǫe p away from a fixed neighborhood of C in such a way that peǫ = O(ǫ). Then pǫ iT0 − peǫ . Hp1 ψǫ = he

(14)

We let ǫ = Ah where A ≫ 1 is independent of h. Then the weight exp(ψǫ /h) is uniformly bounded when h → 0. Indeed, ψǫ = O(h). Using Fourier integral operators with complex phase, we can define a Hilbert space of functions that are “microlocally O(exp(ψǫ /h)) in the L2

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sense”. The norm is uniformly equivalent to the one of L2 , but the natural leading symbol of P , acting in the new space, becomes p(exp(iHψǫ )(ρ)), ρ ∈ T ∗ Rn

(15)

which by Taylor expansion has real part ≈ p2 (ρ) + p0 (ρ) + he pǫ i − peǫ . Very roughly, the real part of the new symbol is ≥ ǫ away from C and √ behaves like dist (ρ, C)2 in a ǫ-neighborhood of C. This can be used to show that the spectrum of P (viewed as an operator on the weighted space) in √ a band 0 ≤ ℜz < ǫ/C comes from an ǫ-neighborhood of C. In such a neighborhood, we can treat P as an elliptic operator and the spectrum is to leading order determined by the quadratic approximation of the dilated symbol (15). This gives Theorem 2.4. We next turn to the proof of Theorem 2.5, and we work near a point ρj = (xj , ξj ) ∈ C. Recall that Λ+ : ξ = φ′+ (x) is the unstable manifold for the Hq -flow, where q(x, ξ) = −p(x, iξ). We have q(x, φ′+ (x)) = 0. In general, if ψ ∈ C ∞ is real, then Pψ := eψ/h ◦P ◦e−ψ/h has the symbol pψ (x, ξ) = p2 (x, ξ)−q(x, ψ ′ (x)) + i(qξ′ (x, ψ ′ (x)) · ξ

(16)

• As long as q(x, ψ ′ (x)) ≤ 0, we have ℜpψ ≥ 0 and we may hope to establish good apriori estimates for Pψ . • This is the case for ψ = 0 and for ψ = φ+ . Using the convexity of q(x, ·), we get suitable weights ψ with q(x, ψ ′ (x)) ≤ 0, equal to φ+ (x) near xj , strictly positive away from xj and constant outside a neighborhood of that point. • It follows that the eigenfunction in Theorem 2.5 is (roughly) O(e−φ+ (x)/h ) near xj in the L2 sense. • On the other hand, we have quasi-modes of the form a(x; h)e−φ+ (x)/h as in [23]. • Applying the exponentially weighted estimates, indicated above, to the difference of the eigenfunction and the quasi-mode, we then get Theorem 2.5. 2.5. Supersymmetry and the proof of Theorem 2.2 We review the supersymmetry from [1], [51], see also G. Lebeau [36]. Let A(x) : Tx∗ Rn → Tx Rn be linear, invertible and smooth in x. Then we have the nondegenerate bilinear form hu|viA(x) = h∧k A(x)u|vi, u, v ∈ ∧k Tx∗ Rn , and we also write (u|v)A(x) = hu|viA(x) .

Some Results on Nonselfadjoint Operators: A Survey

55

If u, v are smooth k-forms with compact support, put Z (u|v)A = (u(x)|v(x))A(x) dx.

The formal “adjoint” QA,∗ of an operator Q is then given by (Qu|v)A = (u|QA,∗ v)A . Let φ : Rn → R be a smooth Morse function with ∂ α φ bounded for |α| ≥ 2 and with |∇φ| ≥ 1/C for |x| ≥ C. Introduce the Witten-De Rham complex: X φ φ (h∂xj + ∂xj φ) ◦ dx∧ dφ = e− h ◦ hd ◦ e h = j, j

where d denotes exterior differentiation and dx∧ j left exterior multiplication A,∗ with dxj . The corresponding Laplacian is then: −∆A = dA,∗ φ dφ + dφ dφ . (q)

Its restriction to q-forms will be denoted by −∆A . Notice that: (0)

−∆A (e−φ/h ) = 0.

Write A = B + C with B t = B, C t = −C. −∆A is a second order differential operator with scalar principal symbol in the semi-classical sense ∂ ( hi ∂x 7→ ξj ) of the form: j X X p(x, ξ) = bj,k (ξj ξk + ∂xj φ∂xk φ) + 2i cj,k ∂xk φ ξj . j,k

j,k

Example 2.1. Replace n by 2n, x by (x, y), let   1 0 I . A= 2 −I γ

Then

(0)

−∆A = h(φ′y · ∂x − φ′x · ∂y ) γX + (−h∂yj + ∂yj φ)(h∂yj + ∂yj φ). 2 j

When φ = y 2 /2 + V (x) we recover the KFP operator (4).

The results of Subsection 2.3 apply, if we make the additional dynamical (q) assumptions there; −∆A has an asymptotic eigenvalue = o(h) associated to the critical point xj precisely when the index of xj is equal to q (as for the Witten complex and analogous complexes in several complex variables). In order to cover the cases q > 0 we also assume that A = Const.

(17)

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The Double well case. Keep the assumption (17). Assume that φ is a Morse function with |∇φ| ≥ 1/C for |x| ≥ C such that −∆A satisfies the extra dynamical conditions of Subsection 2.3 and having precisely three critical points, two local minima U±1 and a saddle point U0 of index 1. (1) (0) Then −∆A has precisely 2 eigenvalues: 0, µ that are o(h) while −∆A has precisely one such eigenvalue: µ. (Here we use as in the study of the Witten complex, that dφ and dA,∗ intertwine our Laplacians in degeree φ 0 and 1. The detailed justification is more complicated however.) e−φ/h (0) is the eigenfunction of ∆A corresponding to the eigenvalue 0. Let Sj = φ(U0 ) − φ(Uj ), j = ±1, and let Dj be the connected component of {x ∈ Rn ; φ(x) < φ(U0 )} containing Uj in its interior. Let E (q) be the corresponding spectral subspaces so that dim E (0) = 2, dim E (1) = 1. Truncated versions of the function e−φ(x)/h can be used as approximate eigenfunctions, and we can show the following. Proposition 2.1. E (0) has a basis e1 , e−1 , where 1

1

ej = χj (x)e− h (φ(x)−φ(Uj )) + O(e− h (Sj −ǫ) ), in the L2 -sense.

Here, we let χj ∈ C0∞ (Dj ) be equal to 1 on {x ∈ Dj ; φ(x) ≤ φ(U0 ) − ǫ}. (1)

Theorems 2.4, 2.5 can be adapted to −∆A and lead to the following. Proposition 2.2. E (1) = Ce0 , where 1

e0 (x) = χ0 (x)a0 (x; h)e− h φ+ (x) + O(e−ǫ0 /h ),

φ+ (x) ∼ (x − U0 )2 , ǫ0 > 0 is small enough, a0 is an elliptic symbol, χ0 ∈ C0∞ (Rn ), χ0 = 1 near U0 . Let the matrices of dφ : E (0) → E (1) and dA,∗ : E (1) → E (0) with φ respect to the bases {e−1 , e1 } and {e0 } be  ∗ 
λ−1 respectively. λ−1 λ1 and λ∗1 Using the preceding two results in the spirit of tunneling estimates and computations of Helffer–Sj¨ ostrand ([23,24]) we can show the following. Proposition 2.3. Put Sj = φ(U0 ) − φ(Uj ), j = ±1. Then we have     1 1 λ−1 ℓ (h)e−S−1 /h = h 2 (I + O(e− Ch )) −1 , λ1 ℓ1 (h)e−S1 /h 

λ∗−1 λ∗1



1 2

= h (I + O(e

1 − Ch

))



ℓ∗−1 (h)e−S−1 /h ℓ∗1 (h)e−S1 /h



,

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57

where ℓ±1 , ℓ∗±1 are real elliptic symbols of order 0 such that ℓj ℓ∗j > 0, j = ±1. From this we get Theorem 6, since µ = λ∗−1 λ−1 + λ∗1 λ1 .



Thanks to the fact that we have only two local minima, certain simplifications were possible in the proof. In particular it was sufficent to control the exponential decay of general eigenfunctions in some small neighborhood of the critical points. For more general configurations, it might be necessary to get such a control also further away and this seems to lead to interesting questions, involving degenerate and non-symmetric Finsler distances. 2.6. Return to equilibrium, ideas of the proof of Theorem 2.3 Keeping the same assumptions, let Π0 , Π1 be the rank 1 spectral projections (0) corresponding to the eigenvalues µ0 := 0, µ1 := µ of −∆A and put Π = Π0 + Π1 . Then e−1, , e1 is a basis for R(Π) and the restriction of P to this range, has the matrix   ∗   ∗
λ−1 λ−1 λ−1 λ∗−1 λ1 (18) λ−1 λ1 = λ∗1 λ∗1 λ−1 λ∗1 λ1 with the eigenvalues 0 and µ = λ∗−1 λ−1 + λ∗1 λ1 . A corresponding basis of eigenvectors is given by 1 (19) v0 = √ (λ1 e−1 − λ−1 e−1 ) µ1 1 v1 = √ (λ∗−1 e−1 + λ∗1 e−1 ). µ1

The corresponding dual basis of eigenfunctions of P ∗ is given by 1 v0∗ = √ (λ∗1 e∗−1 − λ∗−1 e∗−1 ) µ 1 v1∗ = √ (λ−1 e∗−1 + λ1 e∗1 ), µ

(20)

where e∗−1 , e∗1 ∈ R(Π∗ ) is the basis that is dual to e−1 , e1 . It follows that vj , vj∗ = O(1) in L2 , when h → 0. From this discussion we conclude that Πj = (·|vj∗ )vj , are uniformly bounded when h → 0. A non-trivial fact, based on the analysis described in Subsections 2.3, 2.4, is that after replacing the standard norm and scalar product on L2 by certain uniformly equivalent ones, we have h e ℜ(P u|u) ≥ kuk2 , ∀u ∈ R(1 − Π), (21) C

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e is the spectral projection corresponding to the spectrum of P in where Π D(0, Bh) for some B ≫ 1. This can be applied to the study of u(t) := e−tP/h u(0), where the initial state u(0) ∈ L2 is arbitrary: Write u(0) = Π0 u(0) + Π1 u(0) + (1 − Π)u(0) =: u0 + u1 + u⊥ .

(22)

Then ku0 k, ku1 k, ku⊥ k ≤ O(1)ku(0)k ke

−tP/h ⊥

e

u k ≤ Ce

−tP/h

uj = e

−t/C

−tµj /h

ku(0)k

uj , j = 0, 1.

(23) (24) (25)

e + (Π − Π)u, e Here (24) follows if we write u⊥ = (1 − Π)u apply (21) to the evolution of the first term, and use that the last term is the (bounded) spectral projection of u to a finite dimensional spectral subspace of P , for which the corresponding eigenvalues all have real part ≥ h/C.  3. Spectral asymptotics in 2 dimensions 3.1. Introduction This section is mainly based on recent joint works with S. V˜ u Ngo.c and M. Hitrik [29] [30], but we shall start by recalling some earlier results that we obtained with A. Melin [40] where we discovered that in the two dimensional case one often can have Bohr-Sommerfeld conditions to determine all the individual eigenvalues in some region of the spectral plane, provided that we have analyticity. This was first a surprise for us since in the self-adjoint case such results are known only in 1 dimension and in very special cases for higher dimensions. Subsequently, with M. Hitrik we have studied small perturbations of self-adjoint operators. First we studied the case when the classical flow of the unperturbed operator is periodic, then also with S. V˜ u Ngo.c we looked at the more general case when it is completely integrable, or just when the energy surface contains some invariant diophantine Lagrangian tori. 3.2. Bohr-Sommerfeld rules in two dimensions For (pseudo-)differential operators in dimension 1, we often have a BohrSommerfeld rule to determine the asymptotic behaviour of the eigenvalues. Consider for instance the semi-classical Schr¨ odinger operator P = −h2

d2 + V (x), with symbol p(x, ξ) = ξ 2 + V (x), dx2

Some Results on Nonselfadjoint Operators: A Survey

59

where we assume that V ∈ C ∞ (R; R) and V (x) → +∞, |x| → ∞. Let E0 ∈ R be a non-critical value of V such that (for simplicity) {x ∈ R; V (x) ≤ E0 } is an interval. Then in some small fixed neighborhood of E0 and for h > 0 small enough, the eigenvalues of P are of the form E = Ek , k ∈ Z, where Z I(E) ξ · dx, θ(E; h) ∼ θ0 (E) + θ1 (E)h + ... = k − θ(E; h), I(E) = 2πh p−1 (E) In the non-self-adjoint case we get the same results, provided that ℑV is small and V is analytic. The eigenvalues will then be on a curve close to the real axis. For self-adjoint operators in dimension ≥ 2 it is generally admitted that Bohr-Sommerfeld rules do not give all eigenvalues in any fixed domain except in certain (completely integrable) cases. Using the KAM theorem one can sometimes describe some fraction of the eigenvalues. With A. Melin [40]: we considered an h-pseudodifferential operator with leading symbol p(x, ξ) that is bounded and holomorphic in a tubular neighborhood of R4 in C4 = C2x × C2ξ . Assume that R4 ∩ p−1 (0) 6= ∅ is connected.

(26)

On R4 we have |p(x, ξ)| ≥ 1/C, for |(x, ξ)| ≥ C,

(27)

for some C > 0, dℜp(x, ξ), dℑp(x, ξ) are linearly independent for all (x, ξ) ∈ p−1 (0) ∩ R4 . (28) (Here the boundedness assumption near ∞ and (27) can be replaced by a suitable ellipticity assumption.) It follows that p−1 (0) ∩ R4 is a compact (2-dimensional) surface. Also assume that |{ℜp, ℑp}| is sufficiently small on p−1 (0) ∩ R4 .

(29)

Here “sufficiently small” refers to some positive bound that can be defined whenever the the other conditions are satisfied uniformly. When the Poisson bracket vanishes on p−1 (0), this set becomes a Lagrangian torus, and more generally it is a torus. The following is a complex version of the KAM theorem without small divisors (cf T.W. Cherry [3](1928), J. Moser [41](1958)). Theorem 3.1. ([40]) There exists a smooth 2-dimensional torus Γ ⊂ p−1 (0)∩C4 , close to p−1 (0)∩R4 such that σ| Γ = 0 and Ij (Γ) ∈ R, j = 1, 2.

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J. Sj¨ ostrand

R

ξ · dx are the actions along the two fundamental cycles P γ1 , γ2 ⊂ Γ, and σ = 21 dξj ∧ dxj is the complex symplectic (2,0)-form. Here Ij (Γ) :=

γj

Replacing p by p − z for z in a neighborhood of 0 ∈ C, we get tori Γ(z) depending smoothly on z and a corresponding smooth action function I(z) = (I1 (Γ(z)), I2 (Γ(z))), which are important in the Bohr-Sommerfeld rule for the eigen-values near 0 in the semi-classical limit h → 0:

Theorem 3.2. ([40]) Under the above assumptions, there exists θ0 ∈ ( 21 Z)2 and θ(z; h) ∼ θ0 + θ1 (z)h + θ2 (z)h2 + .. in C ∞ (neigh (0, C)), such that for z in an h-independent neighborhood of 0 and for h > 0 sufficiently small, we have that z is an eigenvalue of P = p(x, hDx ) iff I(z) = k − θ(z; h), for some k ∈ Z2 . 2πh

(BS)

Recently, a similar result was obtained by S. Graffi, C. Villegas Bas [13]. An application of this result is that we get all resonances (scattering poles) in a fixed neighborhood of 0 ∈ C for −h2 ∆ + V (x) if V is an analytic real potential on R2 with a nondegenerate saddle point at x = 0, satisfying V (0) = 0 and having {(x, ξ) = (0, 0)} as its classically trapped set in the energy surface {p(x, ξ) = 0}. 3.3. Diophantine case In this and the next subsection we describe a result from [30] and the main result of [29] about individual eigenvalues for small perturbations of a selfadjoint operator with a completely integrable leading symbol. We start with the case when only Diophantine tori play a role. Let Pǫ (x, hD; h) on R2 have the leading symbol pǫ (x, ξ) = p(x, ξ) + iǫq(x, ξ) where p, q are real and extend to bounded holomorphic functions on a tubular neighborhood of R4 . Assume that p fulfills the ellipticity condition (27) near infinity and that Pǫ=0 = P (x, hD)

(30)

is self-adjoint. (The conditions near infinity can be modified and we can also replace R2x by a compact 2-dimensional analytic manifold.) Also, assume that Pǫ (x, ξ; h) depends smoothly on 0 ≤ ǫ ≤ ǫ0 with values in the space of bounded holomorphic functions in a tubular neighborhood of R4 , and Pǫ ∼ pǫ + hp1,ǫ + h2 p2,ǫ + ..., when h → 0. Assume p−1 (0) is connected and dp 6= 0 on that set.

(31)

Some Results on Nonselfadjoint Operators: A Survey

61

Assume complete integrability for p: There exists an analytic real valued function f on T ∗ R2 such that Hp f = 0, with the differentials df and dp ∂ − being linearly independent almost everywhere on p−1 (0). (Hp = p′ξ · ∂x ∂ ′ px · ∂ξ is the Hamilton field.) Then we have a disjoint union decomposition [ p−1 (0) ∩ T ∗ R2 = Λ, (32) Λ∈J

where Λ are compact connected sets, invariant under the Hp flow. We assume (for simplicity) that J has a natural structure of a graph whose edges correspond to families of regular leaves; Lagrangian tori (by the ArnoldMineur-Liouville theorem). The union of edges J \ S possesses a natural real analytic structure. Each torus Λ ∈ J \ S carries real analytic coordinates x1 , x2 identifying Λ with T2 = R2 /2πZ2 , so that along Λ, we have Hp = a 1

∂ ∂ + a2 , ∂x1 ∂x2

(33)

where a1 , a2 ∈ R. The rotation number is defined as the ratio ω(Λ) = [a1 : a2 ] ∈ RP1 , and it depends analytically on Λ ∈ J \ S. We assume that ω(Λ) is not identically constant on any open edge. We say that Λ ∈ J \ S is respectively rational, irrational, diophantine if a1 /a2 has the corresponding property. Diophantine means that there exist α > 0, d > 0 such that |(a1 , a2 ) · k| ≥

α , 0 6= k ∈ Z2 . |k|2+d

(34)

We introduce hqiT =

1 T

Z

T /2

−T /2

q ◦ exp(tHp )dt, T > 0,

(35)

and consider the compact intervals Q∞ (Λ) ⊂ R, Λ ∈ J, defined by, Q∞ (Λ) = [ lim inf hqiT , lim suphqiT ]. T →∞ Λ

T →∞ Λ

(36)

A first localization of the spectrum σ(Pǫ (x, hDx ; h)) ([30]) is given by [ [ ℑ(σ(Pǫ ) ∩ {z; |ℜz| ≤ δ}) ⊂ ǫ[inf Q∞ (Λ) − o(1), sup Q∞ (Λ) + o(1)], Λ∈J

Λ∈J

(37)

when δ, ǫ, h → 0.

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For each torus Λ ∈ J \S, we let hqi(Λ) be the average of q| Λ with respect to the natural smooth measure on Λ, and assume that the analytic function J \ S ∋ Λ 7→ hqi(Λ) is not identically constant on any open edge. By combining (33) with the Fourier series representation of q, we see that when Λ is irrational then Q∞ (Λ) = {hqi(Λ)}, while in the rational case, Q∞ (Λ) ⊂ hqi(Λ) + O(

1 )[−1, 1], (|n| + |m|)∞

(38)

dΛ hqi(Λd ) 6= 0.

(39)

when ω(Λ) = m n and m ∈ Z, n ∈ N are relatively prime. Let F0 ∈ ∪Λ∈J Q∞ (Λ) and assume that there exists a Diophantine torus Λd (or finitely many), such that hqi(Λd ) = F0 ,

With M. Hitrik and S. V˜ u Ngo.c we obtained: Theorem 3.3. ([30]) Assume also that F0 does not belong to Q∞ (Λ) for any other Λ ∈ J. Let 0 < δ < K < ∞. Then ∃C > 0 such that for h > 0 small enough, and k K ≤ ǫ ≤ hδ , the eigenvalues of Pǫ in the rectangle |ℜz| < hδ /C, |ℑz − ǫℜF0 | < ǫhδ /C are given by P (∞) (h(k −

k0 S )− , ǫ; h) + O(h∞ ), k ∈ Z2 . 4 2π

Here P (∞) (ξ, ǫ; h) is smooth, real-valued for ǫ = 0 and when h → 0 we have P (∞) (ξ, ǫ; h) ∼

∞ X

(∞)

hℓ p ℓ

(∞)

(ξ, ǫ), p0

ℓ=0

= p∞ (ξ) + iǫhqi(ξ) + O(ǫ2 ),

(40)

corresponding to action angle coordinates. In [30] we also considered applications to small non-self-adjoint perturbations of the Laplacian on a surface of revolution. Thanks to (38) the total measure of the union of all Q∞ (Λ) over the rational tori is finite and sometimes small, and we could then show that there are plenty of values F0 , fulfilling the assumptions in the theorem. 3.4. The case with rational tori Let F0 be as in (39) but now also allow for the possibility that there is a rational torus (or finitely many) Λr , such that F0 ∈ Q∞ (Λr ),

F0 6= hqi(Λr ),

(41)

Some Results on Nonselfadjoint Operators: A Survey

63

dΛ (hqi)(Λr ) 6= 0, dΛ (ω)(Λr ) 6= 0.

(42)

F0 6∈ Q∞ (Λ), for all Λ ∈ J \ {Λd , Λr }.

(43)

Assume also that

With M. Hitrik we showed the following result: 2

Theorem 3.4. ([29]) Let δ > 0 be small and assume that h ≪ ǫ ≤ h 3 +δ , 2 or that the subprincipal symbol of P vanishes and that h2 ≪ ǫ ≤ h 3 +δ . Then the spectrum of Pǫ in the rectangle ǫδ ǫδ ǫ ǫ , ] + iǫ[F0 − , F0 + ] C C C C is the union of two sets: Ed ∪ Er , where the elements of Ed form a distorted lattice, given by the Bohr-Sommerfeld rule (40), with horizontal spacing ≍ h and vertical spacing ≍ ǫh. The number of elements #(Er ) of Er is O(ǫ3/2 /h2 ). [−

NB that #(Ed ) ≍ ǫ1+δ /h2 . This result can be applied to the damped wave equation on surfaces of revolution. 3.5. Outline of the proofs of Theorem 3.3 and 3.4 The principal symbol of Pǫ is pǫ = p + iǫq + O(ǫ2 ). Put Z 1 T /2 q ◦ exp(tHp )dt. hqiT = T −T /2 As in Section 2 we will use an averaging of the imaginary part of the symbol. Let J(t) be the piecewise affine function with support in [− 12 , 12 ], solving J ′ (t) = δ(t) − 1[− 12 , 21 ] (t), and introduce the weight GT (t) =

Z

t J(− )q ◦ exp(tHp )dt. T

Then Hp GT = q − hqiT , implying pǫ ◦ exp(iǫHGT ) = p + iǫhqiT + OT (ǫ2 ).

(44)

The left hand side of (44) is the principal symbol of the isospectral ǫ ǫ operator e− h GT (x,hDx ) ◦ Pǫ ◦ e h GT (x,hDx ) and under the assumptions of Theorem 3.3 resp. 3.4 its imaginary part will not take the value iǫF0 on

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J. Sj¨ ostrand

p−1 (0) away from Λd resp. Λd ∪ Λr . This means that we have localized the spectral problem to a neighborhood of Λd resp. Λd ∪ Λr . Near Λd we choose action-angle coordinates so that Λd becomes the zero section in the cotangent space of the 2-torus, and pǫ (x, ξ) = p(ξ) + iǫq(x, ξ) + O(ǫ2 ).

(45)

We follow the quantized Birkhoff normal form procedure in the spirit of V.F. Lazutkin and Y. Colin de Verdi`ere [4,35]: solve first Hp G = q(x, ξ) − hq(·, ξ)i,

(46)

where the bracket indicates that we take the average over the torus with respect to x. Composing with the corresponding complex canonical transformation, we get the new conjugated symbol p(ξ) + iǫhq(·, ξ)i + O(ǫ2 + ξ ∞ ). Here the Diophanticity condition is of course important. Iterating the procedure we get for every N , pǫ ◦ exp(HG(N ) ) = p(ξ) + iǫ(hqi(ξ) + O(ǫ, ξ)) +O((ξ, ǫ)N +1 ) {z } | independent of x

This procedure can be continued on the operator level, and up to a small error we see that Pǫ is microlocally equivalent to an operator Pǫ (hDξ , ǫ; h). At least formally, Theorem 3.3 then follows by considering Fourier series expansions, but in order to get a full proof we also have take into account that we have constructed complex canonical transformations that are quantized by Fourier integral operators with complex phase and study the action of these operators on suitable exponentially weighted spaces. Near Λr we can still use action-angle coordinates as in (45) but the homological equation (46) is no longer solvable. Instead, we use secular perturbation theory (cf the book [37]), which amounts to making a partial Birkhoff reduction. After a linear change of x-variables, we may assume that p(ξ) = ξ2 + O(ξ 2 ) and in order to fix the ideas = ξ2 + ξ12 . Then we can make the averaging procedure only in the x2 -direction and reduce pǫ in (45) to peǫ (x, ξ) = ξ2 + ξ12 + O(ǫ) +O((ǫ, ξ)∞ ), {z } | independent of x2 , 2 +iǫhqi (x ,ξ) ≈ξ2 +ξ1 2 1

where hqi2 (x1 , ξ) denotes the average with respect to x2 .

Some Results on Nonselfadjoint Operators: A Survey

65

Carrying out the reduction on the operator level, we obtain up to small errors an operator Peǫ (x1 , hDx1 , hDx2 ; h) and after passing to Fourier series in x2 , a family of non-self-adjoint operators on Sx11 : Peǫ (x1 , hDx1 , hk; h), k ∈ Z. The non-self-adjointness and the corresponding possible wild growth of the resolvent makes it hard to go all the way to study individual eigenvalues. However, it can be shown that in the region |ξ1 | ≫ ǫ1/2 (inside the energy surface p = 0) we can go further and (as near Λd ) get a sufficiently good elimination of the x-dependence. This leads to the conclusion that the contributions from a vicinity of Λr to the spectrum of Pǫ in the rectangle ǫ1+δ ǫ , |ℑz − ǫF0 | ≤ , C C come from a neighborhood of Λr of phase space volume O(ǫ3/2 ). This explains heuristically why the rational torus will contribute with O(ǫ3/2 /h2 ) eigenvalues in the rectangle. The actual proof is more complicated. We use a Grushin problem reduction in order to reduce the study near Λr to that of a square matrix of size O(ǫ3/2 /h2 ). However, even if we avoid the eigenvalues of such a matrix, the inverse can only be bounded by |ℜz| ≤

exp O(ǫ3/2 /h2 ).

(47)

What saves us is that away from Λr ∪ Λd , we can conjugate the operator with exponential weights and show that the resolvent has an “off-diagonal decay” like exp(−1/(Ch)). This implies that we can confine the growth in (47) to a small neighborhood of Λr , if 3

ǫ2 1 ≫ 2, Ch h 2/3 leading to the assumption ǫ ≪ h in Theorem 3.4. 4. Weyl asymptotics for non-self-adjoint operators 4.1. Introduction For self-adjoint differential (pseudo)differential operators we have (under suitable assumptions) the Weyl law for the asymptotic distribution of eigenvalues, established in higher dimensions by H. Weyl [56] in 1912 in the case of second order elliptic boundary value problems. In the semiclassical setting such results were obtained by J. Chazarain, B. Helffer–D. Robert, V. Ivrii and many others (see [11] and further references there). Under suitable additional assumptions it states that

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if P = P w (x, hDx ; h) is a self-adjoint h-pseudodifferential operator with leading (real) symbol p(x, ξ), then if Ω ⊂ C is a domain intersecting R along a bounded interval, the number of eigenvalues of P in Ω (intersected with R) satisfies

#(σ(P ) ∩ Ω) =

1 (vol (p−1 (Ω)) + o(1)), h → 0. (2πh)n

A simple example is provided by the semiclassical harmonic oscillator + x2 ) on the real line which has the eigenvalues (k + 21 )h, k = 0, 1, .... In the non-self-adjoint case Weyl-asymptotics is known to hold in some cases close to the self-adjoint case or for normal operators. We do not always have Weyl-asymptotics: Following Davies and Boulton (see [7]), we can consider the non-self-adjoint harmonic operator: P = 12 ((hDx )2 + ix2 ) whose eigenvalues are given by eiπ/4 (k + 12 )h, k ∈ N ([46]): The set of values of p = 12 (ξ 2 + ix2 ) is the closed first quadrant and if we choose the open bounded set Ω to intersect the 1st quadrant but not the line ℑz = ℜz, we get vol (p−1 (Ω)) > 0, while there are no eigenvalues in Ω. More generally, h-differential operators with analytic coefficients often have their spectrum determined by complex-geometric quantities, and are likely not to obey the Weyl law. In particular, in the one dimensional case it often happens that the eigenvalues are concentrated to certain curves with branch points. As we have seen in Theorem 1.2 we are often confronted with the pseudospectral phenomenon: On the image of p the resolvent may be very large even far from the spectrum. This causes the eigenvalues to be very sensitive to small perturbations of the operator (by Theorem 1.1). In her thesis M. Hager (see [18]) considered a class of perturbed hpseudodifferential operators on the real line of the form Pδ = P (x, hD; h) + δqω (x), where P is analytic and qω is a random linear combination of the C/h first eigen-functions of an auxiliary operator. She showed that with probability very close to 1 when h → 0, Pδ obeys Weyl asymptotics. Here, we shall discuss a generalization to the multidimensional case obtained with Hager [21]. The results will be much more general in many ways, but the class of perturbations will be slightly different. 1 2 2 ((hDx )

Some Results on Nonselfadjoint Operators: A Survey

67

4.2. The result a) The unperturbed operator. Let m(ρ) ≥ 1, ρ = (x, ξ) be an order function on R2n so that 0 < m(ρ) ≤ C0 hρ − µiN0 m(µ). We may assume that m ∈ S(m) = {u ∈ C ∞ (R2n ); ∂ρα u = Oα (m), ∀α ∈ N2n }. Assume m ≥ 1 and let P (ρ; h) ∼ p(ρ) + hp1 (ρ) + ... in S(m). Assume ∃z0 ∈ C, C0 > 0 such that |p(ρ) − z0 | ≥ m(ρ)/C0 (ellipticity). Let Σ = p(R2n ) = p(R2n ) ∪ Σ∞ , Σ∞ = { lim p(ρj ); R2n ∋ ρj → ∞} j→∞

We write P = P w (x, hD; h). Let Ω ⊂⊂ C \ Σ∞ be open and simply connected containing z0 . Then using the pseudodifferential calculus, it is easy to show: 1) σ(P ) ∩ Ω is discrete when h > 0 is small enough.

2) ∀ǫ > 0, ∃h(ǫ) > 0, such that σ(P ) ∩ Ω ⊂ Σ + D(0, ǫ), 0 < h ≤ h(ǫ). b) The random pertubation. Let 0 < m, e m b ≤ 1 be square integrable order functions, one of which is integrable. Let Se ∈ S(m), e Sb ∈ S(m) b be elliptic symbols. The corresponding operators are Hilbert-Schmidt with e HS , kSk b HS = O(h−n/2 ). Let kSk X e Qω = Sb ◦ αj,k (ω)b ej ee∗k ◦ S, j,k

where αj,k , j, k ∈ N, are independent complex Gaussian random variables with expectation value 0 and variance 1, and (b ej )∞ ej )∞ 1 and (e 1 are orthonor2 n ∗ mal bases in L (R ), ebj eek u = (u|e ek )b ej . −2n Let M = C1 h−n with C1 ≫ 1. Then with probability ≥ 1 − Ce−h /C the Hilbert-Schmidt and trace class norms of Q fulfil 3

kQkHS ≤ M, kQktr ≤ M 2

(48)

Let Γ ⊂⊂ Ω be open with smooth boundary. Theorem 4.1. ([21]) Assume p(ρ) ∈ ∂Γ ⇒ dp(ρ), dp(ρ) are linearly independent.

(49)

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J. Sj¨ ostrand

Let ǫ, δ depend on h with 0 < ǫ ≪ 1, ǫ

1

e− Ch ≤ δ ≪ h3n+ 2 , C ≫ 1

(implying that ǫ ≥ Const.h ln h1 ). Then with probability ≥ 1 − we have √ 1 ǫ −1 |#(σ(Pδ ) ∩ Γ) − vol (p (Γ))| ≤ C . n n (2πh) h

ǫ

C − 2(2πh)n √ e ǫ

,

(50)

There is a similar result giving (50) simultaneously for all Γ in a suitable family. The assumption (49) implies that ∂Γ ∩ ∂Σ = ∅, so if we want to count the eigenvalues near ∂Σ, we need to weaken that assumption. For z ∈ neigh (∂Γ), we put Vz (t) = vol {ρ ∈ R2n ; |p(ρ) − z|2 ≤ t}.

(51)

Introduce the assumption ∃κ ∈]0, 1], s. t. Vz (t) = O(tκ ) uniformly for z ∈ neigh (∂Γ), 0 ≤ t ≪ 1. (52) Example. (49) ⇒ (52) with κ = 1.

Example. The best that can happen when ∂Γ ∩ ∂Σ 6= ∅ is that p(ρ) ∈ ∂Γ ⇒ {p, p}(ρ) 6= 0 or {p, {p, p}}(ρ) 6= 0.

(53)

It is easy to see that (53) implies (52) with κ = 3/4. (53) holds for the non-self-adjoint harmonic oscillator when 0 6∈ ∂Γ. Theorem 4.2. ([21]) We assume (52). Let ǫ, δ depend on h with 0 < ǫ ≪ 1, ǫ

1

e− Chκ ≤ δ ≪ h3n+ 2 , C ≫ 1,

implying that ǫ ≥ Const.hκ ln h1 . Then for 0 < r ≪ 1 we have with proba−n ǫ bility ≥ 1 − Cr e− 2 (2πh) , that 1 vol (p−1 (Γ))| ≤ |#(σ(Pδ ) ∩ Γ) − (2πh)n   
C ǫ 1 N −1 , + CN r + ln( )vol p (∂Γ + D(0, r)) hn r r

(54)

for every fixed N ∈ N.

If κ > 12 , we have vol (p−1 (∂Γ + D(0, r))) = O(r2κ−1 ) with 2κ − 1 > 0 and in all cases we may assume that ln( 1r )vol (p−1 (∂Γ+ D(0, r))) = O(rα0 ), where α0 > 0. Then we choose N ≫ 1, r = ǫ1/(1+α0 ) and the right hand side of (54) becomes O(1)ǫα0 /(1+α0 ) h−n .

Some Results on Nonselfadjoint Operators: A Survey

69

Again we have a similar theorem where the conclusion (54) is valid simultaneously for all Γ in a suitable family. Recently, the author obtained similar results when Qω is an operator of multiplication, see [49]. 4.3. Outline of the proofs We can construct Pe ∈ S(m) such that Pe (ρ; h) = P (ρ; h) for |ρ| ≫ 1 and |Pe (ρ; h) − z| ≥ m(ρ)/C for ρ ∈ R2n , z ∈ Ω. The eigenvalues of P in Ω coincide with the zeros of the holomorphic function F (z; h) = det Pz , Pz = (Pe(x, hD; h) − z)−1 (P (x, hD; h) − z)

(55)

The same remark holds for Pδ and Fδ defined as in (55) with P replaced by Pδ , provided that (48) holds. For z ∈ neigh (∂Γ), put Q = Pz∗ Pz . Let 1α (E) = max(E, α), where α = Ch, C ≫ 1. Using semiclassical analysis, we can show that under the assumption (52) (cf [38]) ZZ 1 1 ( ln qdxdξ + O(1)hκ ln ), (56) ln det Q ≤ ln det 1α (Q) = (2πh)n h where q = |pz |2 is the leading symbol of Q. Since ln det Q = ln det Pz∗ Pz = ln |F (z; h)|2 we have ln |F (z; h)| ≤

ZZ 1 1 ( ln |pz |dxdξ + O(1)hκ ln ). n (2πh) h

(57)

For δ > 0 small enough, we get the same upper bound for ln |Fδ (z; h)| (provided that (48) holds). The main step in the proof is to get a corresponding lower bound for each fixed z with a probability close to 1. In the multidimensional case this boils down to a question about random determinants. Let z ∈ neigh (∂Γ). Let e1 , e2 , ... be the first eigen-functions of Q = Pz∗ Pz and let f1 , f2 , ... be the first eigen-functions for Pz Pz∗ . The two operators have the same eigenvalues 0 ≤ λ1 ≤ λ2 ≤ ... We can arrange so that p p Pz ej = λj fj , Pz∗ fj = λj ej .

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J. Sj¨ ostrand

Let N = N (α) = #{j; λj ≤ α} (α = Ch, C ≫ 1). Semiclassical analysis gives that N = O(hκ−n ). Consider   Pz R− 0 P = : L 2 × CN → L 2 × CN , R+ 0 R+ : L2 → CN , R+ u(j) = R− : C

N

0

P has an inverse

√ α(u|ej ),

N √ X → L , R− = α u− (j)fj . 2

1

0

E =



0 E 0 E+ 0 0 E− E−+



1 = O( √ ), α

0 0 0 with E+ , E− , E−+ “explicit”, and

ln | det P 0 |2 = N ln α + det 1α (Pz∗ Pz ).

For Pδ = P + δQω , we form eω , Pzδ = (Pe − z)−1 (P − z + δQω ) = Pz + δ Q

and since δkQω k ≤ δC0 h−n ≪ 1, δ

P := is invertible with inverse δ

E =





Pzδ R− R+ 0

δ E δ E+ δ δ E− E−+





≈ E 0.

Here δ 0 0 e 0 E−+ = E−+ + δE− Qω E+ + “small”,

(58)

and we can show by perturbative arguments that

δ ln det P δ = ln det P 0 + O( √ M 3/2 ), α

leading to

ZZ 1 1 ( ln |pz |dxdξ + O(hκ ln )). ln | det P | = n (2πh) h δ

(59)

On the other hand, computations in [50] can be used to get δ ln | det Pzδ | = ln | det P δ | + ln | det E−+ |

(60)

Some Results on Nonselfadjoint Operators: A Survey

71

δ Using (58), we can view E−+ as a random matrix of size O(hκ−n ), close in a suitable sense to one with independent Gaussian random variables as its entries. This can be used to show: δ For every z ∈ neigh (∂Γ), we have a nice lower bound on ln | det E−+ | with probability close to 1. (60) then gives a corresponding lower bound on ln | det Pzδ |. To complete the proof of Theorem 4.1 we can apply the following result of M. Hager [17,18] with e h = hn :

Proposition 4.1. ([17,18]) Let Γ and Ω be as above. Let φ ∈ C(Ω; R) be smooth near ∂Γ. Let f = f (z; e h) be holomorphic in Ω with e |f (z; e h)| ≤ eφ(z)/h , z ∈ neigh (∂Γ), 0 < e h ≪ 1.

Assume there exist ǫ = ǫ(e h) ≪ 1, zk = zk (e h) ∈ Ω, k ∈ J = J(h), such that [ √ 1 ∂Γ ⊂ D(zk , ǫ), #J = O( √ ), ǫ k∈J

e

|f (zk ; e h)| ≥ e(φ(zk )−ǫ)/h , k ∈ J.

Then, #(f

−1

(0) ∩ Γ) =

1 2πe h

√ ǫ (∆φ)d(ℜz)d(ℑz) + O( ). e h Γ

ZZ

For the proof of Theorem 1.2 we use an improved version of this result, see [21]. 4.4. Comparison with Theorem 3.2 From the example with the non-self-adjoint harmonic oscillator in dimension 1, we have seen that Weyl asymptotics does not always hold for differential operators in one dimension, when the coefficients are analytic. If we add a small random perturbation to the non-self-adjoint harmonic oscillator, the theorems above and the main result in [18] show that with probability close to 1 the eigenvalues will no longer be confined to a halfline but will tend to fill up the range of the principal symbol p with a density that is given by (2πh)−n p∗ (dv), where dv denotes the symplectic volume element on R2n and p∗ (dv) is the direct image under p. From this simple one-dimensional example it is easy to build examples in higher dimension when Weyl asymptotics does not hold. In the 2dimensional case, we can also consider the situation when the unperturbed operator P satisfies the assumptions of Theorem 3.2. It is then natural to

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compare the distribution law given by Theorem 3.2 for P and the one given by the theorems 4.1, 4.2 for the random perturbations. To leading order in h, we get Weyl asymptotics already for P in the (close to normal) case when {p, p} vanishes identically. In general however, we get different asymptotic distributions already to leading order ([48]).

References 1. J. M. Bismut, The hypoelliptic Laplacian on the cotangent bundle, J. Amer. Math. Soc. 18 (2005), 379–476. 2. W. Bordeaux-Montrieux, in preparation 3. T. W. Cherry, On the solution of Hamiltonian systems of differential equations in the neighboorhood of a singular point, Proc. London. Math. Soc. 27 (1928), 151–170. 4. Y. Colin de Verdi`ere, Quasi-modes sur les vari´et´es Riemanniennes, Inv. Math. 43 (1977), 15–52. 5. E. B. Davies, Semi-classical states for non-self-adjoint Schr¨ odinger operators, Comm. Math. Phys. 200 (1999), 35–41. 6. E. B. Davies, Non-self-adjoint differential operators, Bull. London Math. Soc. 34(5) (2002), 513–532. 7. E. B. Davies, Semi-classical analysis and pseudospectra, J. Diff. Eq. 216(1) (2005), 153–187. 8. N. Dencker, The pseudospectrum of systems of semiclassical operators. http://arxiv.org/abs/0705.4561 9. N. Dencker, J. Sj¨ ostrand, Pseudospectra of semiclassical (pseudo-) differential operators, M.Zworski, Comm. Pure Appl. Math. 57(3) (2004), 384–415. 10. L. Desvillettes, C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation, Comm. Pure Appl. Math. 54(1) (2001), 1–42. 11. M. Dimassi, J. Sj¨ ostrand, Spectral asymptotics in the semi-classical limit, London Math. Soc. Lecture Notes Ser., 268, Cambridge Univ. Press, 1999. 12. J. P. Eckmann, M. Hairer, Spectral properties of hypoelliptic operators, Comm. Math. Phys. 235(2) (2003), 233–253. 13. S. Graffi, C. Villegas Blas, A uniform quantum version of the Cherry theorem. http://arxiv.org/abs/math-ph/0702021 14. V. I. Girko, Theory of random determinants, Mathematics and its applications, Kluwer academic publishers, Dordrecht, 1990. 15. I. C. Gohberg, M. G. Krein, Introduction to the theory of linear nonselfadjoint operators, Translations of mathematical monographs, Vol. 18, AMS, Providence, R.I., 1969. 16. A. Grigis, J. Sj¨ ostrand, Microlocal analysis for differential operators, London Math. Soc. Lecture Notes Ser., 196, Cambridge Univ. Press, 1994. 17. M. Hager, Instabilit´e spectrale semiclassique pour des op´erateurs nonautoadjoints. I. Un mod`ele, Ann. Fac. Sci. Toulouse Math. (6)15(2) (2006), 243–280.

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18. M. Hager, Instabilit´e spectrale semiclassique d’op´erateurs non-autoadjoints. II. Ann. Henri Poincar´e 7(6) (2006), 1035–1064. 19. M. Hager, Bound on the number of eigenvalues near the boundary of the pseudospectrum, Proc. Amer. Math. Soc. 135(12) (2007), 3867–3873. 20. M. Hager, E. B. Davies, Perturbations of Jordan matrices. http://arxiv.org/abs/math/0612158 21. M. Hager, J. Sj¨ ostrand, Eigenvalue asymptotics for randomly perturbed nonself-adjoint operators, Math. Ann., to appear. http://arxiv.org/abs/math.SP/0601381 22. B. Helffer, F. Nier, Hypoelliptic estimates and spectral theory for FokkerPlanck operators and Witten Laplacians, Lecture Notes in Mathematics, 1862, Springer-Verlag, Berlin, 2005. 23. B. Helffer, J. Sj¨ ostrand, Multiple wells in the semiclassical limit. I, Comm. Partial Differential Equations 9(4) (1984), 337–408. 24. B. Helffer, J. Sj¨ ostrand, Puits multiples en m´ecanique semi-classique. IV. ´ Etude du complexe de Witten, Comm. Partial Differential Equations 10(3) (1985), 245–340. 25. F. H´erau, M. Hitrik, J. Sj¨ ostrand, Tunnel effect for Kramers-Fokker-Planck operators, Annales H. Poincar´e, to appear. http://arxiv.org/abs/math.SP/0703684 26. F. H´erau, M. Hitrik, J. Sj¨ ostrand, Tunnel effect for Kramers-Fokker-Planck operators: return to equilibrium and applications. http://arxiv.org/abs/0801.3615 27. F. H´erau, F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal. 171(2) (2004), 151–218. 28. F. H´erau, J. Sj¨ ostrand, C. Stolk, Semiclassical analysis for the KramersFokker-Planck equation, Comm. Partial Differential Equations 30(4-6) (2005), 689–760. 29. M. Hitrik, J. Sj¨ ostrand, Rational invariant tori, phase space tunneling, and spectra for non-selfadjoint operators in dimension 2, Ann. Sci. E.N.S., to appear. http://arxiv.org/abs/math/0703394 30. M. Hitrik, J. Sj¨ ostrand, S. V˜ u Ngo.c, Diophantine tori and spectral asymptotics for non-selfadjoint operators Amer. J. Math. 129 (2007), 105–182. 31. L. H¨ ormander, Fourier integral operators I, Acta Math. 127 (1971), 79–183. 32. L. H¨ ormander, Differential operators of principal type, Math. Ann. 140 (1960), 124–146. 33. L. H¨ ormander, Differential equations without solutions, Math. Ann. 140 (1960), 169–173. 34. V. N. Kolokoltsov, Semiclassical analysis for diffusions and stochastic processes, Lecture Notes in Mathematics, 1724, Springer-Verlag, Berlin, 2000. 35. V. F. Lazutkin, KAM theory and semiclassical approximations to eigenfunctions. With an addendum by A.I. Shnirelman. Ergebnisse der Mathematik und ihrer Grenzgebiete, 24, Springer-Verlag, Berlin, 1993. 36. G. Lebeau, Le bismutien, S´em. ´e.d.p., Ecole Pol. 2004–05, I.1–I.15. 37. A. J. Lichtenberg, M. A. Lieberman, Regular and chaotic dynamics, Second

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ed., Springer-Verlag, New York, 1992. 38. A. Melin, J. Sj¨ ostrand, Fourier integral operators with complex-valued phase functions, Fourier integral operators and partial differential equations (Colloq. Internat., Univ. Nice, Nice, 1974), pp. 120–223, Lecture Notes in Math., Vol. 459, Springer, Berlin, 1975. 39. A. Melin, J. Sj¨ ostrand, Determinants of pseudodifferential operators and complex deformations of phase space, Methods Appl. Anal. 9(2) (2002), 177–237. 40. A. Melin, J. Sj¨ ostrand, Bohr-Sommerfeld quantization condition for nonselfadjoint operators in dimension 2, Ast´erisque 284 (2003), 181–244. 41. J. Moser, On the generalization of a theorem of A. Liapounoff, Comm. Pure Appl. Math. 11 (1958), 257–271. 42. K. Pravda Starov, Etude du pseudo-spectre d’operateurs non auto-adjoints, Thesis Rennes 2006. http://tel.archives-ouvertes.fr/tel-00109895 43. S. Roch, B. Silbermann, C ∗ -algebra techniques in numerical analysis, J. Oper. Theory 35 (1996), 241-280. 44. M. Sato, T. Kawai, M. Kashiwara, Microfunctions and pseudo-differential equations. Hyperfunctions and pseudo-differential equations, pp. 265–529, Lecture Notes in Math., Vol. 287, Springer, Berlin, 1973. 45. B. Simon, Semiclassical analysis of low lying eigenvalues. II. Tunneling, Ann. of Math. (2) 120(1) (1984), 89–118. 46. J. Sj¨ ostrand, Parametrices for pseudodifferential operators with multiple characteristics, Ark. Mat. 12(1) (1974), 85–130. 47. J. Sj¨ ostrand, Resonances for bottles and trace formulae, Math. Nachr. 221 (2001), 95–149. 48. J. Sj¨ ostrand, Eigenvalue distributions and Weyl laws for semi-classical nonself-adjoint operators in 2 dimensions, in preparation. 49. J. Sj¨ ostrand, Eigenvalue distribution for non-self-adjoint operators with small multiplicative random perturbations. http://arxiv.org/abs/0802.3584 50. J. Sj¨ ostrand, M. Zworski, Elementary linear algebra for advanced spectral problems, Ann. Inst. Fourier, to appear. http://arxiv.org/abs/math/0312166 51. J. Tailleur, S. Tanase-Nicola, J. Kurchan, Kramers equation and supersymmetry, J. Stat. Phys. 122(4) (2006), 557–595. 52. L. N. Trefethen, Pseudospectra of linear operators, SIAM Review 39(3) (1997), 383–406. 53. L. N. Trefethen, Nonhermitian systems and pseudospectra, s´em. ´e.d.p. Ecole Polytechnique, 2005–06. http://www.math.polytechnique.fr/seminaires/ seminaires-edp/2005-2006/sommaire2005-2006.html 54. L. N. Trefethen, M. Embree, Spectra and pseudospectra, the behaviour of nonnormal matrices and operators, Princeton University Press, 2005. 55. C. Villani, Hypocoercivity, preprint. 56. H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann. 71(4) (1912), 441–479. 57. M. Zworski, A remark on a paper of E. B. Davies, Proc. Amer. Math. Soc. 129 (2001), 2955–2957.

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THE DISTRIBUTION OF PRIMES: CONJECTURES vs. HITHERTO PROVABLES C. Y. YILDIRIM Department of Mathematics, Bo˘ gazi¸ci University & Feza G¨ ursey Institute for Fundamental Sciences, ˙ Istanbul, Turkey E-mail: [email protected] We present the main results, conjectures and ideas concerning the distribution of primes. We recount only the most important milestones and we do not go into technical details. Instead, suggestions for further reading are provided. Keywords: Distribution of prime numbers.

1. Introduction Here I endeavour to give an exposition of what are ‘known’ about the distribution of prime numbers. Of course, there is more than one meaning of ‘known’. In the strictest sense a statement is known if there is a proof agreed upon by all mathematicians; in a lenient sense instead of a proof there could be numerical data or heuristic arguments or evidence of other sorts which indicate the truth of a statement and these may be sufficient for some people to accept the assertion. Number theory is rife with long-standing questions, ranging from completely solved claims to problems about which not even conjectures can be formulated. In what follows we will dwell upon the main results and conjectures concerning the distribution of prime numbers, skipping most of the intermediate developments which along with other relevant questions not mentioned here can be tracked down from the references. Notation For two natural numbers a and b we denote by (a, b) their greatest common divisor. For a natural number n, Euler’s totient φ(n) is the number of positive integers less than or equal to n and relatively prime to n. The letter p will always stand for a prime number. By pn , the n-th largest prime is

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meant (so that p1 = 2, p2 = 3, . . .). On some occasions γ will denote Euler’s constant, and on some other occasions it will denote the imaginary part of a zero of the Riemann zeta-function. In general ǫ will mean an arbitrarily small positive real number, the letters A, C, c, . . . will be used for constants, and these need not have the same value at each occurrence. For two functions f, g we write f (x) = O(g(x)), or equivalently f (x) ≪ g(x), if there is a constant C such that |f (x)| ≤ Cg(x) on the common domain where both f and g are defined. If C depends on a parameter α, we write f = Oα (g) or f ≪α g. When x tends to a limit, f (x) ∼ g(x) means lim f (x)/g(x) = 1, and f (x) = o(g(x)) means lim f (x)/g(x) = 0. We write f (x) = Ω(g(x)) in place of lim sup |f (x)|/g(x) > 0, and f (x) = Ω± (g(x)) if lim sup f (x)/g(x) > 0 and lim inf f (x)/g(x) < 0. 2. The dawn: Up until the 19th century We can’t pinpoint at which epoch and where some person(s) first hit upon the property that some of the counting numbers are prime. The most ancient records we are aware of are the almost 4000 years old clay tablets by mathematicians of the Sumer-Akkad civilization who compiled tables of factors of integers and also studied equations and extraction of square roots involving primes. √ One of the oldest known mathematical proofs, the proof of irrationality of 2 attributed to Pythagoras’s school (6th century B.C.), relies on the concepts of divisibility and primes. Euclid (4th century B.C.), upon giving the fundamental theorem of arithmetic, proved the existence of infinitely many primes which we regard as the oldest result on the distribution of primes. Eratosthenes (3rd century B.C.) described the basic sieve method for determining the primes up to a given bound. Eratosthenes’s method evolved to powerful sieve methods which played key roles in almost all of the strongest results in number theory obtained in the 20th century. A most influential mathematician of antiquity Diophantus (circa 3rd century A.D.) was mostly occupied with finding integer or rational solutions to various equations, but his books were lost in the fire that destroyed the library of Alexandria. Some of Diophantus’s books were found in the 15th century, and then translated into Latin by Bachet (1621). Fermat and Mersenne, upon studying Bachet’s publication, announced new results on divisibility, primes and Diophantine equations. From their works the branches of (as they are now called) elementary number theory, algebraic number theory, Diophantine equations, elliptic curves ... developed. The 20th century witnessed several culminations of these developments, the completion of the proof of Fermat’s Last Theorem by Wiles, and appli-

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cations of primes and factorization to cryptography to name two. As far as we know after Euclid and Eratosthenes the next mathematician to produce results on the distribution of primes was Euler (1737). The only still problematic statement from the times in between is that every even number is a sum of one, two or three primes due to Descartes. Euler’s product identity ∞ Y X 1 1 (1 − s )−1 , = s n p n=1 p: prime

(1)

valid for s > 1, is an analytic way of expressing the unique factorization of natural numbers into a product of primes. Euler had no scruples about using this identity with s = 1, whereupon he deduced that the sum of the reciprocals of the primes diverges. He also corresponded with Goldbach, who conjectured that every number > 2 can be written as a sum of three primes and every even number > 2 is a sum of two primes. (These statements are believed to be true, with the obvious modification to > 5 in the first statement, according to the present understanding that 1 is not a prime number contrary to Goldbach and many mathematicians before him who at times took 1 as a prime. We can immediately notice some properties which distinguish 1 from the primes: 1 is the multiplicative unit of the ring of integers and divisibility by 1 is not a special feature of any integer. Moreover, the value of any multiplicative arithmetic function at 1 is 1, whereas φ(p) = p − 1 for example, so that unnecessary inconvenience in the definitions of arithmetic functions is avoided by excluding 1 from the set of primes. Some information about the status of Goldbach’s still unproved statements will be given in §9). Gauss mentioned much later in his life that around 1792, upon examining tables of primes, he conjectured that a good approximation to X π(x) := 1 (2) p≤x

is given by the logarithmic integral

and that lim

x→∞

Z

x

dt , log t

(3)

π(x) =1 x/ log x

(4)

li(x) :=

2

which is equivalent to pn ∼ n log n. A similar but slightly wrong conjecture was in fact first published in 1798 and 1808 by Legendre, also on empirical

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grounds. Gauss also stated that the number of integers ≤ x which are products of two distinct primes (we call these E2 -numbers) is approximately x log log x/ log x. 3. The launch of rigorous analytic methods for primes Euler stated that an arithmetic progression with the first term 1 contains infinitely many primes. Elementary number theory textbooks contain examples of generalizations of Euclid’s theorem to some arithmetic progressions, i.e. sequences of the form a + kq where a and q are fixed relatively prime positive integers and k runs through all natural numbers, for example, that the progressions 3 + 4k, 1 + 4k, 1 + 3k, 5 + 6k... each contain an infinitude of primes. Some of these are proved by simple modifications in Euclid’s proof, and some require other elementary ingredients. M.R. Murty [88], defining a Euclidean proof for the arithmetic progression a + kq via the existence of an irreducible polynomial f (x) with integer coefficients such that all but finitely many prime divisors of the values f (n), n ∈ Z are either ≡ 1(mod q) or ≡ a(mod q), proved that a Euclidean proof exists if and only if a2 ≡ 1(mod q). (The ‘if’ part of this theorem had been proved by I. Schur). The full generalization of Euclid’s theorem that if a and q are relatively prime natural numbers, then there are infinitely many primes of the form a + kq was conjectured by Legendre (1788) and proved by Dirichlet (18371840) in some memoirs which are regarded as the origin of analytic number theory. Dirichlet’s idea was to adapt Euler’s method to the case when the primes are restricted to the residue class a(mod q). The proof uses the socalled Dirichlet’s L-functions, defined by L(s, χ) :=

∞ X χ(p) χ(n) Y = (1 − s )−1 s n p p n=1

(5)

in s > 1 where the series and the product are absolutely convergent. Here χ is a Dirichlet’s character to the modulus q, a function of an integer variable n which is multiplicative and periodic with period q. It follows that if (n, q) = 1, then χ(n) is a root of unity. For (n, q) > 1, it is apt to define χ(n) = 0. The character χ0 which assumes the value 1 at all n coprime to q is called the principal character in which case (5) differs from (1) by the finite factor Q 1 p|q (1 − ps ). It could be that for values of n coprime to q, the least period of a nonprincipal χ(n) is not q but a divisor of q, in which case χ is called an imprimitive character, and otherwise primitive. There are φ(q) characters in all to the modulus q, which form an abelian group (defining χ1 χ2 (n) =

The Distribution of Primes: Conjectures vs. Hitherto Provables

79

χ1 (n)χ2 (n)) isomorphic to the group of those residue classes which are relatively prime to the modulus q. The characters to the modulus q satisfy  X φ(q) if χ = χ0 , χ(n) = 0 otherwise, n( mod q)

and X

χ(n) =

χ( mod q)



φ(q) 0

if n ≡ 1 (mod q), otherwise.

From a given set of integers those belonging to a particular residue class a(mod q) can be selected by invoking  X 1 1 if n ≡ a (mod q) and (a, q) = 1, χ(a)χ(n) = (6) 0 otherwise. φ(q) χ( mod q)

For nonprincipal χ the series in (5) is conditionally convergent for 0 < s ≤ 1 so that L(s, χ) is regular at s = 1. Dirichlet’s proof of the infinitude of primes of the form a + kq hinges on the fact that for χ nonprincipal, log L(s, χ) is bounded as s → 1+ . Dirichlet’s innovations were wonderful, but the challenge of estimating the number of primes up to x, as x → ∞, was not overcome yet. Around 1849 Chebyshev made advances in this problem. He showed that if limx→∞ π(x)/x log x exists it should be 1, and found lower and upper bounds to this limit as 0.92.. and 1.10.. by an argument based on equating the Stirling formula estimate for log n! with the expression of log n! as a sum over the logarithms of primes up to n. But Chebyshev’s method would not pin down the limit to 1. Mertens (1874) went further along Chebyshev’s lines and established the following asymptotic formulae: X log p = log x + O(1), (7) p p≤x

X1 1 = log log x + A + O( ), p log x

(8)

p≤x

Y

1 (1 − )−1 = eγ log x + O(1). p

(9)

p≤x

P χ(p) Mertens also showed that if χ is a nonprincipal character then p p P converges (Dirichlet had used limσ→1+ p χ(p) is finite), wherefrom he obσ p

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tained the generalization of the second of his formulas given above, a quantified form of Dirichlet’s theorem: X log log x 1 1 = + A(q, a) + O( ). (10) p φ(q) log x p≤x p≡a( mod q)

4. Riemann’s revolution It is indeed interesting that partial summation applied to (7) or (8) is of no avail for obtaining an asymptotic formula for π(x). In between Chebyshev and Mertens there was a development which turned out to be of utmost importance, not only for the goal of settling (4), but also for the impact on the theory of functions of a complex variable. In an article published in 1859, Riemann [98] took the quantity given by either of the two expressions in the Euler product identity (1) as a function of a complex variable s = σ + it ∈ C, σ, t ∈ R. In this way the Riemann zeta-function ζ(s) was defined in the half-plane σ > 1 where both sides of (1) are absolutely convergent Z ∞and therefore make sense. He started from the wellΓ(s) , which upon summing over n known equation e−nx xs−1 dx = ns 0 Z ∞ xs−1 1 dx for σ > 1. Riemann’s remarkable insight gives ζ(s) = Γ(s) 0 ex − 1 to take s ∈ C allowed him to apply the methods of complex integration, and upon using the functional equation for a Jacobi theta function he obtained the representation Z ∞ ∞ X s+1 2 s 1 s − 2s + (x 2 −1 + x− 2 )( π Γ( )ζ(s) = e−n πx ) dx , (11) 2 s(s − 1) 1 n=1

where the integral converges absolutely for any s, and uniformly in compact subsets of the s-plane. Thus Riemann showed that from its original domain of definition ζ(s) can be continued analytically over the whole complex plane as a single-valued and meromorphic function with its only pole at s = 1, a simple pole with residue 1. Since the right-hand side of (11) is unchanged when 1 − s is used in place of s, this also revealed the functional equation of ζ(s):

1−s s s 1−s )ζ(1 − s). (12) π − 2 Γ( )ζ(s) = π − 2 Γ( 2 2 The value of ζ(s) can be calculated at any s with σ > 1 to any desired accuracy from the expressions in (1). Then, by the functional equation, ζ(s) can also be calculated for any s with σ < 0. It is clear from the product

The Distribution of Primes: Conjectures vs. Hitherto Provables

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in (1) that ζ(s) 6= 0 in σ > 1, and it is immediate from (12) that at the s negative even integers where Γ( ) has simple poles, ζ(s) vanishes - these 2 are called the trivial zeros. In the rather mysterious region 0 ≤ σ ≤ 1, the so-called critical strip, one may use (11) or Z ∞ s ζ(s) = (x)x−s−1 dx (σ > 0), (13) −s s−1 1 where (x) is the fractional part of x. In this formula which is obtained by applying partial summation to the series in (1), the integral converges absolutely for σ > 0 and uniformly for σ ≥ δ > 0, so that we have an analytic continuation of ζ(s) to σ > 0. Riemann also made several assertions on the zeros of ζ(s), and provided sketches of the proofs for some. The matter of justifying Riemann’s statements inspired Hadamard’s work on general results for entire func1 tions. Hadamard (1893) showed that the entire function ξ(s) := s(s − 2 s s 1)π − 2 Γ( )ζ(s) is bounded in size by exp(C|s| log |s|) as |s| → ∞, from 2 which he deduced that ζ(s) has infinitely many nontrivial zeros in the critical strip. The nontrivial zeros must be situated symmetrically with respect to the real axis, and with respect to the point 21 . Applying the argument principle to ξ(s), von Mangoldt (1895) gave the proof of Riemann’s assertion that the number of nontrivial zeros ρ = β + iγ with 0 < γ ≤ T is T T log T − + O(log T ), as T → ∞. 2π 2π Riemann’s expression for π(x) in terms of the zeros of ζ(s), and a related formula we will now dwell upon, were also fully proved by von Mangoldt. Logarithmic differentiation of the product expression of ζ(s) gives ∞ X ζ′ Λ(n) (s) = − ζ ns n=2

(σ > 1),

where Λ(n) is von Mangoldt’s function  log p if n = pa , p: prime, a ∈ N , Λ(n) := 0 otherwise. Most researchers find it more convenient to work with X X ψ(x) := Λ(n) = log p n≤x

(14)

(15)

(16)

pm ≤x

and then convert a result involving ψ(x) to a result for π(x) than working straightforwardly with π(x). For instance, (4) is equivalent to ψ(x) ∼ x.

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From (14) one has ψ0 (x) =

1 2πi

Z

c+i∞

[−

c−i∞

xs ζ′ (s)] ds ζ s

(c > 1),

(17)

Λ(x) . Considering the integral from c − iT to c + iT , where ψ0 (x) = ψ(x) − 2 and moving the line of integration all the way to the left in the complex plane one obtains, by the residue theorem, for any x ≥ 2, X xρ ζ′ 1 − (0) − log(1 − x−2 ) (18) ψ0 (x) = x − ρ ζ 2 |γ| 1 − c/ log(|t| + 2) for the zero-free region of ζ(s) yielded the prime number theorem with the error 1 term ψ(x) = x + O(x exp[−c(log x) 2 ]). This can be seen by an argument ′ ζ′ ζ′ ζ which combines the fact that 3 (σ) + ℜ[4 (σ + it) + (σ + 2it)] ≤ 0 (a ζ ζ ζ consequence of 3 + 4 cos θ + cos 2θ ≥ 0, ∀θ ∈ R) with simple inequalities ζ′ obtained from the partial fraction expansion of (s). ζ The hope for an elementary, i.e. without recourse to the theory of functions of a complex variable and in fact to any infinite summation, though not necessarily easy, proof of the prime number theorem seemed to fade.

The Distribution of Primes: Conjectures vs. Hitherto Provables

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Hardy [54] said in 1921 ”No elementary proof is known, and one may ask whether it is reasonable to expect one. Now we know that the theorem is roughly equivalent to a theorem about an analytic function, the theorem that Riemann’s zeta-function has no zeros on a certain line. A proof of such a theorem, not fundamentally dependent upon the ideas of the theory of functions, seems to me extraordinarily unlikely. It is rash to assert that a mathematical theorem cannot be proved in a particular way; but one thing seems quite clear. We have certain views about the logic of the theory; we think that some theorems, as we say, ‘lie deep’, and others nearer to the surface. If anyone produces an elementary proof of the prime number theorem, he will show that these views are wrong, that the subject does not hang together in the way we have supposed, and that it is time for the books to be cast aside and for the theory to be rewritten”. A century after Chebyshev, Selberg [103] succeeded in giving an elementary proof based on Selberg’s identity X X Λ(n) log n + Λ(m)Λ(n) = 2x log x + O(x) (20) n≤x

mn≤x

(and Erd¨ os [20] too; Goldfeld [32] has told the story of the dispute over their roles in the proof). Being a kind of Stirling’s formula of higher degree, (22) leads to achieving more than Chebyshev’s results. There is some folklore to the effect that in fact Selberg might have been inspired by zeta function theory in arriving at his identity (Ingham, in his review [68] of Selberg’s and Erd¨ os’s articles pointed out that it is quicker to obtain the identity starting ′ out from equating coefficients in the Dirichlet series expression for ( ζζ (s))′ + ′

′′

( ζζ (s))2 = ζζ (s)), but Selberg presented an argument which avoids these. Selberg [104] also gave an elementary proof of Dirichlet’s theorem on the infinitude of primes in an arithmetic progression. Using I.M. Vinogradov’s method X for strong majorizations of exponential sums (i.e. sums of the kind g(n)e2πif (n) ), in 1958 Korobov and a1−

c (log(3 +

|t|))2/3 (log log(3

+ |t|))1/3

(21)

is free of zeta zeros. This implies the prime number theorem with the smallest proved error term 3

π(x) = li x + O(x exp[−c

(log x) 5 1

(log log x) 5

]).

(22)

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5. The Riemann Hypothesis In view of the fact that zeta zeros do exist in the critical strip and they are situated symmetrically with respect to the critical line, the strongest improvement on the zero-free region (21) one can ever hope for is σ > 21 . Of all the assertions in Riemann’s memoir the guess that ‘all of the nontrivial zeros lie on the critical line σ = 12 ’ is the only one that still awaits a proof. This statement which has turned out to be very profound is known as the Riemann Hypothesis (RH). Riemann wrote that after some fleeting futile attempts he put it aside, correctly assessing that it wasn’t necessary for his immediate aim of reaching the prime number theorem. Other than the point of view that it is esthetically pleasing and ideally desirable to have all of the zeros of a function which is deeply related to the primes and which has a simple Dirichlet series in the sense that all of the coefficients are 1, lie on one line (and are simple), there is considerable evidence in favour of RH. Beginning with Riemann himself, computations have now reached to cover the first 1013 zeros, as well as millions of zeros in certain intervals at ordinates up to 1024 , and they are all simple zeros on the critical line ([46]). By the theoretical studies of Hardy and Littlewood, Selberg, Levinson, and finally Conrey [10], it is now known that more than 40 % of the zeros of ζ(s) lie on the critical line σ = 12 and are simple. This result was attained by the mollifier method, which a suitable Dirichlet X in an ), producing the effect of polynomial (an expression of the kind ns n≤y

smoothing out the irregularities, is introduced in the integrands involving ζ(s) of the integrals needed for the application of the argument principle. There are also zero-density estimates which are upper bounds for N (σ, T ) := {ρ : ζ(ρ) = 0, σ < β, 0 < γ ≤ T }. Zero-density estimates are useful in tackling many problems of analytic number theory unconditionally (i.e. not assuming the truth of RH or any other unproved hypotheses). It is known that N (σ, T ) ≪ T 3(1−σ)/(2−σ) log5 T uniformly for σ > 21 . (This result due to Ingham (1940) has been improved for some σ by later researchers). The density estimates say that exceptions to RH, if ever they exist, are rare and become rarer as σ moves away from 12 . The importance of ζ(s) is not limited by its connections to the distribution of primes and other applications in analytic number theory by virtue of its Euler product being a key tool in dealing with similar products arising from multiplicative functions. The Riemann zeta-function is the protoype of a class of functions, global L-functions, associated with various algebraic,

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arithmetic or geometric objects (as well as some which are not directly associated as such). These functions are Dirichlet series possessing a product representation of the Euler type, they can be analytically continued from their original domain of definition in keeping with a functional equation, and a version of RH can be stated for them. The existence of these functions and the properties satisfied by them have enormous implications in many areas. In many cases some parts of these conditions have been shown to hold, the rest remain conjectural. There are numerical data concerning some cases which confirm the hypotheses. For no global L-function the truth or falsity of RH has been established. However, for many problems the implications of the relevant form of RH have been proved by independent means. Thus global L-functions can be regarded to provide additional belief in the truth of RH by dint of the (mostly hypothetical) coherence in the big picture which includes many theories. Furthermore, for the zeta functions of algebraic varieties over finite fields, the analogue of RH has been proved. Other mostly hypothetical support comes from additional conjectures which predict results that are confirmed by numerical computations. There are even problems for which predictions were not possible before. The predictions from these conjectures on already proved results coincide with them. We will say a little more about these in §7. There are diverse statements, in complex analysis and functional analysis, formulated as tests or equivalent conditions for the truth of RH, and nothing that points to the falsity of RH has come up. The following probabilistic argument supports RH. Consider the M¨ obius function µ(n) defined on the natural numbers as follows. First of all, as all multiplicative arithmetic functions, µ(1) = 1. If n = pα1 · · · pαk is a product of k distinct primes, then µ(n) = (−1)k . If n is divisible by the square of a prime, then µ(n) = 0. It is easy to see that ∞ X µ(n) 1 = , ζ(s) n=1 ns

and that M (x) :=

X

n≤x

1

(σ > 1),

µ(n) = O(x 2 +ǫ ) ⇐⇒ RH is true.

(23)

(24)

The sequence {µ(n)}∞ n=1 seems to be quite random in the long run, and for almost all random sequences of entries 0 or ±1, the summatory func1 tion is bounded as O(x 2 +ǫ ). (We note that Mertens’ original conjecture 1 that |M (x)| ≤ x 2 was refuted by Odlyzko and te Riele [91] who showed

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C. Y. Yıldırım 1

1

that lim sup M (x)x− 2 > 1.06 and lim inf M (x)x− 2 < −1.009. Ingham [67] x→∞ x→∞ X proved that if at most a finite number of sums of the type cn γn , with n≤N

integers ci having greatest common divisor 1, are 0, then the just mentioned lim inf and lim sup are ∞ and −∞ respectively. Here γ1 , γ2 , . . . denote the imaginary parts of the distinct nontrivial zeros of ζ(s). Bateman et. al. [2] weakened the condition demanded by Ingham’s theorem to having |cn | ≤ 2 with at most one of the |cn | = 2 and not all of them 0). If RH is false, then the distribution of primes would have to exhibit irregularities much wilder than expected, and the first zeta zero off the critical line would be a very significant mathematical constant. If RH is true, then all the ρ have real part 12 , and knowing the count of zeta zeros it is easy to deduce from the explicit formula that 1

ψ(x) = x + O(x 2 log2 x)

and

1

π(x) = li x + O(x 2 log x)

(25)

1 2

with a much smaller error term than (22). The exponent in the error term is the best possible because there are zeta zeros on the critical line. 6. More about the error term in the prime number theorem Littlewood (1914) used Dirichlet’s theorem on Diophantine approximation to show that the sum over ρ cannot be very small at all x, and proved that 1

x 2 log log log x ). log x (26) This also answers Riemann’s [98] thoughts on whether it could always be Z ∞ dt for x > 2. However, Pintz [93] proved that that π(x) < Li x := log t 0 Z X (π(t) − Li t)dt < 0 for all sufficiently large X if and only if RH holds, 1

ψ(x) − x = Ω± (x 2 log log log x)

and

π(x) − li x = Ω± (

2

and that π(x) − Li x is negative in a particular average sense. Furthermore, considerations based on some basic beliefs about the zeta zeros suggest that the probability that π(x) > li x is about 2.6 · 10−7 (see [99]). Montgomery [84] suggested on a probabilistic argument, assuming RH and the linear independence over rational numbers of the imaginary parts of the nontrivial zeros above the real axis so that linear forms in the γ don’t take on very small values, that the sharpest form of the prime number theorem can be lim

x

1 2

ψ(x) − x

(log log log x)2

=±

1 , 2π

(x → ∞).

(27)

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Note that even under (27), Legendre’s conjecture that there is always a prime between n2 and (n + 1)2 for any positive integer n is inaccessible. Averages of the error term in the prime number theorem are also of great relevance in studying the distribution of primes. For brevity call R(x) := ψ(x) − x .

(28)

By (26) it is known that, as x → ∞, R(x) changes sign infinitely many times. Cram´er [14] showed that on RH Z X (R(u))2 du = O(X), (29) u 1

and

Z

1

X

(

R(u) 2 ) du ∼ C log X . u

(30)

Gallagher’s article [31] contains compact proofs of such results. Pintz [92] has shown that for all sufficiently large X Z X 3 X2 3 ≤ |R(u)| du ≤ X 2 , (31) 400 1 where the lower bound is unconditional and the upper bound depends essentially on RH. Jurkat [73], by developing concepts on almost-periodic functions proved upon RH that, x Z x+ log log x R(u) log log log x ) (32) du = Ω± ( 3 log log x 2 u x

(with the implied constants ± 21 ), and that this cannot be improved upon much for he also showed that the left-hand side of (32) is O((log log log x)2 / log log x). This result contains Littlewood’s result (26). From (31) and (25) we see that as x → ∞, |R(x)| spends most of its time 1 roughly around the value x 2 (instead of much smaller values), and (32) reveals the existence of quite long intervals throughout which |R(x)| is almost as large as possible. 7. Pair correlation of zeta zeros and primes In 1972 H. L. Montgomery [83], assuming RH, found a way of looking into the distribution of the nontrivial zeros on the line σ = 21 . Upon developing a new version of the explicit formula, Montgomery was led to define X ′ 4 T T iα(γ−γ ) , (33) log T )−1 F (α, T ) = ( ′ )2 2π 4 + (γ − γ ′ 0T α n≤T α

For α > 1 the estimate for the contribution of the nondiagonal terms become large enough to prevent an asymptotic estimate. Montgomery, drawX ing upon the conjectured size of expressions of the kind Λ(n)Λ(n + h) n≤x

(see §9), conjectured that

F (α, T ) = 1 + o(1),

(1 ≤ α ≤ A).

(36)

In (35) and (36) the estimates are uniform in the respective domains of α, and A > 1 is arbitrary but fixed. Here the condition that A is fixed cannot be relaxed, since from Dirichlet’s theorem on Diophantine approximation it can be seen that there exist large α = α(T ) for which F (α, T ) is close to F (0, T ) ∼ log T . We expect that F (α, T ) ≪ 1 holds on average, for Goldston [34] proved, assuming RH, that for sufficiently large T , Z c+1 Z c+1 2 8 F (α, T ) > − ǫ, F (α, T ) < + ǫ, 3 3 c−1 c uniformly for any c ∈ R (c may even be a function of T ). Using convolutions of F (α, T ) with appropriate kernels rˆ(α) so that Z ∞ X log T T r((γ − γ ′ ) )w(γ − γ ′ ) = ( log T ) F (α, T )ˆ r (α) dα 2π 2π −∞ ′ 0 x, (64) says more than (58) which depends upon GRH. There has been considerable effort for proving (64) with smaller values of c (the theorem should be true with any c > 1). The critical barrier of c = 2 has been overcome for xα < q < xβ for any fixed 0 < α < β < 1 (see Friedlander-Iwaniec [28]). From a similar improvement for small q, the nonexistence of exceptional zeros would follow as can be seen from (56). Another famous problem in this topic is determining the size of the least prime in an arithmetic progression, denoted as pmin (a, q) := min{p : p ≡ a(mod q)}. The Siegel-Walfisz theorem gives a very weak estimate. Upon discounting the small contribution of proper prime powers to ψ(x; q, a) in (58), we see that GRH implies pmin (a, q) ≪ q 2 log5 q. In a pioneering deep work Linnik obtained pmin (a, q) ≪ q C with effectively computable constant. The best known value for the Linnik constant is C = 5.5 due to HeathBrown [60]. It is conjectured that C = 1 + ǫ would work; it is easy to see that pmin (a, q) ≪ o(q log q) is false. 9. Additive questions about the primes It has long been believed but not yet proved that there exist infinitely many twin primes, i.e. p and p + 2 both prime. Brun (1919) showed that the series formed by the reciprocals of twin primes is convergent. His method involved the truncation of certain sums which arise in Legendre’s formulation of Eratosthenes’s sieve. This not only rendered the Eratosthenes sieve usable for his purpose, but also opened the way for the development of various sieve methods to be applied to many formerly unapproachable problems. For example, it is now known that π(0,2) (x), the number of prime p ≤ x Y x 1 ) . (Of course such that p + 2 is also prime, is . 6.84 (1 − 2 (p − 1) (log x)2 p>2 for the twin primes problem one needs a lower bound which tends to infinity with x. A curious result due to Heath-Brown [58] is that the existence (in an appropriate sense) of exceptional zeros implies the infinitude of twin primes).

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Hardy and Littlewood’s [55] prime r-tuples conjecture for the number πd (N ) of positive integers n ≤ N for which n + d1 , . . . , n + dr are all prime (here r is fixed, d1 , . . . , dr are distinct integers and d = (d1 , . . . , dr )) in the form of an asymptotic formula is πd (N ) ∼ S(d) when S(d) 6= 0, where S(d) =

N , logr N

(N → ∞)

Y pr−1 (p − νd (p)) p

(p − 1)r

(65)

,

(66)

and νd (p) is the number of distinct residue classes modulo p occupied by d1 , . . . , dr . A strong form of the conjecture with an error term is N Y r X

n=1 i=1

Λ(n + di ) = S(d)N + Er (N, d),

1

Er (N, d) ≪ǫ,r N 2 +ǫ ,

(67)

uniformly for |di | ≤ N . The r = 1 case of (65) is the prime number theorem. For r ≥ 2 the conjecture remains unproved for any d. The heuristics for arriving at the r-tuples conjecture in the case r = 2, d1 = 0, d2 = 2 (see Hardy and Wright [56, §22.20]), is based on counting the number of positive Q integers ≤ x which are relatively prime to p≤√x p in two ways, using Mertens’s result (9) and using the prime number theorem, that leads to a discrepancy by a factor of 2e−γ . The error lies in the calculation that uses (9), because it involves the assumption that the number of such integers in an interval is always proportional to the length of the interval, even if the interval is very short as in this application. The conjectural step comes in when this reasoning is adopted for pairs of such integers, and the square of the correction factor eγ /2 is introduced. Let us also mention in this context, that Hensley and Richards [62] proved that the conjecture π(x + y) ≤ π(x) + π(y) is incompatible with the prime r-tuples conjecture. Assuming that for each r, (65) holds uniformly for 1 ≤ d1 , . . . , dr ≤ h, Gallagher [30] showed that if Pk (h, N ) is the number of integers n ≤ N for which the interval (n, n + h] contains exactly k primes, then −λ k Pk (λ log N, N ) ∼ N e k!λ as N → ∞, i.e. the distribution tends to the Poisson distribution with parameter λ. Numerical studies [109] indicate that small primes seem to obey Gaussian Orthogonal Ensemble statistics and as more primes are included there is a transition towards Poisson statistics. Assuming the conjecture (67) for 1 ≤ r ≤ K, Montgomery and Soundararajan [85] calculated the K-th moment for the primes in short

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intervals as N X

(ψ(n + h) − ψ(n) − h)

K

= µK h

K 2

Z

N

(log(

1

n=1

K

K x + b) 2 dx h K

+O(N (log N ) 2 h 2 (

(68)

1 h − 1 ) 8K + hK N 2 +ǫ ), log N

1

uniformly for log N ≤ h ≤ N K , where µK = 1 · 3 · · · (K − 1) if K is even, and µK = 0 if K is odd, and b = 1 − γ − log 2π. We notice that in the case K = 1 the assumption is equivalent to RH, and for K = 2 the assumption is equivalent to a strong form of PC. Later on Chan [8] obtained the same result under the weaker assumption that for N ≥ h, X Er (N, d)2 ≪r N 1+ǫ hr . 1≤di ≤h i=1,...,r

Montgomery and Soundararajan conjectured that the right-hand side of (68) continues to be (µK +o(1))N (h log N/h)K/2 uniformly for (log N )1+δ ≤ h ≤ N 1−δ , and that this conjecture is equivalent to Z X X k T log T ) 2 , ( cos(γ log x))k dx = (µk + o(1))X( 4π 1 0 2 is the sum of two prime numbers. Beginning with the work of Brun on sieve methods the furthest achievement in this direction is the theorem of Chen from 1966 [9], that every sufficiently large even number can be expressed as the sum of a prime and a number which has at most two prime factors - counted with multiplicity. Pintz [94], making a great improvement over former results, has shown that the number of even integers ≤ x which are not expressible as a sum 2 of two primes is O(x 3 ). In this context we note that Schnirelman (1930) proved by developing a quite simple method that there exists a constant C such that every natural number > 1 can be expressed as a sum of at most C primes. Ramar´e’s work [96] shows that C = 7 suffices. Vinogradov (1937), using his method of estimating exponential sums, proved that every sufficiently large odd number can be expressed as a sum of three primes (see [16]), from which it immediately follows that every sufficiently large integer can be expressed as a sum of at most 4 primes. As for conditional results, it is known that RH implies C = 6 is fine, while GRH allows Vinogradov’s result for all odd numbers > 5 and this makes C = 4 sufficient.

The Distribution of Primes: Conjectures vs. Hitherto Provables

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The root of the difficulty in such conjectures is that they involve constraints of an additive nature on the primes which are defined in terms of the operation of multiplication. So what needs to be understood deeply is the relation between the operations of addition and multiplication, which are linked only by the distributive law. Some mathematicians, e.g. Knuth [75], think that Goldbach’s conjecture may well be true but there is no rigourous way to prove it. It might be one of the unprovable assertions that G¨ odel showed exist (and it is known that in some sense almost all correct statements about mathematics are unprovable). Schinzel formulated two grand conjectures which embody the Goldbach and twin-prime conjectures as well as many other questions. Let Pi (n) (i = 1, . . . , r) be distinct irreducible polynomials in Z[x] having positive leading coefficients such that the product P = P1 · · · Pr has no fixed prime divisor. The first conjecture is: There exist infinitely many n ∈ Z such that each Pi (n) is prime. The second conjecture is: Let N ∈ N and G ∈ Z[x] be another polynomial with positive leading coefficient such that N − G is irreducible, and also such that (N − G)P has no fixed prime divisor. + For all sufficiently large N , there exists n ∈ Z such that N − G(n) > 0 and each of N − G(n), P1 (n), . . . , Pr (n) is prime. Iwaniec [70] showed that any inhomogeneous irreducible quadratic polynomial in two variables which depends essentially on both of the variables, and whose coefficients are integers without a common factor takes infinitely many prime values. Iwaniec also proved an asymptotic formula for the number of these primes ≤ z as z → ∞. In this case, as well as in the previously dealt cases of the prime number theorem and binary quadratic forms, the number N (z) of the values of the argument of the polynomial for which the value of the polynomial is ≤ z satisfies N (z) ≫ z. Fouvry and Iwaniec [21] proved an asymptotic formula for the number of primes represented by x2 + p2 (where p is a prime), in which case N (z) ≪ z/ log z. In a later development, Friedlander and Iwaniec [29] settled the problem for x2 + y 4 , which is a quite sparse 3 sequence with N (z) = O(z 4 ). More recently Heath-Brown [61] worked out 2 the problem for the even sparser sequence x3 + 2y 3 with N (z) = O(z 3 ) in which case the homogeneity of the polynomial is helpful. In the case of x2 + 1, Iwaniec [71] showed that it is infinitely often a number having at most two (not necessarily distinct) prime factors. A recent major development in a question of additive nature took place when Green and Tao [51] showed that the primes contain arbitrarily long arithmetic progressions. Their proof makes use of combinatorial methods, harmonic analysis and ergodic theory in addition to methods of analytic

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number theory. 10. Primes in short intervals After the prime number theorem it is natural to ask for which functions Φ(x), as x → ∞, π(x + Φ(x)) − π(x) ∼

Φ(x) . log x

(69)

Here one would try to find Φ(x) as slowly increasing as possible. Heath7 Brown [59] proved that one can take Φ(x) = x 12 −ǫ(x) (ǫ(x) → 0, as x → ∞), 1 and Φ(x) = x 2 +ǫ is allowed if RH is assumed. The gap between these upperbounds and the known lower-bounds is huge. We know due to Rankin [97] that (69) doesn’t hold with Φ(x) = c

log x log log x log log log log x , (log log log x)2

(70)

because Rankin showed that there exists a sequence of values of x tending to ∞ with intervals around x of length given in (70) which don’t contain a prime. This is the best proved order of magnitude for the largest gaps between consecutive primes. Maier [77] generalized this to the existence, for any fixed k, of k consecutive gaps between primes each of size at the order of magnitude (70). (This in turn was generalized to the case of arithmetic progressions by Shiu [105]). On the other hand Selberg [102] showed Φ(x) → ∞ as x → ∞. assuming RH that, (69) holds for almost all x if (log x)2 Here what is meant by ‘almost all x’ is that, while X → ∞ the measure of the set of x ∈ [0, X] for which (69) doesn’t hold is o(X). Without assuming 1 RH, this almost-all result is known to hold with Φ(x) = x 6 −ǫ (Zaccagnini [114]). Whether Selberg’s result is true without exceptions or not was answered by Maier [78] who showed that exceptions to (69) exist with Φ(x) as large as (log x)λ with any fixed λ > 1, i.e. π(x + (log x)λ ) − π(x) π(x + (log x)λ ) − π(x) > 1, lim inf < 1. x→∞ (log x)λ−1 (log x)λ−1 x→∞ (71) By the prime number theorem the difference between consectuive primes, pn+1 − pn is on average ∼ log pn . Cram´er [15] conjectured upon probabilistic reasoning that the largest possible gap between consecutive pn+1 − pn = 1, but now it is widely thought that primes satisfies lim sup (log pn )2 n→∞ lim sup

pn+1 − pn = O(log2 pn )

(72)

The Distribution of Primes: Conjectures vs. Hitherto Provables

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is a safer conjecture. Cram´er’s reasoning was based on the simple model that the probability of n being a prime is approximately 1/ log n, and that this can be taken into account independently for different integers (which is the basic flaw in Cram´er’s model; for example, n and n + 2 both being primes are not independent events: if n is even we automatically know that n + 2 can not be prime). Since Rankin’s estimate (70), only the value of the constant c in (70) has been improved (see [80]). The estimates in (51) and (52), even under very strong conjectures for the zeta zeros, fall dismally short of (72). Naturally the more that is known or assumed about the zeta zeros, the stronger results are deduced on the distribution of primes, notwithstanding the zeta zeros by themselves do not pertain steadfastly to primes as ζ ∗ (s) =

∞ Y

(1 −

n=1

1 −1 ) qns

(pn ≤ qn ≤ pn+1 ),

considered by Grosswald and Schnitzer [52], reveals. This product converges absolutely for σ > 1 where it does not vanish, it can be analytically continued to σ > 0 and then it is seen that ζ ∗ (s) has the same zeros as ζ(s) in σ > 0. But some significant properties of ζ(s) are not valid for ζ ∗ (s), namely the analytic continuation is not possible beyond σ > 0 so that for ζ ∗ (s) there is no functional equation, and it has a simple pole at s = 1 with residue r, 21 ≤ r ≤ 1. As for small gaps between primes, there has been considerable progress recently. In the 1990’s Goldston [35] developed a method which gives lower bounds of the correct order of magnitude in many problems about the distribution of primes. This method rested upon using short divisor sum approximations to the von Mangoldt function such as X R µ(d) log( ), ΛR (n) := d d|n, d≤R

with a parameter R to be chosen as large as possible in an application because without the condition d ≤ R this sum is equal to Λ(n) for n > 1. After a sequence of works by Goldston et. al. either concerning various applications or for improving the results by employing better approximants, Goldston, Pintz and the author in [40] and [42] have reached √ pn+r − pn (73) lim inf ≤ e−γ ( r − 1)2 n→∞ log pn for any fixed positive integer r. With r = 1 this shows that for any fixed ǫ > 0, there are infinitely many consecutive primes differing by less than ǫ

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times the average gap. Through a more refined analysis in [41] the sharper result pn+1 − pn lim inf √ ≤∞ (74) n→∞ log pn (log log pn )2 was obtained. The method also allows a generalization of (73) to primes in arithmetic progressions where the modulus q can grow with the size N of the primes subject to q ≪A (log log N )A . Furthermore, it turns out somewhat surprisingly, the assumption that (59) holds with Q = xθ−ǫ for any fixed θ ∈ ( 21 , 1] implies that there exist bounded gaps between primes infinitely often, the size of the gaps being a function of θ. This is an instance revealing the importance and strength of the Bombieri-Vinogradov theorem, and that showing its result continues to hold beyond θ = 21 will be an utmost breakthrough. Together with Graham the same authors applied their methods to the problem of small gaps between E2 -numbers qn . Corresponding to (73), it was shown in [36] that lim inf qn+r − qn ≤ C(r) n→∞

(75)

for a constant C(r), in particular C(1) = 6. The ideas involved in this work also yielded in [37] stronger variants of the Erd¨ os-Mirsky conjecture: There are infinitely many integers n which simultaneously satisfy d(n) = d(n + 1), Ω(n) = Ω(n + 1), ω(n) = ω(n + 1), even by specifying the value of these functions along with generalizations to shifts n + b with an arbitrary positive integer b (here d(n), Ω(n), ω(n) denote respectively the number of positive integer divisors of n, the number of prime divisors of n counted with multiplicity, and the number of distinct prime divisors of n). Suggestions for further reading In addition to the articles and books cited in the text we suggest the following works for a presentation of the topics and the proofs. Section 2: Dickson’s three volume set [17] is a compendium of results in number theory up to approximately 1920. The book by Hardy and Wright [56] is a standard reference for most topics of elementary number theory. Sections 3, 4 & 8: For detailed introductions to the topics of these sections we refer the reader to the books by Davenport [16], and by Montgomery and Vaughan [87]. Sections 4 & 5: Titchmarsh’s book (with notes added by Heath-Brown for the 2nd edition) [110] is a classic treatise on the Riemann zeta-function.

The Distribution of Primes: Conjectures vs. Hitherto Provables

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There are other treatises devoted to the theory of ζ(s), notably those of Edwards [18], Ivic [69], Karatsuba and Voronin [74]. The last two books contain a proof of (21) and (22). Elementary proofs of the prime number theorem are given in Hardy and Wright [56], and in the introductory level book by Tenenbaum and Mendes-France [108]. The articles by Bombieri [6], Sarnak [101], and Conrey [11] are recent essays on the Riemann Hypothesis by three of the leading experts in the field. Section 6: A classical book on the subject which includes the proof of (26) is Ingham’s tract [66]. Section 7: For further details on the pair correlation conjecture and prime numbers we refer the reader to Goldston’s survey article [33]. The nlevel correlations of zeros of ζ(s) were studied by Bogomolny and Keating [4] using heuristic arguments and based on the Hardy-Littlewood conjecture (equations (65)-(66) above with r = 2), and rigorously by Rudnick and Sarnak [100]. For the theory of random matrices, correlation functions and relations of zeta zeros to eigenvalues of a Hermitian operator the reader may consult the book by Mehta [81]. Section 8: The survey article by Iwaniec [72] gives an overview of the current directions in prime number theory. In a series of three papers, the last being [7], Bombieri, Friedlander and Iwaniec obtained improvements in some directions (not including the version stated in §8) on the BombieriVinogradov theorem. Concerning the Barban-Davenport-Halberstam theorem there are the works of Hooley titled ‘On the Barban - Davenport Halberstam theorem I – XVIII’), and of Friedlander and Goldston [26]. The books by Montgomery [82], and Bombieri [5] contain further treatment of most of the topics of this section. For the Brun-Titchmarsh theorem see Friedlander’s survey article [22]. Section 9: The methods of attacking problems having an additive nature, sieve methods, and the conjectures on the distribution of primes mentioned in this section can be found at an introductory level in Friedlander [24], Greaves [49] and Nathanson [89]. At more advanced levels there are the books of Greaves [50], Halberstam and Richert [53], and Vaughan [112]. The article by Kra [76] presents an introduction to the ideas in Green and Tao’s work. Section 10: Maier’s papers listed in the references brought forth important developments in prime number theory. For comments on the error terms in the prime number theorem for arithmetic progressions, and for the consequences of Maier’s work we refer the reader to Friedlander’s survey article [23]. For Cram´er’s model, Maier’s method and related matters on

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the distribution of primes we refer the reader to the articles by Granville [47], [48], Pintz [95] and Soundararajan [106], [107]. The last of these articles is mainly concerned with the problem of small gaps between primes. The developments leading to the results of Goldston, Pintz and Yıldırım are recounted in [43]. It is much easier to find the articles of Riemann, Hardy, Littlewood and Selberg from their collected works. Some of the original articles which have shaped the development of the theory in the 20th century have been collected in a book edited by Wang [113]. And most of the recent articles can be found in the websites of the authors. References 1. R. C. Baker, G. Harman and J. Pintz, Proc. London Math. Soc. (3) 83, 532-562 (2001). 2. P. T. Bateman, J. W. Brown, R. S. Hall, K. E. Kloss and R. M. Stemmler, Linear relations connecting the imaginary parts of the zeros of the zeta function, in Computers in Number Theory (Proc. Sci. Res. Council Atlas Sympos. No. 2, Oxford, 1969), Academic Press, London, 1971, pp. 1-19. 3. C. Bays, K. Ford, R. H. Hudson and M. Rubinstein, J. Number Theory 87, 54-76 (2001). 4. E. B. Bogomolny and J. P. Keating, Nonlinearity 9, 911-935 (1996). 5. E. Bombieri, Le grand crible dans la th´eorie analytique des nombres; Ast´erisque 18 (1987/1974), (Soci´et´e Math´ematique de France, 1987). 6. E. Bombieri, (2000), http://www.claymath.org/library/MPP.pdf, pp. 99111. 7. E. Bombieri, J. B. Friedlander and H. Iwaniec, J. Amer. Math. Soc. 2, no. 2, 215-224 (1989). 8. T. H. Chan, Int. J. Number Theory 2, no.1, 105-110 (2006). 9. J. Chen, Sci. Sinica 16, 157-176 (1973). 10. J. B. Conrey, J. Reine Angew. Math. 399, 1-26 (1989). 11. J. B. Conrey, Notices Amer. Math. Soc. 50, no.3, 341-353 (2003). 12. J. B. Conrey, L-functions and random matrices, in Mathematics unlimited - 2001 and beyond, Springer-Verlag, Berlin, 2001, pp. 331-352. 13. J. B. Conrey, Notes on L-functions and random matrix theory, in Frontiers in Number Theory, Physics and Geometry I, Springer-Verlag, Berlin, 2006, pp. 107-162. 14. H. Cram´er, Ark. Mat. Astronom. Fys. 15, 1-32 (1920). 15. H. Cram´er, Acta Arith. 2, 23-46 (1936). 16. H. Davenport, Multiplicative number theory, 3rd edn., revised and with a preface by H. L. Montgomery, Springer-Verlag, New York, 2000. 17. L. E. Dickson, History of the theory of numbers (3 vol.s), 1923, reprinted, Amer. Math. Soc., Providence, 2002. 18. H. M. Edwards, Riemann’s zeta function, 1974, reprinted, Dover, New York, 2001.

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19. P. D. T. A. Elliott and H. Halberstam, Symposia Mathematica 4 (INDAM, Rome, 1968/69), Academic Press, London, 1970, pp. 59-72. 20. P. Erd¨ os, Proc. Nat. Acad. Sci. 35, 374-384 (1949). 21. E. Fouvry and H. Iwaniec, Acta Arith. 79, no. 3, 249-287 (1997). 22. J. B. Friedlander, On the Brun-Titchmarsh theorem, in Number Theory, Trace Formulas and Discrete Groups (Selberg Symposium, Oslo, 1987) (eds. Aubert, Bombieri and Goldfeld), Academic Press, Boston, 1989, pp 219-228. 23. J. B. Friedlander, Irregularities in the distribution of primes, in Advances in number theory (Kingston, ON, 1991), (eds. Gouvˆea and Yui) Oxford Univ. Press, New York, 1993, pp. 17-30. 24. J. B. Friedlander, Topics in analytic number theory, in Number Theory and its Applications (Ankara, 1996), (eds. Yıldırım and Stepanov), Lecture Notes in Pure and Applied Math. 204, Marcel Dekker, New York, 1999, pp. 47-64. 25. J. B. Friedlander and D. A. Goldston, Quart. J. Math. Oxford (2) 47, 313336 (1995). 26. J. B. Friedlander and D. A. Goldston, Note on a variance in the distribution of primes, in Number theory in progress, Vol. 2 (Zakopane, 1997), (eds. Gy¨ ory, Iwaniec and Urbanowicz), de Gruyter, Berlin, 1999, pp. 841-848. 27. J. B. Friedlander and A. Granville, Ann. of Math.(2) 129, no. 2, 363-382 (1989). 28. J. B. Friedlander and H. Iwaniec, The Brun-Titchmarsh theorem, in Analytic number theory (Kyoto 1996), London Math. Soc. Lect. Note Ser. 247, Cambridge Univ. Press, 1997, pp. 85-93. 29. J. B. Friedlander and H. Iwaniec, Ann. of Math. (2) 148, no. 3, 945-1040 (1998). 30. P. X. Gallagher, Mathematika 23, 4-9 (1976). 31. P. X. Gallagher, Acta Arithmetica 37, 339-343 (1980). 32. D. Goldfeld, http://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf 33. D. A. Goldston, http://aimath.org/preprints.html, Preprint 2004-28. 34. D. A. Goldston, J. Number Theory 27, no. 2, 149-177 (1987). 35. D. A. Goldston, Expo. Math. 13, 366-376 (1995). 36. D. A. Goldston, S. W. Graham, J. Pintz and C. Y. Yıldırım, http://arxiv.org/PS_cache/arxiv/pdf/0609/0609615v1.pdf 37. D. A. Goldston, S. W. Graham, J. Pintz and C. Y. Yıldırım, http://arxiv.org/PS_cache/arxiv/pdf/0803/0803.2636v1.pdf 38. D. A. Goldston and D. R. Heath-Brown, Math. Ann. 266, 317-320 (1984). 39. D. A. Goldston and H. L. Montgomery, Pair correlation of zeros and primes in short intervals, in Analytic Number Theory and Diophantine Problems, Proc. of a conference at Oklahoma State Univ. 1984, (eds. Adolphson, Conrey, Ghosh and Yager), Birkh¨ auser Boston 1987, pp. 183-203. 40. D. A. Goldston, J. Pintz and C. Y. Yıldırım, to appear in Ann. of Math. http://arxiv.org/PS_cache/arxiv/pdf/0508/0508185v1.pdf 41. D. A. Goldston, J. Pintz and C. Y. Yıldırım, http://arxiv.org/PS_cache/arxiv/pdf/0710/0710.2728v1.pdf 42. D. A. Goldston, J. Pintz and C. Y. Yıldırım, Funct. Approx. Comment.

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Math. 35, 79-89 (2006). 43. D. A. Goldston, J. Pintz and C. Y. Yıldırım, The path to recent progress on small gaps between primes, in Analytic Number Theory, Clay Math. Proc. 7, Amer. Math. Soc., Providence, 2007, pp. 129-139. 44. D. A. Goldston and R. C. Vaughan, On the Montgomery-Hooley asymptotic formula, in Sieve methods, exponential sums, and their applications to number theory (Cardiff, 1995) London Math. Soc. Lecture Note Ser. 237, Cambridge Univ. Press, 1997, pp. 117-142. 45. S. M. Gonek, An explicit formula of Landau and its applications to the theory of the zeta-function, in A tribute to Emil Grosswald: Number theory and related analysis (eds. Knopp and Sheingorn) Contemporary Math. 143, Amer. Math. Soc., Providence, 1993, pp. 395-413. 46. X. Gourdon, (2004), http://numbers.computation.free.fr/Constants/ Miscellaneous/zetazeros1e13-1e24.pdf 47. A. Granville, Scand. Actuarial J., no.1, 12-28 (1995). 48. A. Granville, Unexpected irregularities in the distribution of prime numbers, in Proc. International Congress of Mathematicians, Zurich 1994, Birk¨ auser, Basel, 1995, pp. 388-399. 49. G. Greaves, Sieve methods, in Number Theory and its Applications (Ankara, 1996), (eds. Yıldırım and Stepanov), Lecture Notes in Pure and Applied Math. 204, Marcel Dekker, NY, 1999, 65-107. 50. G. Greaves, Sieves in Number Theory, Springer-Verlag, Berlin, 2001. 51. B. Green and T. Tao, Ann. of Math. (2) 167, no. 2, 481-547 (2008). 52. E. Grosswald and F. J. Schnitzer, Pacific J. Math. 74, no.2, 357-364 (1978). 53. H. Halberstam and H.-E. Richert, Sieve Methods, Academic Press, London, 1974. 54. G. H. Hardy, Matematisk Tidsskrift B, 1-16 (1922). 55. G. H. Hardy and J. E. Littlewood, Acta Mathematica 44, 1-70 (1922). 56. G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., Oxford Univ. Press, 1985. 57. D. R. Heath-Brown, Acta Arithmetica XLI, 85-99 (1982). 58. D. R. Heath-Brown, Proc. London Math. Soc. (3) 47(2), 193-224 (1983). 59. D. R. Heath-Brown, J. Reine Angew. Math. 389, 22-63 (1988). 60. D. R. Heath-Brown, Proc. London Math. Soc. (3) 64(2), 265-338 (1992). 61. D. R. Heath-Brown, Acta Math. 186 no. 1, 1-84 (2001). 62. D. Hensley and I. Richards, On the incompatibility of two conjectures concerning primes, in Proc. Symp. Pure Math. 24, Amer. Math. Soc., Providence, 1973, pp. 181-193. 63. C. Hooley, J. London Math. Soc. (2) 11, 399-407 (1975). 64. C. Hooley, J. London Math. Soc. (2) 13, 57-64 (1976). 65. C. Hooley, J. Reine Angew. Math. 499, 1-46 (1998). 66. A. E. Ingham, The distribution of prime numbers, 1932, reprinted with a foreword by R. C. Vaughan, Cambridge Univ. Press, 1992. 67. A. E. Ingham, Amer. J. Math. 64, 313-319 (1942). 68. A. E. Ingham, (1949), http://www.ams.org/mathscinet/pdf/29411.pdf 69. A. Ivic, The Riemann zeta-function, 1985, reprinted, Dover, New York, 2003.

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70. H. Iwaniec, Acta Arith. 24, 435-459, (1973/74). 71. H. Iwaniec, Invent. Math. 47 no. , 171-188, (1973/74). 72. H. Iwaniec, Prime numbers and L-functions, in Proc. International Congress of Mathematicians, Madrid, 2006, Eur. Math. Soc., Zurich, 2007, pp. 279306. 73. W. B. Jurkat, The Mertens conjecture and related general Ω-theorems, in Proc. Symp. Pure Math. 24 (Amer. Math. Soc., Providence, 1973), pp. 147158. 74. A. A. Karatsuba and S. M. Voronin, The Riemann zeta-function (translated from the Russian by N. Koblitz), de Gruyter, Berlin, 1992. 75. D. Knuth, Notices Amer. Math. Soc. 49, no. 3, 318-324 (2002). 76. B. Kra, Bull. Amer. Math. Soc. (N.S.) 43, no. 1, 3-23 (2006). 77. H. Maier, Adv. in Math. 39, no. 3, 257-269 (1981). 78. H. Maier, Michigan Math. J. 32, no. 2, 221-225 (1985). 79. H. Maier, Michigan Math. J. 35, (1988), no. 3, 323-344. 80. H. Maier and C. Pomerance, Trans. Amer. Math. Soc. 322, no. 1, 201-237 (1990). 81. M. L. Mehta, Random Matrices, 3rd ed., Elsevier/Academic Press, 2004. 82. H. L. Montgomery, Topics in multiplicative number theory, Lecture Notes in Math. 227, Springer, Berlin, 1971. 83. H. L. Montgomery, The pair correlation of zeros of the zeta function, in Proc. Symp. Pure Math. 24, Amer. Math. Soc., Providence, 1973, pp. 181-193. 84. H. L. Montgomery, The zeta function and prime numbers in Proc. Queen’s Number Theory Conference, 1979, Queen’s Papers in Pure and Appl. Math. 54, Queen’s University, Kingston, Ont., 1980, pp. 1-31. 85. H. L. Montgomery and K. Soundararajan, Commun. Math. Phys. 252, 589617 (2004). 86. H. L. Montgomery and R. C. Vaughan, J. London Math. Soc. (2) 8, 73-82, (1974). 87. H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory I. Classical Theory, Cambridge University Press, 2007. 88. M. R. Murty, J. Madras Univ., 161–169 (1988); or http://www.math.mast.queensu.ca/~murty/index2.html 89. M. B. Nathanson, Additive Number Theory, The Classical Bases, SpringerVerlag, New York, 1996. 90. A. M. Odlyzko, http://www.dtc.umn.edu/~odlyzko/ 91. A. M. Odlyzko and H. J. J. te Riele, J. Reine Angew. Math. 357, 138-160 (1985). 92. J. Pintz, On the remainder term of the prime number formula and the zeros of Riemann’s zeta-function, in Lec. Notes in Math. 1068 (ed. H. Jager), Springer-Verlag, Berlin, 1984, pp. 186-197. 93. J. Pintz, Acta Math. Hungar. 58 no. 3-4, 383-387 (1991). 94. J. Pintz, Approximations to the Goldbach and twin prime problem and gaps between consecutive primes, in Adv. Studies in Pure Math. 43, Int. Conference on Probability and Number Theory, Kanazawa, 2005, (2006), pp. 1-40.

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95. 96. 97. 98. 99. 100.

J. Pintz, Functiones et Approximatio XXXVII.2, 361-376 (2007). O. Ramar´e, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22, 645-706, (1995). R. A. Rankin, J. London Math. Soc. 13, 242-247 (1938). B. Riemann, Monatsberichte d. Preuss. Akad. d. Wissens., 671–680 (1859). M. Rubinstein and P. Sarnak, Experimental Math. 3, 173-197 (1994). Z. Rudnick and P. Sarnak, C.R. Acad. Sci. Paris 319, S´erie I, 1027-1032 (1994). P. Sarnak, (2004), http://www.claymath.org/library/04report_sarnak.pdf A. Selberg, Arch. Math. Naturvid. 47, no. 6, 87-105 (1943). A. Selberg, Ann. of Math. 50, 305-313 (1949). A. Selberg, Ann. of Math. 50, 297-304 (1949). D. K. L. Shiu, J. London Math. Soc. (2) 61, no. 2, 359-373 (2000). K. Soundararajan, The distribution of prime numbers, in Equidistribution in number theory, an introduction, NATO Sci. Ser. II. Math. Phys. Chem. 237, Springer, Dordrecht, 2007, pp. 59-83. K. Soundararajan, Bull. Amer. Math. Soc. (N.S.) 44, no. 1, 1-18 (2007). G. Tenenbaum and M. Mendes-France, The prime numbers and their distribution (translated from the French), Amer. Math. Soc, Providence, 2000. T. K. Timberlake and J. M. Tucker, (2008), http://arxiv.org/PS_cache/arxiv/pdf/0708/0708.2567v2.pdf E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed. revised by D.R. Heath-Brown, Oxford Univ. Press, 1986. P. Tur´ an, Collected papers of Paul Tur´ an, Vol. 3, ed. P. Erd¨ os Akademiai Kiado, Budapest, 1990. R. C. Vaughan, The Hardy-Littlewood method, 2nd ed., Cambridge Univ. Press, 1997. Y. Wang (editor), The Goldbach Conjecture, 2nd ed., World Scientific, Singapore, 2002. A. Zaccagnini, Acta Arith. 84, no. 3, 225-244, (1998).

101. 102. 103. 104. 105. 106.

107. 108. 109. 110. 111. 112. 113. 114.

Session 01

Analytic Function Spaces and Their Operators

SESSION EDITORS R. Aulaskari H. T. Kaptano˘ glu

University of Joensuu, Joensuu, Finland Bilkent University, Ankara, Turkey

111

A C ∗ -ALGEBRA OF FUNCTIONAL OPERATORS WITH SHIFTS HAVING A NONEMPTY SET OF PERIODIC POINTS M. A. BASTOS Departamento de Matem´ atica, Instituto Superior T´ ecnico Av. Rovisco Pais, 1049-001 Lisboa, Portugal E-mail: [email protected] C. A. FERNANDES Departamento de Matem´ atica, Faculdade de Ciˆ encias e Tecnologia Universidade Nova de Lisboa Quinta da Torre, 2825 Monte de Caparica, Portugal E-mail: [email protected] Y. I. KARLOVICH Facultad de Ciencias, Universidad Aut´ onoma del Estado de Morelos Av. Universidad 1001, Col. Chamilpa, C. P. 62209 Cuernavaca, Morelos, M´ exico E-mail: [email protected] Let A be a C ∗ -algebra of functional operators on the space L2 (T) generated by multiplication operators by piecewise slowly oscillating functions and by a group of unitary operators associated to an amenable group G of orientationpreserving homeomorphisms of the unit circle T onto itself such that all g ∈ G \ {e} have piecewise continuous derivatives and the same nonempty set Λ of periodic points, and there is a point t ∈ Λ with a finite G-orbit. An invertibility criterion for the operators A ∈ A is established. Keywords: Representations of C ∗ -algebras; Amenable group; Local-trajectory method; Functional operator; Invertibility.

1. Introduction Let B(L2 (T)) be the C ∗ -algebra of all bounded linear operators acting on the Lebesgue space L2 (T) where T is the unit circle in C with the length measure and the usual anticlockwise orientation. The aim of this paper is

112

M. A. Bastos, C. A. Fernandes & Y. I. Karlovich

to study the invertibility in the C ∗ -algebra of functional operators A := alg (P SO(T), UG ) ⊂ B(L2 (T))

(1)

generated by all multiplication operators by piecewise slowly oscillating (P SO(T)) functions and by a group UG = {Ug : g ∈ G} of unitary operators where (Ug ϕ)(t) := |g ′ (t)|1/2 ϕ(g(t)) for t ∈ T, and G is an amenable discrete group (see, e.g., [2,5,7]) of orientation-preserving homeomorphisms of T onto itself such that all g ∈ G \ {e} have piecewise continuous derivatives g ′ and the same nonempty set Λ of periodic points, and there is a point t ∈ Λ with the finite G-orbit G(t) := {g(t) : g ∈ G}. Functional operators with shifts and continuous or piecewise continuous coefficients are studied for a long time (see, e.g., [1,2,6,8]). The C ∗ -algebra A = alg (P SO(T), UG ) was recently investigated in [3,4] under the condition that all g ∈ G\{e} have the same (finite, resp. arbitrary) nonempty set Λ of fixed points. Using the local-trajectory method and its generalizations [3,5, 7] (also see [1,2]), we got in [3,4] invertibility criteria for the operators A ∈ A. In the present paper, Λ is an arbitrary nonempty set of common periodic points for all g ∈ G\{e}. Appearance of periodic points of multiplicity n > 1 leads to the necessity to combine local-trajectory methods and the scheme of studying functional operators with Carleman shifts exposed in [8]. The paper is organized as follows. In Section 2 we describe the maximal ideal space of the commutative algebra P SO(T). In Section 3 we study the structure of the group of homeomorphisms G. In Section 4 we describe a reduction of functional operators with shifts having periodic points to matrix functional operators with shifts having fixed points (see [8]). Section 5 contains the main results of the paper. First we reduce the study of invertibility of functional operators A ∈ A to studying the invertibility of their restrictions Aarc , A◦ and A∗ on the spaces L2 (T∗arc ), L2 (Λ◦ ) and L2 (Λ∗ ), respectively, where T∗arc := T \ Λ◦ , Λ◦ is the interior of Λ, and Λ∗ = T \ (T∗arc ∪ Λ◦ ). Then we establish invertibility criteria for Aarc and A◦ , by applying the local-trajectory method and the scheme in Section 4, respectively, and show that the invertibility of Aarc and A◦ implies the invertibility of A∗ . As a result, we obtain the invertibility criterion for A ∈ A. 2. Preliminaries Let C(T) and P C(T) denote the C ∗ -subalgebras of L∞ (T) consisting, respectively, of the functions continuous on T and the functions having finite one-sided limits at each point of T.

A C ∗ -Algebra of Functional Operators with Shifts

113

A function f ∈ L∞ (T) is called slowly oscillating at a point λ ∈ T if n o lim ess sup |f (z1 ) − f (z2 )| : z1 , z2 ∈ Tε (λ) = 0, ε→0

 where Tε (λ) := z ∈ T : ε/2 ≤ |z − λ| ≤ ε . We denote by SO(T) the C ∗ -subalgebra of L∞ (T) consisting of the functions slowly oscillating at each point of T. Let P SO(T) := alg(SO(T), P C(T)) be the C ∗ -subalgebra of L∞ (T) generated by the C ∗ -algebras P C(T) and SO(T). Let M (A) be the maximal ideal space of a commutative unital C ∗ algebra A. We have (cf. [3]): [ [ Mξ (P SO(T)), M (SO(T)) = Mt (SO(T)), M (P SO(T)) = t∈T

ξ∈M(SO(T))

where the fibers Mt (SO(T)) for t ∈ T are given by  Mt (SO(T)) = ξ ∈ M (SO(T)) : ξ|C(T) = t ,

and, if ξ ∈ Mt (SO(T)) with t ∈ T, then Mξ (P SO(T)) = {(ξ, 0), (ξ, 1)}, where for µ ∈ {0, 1}, (ξ, µ)|SO(T) = ξ, (ξ, µ)|C(T) = t, (ξ, µ)|P C(T) = (t, µ). The Gelfand topology on M (P SO(T)) = M (SO(T)) × {0, 1} can be described as follows. If ξ ∈ Mt (SO(T)) (t ∈ T), a base of neighborhoods for (ξ, µ) ∈ M (P SO(T)) consists of all open sets of the form (

− Uξ,t × {0} ∪ Uξ,t × {0, 1} if µ = 0, U(ξ,µ) =

+ Uξ,t × {1} ∪ Uξ,t × {0, 1} if µ = 1,

where Uξ,t = Uξ ∩ Mt (SO(T)), Uξ is an open neighborhood of ξ in − + M (SO(T)), and Uξ,t , Uξ,t consist of all ζ ∈ Uξ such that τ = ζ|C(T) belong, respectively, to the sets (−t, t) := {z ∈ T : −π < arg (z/t) < 0} and (t, −t) := {z ∈ T : 0 < arg (z/t) < π}. 3. Structure of the group of homeomorphisms G Let G be a group of orientation-preserving homeomorphisms of T onto itself such that all g ∈ G \ {e} have the same nonempty set Λ of periodic points. The group operation in G is given by (gh)(t) = h(g(t)) for g, h ∈ G, t ∈ T. Lemma 3.1. If there is t0 ∈ Λ with finite G-orbit then, for all t ∈ Λ, the cardinalities card G(t) of the G-orbits G(t) = {g(t) : g ∈ G} coincide.

Proof. Let n := card G(t0 ) < ∞ for a point t0 ∈ Λ. If n = 1, then t0 and hence [8] all t ∈ Λ are fixed points for each g ∈ G, that is, card G(t) = 1 for

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M. A. Bastos, C. A. Fernandes & Y. I. Karlovich

all t ∈ Λ. Let n > 1 and let t0 ≺ t1 ≺ t2 ≺ . . . ≺ tn−1 (≺ t0 ) be the points in G(t0 ) enumerated anticlockwise. Fix α ∈ G such that α(t0 ) = t1 . Then αk (t0 ) = tk for k = 0, 1, . . . , n where tn := t0 .

(2)

Indeed, if α2 (t0 ) 6= t2 , then α2 (t0 ) ∈ {t3 , t4 , . . . , tn−1 , t0 }. In that case, since α preserves the orientation of T and α(t0 ) = t1 ≺ t2 ≺ α2 (t0 ) = α(t1 ), we conclude that α−1 (t2 ) belongs to the open arc (t0 , t1 ). But there are no points of G(t0 ) between t0 and t1 . Thus, we arrive at a contradiction, and hence α2 (t0 ) = t2 . Analogously we obtain all the other relations in (2). Consequently, t0 is a periodic point of α of multiplicity n = card G(t0 ), and then every point t ∈ Λ is a periodic point of α of the same multiplicity (see Chapter 1 in [8]): for all t ∈ Λ, αk (t) 6= t (k = 1, 2, . . . , n − 1) and αn (t) = t. Fix g ∈ G \ {e}. Clearly g(t0 ) = tj for some j = 0, 1, . . . , n−1. Then we put gb := αj g −1 (∈ G). −1

j

(3) −1

By (2) and (3), we obtain gb(t0 ) = g (α (t0 )) = g (tj ) = t0 . Hence, gb(t) = t for all t ∈ Λ (see again [8]), and from (3) it follows that g(t) = αj (t) for all t ∈ Λ. Thus, for every g ∈ G there is a number k(g) ∈ {0, 1, . . . , n−1} such that g(t) = αk(g) (t) for all t ∈ Λ, which implies that the G-orbit of each point t ∈ Λ is given by G(t) = {αj (t) : j = 0, 1, . . . , n − 1}. Setting ge := gb−1 in (3), we get by the proof of Lemma 3.1 the following.

Proposition 3.1. If G is the group of orientation-preserving homeomorphism of T onto itself such that all g ∈ G \ {e} have the same nonempty set Λ of periodic points and card G(t0 ) < ∞ for some t0 ∈ Λ, then there exists a shift α ∈ G for which all points t ∈ Λ have the multiplicity n = card G(t 0 ), and for any shift g ∈ G there are a number k(g) ∈ {0, 1, . . . , n − 1} and a shift ge ∈ G with the set Λ of fixed points such that g = e g αk(g) . For j = 0, 1, . . . , n − 1, we consider the sets

Gj := {g ∈ G : k(g) = j} ⊂ G.

(4) k(g)

It is easily seen that G0 is a normal subgroup of G. Further, g = ge α and (4) imply that Gj = G0 αj for j = 0, 1, . . . , n − 1, and hence the sets Gj are exactly the cosets of the quotient group G/G0 . 4. Reduction to the case of fixed points Consider the following reduction procedure (see Chapter 3 in [8]). Let Ab P be the dense subalgebra of A consisting of all operators g∈F ag Ug with

A C ∗ -Algebra of Functional Operators with Shifts

115

finite sets F ⊂ G. By (4) and g = ge αk(g) , any A ∈ Ab is represented in the form n−1 X X X A= ag U g = Bk Uαk , with Bk = ag Uge , (5) g∈F

k=0

g∈Gk ∩F

n

where ge, α ∈ G0 . By direct computation, we get the following (cf. [8]).

Lemma 4.1. If B0 , B1 , . . . , Bn−1 are given by (5) and ε
= e2πi/n , then (6) C −1 BA C = DA := diag (A0 , A1 , . . . , An−1 ,    kj k n−1 n−1 C := ε Uα k,j=0 , C −1 := n−1 ε−kj Uα−j k,j=0 , Aj :=

Hk,j :=

n−1 X

 n−1 εkj Bk Uαk (j = 0, 1, . . . , n − 1), BA := Hk,j k,j=0 ,

k=0 Uαk Bj−k Uα−k

(k ≤ j), Hk,j := Uαk Bn+j−k Uαn−k (k > j).

(7)

(8)

Fix t0 ∈ Λ and suppose that the points ts := αs (t0 ) for s = 0, 1, . . . , n−1 are ordered anticlockwise: t0 ≺ t1 ≺ t2 ≺ . . . ≺ tn−1 ≺ tn := t0 . Define the invertible piecewise constant function b on T by b(t) = ε−s if t ∈ (ts , ts+1 ) and s = 0, 1, . . . , n − 1. Since b = b ◦ ge for all ge ∈ G0 , we infer that bj (t)b−j [(e g αk )(t)] = εjk for all t ∈ T and all j, k = 0, 1, . . . , n − 1. Hence, j Aj = b Ab−j I, these operators are invertible only simultaneously, and
DA = D diag A, . . . , A D−1 , with D := diag(I, bI, . . . , bn−1 I). (9) In view of (6) and (9), the map defined on Ab by X Θ:A= ag Ug 7→ BA = CD diag(A, . . . , A)D −1 C −1

(10)

g∈F

extends by continuity to a C ∗ -algebra homomorphism Θ : A → B(L2n (T)), where L2n (T) is the Hilbert space of vector functions u e = [uk ]nk=1 with uk ∈ L2 (T). Applying (10), we immediately obtain the following criterion. Theorem 4.1. Any operator A ∈ A with shifts g ∈ G is invertible on the space L2 (T) if and only if the operator BA := Θ(A) with shifts g ∈ G0 is invertible on the space L2n (T). 5. Invertibility in the C ∗ -algebra A 5.1. Decomposition Using a G-invariant decomposition of T, we will decompose the C ∗ -algebra A defined by (1) into an orthogonal sum of three operator C ∗ -algebras Aarc , A◦ and A∗ , and study separately the invertibility in these new algebras.

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Let Λ◦ := Int Λ be the interior of Λ, ∂Λ the boundary of Λ, and let Λ± be the set of all t ∈ ∂Λ that are limit points of the sets γt± ∩ Λ◦ , respectively, where γt+ (γt− ) is a right (left) semi-neighborhood of t on T. Then the closure Λ◦ of Λ◦ is given by Λ◦ = Λ◦ ∪ Λ+ ∪ Λ− , and Λ◦r ⊂ Λ− , Λ◦l ⊂ Λ+ , where Λ◦l and Λ◦r denotes, respectively, the at most countable set of the initial and final points of all open arcs which compose the set Λ◦ . Let χarc , χ◦ and χ∗ be the characteristic functions of the sets T∗arc := T \ Λ◦ , Λ◦ and Λ∗ := Λ+ ∪ Λ− , respectively. Since T∗arc ∪ Λ◦ ∪ Λ∗ is a G-invariant partition of T, we obtain the following decomposition result. Lemma 5.1. An operator A ∈ A is invertible on L2 (T) if and only if: (i) the operator Aarc := χarc A is invertible on L2 (T∗arc ), (ii) the operator A◦ := χ◦ A is invertible on L2 (Λ◦ ), (iii) the operator A∗ := χ∗ A is invertible on L2 (Λ∗ ) if mes Λ∗ > 0. For every A ∈ A, the operators Aarc , A◦ and A∗ belong to the C ∗ algebras Aarc := χarc A ⊂ B(L2 (T∗arc )), A◦ := χ◦ A ⊂ B(L2 (Λ◦ )) and A∗ := χ∗ A ⊂ B(L2 (Λ∗ )), respectively. 5.2. Invertibility in the C ∗ -algebra Aarc Consider Aarc as the C ∗ -algebra alg (χarc P SO(T), χarc UG ) ⊂ B(L2 (T∗arc )). Since the set T∗arc is open, the commutative C ∗ -algebra Zarc := {χarc aI : f∗ ) of all a ∈ P SO(T)} is isometrically isomorphic to the C ∗ -algebra C(M arc ∗ f continuous functions on the set Marc given by [  f∗ := M∗ = M (SO(T)) × {0, 1} M t arc arc t∈T∗ arc [  [  ∪ Mt (SO(T))×{0} ∪ Mt (SO(T))×{1} t∈Λ+ \Λ− t∈Λ− \Λ+ [  [  arc arc ∪ Mt,+ × {1} ∪ Mt,− × {0} , (11) ◦ ◦ t∈Λ+ \Λl

arc Mt,+

t∈Λ− \Λr

Λ◦l

where for t ∈ Λ+ \ denotes the closed set of all ξ ∈ M (SO(T)) arc for which are limits of nets δtα where tα → t and tα ∈ γt+ ∩ Tarc ; Mt,− ◦ t ∈ Λ− \ Λr is the closed set of all ξ ∈ M (SO(T)) which are limits of nets δtα where tα → t and tα ∈ γt− ∩ Tarc (see [4]). For every g ∈ G, the ∗ -automorphism αg : a 7→ Ug aUg−1 of Zarc induces the homeomorphisms βg : (ξ, µ) 7→ (g(ξ), µ) on M (P SO(T)) where ξ 7→ g(ξ) are homeomorphisms on M (SO(T)) given by a(g(ξ)) = (a ◦ g)(ξ) for all a ∈ SO(T) and all ξ ∈ M (SO(T)) (as usual, a(ξ) := ξ(a)). Hence, in view of the topologically free action (see [1–4]) of the group G on T \ Λ◦ and the group {βg : g ∈ G}

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117

f∗ because g(ξ) ∈ Mg(t) (SO(T)) for ξ ∈ Mt (SO(T)), we obtain an on M arc invertibility criterion for the operators in the C ∗ -algebra Aarc by analogy with [4], by applying the local-trajectory method and the techniques of spectral measures [3,5,7]. Let Oarc be a subset of Tarc = T \ Λ containing exactly one point in each G-orbit defined by the action of the group G on Tarc , and let  Rarc := (ξ, µ) ∈ M (P SO(T)) : ξ ∈ Mτ (SO(T)), τ ∈ Oarc , µ = 0, 1 . f∗ we associate the representation With each ideal (ξ, µ) ∈ M arc Π(ξ,µ) : A → B(l2 (G)), A 7→ A(ξ,µ) ,

given for the operators A = (A(ξ,µ) f )(h) =

X

g∈F

P

g∈F

(12)

b with finite sets F ⊂ G, by ag Ug ∈ A,

[(ag ◦h)(ξ, µ)]f (hg)

(h, g ∈ G, f ∈ l 2 (G)),

(13)

where (ag ◦h)(ξ, µ) means the value of the Gelfand transform of the function ag ◦ h ∈ P SO(T) at the point (ξ, µ) ∈ M (P SO(T)). Theorem 5.1. For each functional operator A ∈ A, the operator Aarc is invertible on the space L2 (T∗arc ) if and only if for all (ξ, µ) ∈ Rarc the operators A(ξ,µ) given by (12)–(13) are invertible on the space l 2 (G) and

sup (A(ξ,µ) )−1 < ∞. (14) (ξ,µ)∈Rarc

5.3. Invertibility in the C ∗ -algebra A◦ Consider A◦ as the C ∗ -subalgebra alg (χ◦ P SO(T), χ◦ UG ) of B(L2 (Λ◦ )). Let g ∈ G. By Proposition 3.1, g = e g αk(g) , where α ∈ G, ge ∈ G0 and k(g) for k(g) ∈ {0, 1, . . . , n − 1}. Consequently, χ◦ Ug = χ◦ Uge Uαk(g) = χ◦ Uα P each g = ge αk(g) ∈ G. Then for every A = g∈F ag Ug ∈ Ab we get χ◦ A = χ ◦

X

g∈F

ag U g =

n−1 X

k χ◦ b A k Uα ,

k=0

where the functions bA k ∈ P SO(T) are given by X bA ag (k = 0, 1, . . . , n − 1). k = g∈Gk ∩F

(15)

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P According to (7)–(8), with every operator A◦ = χ◦ g∈F ag Ug ∈ Ab we ◦ associate the operator matrix BA := χ◦ BA of the form   χ◦ b A χ◦ b A χ◦ b A ... χ◦ bA 0I 1I 2I n−1 I ◦ A   χ◦ (bA χ◦ (bA χ◦ (bA 0 ◦ α)I 1 ◦ α)I . . . χ (bn−2 ◦ α)I   ◦ An−1 ◦ α)I 2 ◦ A 2 ◦ A 2 ◦ A 2  ◦  χ (b ◦ α )I χ (b ◦ α )I χ (b ◦ α )I . . . χ (b ◦ α )I n−2 n−1 0 n−3 BA = ,   .. .. .. .. ..   . . . . .

n−1 n−1 n−1 n−1 )I )I . . . χ◦ (bA χ◦ (bA )I χ◦ (bA )I χ◦ (bA 0 ◦α 1 ◦α 2 ◦α 3 ◦α (16) where the functions bA k are given by (15). b we deduce from (10) that B ◦ := χ◦ Θ(A) ∈ B(L2 (Λ◦ )). Fix For A ∈ A, n A ◦ ◦ for every sequence {Am } ⊂ Ab convergent A ∈ A. Hence, BA = lim BA m m→∞

◦ to the operator A ∈ A, which defines BA ∈ B(L2n (Λ◦ )) for all A ∈ A. Since ◦ BA := χ◦ BA for all A ∈ A, Theorem 4.1 immediately implies the following.

Theorem 5.2. For A ∈ A, the operator A◦ := χ◦ A is invertible on the ◦ space L2 (Λ◦ ) if and only if the operator BA is invertible on the space L2n (Λ◦ ). Consider the commutative C ∗ -subalgebra Z ◦ := {χ◦ aI : a ∈ P SO(T)} of A◦ . By [4], the maximal ideal space of Z ◦ is given by [  f◦ := M◦ = M Mt (SO(T)) × {0, 1} ◦ t∈Λ [  [  ∪ M (SO(T)) × {1} ∪ M (SO(T)) × {0} t t t∈Λ◦ t∈Λ◦ r [ l  [  ◦ ◦ ∪ M × {1} ∪ M × {0} , (17) t,+ t,− ◦ ◦ t∈Λ+ \Λl

t∈Λ− \Λr

◦ where Mt,+ for t ∈ Λ+ \ Λ◦l denotes the closed set of all ξ ∈ M (SO(T)) ◦ for which are limits of nets δtα where tα → t and tα ∈ γt+ ∩ Λ◦ ; Mt,− ◦ t ∈ Λ− \ Λr is the closed set of all ξ ∈ M (SO(T)) which are limits of nets δtα where tα → t and tα ∈ γt− ∩ Λ◦ . f◦ ) are isometrically isomorphic, we Since the C ∗ -algebras Z ◦ and C(M get the following invertibility criterion for the operators A◦ ∈ A◦ .

Theorem 5.3. For any A ∈ A, the operator A◦ := χ◦ A has the form A◦ =

n−1 X

k χ◦ b A k Uα

(18)

k=0

◦ where bA k ∈ P SO(T) for all k = 0, 1, . . . , n−1. The operator A is invertible on the space L2 (Λ◦ ) if and only if

det BA (ξ, µ) 6= 0

f◦ , for all (ξ, µ) ∈ M

(19)

A C ∗ -Algebra of Functional Operators with Shifts

where BA (ξ, µ) :=  bA bA bA 0 (ξ, µ) 1 (ξ, µ) 2 (ξ, µ) A A  (bA µ) (b0 ◦ α)(ξ, µ) (b1 ◦ α)(ξ, µ)  An−1 ◦ α)(ξ, 2 A 2 A 2  (b ◦ α )(ξ, µ) (b ◦ α )(ξ, µ) (b n−2 n−1 0 ◦ α )(ξ, µ)   .. .. ..  . . .

119

 ... bA n−1 (ξ, µ)  . . . (bA n−2 ◦ α)(ξ, µ)  A 2 . . . (bn−3 ◦ α )(ξ, µ)  .  .. ..  . .

n−1 n−1 n−1 n−1 )(ξ, µ) )(ξ, µ) . . . (bA (bA )(ξ, µ) (bA )(ξ, µ) (bA 0 ◦α 1 ◦α 2 ◦α 3 ◦α (20)

Proof. Fix A ∈ A. Let {Am } be a sequence of operators in Ab such that A = Xn−1 k m χ◦ b A lim Am . Then χ◦ A = lim k Uα where, for k = 0, 1, . . . , n−1, m→∞

m→∞

k=0

m bA ∈ P SO(T). By (10) and (16), there are constants Ck ∈ (0, ∞) such that k

A

b m ∞ ◦ ≤ BAm 2 ◦ ≤ Ck kAm kB(L2 (Λ◦ )) (m ∈ N). k L (Λ ) B(Ln (Λ ))  ◦ Am Hence, the restrictions χ bk can be considered as convergent sequences ◦ ◦ f f in C(M ). Since M is a closed subset of M (P SO(T)), there exist functions ◦ Am A A ◦ A bA 0 , b1 , . . . , bn−1 ∈ P SO(T) such that χ bk = lim χ bk , which gives

m→∞

(18). Because A ∈ A is given by (18) with bA k ∈ P SO(T), we conclude that ◦ BA := Θ(A◦ ) is given by (16) with entries identified with their Gelfand f◦ ). Hence, the operator B ◦ is invertible on the space transforms in C(M A L2n (Λ◦ ) if and only if (19) holds. It remains to apply Theorem 5.2.

5.4. Invertibility in the C ∗ -algebra A∗ Let us study the invertibility of the operator A∗ := χ∗ A on the space L2 (Λ∗ ). Consider the C ∗ -algebra A0 := alg (P SO(T), UG0 ) ⊂ B(L2 (T)). Let [ M∗ := Mt (SO(T)) × {0, 1} ⊂ M (P SO(T)). t∈Λ∗

e ξ,µ : A0 → B(l2 (G0 )), To every (ξ, µ) ∈ M∗ we assign the representation Π P A 7→ Aξ,µ , given on the dense set of operators A = g∈F ag Ug ∈ A0 with finite sets F ⊂ G0 by X (Aξ,µ f )(h) = ag (ξ, µ)f (h) (h, g ∈ G0 , f ∈ l2 (G0 )). (21) g∈F

e ξ,µ (A0 ) ⊂ B(l2 (G0 )) is a C ∗ -algebra, applying the homomorphism Since Π X X e ξ,µ (A0 ) → C, Aξ,µ = Π: Π ag (ξ, µ)I 7→ ag (ξ, µ), g∈F

g∈F

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we infer that for every (ξ, µ) ∈ M∗ the composition map X X e ξ,µ : Π∗ξ,µ = Π ◦ Π ag Ug 7→ ag (ξ, µ) g∈F

g∈F

∗

extends to a C -algebra homomorphism of A0 onto C. Since ge, αn ∈ G0 , we deduce from (7)–(8) that each entry of BA belongs to A0 . Applying the map ∗ Π∗ξ,µ to BA entry-wise, we obtain the representations νξ,µ : BA 7→ BA (ξ, µ) for all (ξ, µ) ∈ M∗ , where BA (ξ, µ) for these (ξ, µ) is also defined by (20). ∗ Since BA := χ∗ BA = Θ(χ∗ A) is given by (16) with χ◦ replaced by χ∗ , ∗ and therefore BA is the operator of multiplication by the restriction of a ∗ P SO(T) matrix function to Λ∗ , it follows that BA is invertible on the space 2 Ln (Λ∗ ) if det BA (ξ, µ) 6= 0 for all (ξ, µ) ∈ M∗ . Thus, we get the following. Lemma 5.2. For any operator A ∈ A, if det BA (ξ, µ) 6= 0 for all (ξ, µ) ∈ M∗ , then the operator A∗ := χ∗ A is invertible on the space L2 (Λ∗ ). f∗ ∩ M∗ we associate the representation By [7], with every (ξ, µ) ∈ M arc

2 e Πarc ξ,µ : χarc A0 → Πξ,µ (A0 ) (⊂ B(l (G0 ))), Aarc 7→ Aξ,µ , P given for Aarc = χarc g∈F ag Ug ∈ A0 with finite sets F ⊂ G0 by (21). Applying now Π ◦ Πarc ξ,µ to χarc BA entry-wise, we obtain the representation arc f∗ ∩ M∗ along with ν ∗ . : χarc BA 7→ BA (ξ, µ) for every (ξ, µ) ∈ M ν ξ,µ

arc

ξ,µ

Theorem 5.4. If A ∈ A and the operators Aarc and A◦ are invertible, then the operator A∗ is also invertible.

Proof. If the operator Aarc is invertible on the space L2 (T∗arc ) then, by Theorem 4.1, the operator χarc BA = Θ(χarc A) is invertible on the arc (χarc BA )I is also invertible. space L2n (T∗arc ), and hence the operator νξ,µ f∗arc ∩ M∗ . On the other Therefore, det BA (ξ, µ) 6= 0 for all (ξ, µ) ∈ M

hand, by (19), the invertibility of A◦ implies that det BA (ξ, µ) 6= 0 for all f◦ ∩ M∗ . But (M f∗arc ∪ M f◦ ) ∩ M∗ = M∗ due to (11) and (17). (ξ, µ) ∈ M Hence det BA (ξ, µ) 6= 0 for all (ξ, µ) ∈ M∗ , and then, by Lemma 5.2, the operator A∗ is invertible on the space L2 (Λ∗ ).

5.5. Main result Finally, combining Lemma 5.1 and Theorems 5.1, 5.3 and 5.4, we get the following invertibility criterion in the C ∗ -algebra A. Theorem 5.5. An operator A ∈ A is invertible on L2 (T) if and only if

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f◦ and BA (ξ, µ) given by (20), det BA (ξ, µ) 6= 0; (i) for all (ξ, µ) ∈ M (ii) for all (ξ, µ) ∈ Rarc , the operators A(ξ,µ) given by (13) are invertible on the space l 2 (G) and (14) holds. References 1. A. B. Antonevich, Linear Functional Equations. Operator Approach, Oper. Theory Adv. Appl., vol. 83, Birkh¨ auser, Basel, 1996. Russian original: University Press, Minsk, 1988. 2. A. Antonevich, A. Lebedev, Functional Differential Equations: I. C ∗ -Theory, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 70, Longman Scientific & Technical, Harlow, 1994. 3. M. A. Bastos, C. A. Fernandes, and Y. I. Karlovich, J. Funct. Anal. 242, 86–126 (2007). 4. M. A. Bastos, C. A. Fernandes, and Y. I. Karlovich, C ∗ -algebras of singular integral operators with shifts having the same nonempty set of fixed points, Complex Anal. Oper. Theory 2, 241–272 (2008). 5. Y. I. Karlovich, Soviet Math. Dokl. 37, 407–411 (1988). 6. Y. I. Karlovich, Soviet Math. Dokl. 38, 301–307 (1989). 7. Y. I. Karlovich, A local-trajectory method and isomorphism theorems for nonlocal C ∗ -algebras, In: Oper. Theory Adv. Appl., vol. 170, Birkh¨ auser, Basel, 2006, pp. 137–166. 8. V. G. Kravchenko, G. S. Litvinchuk, Introduction to the Theory of Singular Integral Operators with Shift, Kluwer, Dordrecht, 1994.

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ON BLOCH AND NORMAL FUNCTIONS ON COMPLEX BANACH MANIFOLDS P. V. DOVBUSH Institute of Mathematics and Computer Science of Academy of Sciences of Moldova, 5 Academy Street, Kishinev, MD-2028, Republic of Moldova E-mail: [email protected] Let X be a complex Banach manifold. A holomorphic function f : X → C is called a Bloch function (resp., a normal function) if the family Ff = {f ◦ ϕ − f (ϕ(0)) : ϕ ∈ O(∆, X)} (resp., F = {f ◦ ϕ : ϕ ∈ O(∆, X)}) forms a normal family in the sense of Montel, where O(∆, X) denotes the set of holomorphic maps from the complex unit disc to X. Characterizations of normal and Bloch functions are presented. A sufficient condition for the sum of a normal function and a nonnormal function to be nonnormal is given. Criteria for a holomorphic function to be non-normal are obtained. Keywords: Kobayashi pseudometric, infinitesimal Kobayashi pseudometric, Bloch function, normal function, Banach space, complex Banach manifold.

1. Introduction The concept of a Bloch function was introduced in 1952 by Hayman [1] and has been widely studied. (See e.g. [2], p. 269 et seq.) The idea of associating to a meromorphic function f on the complex unit disc ∆ = {z ∈ C : |z| < 1} a family F = {f ◦ g : g ∈ Aut(∆)}, where Aut(∆) is the group of biholomorphic automorphisms of ∆, and of ascribing to the function f properties of the family F apparently arose in work of Yosida [3] in 1934 and was considered by Noshiro [4] in 1937. In 1957, Lehto and Virtanen [5] defined “normal functions” to be those meromorphic functions f whose associated families F were normal in the sense of Montel. Since that time, the subject of normal functions and Bloch functions has been studied extensively, resulting in substantial development in the single complex variable context and in generalizations to the setting of several complex variables (see lists of references in [6–8]). In this paper, we generalize the theory of normal holomorphic functions to the case of holomorphic functions on a complex Banach manifold.

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2. Basic Definitions We refer the reader to the paper [9] and the books [10,11] for background on complex analysis in infinite dimension. Let X be a complex Banach manifold modelled on a complex Banach space of positive, possibly infinite, dimension; X assumed to be a connected Hausdorff space. For each x in X the tangent space to X at point x will be denoted by Tx (X). The tangent bundle T (X) of X consist of the ordered pairs (y, v) such that x ∈ X and v ∈ Tx (X). We shall denote the space of all holomorphic maps from the unit disk ∆ into X by O(∆, X). The infinitesimal Kobayashi pseudometric on the complex Banach manifold X is the function kX on T (X) defined by the formula kX (x, v) = inf{|a| : ∃ ϕ ∈ O(∆, X), ϕ(0) = x, ϕ∗ (0)a = v} where ϕ∗ (0) is the linear map induced by ϕ from T0 (∆) to Tϕ(0) (X). We say that kX is a metric if kX (x, v) > 0 for all (x, v) ∈ Tx (X), v 6= 0. The Kobayashi length of a piecewise C 1 curve γ : [0, 1] → X in X is to be the upper Riemann integral Z 1 kX (γ(t), γ ′ (t)) dt Lk (γ) = 0

e X (x, y) is the infimum of the lengths of all piecewise and the pseudometric K 1 C curves joining x to y in X. The Kobayashi pseudometric between two points x, y in X is defined as follows. Consider all finite sequences of points p0 = x, p1 , . . . , pk−1 , pk = y of X such that there exist points z1 , . . . , zk , ze1 , . . . , zek of ∆ and maps ϕ1 , . . . , ϕk ∈ O(∆, X) satisfying ϕj (zj ) = pj−1 and ϕj (e zj ) = pj , j = 1, . . . , k. The Kobayashi pseudometric KX (x, y) is, by definition, k X ej −1 zj − z d∆ (zj , zej ) where d∆ (zj , zej ) = tanh KX (x, y) = inf 1 − zj zej j=1

is the Poincar´e metric on ∆ and the infimum is taken over all possible choices of points and maps. The spherical arc length element ds on the Riemann sphere C is given by ds(z, dz) = The spherical length s(γ) =

Z

γ

|dz| . 1 + |z|2

ds(z, dz)

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of a curve γ in C induces a metric in the following manner. Given distinct points a, b on the Riemann sphere, define s(a, b) = inf γ {s(γ)} where the infimum is taken over all piecewise C 1 curves on C which join a with b. Then s(a, b) defines a metric on the sphere known as the spherical metric. Denote by O(X) the set of all holomorphic functions on X. A family F ⊆ O(∆) is said to be normal in ∆ if every sequence {fn } ⊂ F has a subsequence which converges uniformly (with respect to the Euclidean metric) on compacta in ∆ or diverges uniformly to ∞ on compacta in ∆. Definition 2.1. A holomorphic function f : X → C is called a Bloch function (resp., a normal function) if the family Ff = {f ◦ ϕ − f (ϕ(0)) : ϕ ∈ O(∆, X)}, (resp., F = {f ◦ϕ : ϕ : ∆ → X is holomorphic }) is normal. A family F ⊆ O(∆) is said to be spherically equicontinuous at a point z0 ∈ ∆ if for each positive number ǫ there is a positive number δ such that s(f (z), f (z0 )) < ǫ for K∆ (z, z0 ) < δ and f in F. Following Gauthier [12] we shall define a sequence {xn } of points in X to be a P -sequence of f ∈ O(X) if there is a sequence {yn } of points in X such that KX (xn , yn ) → 0 as n → ∞ but s(f (xn ), f (yn )) ≥ ǫ for some ǫ > 0 for each positive integer n. 3. Background Results Lemma 3.1 (Zalcman’s Lemma [13]). Let F be a family of analytic functions in ∆. Then F is not normal in ∆ if and only if there exist (i) a number r with 0 < r < 1; (ii) points zn satisfying |zn | < r; (iii) functions fn ∈ F; (iv) positive numbers ρn → 0 as n → ∞; such that fn (zn + ρn ξ) → g(ξ) as n → ∞,

(1)

uniformly on compact subsets of C, where g is a nonconstant entire function in C. The function g may be taken to satisfy the normalization g ♯ (z) < g ♯ (0) = 1 (z ∈ C). Here g ♯ (z) denotes the spherical derivative g ♯ (z) =

|g ′ (z)| . 1 + |g(z)|2

Lemma 3.2. Let f be a normal function on a complex Banach manifold X and suppose kX is a metric. There exists a constant c > 1 such that log(cµ(f, x)) ≤ log(cµ(f, y)) exp2KX (x,y) for all x, y ∈ X. Here µ(f, x) := max{1, |f (x)|}.

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Remark 3.1. This result was originally proved by Zaidenberg [8] in the case of complex manifolds. The proof given in [14], p. 39, extends immediately to the complex Banach manifold case. 4. Preliminary Properties of Normal Functions Theorem 4.1. Let X be a complex Banach manifold and suppose kX is a metric. The following statements are equivalent for f ∈ O(X) : (a) f is normal; (b) there exists a constant Q > 0 such that Qf (x) :=

ds(f (x), f∗ (x)v) < Q for all x ∈ X; kX (x, v) v∈Tx (X)\{0} sup

(2)

(c) there exists a constant L > 0 such that s(f (x), f (y)) ≤ L · KX (x, y) for all x, y ∈ X;

(3)

(d) there is no P -sequence {xn } of points in X possessed by f. Proof. (a) ⇒ (b) : Assume that the family F = {f ◦ ϕ : ϕ ∈ O(∆, X)} is a normal family. By Marty’s theorem [13], p. 75, there exists a constant L > 0 such that |(f ◦ ϕ)′ (0)| < L for all ϕ ∈ O(∆, X). 1 + |f ◦ ϕ(0)|2

(4)

By the definition of kX there exists ψ ∈ O(∆, X) such that ψ(0) = x, ψ∗ (0)a = v for a > 0 and a/2 < kX (x, v) ≤ a. Therefore, from (4), ds(f (x), f∗ (x)v) < 2L · kX (x, v) for all (x, v) ∈ T (X). Namely, Qf ≤ 2L. (b) ⇒ (c) : Let x and y be distinct points of X. It follows readily from (2) that ds(f (x), f∗ (x)v) < Q · kX (x, v). By integrating both sides of the above inequality along the piecewise C 1 curves joining x to y in X and e X (x, y). Since KX is the by the definitions we have s(f (x), f (y)) ≤ L · K integrated form of the infinitesimal metric kX (see [9], Corollary 3) we have (3). (c) ⇒ (d) : This follows immediately. (d) ⇒ (a) : If (d) holds, then the family F = {f ◦ ϕ : ϕ ∈ O(∆, X)} is equicontinuous at each point of ∆, since otherwise there is a point z0 ∈ ∆,

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some ǫ > 0, a sequence {zn } of points in ∆ with zn → z0 , and a sequence {f ◦ ϕn } ⊆ F satisfying s(f ◦ ϕn (zn ), f ◦ ϕn (z0 )) ≥ ǫ, n = 1, 2, . . . .

(5)

Since K∆ = d∆ , and zn → z0 then an application of Proposition 3.2 in [10] gives K∆ (zn , z0 ) → 0, as zn → z0 . By the contracting property of the Kobayashi metric, KX (ϕ(zn ), ϕ(z0 )) ≤ K∆ (zn , z0 ) → 0, as zn → z0 .

(6)

From (5), (6) follows that {ϕn (z0 )} is a P -sequence for f in X which contradicts (d). Therefore, F = {f ◦ ϕ : ϕ ∈ O(∆, X)} is spherically equicontinuous family at each point of ∆, and hence, by Montel’s theorem [13], p. 74, F is normal. This proves (a). Remark 4.1. Hahn [15] has also published a similar characterization of a normal function on a complex hyperbolic manifold M of finite dimension. Unfortunately, Hahn’s argument is incorrect, because the assumption that F = {f ◦ ϕ : ϕ ∈ O(∆, M )} is not a equicontinuous family does not entail that “there exist an ǫ > 0 such that for all n ∈ N there exist sequences {z n } and {wn } in ∆ with K∆ (zn , wn ) < 1/n but s(f (ψ(zn )), f (ψ(wn ))) ≥ ǫ for some ψ ∈ O(∆, M )” (see [15], p. 60). 5. Main Results A sufficient condition for the non-normality of a holomorphic function in ∆ is given by Lappan [16], Lemma 3. For holomorphic functions on a complex Banach manifold we obtain the following criterion. Theorem 5.1. Let X and kX be given as in Theorem 4.1. The following statements are equivalent for f ∈ O(X) : (a) f is not a normal function; (b) There exist sequences {ym }, {xm } in X, and a constant M > 0 such that KX (ym , xm ) < M for all m ≥ 1, limm→∞ f (xm ) = ∞, and limm→∞ f (ym ) = a ∈ C. Proof. (a) ⇒ (b) : Since h is not normal it follows that the family F = {f ◦ ϕ : ϕ ∈ O(∆, X)} is not normal in ∆. Apply the Zalcman Lemma to F with r, zn , ρn , f ◦ ϕn , and g as given therein. Since g is nonconstant entire function in C it follows that there exist a sequence {ξm } ⊂ C such

On Bloch and Normal Functions on Complex Banach Manifolds

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that |g(ξm )| > m. In fact, if this were not the case, g would have been bounded in C. By Liouville’s theorem g ≡ constant, a contradiction with the assumption that g is nonconstant entire function. For fixed ξm chose nm large enough so that (i) |znm + ρnm ξm | < (1 + r)/2; (ii) |f ◦ ϕnm (znm + ρnm ξm )| > m/2. Put wnm = znm + ρnm ξm , xm = ϕnm (wnm ), and ym = ϕnm (znm ). By the contracting property of the Kobayashi metric, KX (ym , xm ) = KX (ϕnm (wnm ), ϕnm (znm )) ≤ d∆ (wnm , znm ). By the triangle inequality d∆ (wnm , znm ) ≤ d∆ (0, znm ) + d∆ (0, wnm ). For any z ∈ ∆ the Poincar´e distance d∆ (0, z) = log((1 + |z|)/(1 − |z|)). Since |znm | < r, and |wnm | < (1 + r)/2, we have d∆ (wnm , znm ) ≤ log((1 + r)/(1 − r)) + log((3 + r)/(1 − r)). Putting the above together we get KX (ym , xm ) ≤ M where M = log((1 + r)/(1 − r)) + log((3 + r)/(1 − r)) < ∞. By (ii), we have that lim f (xm ) = lim f ◦ ϕnm (znm + ρnm ξm ) = ∞.

m→∞

m→∞

By (1), we have that lim f (ym ) = lim f ◦ ϕnm (znm ) = g(0) ∈ C.

m→∞

m→∞

(b) ⇒ (a) : Assume, to get a contradiction, that f is a normal function. By Lemma 3.2, log(cµ(f, xm )) ≤ log(cµ(f, ym )) exp2KX (xm ,ym ) for all m ≥ 1. The left hand side of the above inequality tends to infinity as m → ∞ while the right-hand side tends to a number which is less than log(c max{1, |a|}) exp2r and we have a contradiction. This contradiction proves the theorem. Corollary 5.1. If under the conditions of the theorem, f1 , . . . , fl are a finite number of normal holomorphic function on X such that each sequence {xn } of points in X contains a subsequence {xnm } on which at most one of fj (1 ≤ j ≤ l) is unbounded, then h := Σlj=1 fj is a normal function.

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Proof. Suppose, on the contrary, that h is not a normal function. By Theorem 5.1, we can find two sequences {xn } and {yn } of points in X, and a positive constant M such that KX (xm , ym ) < M for all m ≥ 1, limm→∞ h(xm ) = ∞, and limm→∞ h(ym ) = a ∈ C. Since limm→∞ h(xm ) = ∞ then {xm } contains a subsequence again denoted by {xm } such that at most one of fj , say f1 , is unbounded on {xm }. Since limm→∞ h(ym ) = a ∈ C then {ym } contains a subsequence {ymk } such that either: (i) at least two of fj (1 ≤ j ≤ l) is unbounded on {ymk }; (ii) or limk→∞ fj (ymk ) = αj ∈ C (1 ≤ j ≤ l). The case (i) is excluded by the assumption of the corollary. Hence limk→∞ f1 (xmk ) = ∞, limk→∞ f1 (ymk ) = α1 ∈ C, and KX (xmk , ymk ) < M for all k ≥ 1. By Theorem 5.1 f1 is not a normal function, a contradiction which proves the corollary. Theorem 5.2. Let X and kX be given as in Theorem 4.1. If f ∈ O(X) is a normal function and if h is not a normal holomorphic function on X such that each sequence {xm } of points in X contains a subsequence {xmk } on which at most one of f or h is unbounded, then f + h is not a normal function. Proof. Since h is not normal on X it follows that the family H = {h ◦ ϕ : ϕ ∈ O(∆, X)} is not normal in ∆. By the Zalcman Lemma, we can find h ◦ ϕn ∈ H, zn ⊆ ∆r , and ρn → 0+ such that h ◦ ϕn (zn + ρn ξ) → g(ξ) uniformly on compacta, where g is a nonconstant entire function on C. But then |h∗ (ϕn (zn + ρn ξ))ϕ′n (zn + ρn ξ)| → g ♯ (ξ). ρn 1 + |h(ϕn (zn + ρn ξ))|2 We claim that there exist a sequence {ξm } ⊂ C such that |g(ξm )| > m and g ′ (ξm ) 6= 0. Let us suppose the contrary.

(1) If g ′ (ξ) 6= 0 and g is bounded in C by Liouville’s theorem g ≡ constant which contradicts the assumption that g is nonconstant entire function. (2) Let {am } be the zeroes of g ′ (ξ). Suppose that |g(ξ)| < L < ∞ on C\{ξ ∈ C : g ′ (ξ) = 0}. The zeros of g ′ (ξ) are isolated. Since nonconstant holomorphic function has no local maxima, we conclude that |g(am )| < L. Therefore g is bounded in C, hence constant by Liouville’s theorem. But this contradicts with the assumption that g is nonconstant entire function, and the assertion is proven.

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For fixed ξm choose nm large enough so that (i) |λnm | < (1 + r)/2; (ii) g ♯ (ξm )/ρnm > m/2; (iii) ρnm |h∗ (xm )vm |/(1 + |h(xm )|2 ) > g ♯ (ξm )/2; (iv) |h(xm )| > m/2. Here λnm = znm + ρnm ξm , xm = ϕnm (λnm ), and vm = ϕ′nm (λnm ). If (i), (ii), and (iii) hold, it is easy to see that g ♯ (ξm ) (4 − (1 + r)2 )m 1 |h∗ (xm )vm | . > > · 2 1 + |h(xm )| 2ρnm 16 1 − |λnm |2 By the distance decreasing property of the Kobayashi metric 1 = k∆ (λnm , 1) ≥ kX (ϕnm (λnm ), ϕ′nm (λnm ) · 1). 1 − |λnm |2 Hence |h∗ (xm )vm | (4 − (1 + r)2 )m > · kX (xm , vm ). 2 1 + |h(xm )| 16 Since |h(xm )| → ∞, as m → ∞, by considering a subsequence if necessary, we may assume from the hypothesis of the theorem that f is bounded on {xm }, namely |f (xm )| < M < ∞ for all m ≥ 1, and hence, for m sufficiently large, we have |h∗ (xm )vm | − |f∗ (xm )vm | |(f + h)∗ (xm )vm | ≥ ≥ 2 1 + |f (xm ) + h(xm )| 2(1 + |f (xm )|2 )(1 + |h(xm )|2 )   1 1 |h∗ (xm )vm | |f∗ (xm )vm | . · − 2 1 + |M |2 1 + |h(xm )|2 1 + |f (xm )|2 By hypothesis, f is a normal function on X hence, by Theorem 4.1, there exists Q > 0 such that Qf (xm ) < Q for all m ≥ 1. Thus   1 (4 − (1 + r)2 )m − Q → ∞ as m → ∞. Qf +h (xm ) ≥ 2 16(1 + |M |2 ) By Theorem 4.1, f + h is not a normal function on X. Thus the proof is complete. Theorem 5.3. Let X and kX be given as in Theorem 4.1. If Qf (xm ) → ∞ as m → ∞, then {xm } contains a subsequence which is a P -sequence of f.

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P. V. Dovbush

Proof. Since Qf (xm ) → ∞ as m → ∞, then by (2) there exist sequences {vm }, vm ∈ Txm (X), and {rm } ⊂ R, rm → ∞ as m → ∞, with ds(f (xm ), f∗ (xm )vm ) > rm · kX (xm , vm ).

(7)

By the definition of kX there exists ψm ∈ O(∆, X) such that ψm (0) = xm , ψm∗ (0)am = vm for am > 0 and am /2 < kX (xm , vm ) ≤ am . Hence, inequality (7) gives, kX (xm , vm ) rm |(f ◦ ψm )′ (0)| > rm · > → ∞ as m → ∞. 1 + |f ◦ ψm (0)|2 am 2

By Marty’s theorem [13], p. 75, the family {f ◦ ψm } ⊆ O(∆) is not normal in any disc ∆ n1 = {z ∈ C : |z| < n1 }. Then by a local adaptation of the Zalcman Lemma [13], p. 152, there is a subsequence {f ◦ ϕn } of {f ◦ ψm }, zn → 0, ρn → 0+ and a nonconstant entire function g with f ◦ ϕn (zn + ρn ξ) → g(ξ) normally in C. Passing to a subsequence if necessary, we can assume that {f ◦ ϕn (0)} converges to a point β ∈ C. Set gn (ξ) = f ◦ ϕn (zn + ρn ξ). The sequence of functions {gn } converges locally uniformly to g. Let α be any complex number, α 6= β, for which the equation g(ξ) = α has a solution ξ0 which is not a multiple solution, that is, g ♯ (ξ0 ) 6= 0. By a theorem of Hurwitz [13], p. 9, in each neighborhood of ξ0 all but a finite number of the functions {gn } assume the value α. Thus there exists a sequence of points {ξn } ⊆ C such that ξn → ξ0 and gn (ξn ) = α for n sufficiently large. It follows s(f ◦ϕn (zn +ρn ξn ), f ◦ϕn (0)) ≥ s(α, β)/2 > 0 for n ≥ N0 . The Poincar´e metric d∆ is very closed to the Euclidian metric near 0 so d∆ (0, zn + ρn ξn ) → 0 as n → ∞. Hence KX (ϕn (zn + ρn ξn ), ϕn (0)) ≤ d∆ (zn + ρn ξn , 0) → 0 as n → ∞. Which proves that a subsequence {ϕn (0)}∞ n=N0 of {xm } is a P -sequence of f. The proof is complete. Theorem 5.4. Let X and kX be given as in Theorem 4.1. Then the following statements are mutually equivalent for f ∈ O(X). (a) f is a Bloch function. (b) The quantity sup{|(f ◦ ϕ)′ (0)| : ϕ ∈ O(∆, X)} is finite. (c) There exists a constant L > 0 such that |f∗ (x)v| ≤ L · kX (x, v)

for all (x, v) in T (X),

where f∗ (x) is the linear map induced by f from Tx (X) to Tf (x) (C). (d) There exists a constant L > 0 such that |f (x) − f (y)| ≤ L · KX (x, y)

for all x and y in X.

Remark 5.1. See [17] for the proof of Theorem 5.4.

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References 1. W. K. Hayman, Functions with values in a given domain, Proc. Amer. Math. Soc. 3 (1952), pp. 428–432. 2. C. Pommerenke, Univalent functions, G¨ ottingen, Vandenhoek and Ruprecht, 1975. 3. K. Yosida, On a class of meromorphic functions, Proc. Phys.-Math. Soc. Japan, 3 (1934), ser. 1, pp. 227–235. 4. K. Noshiro, Contributions to the theory of meromorphic functions in the unit circle, J. Fac. Sci. Hokkaido Imp. Univ., Ser. I (1938), pp. 149–159. 5. O. Lehto and K. I. Virtanen, Boundary behaviour and normal meromorphic functions, Acta Math., 97(1957), pp. 47–65. 6. A. J. Lohwater, The boundary behavior of analytic functions, in Current problems in mathematics, Fundamental directions, vol. 10, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1973, pp. 99–259 (in Russian). 7. P. Lappan, Normal families and normal functions: results and techniques, in: Function Spaces and Complex Analysis, Joensuu 1997, Univ. Joensuu, Department of Mathematics Rep. Ser. 2 (1997), pp. 63–78. 8. M. G. Zaidenberg, Schottky–Landau growth estimates for s-normal families of holomorphic mappings, Math. Ann. 293 (1992), pp. 123–141. 9. C. J. Earle, L. A. Harris, J. H. Hubbard, and S. Mitra, Schwarz’s lemma and the Kobayashi and Carath´eodory metrics on complex Banach manifolds, in: Kleinian Groups and Hyperbolic 3-Manifolds, (Cambridge Univ. Press, Cambridge, Lond. Math. Soc. Lec. Notes 299, 2003), pp. 363–384. http://www.ms.uky.edu/~larry/paper.dir/minsky.ps 10. S. Dineen, The Schwarz Lemma, Oxford Mathematical Monographs, Oxford, 1989. 11. T. Franzoni and E. Vesentini, Holomorphic Maps and Invariant Distances, North-Holland Mathematical Studies 40, North-Holland Publishing, Amsterdam, 1980. 12. P. Gauthier, A criterion of normalcy, Nagoya Math. J. 32 (1968), pp. 272– 282. 13. J. L. Shchiff, Normal families, (Springer, New York, 1993). 14. M. H. Kwack, Families of normal maps in several variables and classical theorems in complex analysis Lecture Notes Series 33, Res. Inst. Math., Global Analysis Res. Center, Seoul, Korea, 1996. 15. K. T. Hahn, Non-tangential limit theorems for normal mappings, Pacific J. Math. 135 (1988), pp. 57–64. 16. P. Lappan, Non-normal sums and prodacts of unbounded normal functions, Mich. Math. J. 8 (1961), pp. 187–192. 17. P. V. Dovbush, Bloch functions on complex Banach manifolds, Math. Proc. R. Ir. Acad., to appear.

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STRONG-TYPE ESTIMATES AND CARLESON MEASURES FOR WEIGHTED BESOV-LIPSCHITZ SPACES VAGIF S. GULIYEV Baku State University, Baku, Azerbaijan E-mail: [email protected] ZHIJIAN WU University of Alabama, USA E-mail: [email protected] We establish a capacitory strong-type estimate for weighted inhomogeneous α,ϕ ˙ α,ϕ (homogeneous) Besov-Lipschitz space Bpq (Bpq ). As an application, we characterize the related Carleson measures. Keywords: Weighted Besov-Lipschitz spaces, Carleson measures, Poisson kernel, Hardy-Littlewood maximal function, capacity.

1. Introduction Let Rn be the n-dimensional Euclidean space of points x = (x1 , ..., xn ) with
Pn 2 1/2 . For x ∈ Rn and r > 0, denote by B(x, r) the open norm |x| = i=1 xi ball centered at x of radius r. Let E ⊂ Rn be a measurable set. By a weight function we mean any nonnegative function ϕ that is Lebesgue integrable on any bounded and measurable subset of E. If f is a measurable function on E, we denote Z 1/p kf kLp (E,ϕ) ≡ kf kp,ϕ;E = |f (x)|p ϕ(x)dx , 1 ≤ p < ∞, E

kf kL∞ (E,ϕ) ≡ kf k∞,ϕ;E = esssupx∈E |f (x)| ϕ(x),

p = ∞.

If ϕ ≡ 1 we write kf kLp (E) , and if E = R we write kf kLp,ϕ ≡ kf kp,ϕ . We say that a weight function ϕ satisfies condition Ap , 1 < p < ∞, if there exists a positive number Cp such that for every cube Q ⊂ Rn  p/p′ Z Z 1 1 −p′ /p ϕ(x)dx (ϕ(x)) dx ≤ Cp , |Q| Q |Q| Q n

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133

where |Q| is the Lebesgue measure of Q and 1/p + 1/p′ = 1. Condition Ap , in this form, first appeared in Muckenhoupt’s fundamental paper [8]. For p, q ≥ 1 and 0 < α < 1, the weighted Besov-Lipschitz space α,ϕ Bpq (Rn ) consist of all functions f in Lp,ϕ so that  Z kf (· + h) − f (·)kqp,ϕ α,ϕ = kf k dh . (1) + kf kBpq p,ϕ |h|n+αq Rn α,ϕ For any open set O ⊂ Rn , the Bpq (Rn )-capacity of O is defined by o n
α,ϕ n α,ϕ α,ϕ (O) = cap O; B capBpq pq (R ) = inf kf kBpq ; f ≥ 1 on O .

Let Rn+1 = {(x, t) : x ∈ Rn , t > 0} be the upper half space. For any open set O ⊂ Rn , the T (O) the tent of O in Rn+1 , which is T (O) =  (x, t) ∈ Rn+1 : the open ball centered at x with radius t lies in O . Let t Pt (x) = cn n+1 (|x|2 + t2 ) 2 R be the Poisson kernel, where the constant cn is chosen so that Rn Pt (x)dx = 1. For any function f ∈ Lp,ϕ , the harmonic extension of f onto Rn+1 is the convolution between Pt and f , i.e. Z Pt (x − y)f (y)dy. F (x, t) = Pt ∗ f (x) = Rn

The main results of this note are the following: Theorem 1.1. For 1 ≤ p ≤ q ≤ ∞, 0 < α < 1 and for weighted function ϕ, there is a constant C > 0 such that the following strong type estimate Z ∞ q q α,ϕ (|f | > t) dt ≤ Ckf k α,ϕ capBpq Bpq 0

holds for all f ∈

α,ϕ Bpq .

Theorem 1.2. Suppose 1 < p ≤ q ≤ ∞, 0 < α < 1, ϕ ∈ Ap . A nonnegative measure µ on Rn+1 satisfies + Z α,ϕ (2) ∀f ∈ Bpq |Pt ∗ f (x)|q dµ(x, t) ≤ Ckf kqBpq α,ϕ , Rn+1 +

if and only if
α,ϕ , µ(T (O)) ≤ C cap O; Bpq

for any bounded open set

O ⊂ Rn . (3)

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V. S. Guliyev & Z. Wu

For ϕ ≡ 1, both Theorems 1.1 and 1.2 are established in [10]. Strong type estimates (Theorem 1.1) for Lp and Sobolev type spaces have been studied systematically in the past. They play an important role in the imbedding theory of function spaces and related areas. We refer the reader to [1,5,7,14] and the references therein. Estimate (2) defines the usual Carleson measure α,ϕ if the space Bpq is replaced by Lq . Therefore, Theorem 1.2 gives a characα,ϕ terization for Carleson measures for weighted Besov-Lipschitz space Bpq with 1 < p ≤ q ≤ ∞, 0 < α < 1 and ϕ ∈ Ap . In general, the bounded open set in characterization (3) cannot be replaced by an open ball as it can in the characterization for usual Carleson measures. Carleson measures play an important role in harmonic analysis and operator theory on function 2 ∂ Pt ∗ f (x) tdxdt spaces. For example, it is well-known that the measure ∂t is a usual Carleson measure if and only if f ∈ BM O. Carleson measures for Dirichlet space (analytic) on the unit disk of complex plane is characterized in [9] (see also [6] for a “single-box condition” characterization). Applications of Carleson measures for Besov space on commutators and related bilinear forms are studied in [11,12]. For the sake of simplicity, the letter C always denotes a positive constant which may change from one step to the next. For two positive functions a and b, we write a ≍ b, if there is a constant C > 0 such that both a ≤ Cb and b ≤ Ca hold. 2. Proof of Theorem 1.1 Let ψ be a nondecreasing function in C0∞ (0, ∞) which satisfies ( 0, if t ≤ 12 ; ψ(t) = 1, if t ≥ 1. ∞

Consider the smooth truncation {Fj }−∞ given by   |f | j Fj (f ) = 2 ψ , j = 0, ±1, ±2, . . . . 2j

D. R. Adams first used this smooth truncation in [1] to prove a strong type estimate for Sobolev space on Rn (see also [9]). The key properties of this smooth truncation are the following.  0 ≤ Fj (f ) ≤ 2j , {Fj (f ) > 0} ⊂ |f | > 2j−1 , and if 2j−1 ≤ |f | < 2j , then

Fk (f ) =

(

2k ,

if

k < j;

0,

if

k > j.

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135

Lemma 2.1. For 1 ≤ p ≤ q ≤ ∞, 0 < α < 1 and for weighted function ϕ ∞ α,ϕ the smooth truncation {Fj }−∞ maps Bpq to itself, and there is a constant C > 0 such that ∞ X q q kFj (f )kBpq α,ϕ ≤ C kf k α,ϕ Bpq j=−∞

holds for all f ∈

α,ϕ Bpq .

Proof. It is easy to check that Z kFj (f )kpLp,ϕ ≤ 2jp

{|f |>2j−1 }

Since q/p ≥ 1, we have therefore ∞ X

j=−∞

kFj (f )kqLp,ϕ ≤


ϕ(x) dx ≡ 2jp ϕ |f | > 2j−1 .

∞ X

2jp ϕ |f | > 2j−1

j=−∞ ∞ X

≤

j=−∞

2jp ϕ |f | > 2

It is standard that kf kpLp,ϕ

=

Z

0

∞

ϕ (|f | > t) dtp ≍

∞ X

j=−∞



q/p

!q/p
j−1

.


2jp ϕ |f | > 2j .

Combining the above two estimate, we obtain Z ∞ q/p ∞ X kFj (f )kqLp,ϕ ≤ C ϕ (|f | > t) dtp = Ckf kqLp,ϕ . j=−∞

0

On the other hand, it is clear that ∞ X

kFl (f (· +

l=−∞

h))−Fl (f (·))kqLp,ϕ

≤

∞ X

kFl (f (· +

!q/p

h))−Fl (f (·))kpLp,ϕ

l=−∞

q/p

∞

X

p |Fl (f (· + h)) − Fl (f (·))| ≤

l=−∞

By the definition of the weighted Besov space that for any x, h ∈ Rn ∞ X

l=−∞

p

L1,ϕ

α,ϕ Bpq ,

it suffices to show

p

|Fl (f (x + h)) − Fl (f (x))| ≤ C |f (x + h) − f (x)| .

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V. S. Guliyev & Z. Wu

This estimate is proved in [10]. The proof is complete. Proof of Theorem 1.1. It is standard that Z ∞ ∞ X k q α,ϕ (|f | > 2 ). α,ϕ (|f | > t) dt ≍ 2kq capBpq capBpq 0

Since 2

−k

k=−∞

Fk (f ) ≥ 1 on the set {|f | > 2k }, and hence q

k −kq α,ϕ (|f | > 2 ) ≤ 2 capBpq kFk (f )kBpq α,ϕ

we have therefore ∞ ∞ X X q q k α,ϕ (|f | > 2 ) ≤ 2kq capBpq kFk (f )kBpq α,ϕ ≤ Ckf k α,ϕ . Bpq k=−∞

k=−∞

The last inequality follows from Lemma 2.1.

α,ϕ Consider the weighted homogeneous Besov-Lipschitz space B˙ pq which ∞ is the completion of C0 under the norm Z  q kf (· + h) + f (·)kp,ϕ α,ϕ kf kB˙ pq = dh . |h|n+αq Rn α,ϕ We can define B˙ pq -capacity similarly. The above discussion proves also the following result.

Theorem 2.1. For 1 ≤ p ≤ q ≤ ∞, 0 < α < 1 and for weighted function ϕ there is a constant C > 0 such that the following strong type estimate Z ∞ q q α,ϕ (|f | > t) dt ≤ Ckf k α,ϕ capB˙ pq ˙ B (Rn ) pq

0

α,ϕ holds for all f ∈ B˙ pq .

3. Proof of Theorem 1.2 For a function f defined on Rn , the Hardy-Littlewood maximal function M (f ) is defined by Z 1 M (f )(x) = sup |f (y)| dy. r>0 |B(x, r)| B(x,r)

The proof of next lemma has its root in [10] (which is for the case ϕ ≡ 1).

Lemma 3.1. For 1 < p < ∞, 0 < α < 1 and ϕ ∈ Ap the Hardy-Littlewood α,ϕ maximal operator is bounded on Bpq . More precisely, there is a constant C > 0 such that α,ϕ ≤ Ckf k α,ϕ kM (f )kBpq Bpq

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137

α,ϕ hold for all f ∈ Bpq .

Proof. It is well-known (see [8]) that M is bounded in Lp,ϕ , i.e. kM (f )kp,ϕ ≤ Ckf kp,ϕ . For h ∈ Rn , let Th be the translation by h, i.e. Th f (x) = f (x + h). By the definition of the weighted Besov-Lipschitz space it suffices to show that kTh M (f ) − M (f )kp,ϕ ≤ CkTh f − f kp,ϕ. It is easy to see that Th commutes with M , i.e. Th M (f ) = M (Th f ). Hence we have |Th M (f ) − M (f )| = |M (Th f ) − M (f )| ≤ M (|Th f − f |). Taking Lp,ϕ norm on both ends of the above inequality, by the boundedness of M on Lp,ϕ , we obtain the desired result. Lemma 3.1 is proved.  For x ∈ Rn denote by Γ(x) = (u, t) ∈ Rn+1 : |u − x| < t , the cone + with vertex at x. For a function g defined on Rn+1 + , the nontangential maximal function of g is a function on Rn defined by N (g)(x) =

sup (u,t)∈Γ(x)

|g(u, τ )|.

The following lemma is proved in [10]. Lemma 3.2. Let µ be a nonnegative Borel measure on Rn+1 + . Then for any n+1 measurable function g on R+ and any t > 0 µ ({|g(x, τ )| > t}) ≤ µ (T ({N (g)(x) > t})) . Proof of Theorem 1.2. The approach in the following has its root in [9] (see also [10]). There Stegenga proved a result for Carleson measures for the (analytic) Dirichlet space on the unit disk of the complex plane which is similar to the case p = q = 2 and ϕ = 1 here. We prove the “only if” part of Theorem 1.2 first. Given a bounded open α,ϕ α,ϕ set O ⊂ Rn , by the definition of Bpq -capacity, there is a function f ∈ Bpq
q α,ϕ such that f ≥ 1 on the set O and kf kBpq . We can assume α,ϕ ≤ 2cap O; Bpq n further that f ≥ 0 on R , because it is easy to show (similar to the proof α,ϕ ≤ kf k α,ϕ . Since (x, τ ) ∈ T (O) is equivalent of Lemma 3.1) that k|f |kBpq Bpq to B(x, τ ) ⊂ O, we have for any (x, τ ) ∈ T (O) (see [10]), Z Pt ∗ f (x) ≥ P1 (y) dy = δ. B(0,1)

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Therefore, we obtain Z
q α,ϕ µ (T (O)) ≤ δ |Pτ ∗ f (y)|q dµ(y, τ ) ≤ Ckf kqBpq . α,ϕ ≤ Ccap O; Bpq Rn+1 +

To prove the “if” part of Theorem 1.2, we consider the measure µk = µ|T (B(0,k)) ,

k > 0.

α,ϕ For any f ∈ Bpq , let F (y, τ ) = Pτ ∗ f (y). By Lemma 3.2 and the standard result that N (F )(y) ≤ CM (f )(y), we have

µk ({|F (y, τ )| > t}) ≤ µk (T ({N (F )(y) > t})) ≤ µk (T ({M (f ) > t/C})) .

It is clear that for any pair of open sets O1 , O2 ⊂ Rn , we have α,ϕ (O) ≤ T (O1 ∩ O2 ) = T (O1 ) ∩ T (O2 ) and if O ⊂ O∗ then capBpq α,ϕ (O ∗ ). We can continue the above estimate by capBpq µk ({|F (y, τ )| > t}) ≤ µk (T ({M (f ) > t/C} ∩ B(0, k)))

α,ϕ ({M (f ) > t/C} ∩ B(0, k)) . ≤ capBpq

Hence, by Theorem 1.1 and Lemma 3.1, we obtain Z ∞ Z |Pτ ∗ f (y)|q dµ(y, τ ) = µk ({|Pτ ∗ f (y)| > u}) duq T (B(0,k))

≤

Z

0

∞

0

q α,ϕ ({M (f ) > u/C}) du capBpq

q ≤ CkM (f )kqBpq α,ϕ ≤ Ckf k α,ϕ . Bpq

Theorem 1.2 is proved. 4. Similar Results on Unit Disk Let U be the unit disk of the complex plane, T be the boundary of U. For z ∈ U, we can write z = rζ with r = |z| and ζ ∈ T. Let φ ≥ 0 be a measurable function on T. If f is a measurable function defined on T, then Z 1/p kf kLp (T,φ) ≡ kf kp,φ;T = |f (ζ)|p φ(ζ)dζ , f or 1 ≤ p < ∞, T

kf kL∞ (T,φ) ≡ kf k∞,φ;T = esssupζ∈T|f (ζ)|φ(ζ),

f or p = ∞.

If φ ≡ 1 we write kf kLp (T) ≡ kf kp,T. We say that a weight function φ on T satisfies condition Ap (T), 1 < p < ∞, if there exists a positive number Cp such that for every interval I ⊂ R  Z p/p′ Z 1 1 −p′ /p φ(x) dx (φ(x)) dx ≤ Cp , |I| I |I| I

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where |I| is the Lebesgue measure of I. α,φ For p, q ≥ 1 and 0 < α < 1, the weighted Besov-Lipschitz space Bpq (T) consist of all functions f in Lp (T, φ) so that  Z kf (ei(·+h) ) − f (ei· )kqp,φ;T dh . = kf k + kf kBpq α,φ p,φ;T (T) |h|1+αq T α,φ For any open set O ⊂ T, the Bpq (T)-capacity of O is defined by o n
α,φ α,φ capBpq α,φ (T) ; f ≥ 1 on O . (T) (O) = cap O; Bpq (T) = inf kf kBpq

Denote by D and D ′ the spaces of test functions and distributions on T, respectively. For g ∈ D ′ , let the Fourier series of g be g(ζ) ∼

∞ X

gk ζ k .

k=−∞

We define accordingly (Q0 g)(ζ) = g0 and (Qm g)(ζ) =

m 2X −1

k=2m−1

(gk ζ k + g−k ζ −k ) ,

m = 1, 2, . . . .

It is clear that g can be represented by the weakly convergent series D′

g(ζ) =

∞ X

(Qm g)(ζ).

m=0

In [4] we study the weighted holomorphic Besov space on the unit disk U α,φ and their boundary values on T. There the norm of f in Bpq (T) is defined as

n o

mα

2 kQ gk bα,φ (g, T) =

m p,q Lp (T,φ) , lq

which is equivalent to the norm defined in (4) (see, for example, [2,3] for the case φ ≡ 1. For general φ ∈ Ap (T) the proof of equivalences of these norms is the same.). Under certain restrictions on the weighted function and parameters, we establish in [4] the equivalent norms for holomorphic functions in terms of their boundary functions. For any open set O ⊂ T, the tent of O in U is defined as T (O) = {z ∈ U : the open interval centered at ζ with radius 1 − r is contained in O}. Let Pr (eit ) =

1 1 − r2 π |eit − r|2

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be the Poisson kernel. For any function f ∈ Lp (T, φ), the harmonic extension of f onto U is the convolution between Pr and f , i.e. Z it Pr (ei(t−τ ) )f (eiτ ) dτ, z = reit . F (z) = Pr ∗ f (e ) = T

Similarly we can prove the following results. Theorem 4.1. For 1 ≤ p ≤ q ≤ ∞, 0 < α < 1 and for weighted function φ there is a constant C > 0 such that the following strong type estimate Z ∞ q q capBpq α,φ (T) (|f | > t) dt ≤ Ckf k α,φ Bpq (T)

0

α,φ holds for all f ∈ Bpq (T).

Theorem 4.2. Suppose 1 < p ≤ q ≤ ∞, 0 < α < 1, φ ∈ Ap (T). A nonnegative measure µ on U satisfies (z = reit ) Z α,φ (T), ∀f ∈ Bpq |Pr ∗ f (eit )|q dµ(z) ≤ F kf kq α,φ , U

Bpq (T)

if and only if
α,φ µ(T (O)) ≤ C cap O; Bpq (T) ,

for any bounded open set

O ⊂ T.

Remark 4.1. In Theorem 4.2 the open set O can be replaced by finite union of open intervals. Acknowledgments The authors are thankful to the referee for his/her valuable comments. References 1. D. R. Adams, On the existence of capacitory strong type estimates in R n . Ark. Mat. 14 (1966), 125-140. 2. V. S. Guliev and P. I. Lizorkin, B– and L–spaces of harmonic and holomorphic functions in the disc, and classes of boundary values. (Russian) Dokl. Akad. Nauk SSSR 319 (1991), no. 4, 806-810. English transl. in Soviet Math. Dokl. 44 (1992), no. 1, 215-219. 3. V. S. Guliev and P. I. Lizorkin, Classes of holomorphic and harmonic functions in a polydisk in connection with their boundary values. Trudy Math. Inst. Steklov, 204 (1993), no. 16, 137-159. English trans. in Proc. Steklov Inst. 204 (1994), no. 3, 117-135. 4. V. S. Guliyev and Z. Wu, Weighted holomorphic Besov spaces and their boundary values. Anal. Theory Appl. 21 (2005), 143-156.

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5. K. Hansson, Imbedding theorems of Sobolev type in potential theory. Math. Scand. 45 (1979), 67-109. 6. R. Kerman and E. Sawyer, Carleson measures and multipliers of Dirichlet spaces. Trans. Amer. Math. Soc. 329 (1988), 87-99. 7. V. G. Mazya and T. O. Shaposhnikova, Theory of Multipliers in Spaces of Differentiable Functions. Pitman Advanced Publishing Program, Boston and London. 8. B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal functions. Trans. Amer. Math. Soc. 165 (1972), 207-226. 9. D. Stegenga, Multipliers of the Dirichlet space. Illinois J. Math. 24 (1980), 113-139. 10. Z. Wu, Strong type estimate and Carleson measures for Lipschitz spaces. Proc. Amer. Math. Soc. 137 (1999), no. 11, 3243-3249. 11. Z. Wu, Clifford analysis and commutators on the Besov spaces. J. Funct. Anal. 169 (1999), no. 1, 121-143. 12. Z. Wu, Carleson measures and multipliers for Dirichlet spaces. J. Funct. Anal. 169 (1999), no. 1, 148-164. 13. E. M. Stein, Harmonic Analysis. Princeton University Press, 1993. 14. W. P. Ziemer, Weakly Differentiable Functions. Springer-Verlag, New York, 1989.

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COMPOSITION THEOREMS IN BESOV SPACES, THE VECTOR-VALUED CASE MADANI MOUSSAI Department of Mathematics, Lab. Math. Pure and Applied, University of M’Sila, P.O. Box 166, M’Sila 28000, Algeria E-mail: [email protected] Let Ud be the set of the functions g : R → Rd , g = (g1 , . . . , gd ), such that gi′ = gj′ for i 6= j. For a Lipschitz continuous function f : Rd → R vanishing at the origin, we will show that the composition operator g 7→ Tf (g) = f ◦ g takes s (Ud ) into B s (R) under some conditions on ∂ f and on the Besov space Bp,q j p,q the parameters s, p, q. Keywords: Composition operators, Besov spaces, Lipschitz spaces.

1. Introduction Let Tf : g 7→ f ◦ g be a composition operator defined from a function s f : R → R. In the Besov space Bp,q (Rn ) it is well know the following result (see e.g. [7]): Theorem 1.1. Let 1 + (1/p) < s < (n/p) and f : R → R. Then T f takes s Bp,q (Rn ) into itself if and only if there exists some constant c such that f (t) = c t . The same result holds in the limit case 1 + (1/p) = s < (n/p) as soon as q > 1. s In the space Bp,q (Rn ) ∩ L∞ (Rn ), the composition operator problem becomes nontrivial, in the sense that f : R → R is not automatically linear. Then the extension of Theorem 1.1 to a nonlinear function f needed some conditions on f and on the parameters s, p, q; for example f (0) = 0, f ∈ Bps,ℓoc , q (R) and (1/p) < s − [s], cf. [2,4,5]. In this paper we consider the set

Ud = {g : R → Rd , g = (g1 , . . . , gd ) , such that gi′ = gj′ for i 6= j}

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and we will study the composition operator problem in Besov space of vector-valued case in the context f : Rd → R and g = (g1 , . . . , gd ) ∈ Ud ,

s gi ∈ Bp,q (R),

i = 1, . . . , d.

Notation All functions are assumed to be real-valued. We define the difference operator for an arbitrary function f , by ∆h f (x) := f (x + h) − f (x),

∀h, x ∈ Rd .

For a space E, of functions defined on Rd , the associated local space is defined as E ℓoc = {f ∈ S ′ (Rd ) : ϕf ∈ E, ∀ϕ ∈ D(Rd )} . If s is a real number, then [s] denotes the integer part of s, i.e. the largest integer less than or equal to s. The constants c, c1 , . . . are strictly positive and depend only on the fixed parameters d, s, p and q; their values may vary from line to line. Some Definitions In this section we will define some spaces considered in this work. (i) The Lipschitz and Besov spaces: Definition 1.1. If f ∈ Lip1 (Rd ) , then kf kLip1 (Rd ) = sup x6=y

| f (x) − f (y) | < +∞ . |x − y|

Definition 1.2. Let 1 ≤ p, q ≤ ∞ and 0 < s < 1 . If f ∈ Bps , q (R) , then 1/q Z dh q −sq < +∞ . kf kBps , q (R) = kf kLp(R) + |h| k∆h f kLp (R) |h| |h|≤1 Remark 1.1. The following expressions define equivalent norms in Besov spaces cf. [8, pp. 58-59]. For an integer m ≥ 1, kf kBps , q (R) = kf kLp(R) + kf (m) kBps−m , , q (R)

(m < s < m + 1),

(1)

and kf k

Bpm, q (R)

= kf kLp(R) +

Z

|h|≤1

−q

|h|

dh k∆2h f (m−1) kqLp (R)

|h|

1/q

.

(2)

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Remark 1.2. If f ∈ E(Ud ), then we will identify kf kE(Ud) with kf kE(R,Rd) . (ii) The space Mps , q (Rd ): For f : Rd → R, x = (x1 , . . . , xd−1 ) , d ∈ {2, 3, . . .} and j ∈ {1, . . . , d} we put fj,x (y) = f (x1 , . . . , xj−1 , y, xj , . . . xd−1 ) ,

(∀y ∈ R).

Definition 1.3. Let 1 ≤ p, q ≤ ∞ , 0 < s < 1 and d ∈ {2, 3, . . .}. If f ∈ Mps , q (Rd ) , then kf kMps , q (Rd ) = k Z ∞ Z q/p dt 1/q X −sq < +∞ . t sup k∆h fj,(·) (y)kpL∞ (Rd−1 ) dy t 0 R |h|≤t j=1 Remark 1.3. The extension of Definition 1.3 to the case [s] ≥ 2 requires to replace ∆h by ∆h (∆hL−1 ), where L is an integer ≥ 1 + [s]. This is known as well for Besov spaces, cf. [8,9]. Remark 1.4. We obtain equivalent norm in Mps , q (Rd ), by replacing the integration for t ∈]0, +∞[ by the integration for t ∈]0, 1[, and by replacing the part of the integral for which t ∈]1, +∞[ by k Z X j=1

R

kfj,(·) (y)kpL∞ (Rd−1 ) dy

1/p

.

(iii) Functions of bounded variation: Let h ∈]0, +∞] and 1 ≤ p < ∞. For a function g : R → R, we denote by νp (g, h) the positive number defined by the supremum of X 1/p N |g(tk ) − g(tk−1 )|p k=1

taken on all reals {tk }N k=0 such that t0 < t1 < · · · < tN and |tk − tk−1 | ≤ h; and if h = +∞, we simply put νp (g, ∞) = νp (g). Definition 1.4. Let 1 ≤ p < ∞. 1- A function g is said to be of bounded p-variation if νp (g) < +∞. 2- The space BVp1 (R) is the set of the primitives of functions of bounded p-variation, and endowed with the seminorm kf kBVp1 (R) = inf νp (g) , where the infimum is taken on all functions g whose primitive is f .

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Remark 1.5. We have at our disposal the Peeter embedding type (cf. [3, Thm. 5]) 1+(1/p)

Bp , 1

(R) ֒→ BVp1 (R) ,

(1 ≤ p < ∞).

(3)

2. Statement of the Result For brevity, we will limit our study to the case [s] = 1. Theorem 2.1. Let 1 < p < ∞ , 1 ≤ q ≤ ∞, and a real number s such that p 1 + < s < 2. min(p, q) p

(4)

If a Borel-measurable function f : Rd → R satisfying • f is Lipschitz continuous, • f (0) = 0, s−1,ℓoc • ∂j f ∈ Mp,q (Rd ) , j = 1, . . . , d , s then the superposition operator Tf takes the space Bp,q (Ud ) to Bps , q (R).

Remark 2.1. 1- In the case [s] ≥ 2 : the condition (4) remains (1/p) < s − [s] and

p 1 + − 1 < s − [s] . min(p, q) p

By induction on [s] we can prove this case, that we omit the details, cf. [5]. 2- We obtain [5, Thm. 2] by taking d = 1 in Theorem 2.1. 3- It is possible to extend Theorem 2.1 to the case s − [s] ≤ (1/p), cf. [5]. Examples. We will discuss two examples. 1- For β > 1 we consider the function fβ (x) = |x|β , (x ∈ Rd ). Let us accept s−1,ℓoc for the moment that ∂j fβ ∈ Mp,q (Rd ) if, either 0 < s < β + (1/p) and 1 ≤ q < ∞, or s = β + (1/p) and q = ∞. By Theorem 2.1 we have s g ∈ Bp,q (Ud ) ⇒ |g|β ∈ Bps , q (R) ,

as soon as s, p and q satisfy (4) and, either 0 < s < β + (1/p) and 1 ≤ q < ∞, or s = β + (1/p) and q = ∞ . Furthermore, the following elementary inequality (aµ + bµ )1/µ ≤ a + b ≤ 21−(1/µ) (aµ + bµ )1/µ ,

(a , b > 0 , µ > 1) ,

s/β

and the embedding Bps , q (R) ֒→ Bβp , βq (R) yield k |g|β kBps , q (R) ≤ c kgkβB s

p,q (U

d)

,

s ∀g ∈ Bp,q (Ud )

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M. Moussai

with some constant c > 0 independent of g . We now turn to the proof of the following: The function Φ (x) = β |x| ψ (x) defined on Rd , such that ψ ∈ D(Rd ) and β ∈ R, belongs to Mps , q (Rd ) if, either 0 < s < β + (1/p) and 1 ≤ q < ∞, or s = β + (1/p) and q = ∞. Indeed, we put L = 1 + [β + (1/p)], then we have Z

p

sup △L h Φj,(·) (y) L (Rd−1 ) dy ∞

|y|≤2t |h|≤t

≤ (L + 1)p

Z

|y|≤(L+2)t



Φj,(·) (y) p dy ≤ c tβp+1 ; L∞ (Rd−1 )

and by the mean value theorem we have Z Z

L

p Lp

sup △h Φj,(·) (y) L (Rd−1 ) dy ≤ c1 t |y|≥2t |h|≤t

∞

|y|≥2t

p(β−L)

|y|

dy

≤ c2 tβp+1 .

2- For β > 1 and δ > 0 we define fβ,δ (x) = ψ(x) |x|β (− log |x|)−δ ,

(x ∈ Rd ) ,

where ψ ∈ D(Rd ) with ψ(0) 6= 0. As in the first example, we have ∂j fβ,δ ∈ s−1,ℓoc Mp,q (Rd ) if, either s < β + (1/p), or s = β + (1/p) and δ > 1/q, cf. [7, 2.3.1]. Furthermore if the parameters s, p and q satisfy (4), then s fβ,δ ◦ g ∈ Bps , q (R) for all g ∈ Bp,q (Ud ). 3. Proof of Theorem 2.1 We propose the following formulation more precise than Theorem 2.1. Proposition 3.1. Let p , q , s be a real numbers as in Theorem 2.1 . There exists a constant c = c(s, p, q, d) > 0 such that the estimate kf ◦ gkBps , q (R) s−(1/p)   1 + kgk ≤ c kf kLip1 (Rd ) + k∇f kMps−1 s (Ud ) d) B (R p,q ,q

(5)

s holds, for all g ∈ Bp,q (Ud ) and all Lipschitz continuous function f : Rd → R d such that f (0) = 0 and ∂j f ∈ Mps−1 , q (R ) , (j = 1, . . . , d) .

If d = 1, Proposition 3.1 is given by [4,5]. Also its proof is based on the following lemma, which is proved in [4, Prop. 2]:

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Lemma 3.1. For 1 < p < ∞, (1/p) < s < 1, and 1 ≤ q ≤ ∞, there exists a constant c = c(s, p, q) > 0 such that Z ∞  1/q q dh νp (g, h) ≤ c kgkBps , q (R) , ∀g ∈ Bps , q (R). (6) h hs−(1/p) 0 Remark 3.1. The exponent s − (1/p) in (5) is optimal for all functions f except the polynomials. Then as in [3, Prop., 11] we can easily obtain the following assertion: Let s ≥ 1 and 0 < α < s − (1/p), let f : Rd → R be a continuous function. If there exists a constant c = c(f ) > 0 such that the inequality  α kf ◦ gkBps , q (R) ≤ c 1 + kgkBp,q s (Ud ) holds for all g ∈ D(R, Rd ), then f is a polynomial of degree less than [s] .

Proof of Proposition 3.1. The proof will be done on two steps. We will s use, in the first, the real analytic functions in Bp,q (Ud ), cf. [3, proof of Thm. 7]. In the second, the general case will be obtained by the property of Fatou (cf. [5,6]). s Step 1. We take d = 2. Let g = (g1 , g2 ) ∈ Bp,q (U2 ) real analytic function. By Remark 1.1 (1) we have to to estimate Z

1

h−(s−1)q

0

′

Z

q/p dh p/q , |∆h (f ◦ g)′ (x)|p dx h R

where (f ◦ g) = (∂1 f ◦ g + ∂2 f ◦ g) g1′ . The Lipschitz continuity of f and the assumption f (0) = 0 yield kf ◦ gkLp(R) ≤ kf kLip1 (R2 ) kgkLp(U2 ) .

(7)

Now from the following decomposition ∆h (f ◦ g)′ (x) =

2   X (∂j f ◦ g)(x + h) ∆h g1′ (x) + g1′ (x) ∆h (∂j f ◦ g)(x) , j=1

s (U2 ) , we have s−1 and (7), and k∂gkBp,q (U2 ) ≤ c1 kgkBp,q
s (U2 ) + V (f ; g) kf ◦ gkBps , q (R) ≤ c2 kf kLip1 (R2 ) + k∇f kL∞ (R2 ) kgkBp,q

where V = V1 + V2 , with Z ∞ Z q/p dh 1/q −(s−1)q p ′ p Vj (f ; g) = h |∆h (∂j f ◦ g)(x)| |g1 (x)| dx , h 0 R

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M. Moussai

for j = 1, 2. We note that the estimate of V1 and of V2 are similar. Again using the following decomposition Z 1/p |∆h (∂1 f ◦ g)(x)|p |g1′ (x)|p dx R Z 1/p ≤ |∂1 f (g1 (x), g2 (x + h)) − ∂1 f (g1 (x), g2 (x))|p |g1′ (x)|p dx R Z 1/p + |∂1 f (g1 (x + h), g2 (x + h)) − ∂1 f (g1 (x), g2 (x + h))|p |g1′ (x)|p dx , R

then we have V1 ≤ V3 + V4 , where Vj+2 (f ; g) = Z ∞ Z q/p dh 1/q −(s−1)q h k∆h (∂1 fj,(·) ◦ gj )(x)kpL∞ (R) |gj′ (x)|p dx h 0 R for j = 1, 2, and also their estimates are similar. Estimate of V3 . Let {Il }l a family of nonempty open disjoint intervals deS fined such that the complement of l Il in R is the discrete set {x ∈ R : g1′ (x) = 0}. For any h > 0 we set ′ ′′ ′ = {x ∈ Il : dist(x, the right endpoint of Il ) ≥ h} and Il,h = Il \ Il,h . Il,h ′ (Il,h is possibly empty) . Then we write V3 ≤ V5 + V6 , where

V5 (f ; g) =

Z

∞

0

h−(s−1)q

XZ l

...

′ Il,h

q/p dh 1/q h

,

′′ and V6 (f ; g) is the part corresponding to Il,h in the integral with respect to h. Estimate of V5 . We put α = min(p, q) , al = supt∈Il |g1′ (t)| and

Ωp,j (f, t) =

Z

sup k∆v fj,(·) (y)kpL∞ (R) dy

R |v|≤t

1/p

,

j = 1, 2 .

The fact that the function g1|Il is a diffeomorphism of Il into g1 (Il ), then we have −1 |g1 (g1| (y) + h) − y| ≤ al h I l

for

′ ), y ∈ g1|Il (Il,h

(8)

−1 where g1| denotes the inverse function of g1|Il . By (8), and by a change of Il ′ , and by the embedding ℓα/p ֒→ ℓp , and variable g1|Il (x) = y on each Il,h

Composition Theorems in Besov Spaces, the Vector-Valued Case

149

by Minkowski’s inequality with respect to Lq/α , successively, we find Z ∞ X q/p dh q h−(s−1)q V5 (f ; g) ≤ alp−1 Ωpp,1 (∂1 f, al h) h 0 l Z ∞ X q/α dh (p−1)α/p α h−(s−1)q ≤ al Ωp,1 (∂1 f, al h) h 0 l Z X  ∞ dh α/q q/α (p−1)α/p ≤ al h−(s−1)q Ωqp,1 (∂1 f, al h) h 0 l X q/α α(s−(1/p)) = k∂1 f kqM s−1 (R2 ) al . p,q

l

g1′ (βl−1 )

We put Il =]βl−1 , βl [, with least one point ξl ∈ Il such that

= g1′ (βl ) = 0 . In addition, there is at

|g1′ (ξl )| = sup |g1′ (t)| . t∈Il

Hence X l

sup |g1′ (t)|u = t∈Il

X l

|g1′ (ξl ) − g1′ (βl )|u

≤ νu (g1′ )u ,

with

u = α(s − (1/p)) .

(9)

Now by (3) we have 1+1/u

Bps , q (R) ֒→ Bu , 1

(R) ֒→ BVu1 (R) .

We combine this chain of embeddings with (9) then we obtain s−(1/p)

V5 (f ; g) ≤ c1 k∂1 f kMps−1 2 kg1 kBV 1 (R) , q (R ) u

≤

s−(1/p) c2 k∂1 f kMps−1 2 kg1 kB s , q (R ) p , q (R)

.

′′ have length equal to h, Estimate of V6 . Since the nonempty intervals Il,h we have Z k∆h (∂1 f1,(·) ◦ g1 )(x)kpL∞ (R) |g1′ (x)|p dx ′′ Il,h

≤ h (2 k∂1 f k∞ )p sup |g1′ (t)|p .

(10)

′′ t∈Il,h

′′ ′′ As above, the right endpoint of Il,h is βl , also there exists ξl ∈ Il,h such that

ξl < βl

and |g1′ (ξl )| = sup |g1′ (t)| . ′′ t∈Il,h

150

M. Moussai

Then we have X l

sup |g1′ |p = ′′ Il,h

X l

|g1′ (ξl ) − g1′ (βl )|p ≤ νpp (g1′ , h) .

Now by (10), and by Lemma 3.1, we conclude that V6 (f ; g) ≤ c k∂1 f k∞ kg1 kBps , q (R) . s Step 2. Let g be any function in Bp,q (Ud ). We consider a sequence {φj }j∈N of the functions defined as φj (x) = jφ(jx) , ∀x ∈ R , where φb ∈ D(R) and b = 1 . We put φ(0) d

gej = (g1 ∗ φj , . . . , gd ∗ φj ),

(∀j ∈ N) .

Clearly gej ∈ U , and it is easy to obtain

kgℓ ∗ φj kBps , q (R) ≤ kφkL1 (R) kgℓ kBps , q (R) ,

∀ℓ ∈ {1, . . . , d} .

Then the sequence {e gj }j∈N is of real analytic functions, and satisfies sup ke gj kBp,q s (Ud ) < +∞

and

j≥0

Now, the Lipschitz continuity of f yields

lim k e gj − g kLp (Rd ) = 0 .

j→∞

lim k f ◦ gej − f ◦ g kLp (R) = 0 .

j→∞

Then by Step 1 and by Fatou’s property, we have both f ◦ g ∈ Bps , q (R) and (5). Proof of Theorem 2.1. Let ρ be a real-valued function in D(R2 ) such that ρ(x) = 1 if |x| ≤ 1 and ρ(x) = 0 if |x| ≥ 2. We put ρt (x) = ρ(x/t) for t > 0. Since f ◦ g = (f ρt ) ◦ g

with t = kgkL∞ (R) ,

the result follows from Proposition 3.1. The condition “f is Lipschitz continuous” in Theorem 2.1 must be necessary at least locally. This is proved recently by S. E. Allaoui, cf. [1]: s Theorem 3.1. If Tf takes Bp,q (R, Rd ) ∩ L∞ (Rd ) to Bps , ∞ (Rd ), then f is locally Lipschitz continuous.

Remark 3.2. The necessity of condition f (0) = 0 is obvious. However the s−1,ℓoc necessity of ∂j f ∈ Mp,q (Rd ) still open (for d = 1, it has been known before cf. [7, Thm. 1, p. 292]). This remark was reported by the referee in the first version of this paper. I would like to thank him.

Composition Theorems in Besov Spaces, the Vector-Valued Case

151

References 1. S. E. Allaoui. Remarques sur le calcul symbolique dans certains espaces de Besov a ` valeurs vectorielles (submitted). 2. G. Bourdaud. Une propri´et´e de composition dans l’espace H s (II). C. R. Acad. Sci. Paris, Ser. I 342 (2006), 243–246. 3. G. Bourdaud, M. Lanza de Cristoforis, W. Sickel. Superposition operators and functions of bounded p-variation. Rev. Mat. Iberoamer. 22 (2006), 455–485. 4. G. Bourdaud, M. Moussai, W. Sickel. An optimal symbolic calculus on Besov algebras. Ann. Inst. H. Poincar´e (C) Anal. Non Lin´earire 23 (2006), 949–956. 5. G. Bourdaud, M. Moussai, W. Sickel, Towards sharp superposition theorems in Besov and Lizorkin-Triebel spaces. Nonlinear Anal. 68 (2008), 2889–2912. 6. J. Franke. On the spaces Fps , q (R) of Triebel-Lizorkin type: Pointwise multipliers and spaces on domains. Math. Nachr. 125 (1986), 29–68. 7. T. Runst, W. Sickel. Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations. De Gruyter, Berlin, 1996. 8. H. Triebel. Theory of Function Spaces. Birkh¨ auser, Basel, 1983. 9. H. Triebel. Theory of Function Spaces II. Birkh¨ auser, Basel, 1992.

152

POINTWISE APPROXIMATION FOR CERTAIN MIXED ´ SZASZ-BETA OPERATORS QIULAN QI∗ and YUPING ZHANG† College of Mathematics and Information Science Hebei Normal University Shijiazhuang 050016, People’s Republic of China ∗ E-mail: [email protected] † E-mail: [email protected]. We study the mixed summation-integral type operators having Sz´ asz and beta basis functions. We obtain the pointwise approximation equivalence theorems using the unified modulus of smoothness. Keywords: Sz´ asz-Beta operators, Approximation, Modulus of smoothness, Kfunctional.

1. Introduction In recent years, there are many results for summation-integral type operators (see [3, 4, 7–11]) and Sz´ asz type operators (see [5]). Recently, Z. Finta [3] and V. Gupta [9] introduced a mixed sequence of summation-integral type operators defined for x ∈ [0, ∞) by Z ∞ ∞ X sn,v (x) bn,v (t)f (t)dt + sn,0 (x)f (0), (1) Sn (f, x) = v=1

0

where δ(t) is the Dirac delta function, and sn,v (x) = e−nx bn,v (t) =

(nx)v , v!

tv−1 tv−1 (n + v)! 1 = n+v+1 B(n + 1, v) (1 + t) (v − 1)!n! (1 + t)n+v+1

are respectively Sz´ asz and beta basis functions. We define sn,v (x) = 0 (v < 0),

bn,v (t) = 0 (v < 1).

It is easily verified that the operators (1) are linear positive.

Pointwise Approximation for Certain Mixed Sz´ asz-Beta Operators

153

As far as approximation properties of the operators Sn are concerned, in [4] the authors extended the study of [9] and obtained some direct results and global direct approximation theorems using ωϕ2 (f, t)∞ . Some other operators of type Sn were recently considered in [7,8]. Ditzian [1] introduced ωϕ2 λ (f, t) (0 ≤ λ ≤ 1) and obtained a direct theorem with ωϕ2 λ (f, t) for the Bernstein operator. In this paper, we introduce the idea of Ditzian to the operators Sn with the unified modulus of smoothness ωϕ2 λ (f, t), and obtain direct, inverse and equivalence theorems. To state our results, we give some notation: If 0 ≤ λ ≤ 1, let ωϕ2 λ (f, t) = sup

0 t, observe that the function λ g(x) = x/δn2λ (x) is monotone increasing in Enc and combine it with |t − u|x ≤ |t − x|u. Lemma 2.4. Sn ((2 + t)−2 , x) ≤ 20(2 + x)−2 for n ≥ 2. Proof. Since 1 + x ∼ 2 + x, it is sufficient to show that (1 + x)2 Sn ((1 + t)−2 , x) ≤ 20. (1 + x)2 Sn ((1 + t)−2 , x) Z ∞ ∞ X = (1+x)2 sn,v (x) 0

v=1

(n + 1)(n + 2) bn+2,v (t)dt + e−nx (n+v+1)(n+v+2) (n+1)(n+2) (n+v+1)(n+v+2)

Case I: x ≤ 1. By (1 + x)2 ≤ 4, I≤4

∞ X v=1

sn,v (x)

Z

∞ 0

!

≤ 1, we have !

bn+2,v (t)dt + e−nx

= 4.

Case II: x > 1. Using (1 + x)2 ≤ 4x2 ,

(n + 1)(n + 2) (v + 1)(v + 2) ≤ 4, (n + v + 1)(n + v + 2) n2

=: I.

(2)

Pointwise Approximation for Certain Mixed Sz´ asz-Beta Operators

for n ≥ 2, we have I≤4 ≤ 20.

∞ X

sn,v+2 (x)

v=1

Z

∞

155

(n+1)(n+2)(v+1)(v+2) bn+2,v (t)dt + x2 e−nx (n+v+1)(n + v + 2)n2

0

!

(3)

The proof of Lemma 2.4 is completed from (2) and (3). Proof of Theorem 1.1. By the definition of K ϕλ (f, t2 ) and the equiva2

2

lence of ωϕ2 λ (f, t) and K ϕλ (f, t2 ), we can choose g ∈ D λ such that 4 2  2−λ  1  1 kg ′′ k kf − gk + n− 2 δn1−λ (x) kϕ2λ g ′′ k + n− 2 δn1−λ (x) 1

2 ≤ Cωϕλ (f, n− 2 δn1−λ (x)).

Using the Taylor expansion of g(t), we will consider x ∈ En and x ∈ Enc respectively. For x ∈ En , δn2 (x) ∼ ϕ2 (x), by Lemmas 2.2–2.4 and the H¨ older inequality, we have Z t  ′′ Sn (t − u)g (u)du, x x     (t − x)2 (t − x)2 2λ ′′ , x kϕ2λ g ′′ k , x kϕ g k + S ≤ Sn n ϕ2λ (x) xλ (2 + t)λ λ/2  1/2   (t−x)4 1 ϕ2(1−λ) (x) 2λ ′′ , x S , x kϕ g k+ Sn kϕ2λ g ′′ k ≤C n n (2+t)2 x2λ
λ/2 −1 2 ϕ2(1−λ) (x) 2λ ′′ ≤C n ϕ (x)x−λ kϕ2λ g ′′ k kϕ g k + (2 + x)−2 n  1 2 2 ≤ C n− 2 ϕ1−λ (x) kϕ2λ g ′′ k ≤ Cωϕλ (f, n−1/2 δn1−λ (x)). Hence, we get for x ∈ En , Enc ,

1

2 |Sn (g, x) − g(x)| ≤ Cωϕλ (f, n− 2 δn1−λ (x)).

δn2 (x) t

For x ∈ ∼ 1/n, by Lemmas 2.2 and 2.3, we get Z    2 ′′ Sn ≤ Sn (t − x) , x kδ 2λ g ′′ k (t − u)g (u)du, x n δn2λ (x) x 2  1 ≤ C n− 2 δn1−λ (x) kδn2λ g ′′ k 4  1  1 2  2−λ ≤ C n− 2 δn1−λ (x) kϕ2λ g ′′ k + C n− 2 δn1−λ (x) kg ′′ k 1

2 ≤ Cωϕλ (f, n− 2 δn1−λ (x)).

(4)

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Q. Qi & Y. Zhang

Therefore 1

2 |Sn (g, x) − g(x)| ≤ Cωϕλ (f, n− 2 δn1−λ (x)).

(5)

From (4), (5) and Lemma 2.1, we can write |Sn (f, x) − f (x)| ≤ C (kf − gk + |Sn (g, x) − g(x)|) 1

≤ Cωϕ2 λ (f, n− 2 δn1−λ (x)).

3. The Inverse Theorem To prove Theorem 1.2, we need some new notation. Let C 0 = {f ∈ CB [0, ∞), f (0) = 0}, C 2 = {f ∈ C 0 , f ′′ ∈ CB [0, ∞)}, kf k0 = α(λ−1) 2+α(λ−1) 0 sup {|δn (x)f (x)|}, kf k2 = sup {|δn (x)f ′′ (x)|}, Cα,λ = x∈[0,∞) x∈[0,∞)   2 f ∈ C 0 , kf k0 < +∞ , Cα,λ = f ∈ C 2 , f ′ ∈ A.C.loc, kf k2 < +∞ , and 2 2 kf − gk0 + t2 kgk2 . Kα,λ (f, t ) = inf2 g∈Cα,λ

Lemma 3.1. Let ϕ21 (t) = t(1 + t), t ∈ En = [1/n, ∞). Then

i 2 X 2 v − 1 − 2t i ϕ1 (t)b′′n,v (t) ≤ C − t ϕ−i n1+ 2 bn,v (t) 1 (t). n i=0

Proof. We use the relations

 v−1 n+2 − t , n n 2 X v − 1 n + 2 i 1 ′′ i |bn,v (t)| ≤ − t , Qi (n, t)n bn,v (t) (t(1 + t))2 i=0 n n b′n,v (t)

n = bn,v (t) t(1 + t)



where Q0 (n, t) = nt(1 + t), Q1 (n, t) = 2t + 1, Q2 (n, t) = 1. On the other hand 1+ 2i  n (t(1 + t))−2 Qi (n, t)ni ≤ C , t(1 + t)

which implies the lemma.

Lemma 3.2. For ϕ21 (x) = x(1 + x), one has Z 2i v − x dx ≤ Cn−i , i ∈ N0 = N ∪ {0}. bn,v (x)ϕ1−2i (x) n En

Pointwise Approximation for Certain Mixed Sz´ asz-Beta Operators

157

R Proof. We use induction since En bn,v (x)dx ≤ 1 is our lemma for the case i = 0 . We assume that Z 2l v − x dx ≤ Cn−l . bn,v (x)ϕ1−2l (x) n En Now we only need to show that Z v 2l+2 bn,v (x)ϕ−2l−2 (x) −x dx ≤ Cn−l−1 . 1 n En

(6)

For v ≥ 10l + 10, using the integration by parts, we write 2l+1  Z 2l bn,v ( n1 ) 1 1 v bn,v (x)  v − x dx = − l l n 2l + 1 ( n1 )l (1 + n1 )l n n En x (1 + x)  Z v 2l+1  v − l − 1 2l + n + 2 bn,v (x) + − x − x dx, l+1 (1 + x)l+1 2l + 1 n 2l + n + 2 En x or Z  v 2l+1  v − l − 1 bn,v (x) −x − x dx l+1 l+1 (1 + x) n 2l + n + 2 En x  2l+1 1 v 1 bn,v ( n ) 1 −l−1 ≤ Cn + − . n ( n1 )l (1 + n1 )l n n Since

bn,v ( n1 ) 1 l ( n ) (1 + n1 )l



1 v − n n

2l+1

≤ Cn−l ,

we can get Z  v 2l+1  v − l − 1 bn,v (x) − x − x dx = O(n−l−1 ). l+1 l+1 (1 + x) n 2l + n + 2 En x

Let [2, pp. 125–129]   v+1 nv − 2(l + 1)(n + v) ⊂ En = [1/n, ∞). 0, − x ≤ C − x . n n + 2l + 2 n n + 2l + 2 We get Z v 2l+2 bn,v (x) − x dx l+1 (1 + x)l+1 n En x Z Z v v 2l+2 2l+2 bn,v (x) bn,v (x) −x dx + −x dx = l+1 (1+x)l+1 n l+1 (1+x)l+1 n x x Gn,v En \Gn,v

=: T1 + T2 , (7)

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Q. Qi & Y. Zhang

 v 2l+1  v − l − 1 bn,v (x) − x − x dx l+1 (1 + x)l+1 n n + 2l + 2 En \Gn,v x Z  v 2l+1  v − l − 1 bn,v (x) −x − x dx ≤C l+1 (1 + x)l+1 x n n + 2l + 2 E Z n v 2l+1 v−l−1 bn,v (x) +C −x dx = O(n−l−1 )+T3 . −x l+1 l+1 (1+x) n n+2l+2 Gn,v x

Z

One obtains


T1 = O n−l−1 + T3 .

(8)

The integrals T2 and T3 are defined on Gn,v . For x ∈ Gn,v , one has       v v−l−1 v v 1 1 , , − x = O 1+ 1+ − x = O n n n n + 2l + 2 n n 1 v  v  ϕ21 (x) ≥ 1+ . 2 n n We can deduce T2 = O(n−l−1 ),

T3 = O(n−l−1 ).

(9)

From (7)–(9), we get the relation (6) for v ≥ 10l + 10. For v < 10l + 10, we have | nv − x| ≤ C(l)|x| and Z Z v 2l+2 xl+1 −2l−2 (x) dx dx ≤ C(l) bn,v (x)ϕ1 −x bn,v (x) n (1 + x)l+1 En En Z (v + l)!(n + v)! bn,v+l+1 (x) ≤ C(l) dx ≤ Cn−l−1 . (n + v + l + 1)!(v − 1)! En Lemma 3.3. If c1 , c2 , c3 , c4 ∈ Z, k = 2 or k = −2, we have Z ∞ ∞ X sn+c1 ,v+c2 (x) bn+c3 ,v+c4 (t)δnk (t)dt ≤ Cδnk (x). 0

v=1

Proof. For k = 2, by direct calculation, we have Z ∞ ∞ X bn+c3 ,v+c4 (t)δn2 (t)dt sn+c1 ,v+c2 (x) 0

v=1

=

∞ X

sn+c1 ,v+c2 (x)

v=1

=

∞ X

sn+c1 ,v+c2 (x)

v=1 ∞ X

+2

v=1

Z

Z

sn+c1 ,v+c2 (x)

∞

bn+c3 ,v+c4 (t)ϕ2 (t)dt +

0 ∞ 0

Z

(v + c4 )(v + c4 + 1) bn+c3 −2,v+c4 +2 (t)dt (n + c3 )(n + c3 − 1)

∞ 0

1 n

1 v + c4 bn+c3 −1,v+c4 +1 (t)dt + n + c3 n

Pointwise Approximation for Certain Mixed Sz´ asz-Beta Operators

=

∞ X

sn+c1 ,v+c2 (x)

v=1

159

∞ X 1 v + c4 (v + c4 )(v + c4 + 1) + +2 sn+c1 ,v+c2 (x) (n + c3 )(n + c3 − 1) n + c n 3 v=1

1 ≤ Cδn2 (x). n For k = −2, by a similar calculation, we complete our proof. ≤ C(x2 + 2x) +

2 Lemma 3.4. For f ∈ Cα,λ , 0 < α < 2, 0 ≤ λ ≤ 1, n ≥ 2, we have kSn f k2 ≤ Ckf k2 .

  2 P 2 sn,v−j (x)(−1)2−j and Proof. Using the expressions s′′n,v (x) = n2 j j=0   2 P 2 b′′n,v (x) = (n + 1)(n + 2) bn+2,v−j (x)(−1)2−j , and also noting that j j=0 2 f ∈ Cα,λ , f (0) = 0, we have Sn′′ (f, x) =

Z ∞ ∞ n X sn,v (x) b′′n−2,v+2 (t)f (t)dt. n − 1 v=1 0

(10)

By integration by parts, Jensen’s inequality and Lemma 3.3, we write 2+α(λ−1) (x)Sn′′ (f, x) δn Z ∞ ∞ X ≤ Ckf k2 δn2+α(λ−1) (x) sn,v (x) bn−2,v+2 (t)δn−2+α(1−λ) (t)dt ≤ Ckf k2 δn2+α(λ−1) (x) ≤ Ckf k2 .

v=1

0

"

Z

∞ X

v=1

sn,v (x)

∞ 0

bn−2,v+2 (t)δn−2 (t)dt

# 2+α(λ−1) 2

0 Lemma 3.5. For f ∈ Cα,λ , 0 < α < 2, 0 ≤ λ ≤ 1, n ≥ 3, we have kSn f k2 ≤ Cnkf k0 .

Proof. For x ∈ En , m = 0, 1, 2, · · · , one has [2] ∞ X v=0

sn,v (x)

v

2m xm ≤ C m, −x n n

i 2 i X ′′ n1+ 2 sn,v (x) v xsn,v (x) ≤ C − x . i/2 n x i=0

(11) (12)

Next we will estimate the lemma in the two cases x ∈ Enc and x ∈ En .

160

Q. Qi & Y. Zhang

Case I: For x ∈ Enc , noting that δn2 (x) ∼ 1/n, using (10), Jensen’s inequality and Lemma 3.3, we have Z ∞ ∞ X 2+α(λ−1) ′′ (x) sn,v (x) bn−2,v+2 (t)f (t)dt δn 0 v=1 Z ∞ ∞ X ≤ Ckf k0 δn2+α(λ−1) (x)n2 sn,v (x) [bn,v+2 +2bn,v+1 +bn,v ]δnα(1−λ) dt 0

v=1

≤

Ckf k0 δn2+α(λ−1) (x)n2

≤ Cnkf k0 .

"∞ X

v=1

#α(12−λ) Z ∞ 2 sn,v (x) (bn,v+2 +2bn,v+1 +bn,v )δn dt 0

Therefore, for x ∈ Enc , we get 2+α(λ−1) (x)Sn′′ (f, x) ≤ Cnkf k0 . δn

(13)

Case II: For x ∈ En , δn2 (x) ∼ ϕ2 (x), we write Z ∞ ∞ 2 X ′′ 2+α(λ−1) 2+α(λ−1) ′′ (x)Sn (f, x) ≤ δn (x) bn,v (t)f (t)dt sn,v (x) δn 2+x v=1 0 Z ∞ ∞ x X ′′ sn,v (x) bn,v (t)f (t)dt =: I1 + I2 . (14) + δn2+α(λ−1) (x) 2+x 0 v=1

In order to complete our proof, we need estimate I1 and I2 . Choosing m ∈ N 1 + α(1−λ) < 1, using (11), (12), Lemma 3.3, H¨ older’s and such that 2m 2 Jensen’s inequalities, we have Z ∞ ∞ 2 X ′′ 2+α(λ−1) sn,v (x) bn,v (t)dt (x) δn 2 + x v=1 0 2 ∞ i Z ∞ X X i i v bn,v f δnα(1−λ) dt ≤ Cϕα(λ−1) (x)kf k0 n1+ 2 sn,v (x)x− 2 −x n 0 v=1 i=0 ) 1 ( 2 ∞ v 2mi 2m X X i i n1+ 2 x− 2 ≤ Cϕα(λ−1) (x)kf k0 sn,v (x) −x n v=1 i=0 ·

(

∞ X v=1

sn,v (x)

Z

∞ 0

≤ Cϕα(λ−1) (x)kf k0

bn,v (t)δn2 (t)dt

2 X i=0

i

i

) α(1−λ) 2

n1+ 2 n− 2 δnα(1−λ) (x) ≤ Cnkf k0 .

Pointwise Approximation for Certain Mixed Sz´ asz-Beta Operators

161

Then I1 ≤ Cnkf k0 .

(15)

Noting that when x ≤ 1, I2 ≤ I1 , from (13)–(15), we can get our result. For x > 1, we can split I2 into two parts Z 1 ∞ x X ′′ 2+α(λ−1) I2 ≤ δn (x) sn,v (x) bn,v (t)f (t)dt 2 + x v=1 0 Z ∞ ∞ x X ′′ 2+α(λ−1) sn,v (x) bn,v (t)f (t)dt =: J1 + J2 . (16) + δn (x) 2 + x v=1 1

Using (11), (12), Lemma 3.3, H¨ older’s and Jensen’s inequalities, we have ∞ 2 i Z 1 i v 1 X 1+ i X 2+α(λ−1) sn,v (x)x− 2 − x bn,v (t)f (t)dt (x) n 2 δn 2 + x i=0 n 0 v=1 ∞ 2 i Z 1 v+1 X X sn,v+1 (x) v i −x bn,v δnα(1−λ) dt n1+ 2 ≤ Ckf k0 ϕα(λ−1) (x) i/2 n n x 0 v=1 i=0 i 2 ∞ X X sn,v+1 (x) v + 1 1+ 2i α(λ−1) − x n ≤ Ckf k0 ϕ (x) i/2 n x v=1 i=0 Z 1 [bn+1,v−1 (t) + bn,v (t)]δnα(1−λ) (t)dt · 0

α(λ−1)

+ Ckf k0 ϕ

α(λ−1)

≤ Ckf k0 ϕ ·

(

∞ X

v=1

(x)

2 X

n

Z ∞ X sn,v+1 (x)

1+ 2i

v=1

i=0

(x)

sn,v+1 (x)

2 X

n

1+ 2i

− 2i

x

1 0

(∞ X

v=1

i=0

Z

xi/2 ni

0

1

[bn+1,v−1 +bn,v ]δnα(1−λ) dt

sn,v+1 (x)



[bn+1,v−1 (t) + bn,v (t)]δn2 (t)dt

1 2mi ) 2m v+1 −x n

) α(1−λ) 2

"∞ #α(12−λ) Z 1 2 1+ 2i X X n sn,v+1 (x) [bn+1,v−1 +bn,v ]δn2 dt + Ckf k0 ϕα(λ−1) (x) i/2 ni x 0 v=1 i=0 1   2 X xmi 2m α(1−λ) i i n1+ 2 x− 2 δn (x) ≤ Ckf k0 ϕα(λ−1) (x) nmi i=0 + Ckf k0 ϕα(λ−1) (x)

2 X i=0

i

i

n1+ 2 n 2 n−i δnα(1−λ) (x) ≤ Cnkf k0 .

162

Q. Qi & Y. Zhang

We now get J1 ≤ Cnkf k0 .

(17)

Using (11), (12) and Lemma 3.1, we write Z ∞ ∞ x X 2+α(λ−1) ′′ J2 ≤ δn (x) sn,v (x) bn−2,v+2 (t)f (t)dt 2 + x v=1 1 i α(1−λ) Z ∞X i 2 ∞ X n1+2 bn−2,v+2 v+1−2t δn α(λ−1) 2 ≤ Ckf k0 ϕ (x)x sn,v (x) n − 2 −t t(1+t) dt ϕi1 (t) 1 i=0 v=1 i Z ∞X i ∞ 2 X n1+2 bn,v(t) v+1−2t α(1−λ) α(λ−1) (t)dt −t δn ≤ Ckf k0 ϕ (x) sn,v+2 (x) ϕi1 (t) n − 2 1 i=0 v=1 Z ∞X i ∞ 2 X n1+ 2 bn,v (t) v i α(1−λ) (t)dt ≤ Ckf k0 ϕα(λ−1) (x) sn,v+2 (x) −t δn n ϕi1 (t) 1 i=0 v=1  i Z ∞X i ∞ 2 X n1+ 2 bn,v (t) 2v + Ckf k0 ϕα(λ−1) (x) sn,v+2 (x) δnα(1−λ) (t)dt i (t) n(n−2) ϕ 1 1 v=1 i=0 i  Z ∞X i ∞ 2 X n1+ 2 bn,v (t) 2t−1 α(λ−1) δnα(1−λ) (t)dt + Ckf k0 ϕ (x) sn,v+2 (x) i (t) n−2 ϕ 1 1 v=1 i=0 =: L1 + L2 + L3 .

Similar to I2 and Lemma 3.2, we can get α(λ−1)

kf k0 ϕ ·

(

∞ X

(x)

2 X

n

1+ 2i

i=0

sn,v+2 (x)

v=1

Z

≤ Ckf k0 ϕα(λ−1) (x)

∞ 1

(

∞ X

sn,v+2 (x)

v=1

bn,v (t)δn2 (t)dt

2 X i=0

i

n1+ 2 n−mi

Z

∞ 1

) α(1−λ) 2

1
2m

2m bn,v (t)  v dt −t 2m ϕ1 (t) n

1 ) 2m

δnα(1−λ) (x) ≤ Cnkf k0 .

Therefore, one has L1 ≤ Cnkf k0 .

(18)

Similarly we can get the estimates for L2 , L3 . Combining (16)–(18), we obtain I2 ≤ Cnkf k0 for x > 1.

Pointwise Approximation for Certain Mixed Sz´ asz-Beta Operators

163

Lemma 3.6. [6, Theorem 3.2] If f ∈ CB [0, ∞), 0 < α < 2, 0 ≤ λ ≤ 1, we 2 have |ϕα(λ−1) (x)∆2hϕλ f (x)| ≤ CKα,λ (f, h2 ϕ2(λ−1) (x)), where ∆2hϕλ f (x) = λ λ f (x + hϕ (x)) − 2f (x) + f (x − hϕ (x)). 2 Proof of Theorem 1.2. By the definition of Kα,λ (f, t2 ), we can choose 2 −1 2 g ∈ Cα,λ such that kf − gk0 + n kgk2 ≤ 2Kα,λ (f, n−1 ). The condition 1 |Sn (f, x) − f (x)| = O((n− 2 δn (x)1−λ ))α implies that kSn (f, x) − f (x)k0 = α O(n− 2 ). Using Lemmas 3.4 and 3.5, we have α

2 Kα,λ (f, t2 ) ≤ kf −Snf k0 + t2 kSn f k2 ≤ Cn− 2 + t2 (kSn (f −g)k2 +kSn gk2 ) α

≤ Cn− 2 + t2 (Cnkf − gk0 + Ckgk2 ) 2    α t 2 √ Kα,λ (f, n−1 ) . ≤ C n− 2 + 1/ n

2 Using the Berens-Lorentz Lemma [2], we get Kα,λ (f, t2 ) ≤ Ctα . From Lemma 3.6, one has ωϕ2 λ (f, t) = O(tα ).

References 1. Z. Ditzian, Direct estimate for Bernstein polynomials, J. Approx. Theory 79 (1994), 165–166. 2. Z. Ditzian and V. Totik, Moduli of Smoothness, Springer, New York, 1987. 3. Z. Finta, On converse approximation theorems, J. Math. Anal. Appl. 312 (2005), 159–180. 4. Z. Finta, N. K. Govil and V. Gupta, Some results on modified Sz´ asz-Mirakjan operators, J. Math. Anal. Appl. 327 (2007), 1284–1296. 5. S. Guo and Q. Qi, Simultaneous approximation for Sz´ asz-Mirakin quasiinterpolants, Czechoslovak Math. J. 56 (2006), 789–803. 6. S. Guo and Q. Qi, Strong converse inequalities for Baskakov operators, J. Approx. Theory 124 (2003), 219–231. 7. V. Gupta, Error estimate for mixed summation-integral type operators, J. Math. Anal. Appl. 313 (2006), 632–641. 8. V. Gupta, R. N. Mohapatra and Z. Finta, A certain family of mixed summation-integral type operators, Math. Comput. Modelling 42 (2005), 181–191. 9. V. Gupta and M. A. Noor, Convergence of derivatives for certain mixed Sz´ asz-beta operators, J. Math. Anal. Appl. 321 (2006), 1–9. 10. M. K. Gupta and V. Vasishtha, The iterative combinations of a new sequence of linear positive operators, Math. Comput. Modelling 39 (2004), 521–527. 11. H. M. Srivastava and V. Gupta, A certain family of summation-integral type operators, Math. Comput. Modelling 37 (2003), 1307–1315.

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Q. Qi & Y. Zhang

Session 02

Clifford and Quaternion Analysis

SESSION EDITORS I. Sabadini M. Shapiro F. Sommen

Politecnico di Milano, Milano, Italy Instituto Polit´ecnico Nacional, M´exico, Mexico Ghent University, Ghent, Belgium

167

CRITERIA FOR MONOGENICITY OF CLIFFORD-ALGEBRA-VALUED FUNCTIONS RICARDO ABREU BLAYA Facultad de Inform´ atica y Matem´ atica, Universidad de Holgu´ın, Holgu´ın 80100, Cuba E-mail: [email protected] JUAN BORY REYES Department of Mathematics, Universidad de Oriente, Santiago de Cuba 90500, Cuba E-mail: [email protected] ˜ PENA ˜ ∗ and FRANK SOMMEN† DIXAN PENA Department of Mathematical Analysis, Ghent University, 9000 Gent, Belgium ∗ E-mail: [email protected] † E-mail: [email protected] A differential and integral criterion for monogenicity is presented within the framework of Clifford analysis. The results obtained provide characterizations for the holomorphic and the two-sided biregular functions. Keywords: Clifford analysis, Dirac operator, Cauchy transform.

1. Preliminaries Let (e1 , . . . , em ) be an orthonormal basis of the Euclidean space Rm . Let Cm be the complex Clifford algebra constructed over Rm . The non-commutative multiplication in Cm is governed by the rules e2j = −1,

j = 1, 2, . . . , m and ej ek + ek ej = 0,

1 ≤ j 6= k ≤ m.

The Clifford algebra Cm is generated additively by elements of the form eA = ej1 . . . ejk , where A = {j1 , . . . , jk } ⊂ {1, . . . , m} is such that j1 < · · · < jk . For the empty set ∅, we put e∅ = 1, the latter being the identity element.

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R. Abreu Blaya, J. Bory Reyes, D. Pe˜ na Pe˜ na & F. Sommen

Any Clifford number a ∈ Cm may thus be written as X a= aA eA , aA ∈ C, A

or still as a =

Pm

k=0 [a]k ,

where [a]k =

P

|A|=k

aA eA is a so-called k-vector (k)

(k = 0, 1, . . . , m). Denoting the space of k-vectors by Cm , we have that Cm =

m M

C(k) m .

k=0

The 0-vectors and 1-vectors are simply called scalars and vectors respectively. The product of two Clifford vectors splits up into a scalar part and a 2-vector or a so-called bivector part a b = − ha, bi + a ∧ b where ha, bi =

m X

aj b j

j=1

and a∧b=

X j 0 such that for all x ∈ Σ and all 0 < r ≤ diam Σ, c−1 rm−1 ≤ Hm−1 (Σ ∩ {|y − x| ≤ r}) ≤ c rm−1 .

We also assume that f belongs to the H¨ older space C 0,α (Σ), 0 < α < 1. The Cauchy transform CΣ and the principal value Hilbert transform HΣ of f are defined respectively by Z Em (y − x)ν(y)f (y)dHm−1 (y), x ∈ Rm \ Σ, CΣ f (x) = Σ

HΣ f (x) =

Z

Σ

Em (y − x)ν(y)(f (y) − f (x))dHm−1 (y),

x ∈ Σ.

Here and subsequently, ν(y) stands for the unit normal vector on Σ at the point y introduced by Federer (see [13]). We notice that CΣ f is left monogenic in Rm \ Σ. Let us now formulate some important properties of CΣ f and HΣ f . For their proofs we refer the reader to [1] and [3].

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R. Abreu Blaya, J. Bory Reyes, D. Pe˜ na Pe˜ na & F. Sommen

(A) HΣ f ∈ C 0,α (Σ). (B) (Sokhotski-Plemelj formula) For z ∈ Σ, lim CΣ f (x) = HΣ f (z) + f (z).

Ω∋x→z

Theorem 3.1. Let F : Ω → Cm be a continuous function such that f = F |Σ belongs to C 0,α (Σ) where Σ is a Ahlfors-David regular surface. Then F is left monogenic in Ω iff F is harmonic in Ω and HΣ f = 0. Proof. Suppose that F is left monogenic in Ω. By Cauchy’s integral formula (see [5]), we have F (x) = CΣ f (x) for x ∈ Ω. Now (B) yields f (x) = HΣ f (x) + f (x),

x ∈ Σ.

Consequently, HΣ f (x) = 0 for all x ∈ Σ. Conversely, assume that F is harmonic in Ω and HΣ f = 0. Let us define  CΣ f (x), x ∈ Ω G(x) = x ∈ Σ. f (x), The function G is left monogenic in Ω and hence harmonic in Ω. By (A) and (B), G is also continuous on Ω. As F −G is harmonic in Ω and (F −G)|Σ = 0 it follows that F (x) = CΣ f (x) for x ∈ Ω. Let m = 2n be even and consider in particular F taking values in Cn . As mentioned above F is isotonic in Ω iff F I is left monogenic in Ω. By Theorem 3.1, F I is left monogenic in Ω iff HΣ (f I) = 0 with f = F |Σ , or, equivalently, applying (1): HΣisot f = 0.

(4)

Here HΣisot stands for the isotonic Hilbert transform defined for x ∈ Σ as HΣisot f (x) 1 =− ω2n +

Z " Σ


˜ − f˜(x))ν (y) (y 1 − x1 ) ν 1 (y)(f (y) − f (x)) + i(f(y) 2 |y − x|2n

#
˜ (f (y) − f (x))ν 2 (y) − iν 1 (y)(f˜(y) − f(x)) (y 2 − x2 ) |y − x|2n

dH2n−1 (y),

Pn Pn where ν 1 (x) = j=1 ej νj (x) and ν 2 (x) = j=1 ej νn+j (x) (see [2] and [4]). Using Theorem 3.1 we can state two important consequences related to holomorphic and two-sided biregular functions.

Criteria for Monogenicity of Clifford-Algebra-Valued Functions

173

Corollary 3.1. Let m = 2n and suppose that F takes values in the space of scalars C. Then F is holomorphic in Ω iff F is harmonic in Ω and for all x ∈ Σ the following equalities hold Z

hy 1 − x1 , ν 1 + iν 2 i + hy 2 − x2 , ν 2 − iν 1 i |y − x|2n

Σ

(f (y) − f (x))dH2n−1 (y)

Z (y 1 −x1 ) ∧ (ν 1 +iν 2 ) − (y 2 −x2 ) ∧ (ν 2 −iν 1 ) = (f (y)−f (x))dH2n−1 (y) |y − x|2n Σ = 0.

Proof. The proof follows easily by taking the scalar and the bivector part of (4). Corollary 3.2. Let m = 2n and suppose that F takes values in the real Clifford alegra R0,n . Then F is a two-sided biregular function in Ω iff F is harmonic in Ω and for all x ∈ Σ the following equalities hold Z

Σ

=

(y 1 − x1 )ν 1 (y)(f (y) − f (x)) + (f (y) − f (x))ν 2 (y)(y 2 − x2 ) Z

|y − x|2n

(y 1 −x1 )(f (y)−f (x))ν 2 (y) − ν 1 (y)(f (y)−f (x))(y 2 −x2 ) Σ

|y − x|2n

dH2n−1 (y) dH2n−1 (y)

= 0. Proof. It is sufficient to take the real and the imaginary part of (4).

Acknowledgments This paper was written while the second author was visiting the Department of Mathematical Analysis of Ghent University. He was supported by the Special Research Fund No. 01T13804, obtained for collaboration between the Clifford Research Group in Ghent and the Cuban Research Group in Clifford analysis, on the subject Boundary values theory in Clifford Analysis. J. Bory Reyes wishes to thank all members of this Department for their kind hospitality. D. Pe˜ na Pe˜ na was supported by a Doctoral Grant of the Special Research Fund of Ghent University. He would like to express his sincere gratitude.

174

R. Abreu Blaya, J. Bory Reyes, D. Pe˜ na Pe˜ na & F. Sommen

References 1.

2. 3. 4. 5. 6. 7.

8.

9. 10.

11.

12. 13. 14.

15. 16.

17. 18.

R. Abreu Blaya, J. Bory Reyes and D. Pe˜ na Pe˜ na, Clifford Cauchy type integrals on Ahlfors-David regular surfaces in Rm+1 . Adv. Appl. Clifford Algebras 13 (2003), no. 2, 133–156. R. Abreu Blaya, J. Bory Reyes, D. Pe˜ na Pe˜ na and F. Sommen, The isotonic Cauchy transform. Adv. Appl. Clifford Algebras 17 (2007), no. 2, 145–152. J. Bory Reyes and R. Abreu Blaya, Cauchy transform and rectifiability in Clifford analysis. Z. Anal. Anwendungen 24 (2005), no. 1, 167–178. J. Bory Reyes, D. Pe˜ na Pe˜ na and F. Sommen, A Davydov theorem for the isotonic Cauchy transform. J. Anal. Appl. 5 (2007), no. 2, 109–121. F. Brackx, R. Delanghe and F. Sommen, Clifford analysis. Research Notes in Mathematics, 76. Pitman Advanced Publishing Program, Boston, MA, 1982. F. Brackx, H. De Schepper and F. Sommen, The Hermitean Clifford analysis toolbox, accepted for publication in Proceedings of ICCA7. F. Brackx and W. Pincket, A Bochner-Martinelli formula for the biregular functions of Clifford analysis. Complex Variables Theory Appl. 4 (1984), no. 1, 39–48. F. Brackx and W. Pincket, Two Hartogs theorems for nullsolutions of overdetermined systems in Euclidean space. Complex Variables Theory Appl. 4 (1985), no. 3, 205–222. F. Brackx and W. Pincket, Series expansions for the biregular functions of Clifford analysis. Simon Stevin 60 (1986), no. 1, 41–55. G. David and S. Semmes, Analysis of and on uniformly rectifiable sets. Mathematical Surveys and Monographs, 38. American Mathematical Society, Providence, RI, 1993. R. Delanghe, F. Sommen and V. Souˇcek, Clifford algebra and spinor-valued functions. Mathematics and its Applications, 53. Kluwer Academic Publishers Group, Dordrecht, 1992. K. J. Falconer, The geometry of fractal sets. Cambridge Tracts in Mathematics, 85. Cambridge University Press, Cambridge, 1986. H. Federer, Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York 1969. P. Mattila, Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability. Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. W. Pincket, Biregular functions. Ph.D. Thesis, Ghent University, 1984. R. Rocha-Ch´ avez, M. Shapiro and F. Sommen, Integral theorems for functions and differential forms in Cm . Research Notes in Mathematics, 428. Chapman & Hall/CRC, Boca Raton, FL, 2002. F. Sommen, Tangential Cauchy-Riemann operators in Cm arising in Clifford analysis. Simon Stevin 61 (1987), no. 1, 67–89. F. Sommen and D. Pe˜ na Pe˜ na, Martinelli-Bochner formula using Clifford analysis. Archiv der Mathematik, 88 (2007), no. 4, 358–363.

175

HYPERMONOGENIC FUNCTIONS AND BIHYPERMONOGENIC FUNCTIONS IN REAL CLIFFORD ANALYSIS YUYING QIAO‡ and HEJU YANG∗ College of Mathematics and Information, Hebei Normal University Shijiazhuang , 050016, P. R. China ‡ E-mail: [email protected] XIAOLI BIAN† Department of Mathematics and Physics, Tianjin University of Technology and Education, Tianjin 300222, P. R. China E-mail: [email protected]

We discuss the properties of bihypermonogenic functions in Clifford analysis. Firstly, we obtain equivalent conditions for bihypermonogenic functions. Secondly, we give a Cauchy integral formula and a Plemelj formula for bihypermonogenic functions. Keywords: Bihypermonogenic; Plemelj Formula; Cauchy Integral Formula; Bihypermononic Function.

1. Introduction Clifford algebra is an associative and noncommutative algebraic structure that was introduced in the middle of the 1800s. Clifford analysis is an important branch of modern analysis that studies functions defined on Rn+1 with the values in a Clifford algebra [1]. Clifford analysis possesses important theoretical and applicable value in many fields. Since 1987, Wen, Huang, Qiao, J. Ryan, etc. have done a lot of work on boundary value problems and the properties of monogenic functions in real Clifford analysis Pn ∂ . A solution of the equation [2-5]. In Clifford analysis, let D = i=0 ei ∂x i ∗ Research supported by National Natural Science Fundamental of China (No.10771049,10671207) and National Natural Science Fundamental (No.A2007000225) † Corresponding author.

of

Hebei

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Y. Qiao, H. Yang & X. Bian

Df = 0 is called a monogenic function (or regular function). Monogenic functions are a generalization of analytic functions in complex analysis and their properties have been studied [1] . W. Hengartner and H. Leutwiler have researched hypermonogenic functions of R3 [7-8]. S.-L. Eriksson and H. Leutwiler introduced hypermonogenic functions in Clifford analysis and research some of its properties, discuss the integral representation for hypermonogenic functions [9-10]. Qiao discuss the boundary value problems of hypermonogenic functions [11]. In recent years, Z. Zhang and J. Du research Riemann boundary value problems and singular integral equations in Clifford analysis [12]. They get Laurent expansion and residue theorems in universal Clifford analysis [13]. In real and complex Clifford analysis [14], S. Huang introduced biregular functions and researched some of their properties. In this paper, we discuss the extension theorem of hypermonogenic functions. Using the idea of quasi-permutation created by Sha Huang [6], we give an equivalent definition of bihymonogenic functions. 2. Preliminaries 2.1. Clifford Algebra and Clifford Analysis Let An+1 (R) be a real Clifford algebra over an n + 1-dimensional real vector space Rn+1 with orthogonal basis e := {e0 , e1 , · · · , en }, where e0 = 1 is a unit element in Rn+1 . Then An+1 (R) has its basis e0 = 1, e1 , · · · , en ; e1 e2 , · · · , en−1 en ; · · · ; e1 · · · en . Hence an arbitrary element of the basis may be written as eA = ei1 ,··· ,ik = ei1 ei2 · · · eik , here A = {i1 , i2 , · · · , ik | 1 ≤ i1 < i2 < · · · < ik ≤ n} and when A = ∅ (empty set), eA = e0 .X So real Clifford algebra is composed of the elements having the type x = xA eA , in which xA (∈ R) are real numbers. In general, one A

has e2i = −1, i = 1, · · · , n, and ei ej + ej ei = 0, i, j = 1, · · · , n, i 6= j. Note that the real vector space Rn+1 consists of the elements x := x0 e0 + · · · + xn en . 2.2. Dirac Operator Let Ω ⊂ Rn+1 be an open connected set. A function f defined in Ω with values in An+1 (R) is a C 1 -function. The Dirac operator is defined as Dl f =

n X i=0

n

ei

X ∂f ∂f , Dr f = ei , ∂xi ∂xi i=0

Hypermonogenic and Bihypermonogenic Functions in Real Clifford Analysis

Dl f =

n X i=0

177

n

ei

X ∂f ∂f , Dr f = ei . ∂xi ∂xi i=0

Then f is left regular in Ω, if Dl f (x) = 0 in Ω; and f is right regular in Ω, if Dr f (x) = 0 in Ω. By calculation we can get DD = DD = ∆. 2.3. An Involution and a Decomposition in An+1 (R) The involution ′ : An+1 (R) −→ An+1 (R) is defined by X x′ = xA (−1)|A| eA A

where usually |A| is the number of elements in the set A. Clearly we have e′0 = e0 = 1 and e′i = −ei , if i = 1, · · · , n. Moreover the mapping ′ is an isomorphism since (ab)′ = a′ b′ . Recall that any element x ∈ An+1 (R) may be uniquely decomposed as x = b + cen , for b, c ∈ An (R).Using this decomposition we define the mappings P : An+1 (R) −→ AnX (R) and Q : An+1 (R) −→ An (R) by P x = b and Qx = c. Note that if x = xA eA ∈ An+1 (R) then Px =

X

n∈A /

A

xA eA , Qx =

X

xA eA\{n} .

n∈A

We call P x and Qx be the P part and Q part of x respectively. P part and Q part are similarly with the real part and imaginary part of complex number. 2.4. The First and the Third Class Quasipermutation Let A = {h1 , h2 , · · · , hk }, 1 ≤ h1 < h2 < · · · < hk ≤ n, where hi ∈ N, i = 1, 2, · · · , k. 1 ≤ m ≤ n, m be natural numbers. For the arrangement  A\{m}, m ∈ A; mA = {g1 , g2 , · · · , gk+1 }, m ∈ / A, where gi ∈ A ∪ {m}, 1 ≤ g1 < g2 < · · · < gk+1 . We call mA the first class quasi-permutation for arrangement mA, and define mm = ∅, m0 = m. If there are p natural numbers hi ∈ [1, m], then the sign δmA = δAm = (−1)p is called the sign of the first class quasi-permutation mA. The properties of the first class quasi-permutation: 1. If mA = B, then mB = A; 2. If m = 0, then δmA = 1;

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Y. Qiao, H. Yang & X. Bian

3. If mA = B, then δmA = −δmB ; 4. em eA = δmA emA . Let j be a natural number and D = {h1 , h2 , · · · , hs }, 1 < h1 < h2 < · · · < hs ≤ n. hp {1 ≤ p ≤ s} be natural numbers, the arrangement Dj is called the third class quasi-permutation of arrangement jD, and the sign  (−1)s , ∀p ∈ [1, s], hp 6= j; εjD = (−1)s−1 , ∃p ∈ [1, s], hp = j, ε0D = 1, is called the sign of the third class quasi-permutation. If j, D are as stated above, then ej eD = εjD eD ej . 3. Some Special Functions 3.1. Hypermonogenic Functions l

r

We consider the modified Dirac operators M l , M r , M and M in An+1 (R) defined by Q′ f , xn Qf , M r f (x) = Dr f (x) + (n − 1) xn Q′ f l , M f (x) = Dl f (x) − (n − 1) xn Qf r M f (x) = Dr f (x) − (n − 1) , xn M l f (x) = Dl f (x) + (n − 1)

′

′

where Q f = (Qf ) . Let Ω ⊂ Rn+1 = {x = (x0 , x1 , · · · xn ) | xn > 0}. A + mapping f : Ω → An+1 (R) is called left hypermonogenic (briefly hypermonogenic) if f ∈ C 1 (Ω) and M l f = 0 for any x ∈ Ω. A right hypermonogenic function is defined similarly. Hypermonogenic functions were introduced in [9]. 3.2. Bihypermonogenic Functions Suppose Ω1 and Ω2 be open subset of Rm+1 \{xm ≤ 0} and Rk+1 \{yk ≤ 0} respectively. If  l Mx f (x, y) = 0, Myr f (x, y) = 0, for any (x, y) ∈ Ω1 × Ω2 , then a functionf (x, y) is called bihypermonogenic in Ω1 × Ω2 .

Hypermonogenic and Bihypermonogenic Functions in Real Clifford Analysis

179

4. An Equivalent Condition for Bihypermonogenic Functions Lemma 4.1. Let the set β = {eA | A = {h1 , h2 , · · · , hr }, 1 ≤ h1 < h2 < · · · < hr ≤ n}, r = {ejA | A = {h1 , h2 , · · · , hr }, 1 ≤ h1 < h2 < · · · < hr ≤ n}, where 1 ≤ j ≤ n is a fixed integer, then β = r. Proof. For j, 1 ≤ j ≤ n, and any eA ∈ β, (1) If j ∈ A, we can find eB from r, B = A\{j}, then ejB = eA . (2) If S j∈ / A, we can find eB from r, B = A {j}, then ejB = eA . Summarizing the above discussion, we can get eA ∈ r, so β ⊂ r, similarly r ⊂ β can be obtained, hence β = r. Theorem 4.1. If f (x, y) =

X

X

fA,B (x, y)eA eB ,

A⊂{1,...,m} B⊂{m+1,...,m+k}

where fA,B ∈ C r (Ω) and x ∈ Rm+1 , y ∈ Rk+1 , then the f (x, y) is bihypermonogenic functions if and only if it satisfying the system m X ∂fjA,B ∂fA,B δjA (x, y) − (x, y) = 0, if m ∈ A. ∂x0 ∂xj j=0 m ∂fjA,B m−1 ∂fA,B (x, y) X δjA − (x, y) + (−1)|A| fmA,B (x, y) = 0, ∂x0 ∂x x j m j=0

if m ∈ / A.

k

∂fA(i+k)B ∂fA,B (x, y) X k−1 εB(i+k) δ(i+k)B (x, y) = 0, − (x, y) + f ∂y0 ∂yi yk A,Bk i=0

if m + k ∈ / B. k ∂fA,iB (x, y) ∂fA,B (x, y) X εB(i+k) δ(i+k)B − = 0, if m + k ∈ B. ∂y0 ∂yi i=0 Proof. Assume that f (x, y) is bihypermonogenic. We obtain Dxl f (x, y) =

X

m

eA eB

A,B

A,B

=

X

A,B

∂fA,B (x, y) ∂fA,B (x, y) X X ej eA eB + ∂x0 ∂xj j=1

eA eB

∂fA,B (x, y) + ∂x0

m XXX B

A j=1

δjA ejA eB

∂fA,B (x, y) . ∂xj

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Y. Qiao, H. Yang & X. Bian

Denoting jA = C, we infer A = jC and δjA = −δjC , which implies Dxl f (x, y) =

X

A,B

m

eA eB

∂fjC,B (x, y) ∂fA,B (x, y) X X X δjC eC eB − . (1) ∂x0 ∂xj j=1 B

C

Denoting Q-operator in An+1 (R) by Qx we obtain XX ′ ′ Qx f (x, y) = fA,B (x, y)eA\{m} eB m∈A B

=

XX

|A\{m}|

(−1)

fA,B (x, y)eA\{m} eB

m∈A B

Denoting D = A\{m}, we infer A = mD and X X ′ Qx f (x, y) = (−1)|D| fmD,B (x, y)eD eB .

(2)

B m∈D /

Combining (1) and (2) we deduce m−1 ′ Qx f (x, y) xm m X ∂fjC,B (x, y) ∂fA,B (x, y) X δjA − ) = eA eB ( ∂x0 ∂xj j=1

Mxl f (x, y)=Dxl f (x, y) +

A,B

m−1 X X + (−1)|A| eA eB fmA,B (x, y). xm B m∈A /

Collecting the coefficients of eA eB we obtain that Mxl f (x, y) = 0 is equivalent to the the system ∂fA,B (x,y) ∂x0 ∂fA,B (x,y) ∂x0

− −

Pm

∂fjA,B (x,y) j=1 δjA ∂xj Pm ∂fjA,B (x,y) δ j=1 jA ∂xj

= 0, if m ∈ A,

+

m−1 |A| fmA,B (x, y) = 0, xm (−1)

if m ∈ / A.

Similarly, we obtain the equivalent equations for Myr f (x, y) = 0. The preceeding theorem gives the relation between the system  l Mx f (x, y) = 0 Myr f (x, y) = 0 in Clifford analysis and the system that has 2n real partial differential equations.

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181

5. Cauchy-Riemann-Type Condition In this part we give a kind of Cauchy-Riemann-type condition for hypermonogenic function like holomorphic function in complex variables. Lemma 5.1. [10] Let Ω be stated as above, f ǫC 1 (Ω, An (R)), then 



M lf P ′f n−1 − (n − 1) xn en xn n   f M rf Pf = n−1 − (n − 1) n en Dr xn xnn−1 xn   f l n−1 P (M f ) = xn P Dl xnn−1   M l (Qf en ) Qf en = Q(M l f ) = Q(Dl f ) Dl xnn−1 xnn−1 Dl

f

xnn−1

=

Theorem 5.1. Let Ω, f be stated as above, a function f is a hypermonogenic function if and only if  n−1 ′   ∂P f (x) X ∂Qf (x)  ei + = 0,   ∂xn ∂xi i=0 n−1 X ∂P f (x) ∂Q′ f (x)  Q′ f (x)   ei  − + (n − 1) = 0.  ∂xi ∂xn xn i=0 Proof. Applying Lemma 5.1, since

l M l f (x) = P M l f(x) + Q M f (x) en f (x) + Q (Dl f (x)) en = xnn−1 P Dl xnn−1

Q(Dl f (x)) = QDl (P f (x) + Qf (x)en ) = Q n−1 X

n X i=0

ei

∂(P f (x) + Qf (x)en ) ∂xi

∂P f (x) ∂P f (x) ei =Q + en ∂x ∂xn i i=0 ! n−1 X ∂Qf (x) ∂Qf (x) ei + en + en en ∂xi ∂xn i=0 n−1 ∂P ′ f (x) X ∂Qf (x) + ei . = ∂xn ∂xi i=0

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xnn−1 P D



f xnn−1



= xnn−1 P

n X i=0

∂ ei



f (x) xnn−1 ∂xi



P f (x) + Qf (x)en xnn−1 ei = xnn−1 P ∂xi i=0  P f (x) Qf (x) P f (x) Qf (x) ∂ ∂ ∂ ∂ n−1 n−1 X X  xnn−1 xnn−1 xnn−1 xnn−1 = xnn−1 P  ei +en + en +en en  ei   ∂xi ∂xn ∂xi ∂xn i=0 i=0 n X

= =

n−1 X

i=0 n−1 X i=0

∂

∂

Q′ f (x) xnn−1 ∂xn

ei

∂P f (x) − xnn−1 ∂xi

ei

∂P f (x) ∂Q′ f (x) n − 1 ′ − + Q f (x). ∂xi ∂xn xn

6. The Extension Theorem of Hypermonogenic Functions Lemma 6.1. [10] A function f ǫC 1 (Ω, An (R)) is hypermonogenic in an open connected subset Ω ⊂ Rn+1 \ {xn = 0} if and only if the property Z Z Q(dσ0 f (x))en = 0, P (dσn−1 f (x)) + ∂K

or equivalently Z  ∂K

∂K

 c n−1 − dσ c 0 )fd (dσn−1 + dσ0 )f (x) + (dσ (x) = 0.

holds for any n+1-chain K satisfying K ⊂ Ω.

We call this lemma a Cauchy-Morera theorem for hypermonogenic functions. T ′ Theorem 6.1. Let Ω be as above and Ω = Ω {(x0 , x1 , . . . , xn−1 , an )}, ′ where an > 0 is a constant. If f is a hypermonogenic function in Ω\Ω and f ǫC 1 (Ω, An (R)), then f is a hypermonogenic function in Ω. T e contained in Ω, if Ω e Ω′ = ∅, we obProof. For any Ω tain that f is a hypermonogenic function, since f is a hyperT ′ e Ω′ 6= ∅, we put K1ε = monogenic function in Ω\Ω . If Ω e xn ≥ an + ε}, K2ε = {x = {x = (x0 , x1 , . . . , xn−1 , xn ) | x ∈ Ω,

Hypermonogenic and Bihypermonogenic Functions in Real Clifford Analysis

183

e xn ≤ an − ε}, where an − ε > 0. Using (x0 , x1 , . . . , xn−1 , xn ) | x ∈ Ω, Lemma 6.1, we obtain Z   c n−1 − dσ c 0 )fd (dσn−1 + dσ0 )f (x) + (dσ (x) = 0, ∂K1ε Z   c n−1 − dσ c 0 )fd (dσn−1 + dσ0 )f (x)(dσ (x) = 0. ∂K2ε

Since f is a continuous function in Ω, we have Z   c n−1 − dσ c 0 )fd (dσn−1 + dσ0 )f (x) + (dσ (x) e Z ∂Ω   c n−1 − dσ c 0 )fd = lim+ (dσk + dσ0 )f (x) + (dσ (x) ε→0 ∂K Z1   c n−1 − dσ c 0 )fd (dσn−1 + dσ0 )f (x) + (dσ (x)) + lim ε→0+ ∂K2

= 0.

By Lemma 6.1 again, the result follows. References 1. R. Delanghe, F. Sommen, V. Soucek, Clifford Algebra and Spinor-Valued Functions, Dordrecht, Kluwer, 1992. 2. G. C. Wen, Clifford Analysis and Elliptic Systems, Hyperbolic Systems of First Order Equations, World Scientific, Singapore, 1991, 230-237. 3. H. Sha, Nonlinear Boundary Value Problems for Biregular Functions in Clifford Analysis, Sci. China Ser. A, 39 (1996), 1152-1164. 4. Y. Qiao, Nonlinear Boundary Value Problems of Generalized Biregular Functions with Haseman Shift in Real Clifford Analysis, J. Sys. Sci. Math. Sci., 19 (1999), 484-489. 5. E. Franks and J. Ryan, Bounded Monogenic Functions on Unbounded Domains, Contemp. Math., 1998, American Mathematical Society. 6. S. Huang, Quasi-Permutations and Generalized Regular Functions in Real Clifford Analysis, J. Sys. Sci. Math. Sci., 18 (1998), 380-384. 7. W. Hengartner and H. Leutwiler, Hyperholomorphic Functions in R3 , Adv. Appl. Clifford Algebras, 11 (2001), S1, 247-259. 8. H. Leutwiler, Modified Clifford Analysis, Complex Variables, 17 (1992), 153171. 9. H. Leutwiler, Modified Quaternionic Analysis in R3 , Complex Variables, 20 (1992), 19-51. 10. S.-L. Eriksson and H. Leutwiler, Hypermonogenic Functions, In Clifford Algebras and Their Applications in Mathematical Physics, 2000, Vol. 2, Clifford Analysis, 286-301. 11. S.-L. Eriksson and H. Leutwiler, Hypermonogenic Functions and Their Cauchy-Type Theorems, in Trends in Mathematics, Advances in Analysis and Geometry, Birkh¨ auser, 2004, 97-112.

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12. S.-L. Eriksson, Integral Formulas for Hypermonogenic Functions, Bull. Belgian Math. Soc., 11 (2004), 705-717. 13. Y. Y. Qiao, A Boundary Value Problem for Hypermonogenic Functions in Clifford Analysis, Sci. China Ser. A, 48 (2005), Supp., 324-332. 14. Z. Zhang, J. Du, Laurent Expansions and Residue Theorems in Universal Clifford Analysis, Acta Math. Sci., 23A (2003), 692-703. 15. Z. Zhang, J. Du, On Certain Riemann Boundary Value Problems and Singular Integral Equations in Clifford Analysis, Chinese Ann. Math., 23A (2001), 421-426. 16. S. Huang, Y. Y. Qiao, G. C. Wen, Real and Complex Clifford Analysis, Springer, 2005. 17. R. P. Gilbert, J. L. Buchanan, First Order Elliptic Systems: A Function Theoretic Approach, New York, Academic Press, 1983.

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INITIAL VALUE PROBLEMS FOR REGULAR QUATERNION-VALUED INITIAL FUNCTIONS LE HUNG SON∗ , NGUYEN CANH LUONG† and NGUYEN QUOC HUNG‡ Faculty of Applied Mathematics, Hanoi University of Technology, Hanoi, Vietnam ∗ E-mail: sonlh [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] In the talk at 15th ICFIDCAA, Osaka, Japan, we constructed a method to solve the initial value problem in Clifford and quaternion analysis. On the other hand, in [9] we also constructed first-order operator of functions that are associated to a space of regular functions in quaternionic analysis. We will now find out conditions under which the initial value problem is uniquely solvable in this associated space. Keywords: Initial value problems of Cauchy-Kovalevskaya type; Associated differential operators; Quaternionic analysis; Fixed-point methods.

1. Introduction An initial value problem (IVP) of Cauchy-Kovalevskaya type has the form   ∂w ∂w (1) = L t, x, w, ∂t ∂xj w(0, x) = ϕ(x)

(2)

where the desired function w = w(t, x) depends on the time t and the spacelike variable x, and L is a first order differential operator depending on t, x, w and the derivatives of w with respect to the spacelike variables. The classical approach to IVP (1)-(2) is based on power-series representations. In this case, one assumes that both the right-hand side L and the initial function ϕ as well have power-series representations in their variables. On the other hand, the IVP (1)-(2) can be formulated within the framework of continuously differentiable functions. Therefore it is natural to ask whether or not there exist continuously differentiable solutions in case the righthand side of (1) is only a continuously differentiable function. In [8] H. Lewy

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gives a famous example which shows that there exist such linear first order differential equations with infinitely differentiable coefficients not having any solutions. However, the technique of associated differential operators allows to construct operators L such that the IVP (1)-(2) is solvable for each initial function ϕ satisfying lϕ = 0, where l is an operator which satisfies the interior estimate property. The first step of this technique is M. Nagumo’s integral rewriting of the IVP (1)-(2):  Z t  ∂w(τ, x) dτ (3a) w(t, x) = ϕ(x) + L τ, x, w(τ, x), ∂xj 0 Consequently, the solution of the IVP (1)-(2) is a fixed point of the operator  Z t  ∂w(τ, x) dτ (3b) F w(t, x) = ϕ(x) + L τ, x, w(τ, x), ∂xj 0 W. Walter (see [7]) uses M. Nagumo’s lemma which is such an interior estimate for a weighted supremum norm to prove the classical CauchyKovalevskaya theorem. Later, W. Tutschke ([3,4]) also points out the method of weighted function spaces for solving IVP which is based on the technique of associated differential operators. In [9], we consider a linear differential operator of first order of the form Lw =

3 X

A(j)

j=1

∂w + Bw + C, ∂xj

(3)

where w(t, x) are quaternion-valued functions, the initial funtion ϕ(x) is a regular function (satisfying Dϕ = 0, D being the Dirac operator) and the coefficients A(j) = A(j) (t, x), B = B(t, x), C = C(t, x) are also quaternionvalued functions and x ∈ Ω ⊂ R3 and t ∈ [0, T ] is the time variable. We formulated conditions for the coefficients of L under which L is associated to the Dirac operator D. In the present paper, we will prove an interior estimate for the Dirac operator and use the results of associated pair in [9] to formulate conditions under which the IVP (1)-(2) in quaternionic analysis is uniquely solvable. 2. Preliminaries and notation We denote by e0 = 1, e1 = i, e2 = j, e3 = k, where i, j, k are the units of the (real) quaternionic algebra H. A function f defined in the bounded domain Ω ⊂ R3 and takes value in H is a mapping f :Ω→H

Initial Value Problems for Regular Quaternion-Valued Initial Functions

187

and thus f can be represented in the form f=

3 X

ej fj (x)

j=0

where the fj (x) are real valued functions of x = (x1 , x2 , x3 ) ∈ Ω. Properties such as continuity, differentiability, integrability and so on, which are ascribed to the function f have to be fulfiled by all components fj . We denote by CH (Ω), CHk (Ω),... the corresponding spaces of continuous or k-times continuously differentiable functions. Now let us introduce the Dirac differential operator as D=

3 X

ek

k=1

∂ ∂xk

For other definitions concerning quaternions we refer the reader to [1,2]. Lemma 2.1. ([2]) Let u, v ∈ CH1 (Ω). Then D(u, v) = D(u)v + u ¯(Dv) − 2

3 X j=1

uj

∂v . ∂xj

Definition 2.1. The function u ∈ CH1 (Ω) is called a regular function if Du = 0, and we denote by AH (Ω) the space of regular functions on Ω. Definition 2.2. ([4]) Let L be a first order differential opeator depending on t, x, w = w(t, x) and on the spacelike first order derivatives ∂w , while ∂xj l is any differential operator with respect to the space variables xj whose coefficients do not depend on the time t. Then L is associated to l if L transforms solutions of lw = 0 into solutions of the same equation for fixedly chosen t, i.e, lw = 0 implies l[Lw] = 0. Theorem 2.1. ([9]) Suppose that A(j) (t, x) ∈ CH2 (Ω) for each t ∈ [0, T ] and B(t, x), C(t, x) ∈ CH1 (Ω) for each t ∈ [0, T ]. The operator L (right-hand side of (1)) is associated to the Dirac operator if the following conditions are satisfied: (i)

A0 = A3 = −A2 ,

(1)

(3)

A1 = A2 = A3 ,

A2 = −A1 = A0 ,

(3)

A3 = −A0 = −A1 ,

B, C ∈ AH (Ω)

t ∈ [0, T ].

(1)

(ii) (iii)

(2)

(2)

(1)

(2)

(1)

(2)

(3)

(3)

(DA(1) − 2B1 e0 )e1 = (DA(2) − 2B2 e0 )e2 = (DA(3) − 2B3 e0 )e3 , for each

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With the condition (i) of Theorem 2.1, if A(1) =

3 P

k=0

(1)

Ak ek is given,

then we can get A(2) and A(3) . Furthermore (ii) of Theorem 2.1 gives us one (1) ∂A system with 8 equations for 12 unknowns ∂xk , (k = 0, 1, 2, 3; j = 1, 2, 3). j In [9] we also illustrated this result by giving a general example. 3. Solving the initial value problem 3.1. Exhaustion First, we introduce an exhaustion of Ω by a family of subdomains Ωs , 0 < s < s0 , satisfying the following conditions: • To each point x 6= x0 of Ω where x0 ∈ Ω is fixedly chosen there exists uniquely determined s(x) with 0 < s(x) < s0 such that x ∈ ∂Ωs(x) . • If 0 < s′ < s′′ < s0 , then Ωs′ is a compact subset of Ωs′′ . • There exists a positive constant C1 such that for any s′ , s′′ with 0 < s′ < s′′ < s0 the distance of Ωs′ from ∂Ωs′′ can be estimated by: dist (Ωs′ , ∂Ωs′′ ) ≥ C1 (s′′ − s′ ), where C1 does not depend on the choice of s′ and s′′ . Define, finally s(x0 ) = 0. Then s0 − s(x) is a meansure of the distance of a point x of Ω from the boundary ∂Ωs(x) . Now we consider the conical set M = {(t, x) : x ∈ Ω, 0 ≤ t < η(s0 − s(x))} in the (t, s)-space. The parameter η will describes the height of the conical set and will be fixed later (Theorem 3.1). The base of M is the given domain Ω, whereas its lateral surface is defined by t = η(s0 − s(x))

(4)

The nearer a point x to the boundary ∂Ω, the shorter the coresponding time interval (4). The expression d(t, x) = s0 − s(x) −

t η

(5)

is positive in M , while it vanishes identically on the lateral surface of M . Thus (5) can be interpreted as some pseudo-distance of a point (t, x) of M from the lateral surface of M .

Initial Value Problems for Regular Quaternion-Valued Initial Functions

189

Again let Ωs , 0 < s < s0 , be an exhaution of a given (bounded) domain in R3 . Let Bs be the space of all w(x) ∈ AH (Ωs ) equipped with the norm kwks = kwkΩs = max (sup |wj |). i=0,1,2,3 Ωs

The limit function of a Cauchy sequence of functions belonging to Bs is continuous since convergence with respect to the supremum norm means uniform convergence. In view of Weierstrass convergence theorem, moreover, the limit function of an uniformly convergent sequence of regular functions is regular, too. Thus the space Bs is proved to be complete, i.e., Bs is a Banach space. For a chosen fixed t˜ < ηs0 the intersection of M with the plane t = t˜ in the (t, x)-space is given by 

(t, x) : t = t˜; s(x) < s˜

where s˜ = s0 −

t˜ η

(6)

Let B∗ (M ) be the set of all function w = w(t, x) which are continuous in M . We see w(t˜, x) belongs to Bs(x) for fixed t˜ if only s(x) < s˜ where s˜ is given by (6). And B∗ (M ) is equipped with the norm kwk∗ =

sup kw(t, .)ks(x) d(t, x).

(7)

(t,x)∈M

The definition (7) of the norm k.k∗ implies the estimate kw(t, .)ks(x) ≤

kwk∗ d(t, x)

(8)

for any point (t, x) ∈ M . Proposition 3.1. B∗ (M ) is a Banach space. Proof. Note that the inequality d(t, x) ≥ δ > 0 defines a closed subset M δ of the conical domain M . Each point of M is contained in such a subset Mδ provided δ is suitably chosen. For points (t, x) in Mδ , the definition (7) implies the estimate kw(t, .)ks)(x) ≤

1 kwk∗ δ

Now consider a fundamental sequence w1 , w2 , ... with respect to the norm k.k∗ . Then one has kwn (t, .) − wm (t, .)ks(x) ≤

1 ǫ δ

(9)

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L. H. Son, N. C. Luong & N. Q. Hung

for points in Mδ provided n and m are sufficiently large. This implies also |wn − wm | ≤

1 ǫ δ

for points in Mδ . Consequently, a fundamental sequence converges uniformly in each Mδ , i.e., the wn have a continuous limit function w∗ (t, x) in M . Similarly, estimate (9) shows that for t = t˜ and s(x) < s˜ the limits function belongs to Bs(x) because of the completeness of this space. Carrying out the limiting process m → ∞ in the inequality kwn − wm k∗ < ǫ, it follows, finally, kwn − w∗ k∗ ≤ ǫ and, therefore, kw∗ k∗ is finite. 3.2. Interior estimate Proposition 3.2. If 0 < s′ < s′′ < s0 and w ∈ AH (Ω), then

∂w 3

∂xi ′ ≤ dist (Ωs′ , Ωs′′ ) kwks′′ s

Proof. Let x0 be an arbitrary point of Ωs′ , then BR (x0 ) is a ball with centre x0 and radius R will belong to Ωs′′ if R < dist (Ωs′ , ∂Ωs′′ ). Since w(x) ∈ AH (Ω) ⇒ △w = 0, we have △wk = 0 (k = 0, 1, 2, 3). By using Poisson integral formula for harmonic functions, we get 1 wk (x) = 4πR

Z

|ξ−x0 |=R

wk (ξ)

R2 − |x − x0 |2 dµ |ξ − x|3

(10)

Initial Value Problems for Regular Quaternion-Valued Initial Functions

We denote r = |ξ − x|. Then r = Formula (10) implies



3 P

i=1

(ξi − xi )2

1/2

191

x −ξ and ∂r = i r i . ∂x0

 −2(xi −x0i ) 2 2 xi −ξi dµ − 3(R −|x−x0 | ) 5 wk (ξ) r3 r |ξ−x0 |=R (11) where x0 = (x01 , x02 , x03 ). Substitute x = x0 into (11), one gets |x−x0 | = 0; |xi − x0i | = 0. Thus Z 1 (x0i − ξi ) ∂wk (x) = dµ wk (ξ)(−3)R2 ∂xi 4πR |ξ−x0 |=R R5 ∂wk (x) 1 = ∂xi 4πR

Z



x − ξ On the other hand, 0i R i ≤ 1; thus

Z ∂wk 3 ≤ 1 (x ) implies 0 ∂xi 4πR |wk (ξ)| R2 dµ ∂wk 3 1 3 ∂xi (x0 ) ≤ 4πR sup |wk (ξ)| R2 4πR . |ξ−x0 |=R

By maximum principle (see [2]). we get

sup

|x−x0 |≤R

|wk | =

sup

|ξ−x0 |=R

∂wk 3 ∂xi (x0 ) ≤ R sup |wk |. |x−x0 |≤R

|wk |. Hence (12)

In view of the definition of the norm in B(Ω), (12) leads to

∂w 3

∂xi ′ ≤ R kwks′′ . s

Let R → dist (Ωs′ , ∂Ωs′′ ); we get

∂w 3

∂xi ′ ≤ dist (Ωs′ , ∂Ωs′′ ) kwks′′ . s

But dist (Ωs′ , ∂Ωs′′ ) ≥ C1 (s′′ − s′ ) provided s′ < s′′ . Thus

∂w 3

∂xi ′ ≤ C1 (s′′ − s′ ) kwks′′ . s

(13) is called an interior estimate.

(13)

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L. H. Son, N. C. Luong & N. Q. Hung

3.3. The main theorem Theorem 3.1. The initial value problem (1)-(2) is uniquely solvable, and the solution w = w(t, x) satisfies the side condition Dw = 0 for each t if • hypotheses (i), (ii), (iii) of Theorem 2.1 are satisfied; • there exists positive constants K0 , K1 and K2 such that max (

sup

i=0,1,2,3 Ω×[0;+∞)

max (

(1)

sup

i=0,1,2,3 Ω×[0;+∞)

max (

|Bi |) ≤ K0

|Ai |) ≤ K1

sup

i=0,1,2,3 Ω×[0;+∞)

|Ci |) ≤ K2 ;

• the parameter η (describing the height of the conical set M ) satisfies η
0. η

Define s˜ = s(x) + 12 d(t, x) implying 1 1 1 s˜ ≤ s(x) + (s0 − s(x)) = s(x) + s0 < s0 . 2 2 2 Thus there exists a point x ˜ with s(˜ x) = s˜, i.e., x˜ ∈ ∂Ωs˜. One has d(t, x˜) = s0 − s(˜ x) −

t 1 = d(t, x). η 2

Taking into account the estimate (8), the last relation gives kw(t, .)ks˜ ≤

2kwk∗ kwk∗ ≤ . d(t, x ˜) d(t, x)

In view of (13) one gets, therefore,

∂w 3 4 3

≤ kwks˜ ≤ kwk∗ .

∂xi 2 (t, x) C (˜ s − s) C d 1 1 s(x)

(14)

Initial Value Problems for Regular Quaternion-Valued Initial Functions

193

  ∂w by Lw. In particular, one has Lθ = To be brief, denote L t, x, w, ∂x j

L(t, x, 0, 0). Now, we have to estimate the norm of Lw. We get   3 X ∂v ∂w (j) + B(w − v)ks − A kLw − Lvks = k ∂xj ∂xj j=1 3 X ∂w ∂v k − ks ≤ K0 kw − vks + K1 ∂x ∂x j j j=1

(15)

Using (8) and (14), condition (15) implies kLwks(x) − kLθks(x) ≤ kLw − Lθks(x) ≤ K0 kwks(x) + K1

3 X ∂w k ks(x) ∂x j j=1

and kLwks(x) ≤ kLθks(x) + K0 kwks(x) + K1 ≤ kC(t, x)k∗

3 X ∂w k ks(x) ∂xj j=1

s0 1 s0 12 + K0 kwk∗ 2 + 3 K1 kwk∗ 2 , d2 (t, x) d (t, x) C1 d (t, x)

because d(t, x) < s0 . The definition (5) of the weight function d(t, x) implies Z t 1 η dτ < , and thus it follows that 2 (τ, x) d d(t, x) 0

Z t

η ∂w(τ, x)

)dτ ≤ (K2 .s0 + C2 kwk∗ ) (16) L(τ, x, w(τ, x),

∂x d(t, x) i 0 s(x) 12 K . where C2 = s0 K0 + 3 C 1 1 The estimate (16) of the s(x)-norm yields

Z t  

∂w(τ, x)

dτ L τ, x, w(τ, x),

≤ η (K2 s0 + C2 kwk∗ ) .

∂x i 0 ∗

From (3), (3b), it follows that

kF k∗ ≤ kϕk∗ + η(K2 s0 + C2 kwk∗ ).

(17)

The conditions (17) and (15) imply

  12 kF w − F vk∗ ≤ ηC2 kw − vk∗ = η s0 K0 + 3 K1 kw − vk∗ . C1

Now we consider the space B∗D (M ) = { w ∈ B∗ (M ) : Dw(t, ·) = 0 for each fixed t }.

(18)

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We see that B∗D (M ) is closed subset of B∗ (M ) in view of Weierstrass convergence theorem. Because of Theorem 2.1, if w ∈ B∗D (M ), that means Dw(t, ·) = 0 for each fixed t which implies D[F w(t, ·)] = 0, because Dϕ = 0 and D(Lw) = 0. So the operator F maps the subspace B∗D (M ) into itself. From the hypothesis of Theorem 3.1, we have η < s0 K0 +
−1 , which together with (18) imply that F is a contractive 3(12)K1 /C1 mapping. Theorem 3.1 is now proved. References 1. F. Brackx, R. Delanghe and F. Sommen, Clifford analysis, Pitman Advanced Publishing Program, Boston-London-Melbourne, 1982. 2. K. G¨ urlebeck and W. Spr¨ ossig, Quaternionic analysis and elliptic boundary value problems, Akademie-Verlag, Berlin, 1989. 3. W. Tutschke, Solution of initial value problems in classes of generalized analytic functions, Teubner, Leipzig and Springer Verlag, 1989. 4. H. Florian, N. Ortner, F. J. Schnitzer, and W. Tutschke, Functional analytic and complex methods, their interactions, and applications to partial differential equations, World Scientific, 2001, 75–90. 5. L. H. Son and W. Tutschke (2003), First order differential operators associated to the Cauchy-Riemann operator in the plane, Complex Variables, 48, 797–801. 6. L. H. Son and N. Q. Hung, The inital value problems in Clifford and quaternion analysis, Proceedings of the 15th ICFIDCAA, Osaka Municipal Universities Press 3, 2008, 317–323. 7. W. Walter (1985), An elementary proof of the Cauchy-Kovalevskaya theorem, Amer. Math. Monthly, 92, 115–125. 8. H. Lewy (1957), An example of a smooth linear partial differential equaiton without solution, Ann. of Math., 66, 155–158. 9. N. Q. Hung and N. C. Luong, First order differential operators associated to the Dirac operator in quaternionic analysis, Proceedings of the 2004 International Conference on Applied Mathematics, SAS International Publications, Delhi, 369–378.

Session 03

Complex Analysis and Potential Theory

SESSION EDITORS T. Aliyev Azero˘ glu M. Lanza de Cristoforis P. M. Tamrazov

Gebze Institute of Technology, Kocaeli, Turkey Universit` a di Padova, Padova, Italy National Academy of Sciences, Kiev, Ukraine

197

THE GEOMETRY OF BLASCHKE PRODUCTS MAPPINGS ILIE BARZA Karlstad University, Department of Mathematics, S-651 88 Karlstad, Sweden E-mail: [email protected] DORIN GHISA York University, Glendon College, Department of Mathematics, 2275-Bayview Av., Toronto, Canada, M4N3M6 E-mail: [email protected] A Blaschke product B generates a covering Riemann surface (W, B) of the complex plane. The study of such surfaces is undertaken here in a very general context, where the cluster points of the zeros of B form a (generalized) Cantor set. Explicit forms and fundamental domains for the covering transformations are revealed for a particular case of Blaschke products. Keywords: Blaschke product, covering surface, covering transformation, fundamental domain, Cantor set, conformal module.

1. Blaschke Products A Blaschke product is an expression of the form w = B(z) =

n≤∞ Y

bk (z),

(1)

k=1

where bk (z) =

a ¯ k ak − z |ak | 1 − a ¯k z

(2)

where ak ∈ D := {z ∈ C| |z| < 1}. The Moebius transformations (2) are called the Blaschke factors of B and the complex numbers ak are called the zeros of B. The product B can have a finite number of factors and then it is called finite Blaschke product, or infinitely many factors and then it is called infinite Blaschke product. If B has m Blaschke factors (counted with their multiplicities), we say that

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B is a Blaschke product of order m. In the infinite case, we understand by B(z) the limit (if it exists) of the sequence (Bn (z)), where Bn (z) =

n Y

bk (z).

k=1

It is known (see [6]) that under the following Blaschke condition: ∞ X

(1 − |an |) < ∞

(3)

n=1

the sequence (Bn ) converges uniformly on compact subsets of W = C\ (E ∪ A), where E is the set of cluster points of zeros of B (necessarily a subset of ∂D) and A = {z ∈ C | z =

1 , k = 1, 2, ...}. a ¯k

While Bn (z) are meromorphic functions in C, every point of E is an essential singularity of B. Moreover,these points are non isolated essential 1 . There are singularities, since they are also cluster points of the poles a ¯k known examples (see [3]) where E = ∂D, and therefore the domain of B is not a connected subset of C. However, if ∂D \ E contains an arc of the unit circle, then W is connected and (W, B), where W is endowed with the conformal structure induced by the identity mapping, represents a (branched) Riemann covering surface of the complex plane (see [2], page 119). The most general setting in which we will treat the infinite Blaschke products in this paper is that where E is the union of an usual Cantor set (see [5], page 41) and a discrete subset of the unit circle ∂D. We will call such a set generalized Cantor set, since the last subset can also be obtained by arc removal, namely when the removed open arcs are contiguous. This kind of generalized Cantor set might not be any more perfect. We consider that E is the empty set when B is finite. 2. The Behavior of a Blaschke Product on the Unit Circle We would like to know what is the group of covering transformations of (W, B) over C, and determine the fundamental domains of that group. Therefore, we are looking for conformal mappings U : W −→ W such that B ◦ U (z) = B(z), for every z ∈ W.

(4)

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Functions U defined in a neighborhood of the unit circle and fulfilling (4) have been found in [4] in the case of finite Blaschke products and this result has been generalized to some infinite Blaschke products in [3]. The main fact proven in [3] is the following: If the Blaschke sequence (an ) has a unique cluster point eiθ0 , then there are infinitely many disjoint half-open arcs Γn = {z = eiθ | αn ≤ θ < αn+1 }, n ∈ Z, limn−>±∞αn = θ0 (5) S such that Γn = ∂D \ {eiθ0 } and B maps every Γn continuously and bijectively on the unit circle in the w- plane. There are continuous functions Uk : ∂D \ {eiθ0 } −→ ∂D \ {eiθ0 }

such that Uk (Γn ) = Γm , k, m, n ∈ Z and B ◦ Uk (ζ) = B(ζ) for every ζ ∈ ∂D \ {eiθ0 }. The set of these functions form an infinite cyclic group. Moreover, the functions Uk can be extended to analytic functions in a neighborhood (in C) of every compact set included in ∂D \ {eiθ0 }. Theorem 2.1. If E is a generalized Cantor subset of ∂D, then every removed open arc I = {z = eiθ | α < θ < β} contains infinitely many disjoint half-open sub-arcs Γn = {z = eiθ | αn ≤ θ < αn+1 }, n ∈ Z, S+∞ limn→+∞ αn = α, limn→−∞ αn = β, and I = n=−∞ Γn , such that B maps every subarc Γn continuously and bijectively on the unit circle in the w-plane. Proof. We know (see [3]) that, for every n ∈ N∗ , the equation Bn (z) = 1 has exactly n distinct solutions, no matter if the zeros ak of Bn (z) are distinct or not. Let us notice that B(z) =1 (6) lim n→∞ Bn (z) uniformly on compact subsets of C \ E ∪ A. This means that, for every ε > 0 and δ > 0, there is n0 ∈ N∗ such that n ≥ n0 and | z − eiα |≥ δ, where eiα ∈ E imply |

B(z) − 1 |< ε. Bn (z)

Therefore, if ζ is a solution of the equation Bn (z) = 1, n ≥ n0 and | ζ − eiα |≥ δ, then | B(ζ) − 1 |< ε. Due to the continuity of the local inverse of B at B(ζ), given η > 0, we can choose ǫ > 0 such that | B(ζ) − 1 |< ǫ implies | ζ − B −1 (1) |< η. Therefore,

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there is ζ ′ such that B(ζ ′ ) = 1 and | ζ − ζ ′ |< η. In other words, the solutions of the equation B(z) = 1 are as close as we want of the solutions of Bn (z) = 1, if n is big enough. This means that the equation B(z) = 1 has infinitely many solutions. They cannot cluster to any point ζ0 ∈ I since then the function B(z) − 1 which is analytic in a neighborhood V of ζ0 , would be identically zero in V , which is impossible. Consequently, they accumulate to the ends of those arcs. Let us show that the end of every removed arc is a cluster point of the roots of the equation B(z) = 1. Suppose that the contrary happens, i.e. there is a point eiα such that a subsequence (ank )k∈N of (an )n∈N converges to eiα and there is a neighborhood V of eiα such that B(z) 6= 1 in V . Then ϕ(z) = 1/ [B(z) − 1] is analytic in V , and ϕ(1/ank ) = 0, k = 1, 2, ... This would imply that ϕ(z)is identically equal to 0 in V since lim 1/ank = eiα ∈ V, which is a k→∞

contradiction. Now the conclusion of the theorem is obvious.

Remark 2.1. Actually, even more can be said about the behavior of B in any neighborhood of an eiα ∈ E. We can replace 1 at the denominator of ϕ by any complex number w 6= 0. Then again ϕ would be analytic in V and ϕ(1/ank ) = 0, k = 1, 2, ..., which contradicts the fact that ϕ is not identically 0 in V . Consequently, in any neighborhood V of eiα ∈ E, the Blaschke product B takes every complex value (zero included, for an obvious reason!) infinitely many times. We will show in the following that this happens in a very orderly way, namely there are infinitely many regions b , the mapping being conformal in V which are mapped by B bijectively on C in the interior of each one of these regions. 3. Simultaneous Continuation In order to describe the geometry of the mappings realized by a Blaschke product B, we need to perform a partition of the domain of B in subsets in which B is injective. The following example illustrates this idea in a very simple case, namely when the Blaschke product is of the form  n a ¯ a−z , (7) w = B(z) = |a| 1 − a ¯z

where a = reiθ , 0 ≤ r < 1 and n is a natural number greater than 1. Let 0 ≤ λ ≤ 1 and let us solve the equation B(z) = λn for the unknown z. We

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get the solutions of the form: zk (λ) =

ωk λ − r iθ e , k = 0, 1, 2, ..., n − 1, ωk λr − 1

where ωk are the roots of order n of the unity. It can be easily checked that, for λ 6= 0, zk (λ) are distinct numbers, while zk (0) = a for every k. Moreover, as λ varies from o to 1, zk (λ) describes a Jordan arc γk connecting the point a with ζk =

ωk − r iθ e . ωk r − 1

The points ζk are all distinct points on the unit circle and consequently the arcs γk have no other common point except for a. We notice that in this particular example the arcs γk , as well as their symmetric arcs with respect to the unit circle should be arcs of circle or line segments. Indeed, zk (λ) are Moebius transformations and λ varies on an interval. Let us denote by Γk the arc of the unit circle between ζk and ζk+1 . Then Ck = γk + Γk − γk+1 is a Jordan curve bounding a domain Dk . The image of Ck in the w− plane is a curve obtained by adding to the unit circle the superior border and subtracting the inferior border of the slit alongside the interval [0, 1] of the real axis. Since B ′ (z) 6= 0 in Dk , the function B represents conformally Dk on the slit unit disc. This map can be extended by symmetry to the domain [ [ ck Ωk = D k D Γ0k ,

ck is the symmetric of Dk with respect to the unit circle and Γ0 = where D k Γk \ {ζk , ζk+1 }. The function B |Ωk realizes a conformal mapping of Ωk on the domain obtained from the complex w-plane by removing the positive real half-axis. For k 6= n − 1, there is a continuous passage from B |Ωk to B |Ωk+1 , when crossing γk+1 as well as from B |Ωn−1 to B |Ω0 when crossing γ0 . In fact the function B is locally injective in W = C \ {a; 1/¯ a} and (W, B) is a covering Riemann surface of the complex w− plane. The example (7) is useful also from another point of view, namely because we can find explicitly in that case the relation between the points having the same image by B. Indeed, suppose that B(z ′ ) = B(z), i.e. n n   a ¯ a−z a ¯ a − z′ . = |a| 1 − a ¯z ′ |a| 1 − a ¯z

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a ¯ a − z′ a ¯ a−z = ωk , where ωk , k = 0, 1, 2, ..., n − 1 are the |a| 1 − a ¯z ′ |a| 1 − a ¯z roots of order n of the unity.This gives

Then,

z′ =

a(1 − ωk ) − (| a |2 −ωk )z . 1− | a |2 ωk − a ¯ (1 − ωk )z

If we denote z ′ = Tk (z), we can easily see that {Tk } is a group of order n with respect to the usual composition of functions. Indeed, T0 (z) = z and for arbitrary k and k ′ we have Tk ◦ Tk′ = Tk+k′ (modn) . Particularly,Tk ◦ Tn−k = T0 . These formulas imply that a, then Tk (Ωk′ ) = Ωk+k′ (modn) . We notice also that if z 6= a, and z 6= 1/¯ Tk (z) 6= z and Tk (a) = a, Tk (1/¯ a) = 1/¯ a, for every k. Therefore, the Moebius transformations Tk are covering transformations of (W, B) over C. b B) as being a branched covering surface of C, b We can also think of (C, with the branch points a and 1/¯ a, both of them of multiplicity n. Obviously, b B) over C b is the same, if we the group of covering transformations of (C, b consider Tk as being this time self-mappings of C. The purpose of this section is to show that similar results can be obtained in a much more general context. The partition of the corresponding W will have infinitely many subsets when B is an infinite product and it can be realized by a technique that we call simultaneous continuation. Theorem 3.1. For every Blaschke product B, with E a generalized Cantor subset of ∂D, there is a partition of W into connected sets, such that the interior of each one of these sets is mapped by B conformally onto a slit w-plane. There is a continuous passage from the mapping of every set to that of a contiguous set in the partition. Proof. Let us deal first, for simplicity, with the case of a finite Blaschke product B. Let b1 , b2 , ..., bp be the branch points of B situated in the unit disc. If q1 , q2 , ..., qp are their order of multiplicity, then an easy computation shows that q1 + q2 + ... + qp = n. We might have wj = wj ′ , for bj 6= bj ′ ; particularly wj = 0, for all multiple zeros of B, which are obviously brunch points. Let wj = B(bj ), j = 1, 2, ..., p. We connect all the points wj by a polygonal line with no self-intersection, in such a way that one of its ends can be connected with w = 1 by a segment not intersecting the polygonal line. Let η be the new polygonal line which is a Jordan arc. Now we do continuations over η from every root eiθk of the equation B(z) = 1. There are exactly n such arcs and an easy topological argument

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shows that they must all end in points bj . Moreover, every bj should be reached at some moment by at least two consecutive arcs γj and γj+1 . If the point bj is a branch point of multiplicity qj , then exactly qj consecutive arcs γk will end in bj . Indeed, a corollary of the open mapping theorem (see [9]) guarantees that there is a local parameter at bj , z = z(u), z(0) = bj such that w = B(z(u)) = wj + uqj and there is a disc centered at 0 in the u-plane such that B(z(u)) maps p rays of that disc on η with B(z(0)) = wj . The images by z(u) of these rays are sub-arcs of γk converging to bj , and they are all mapped by B on η. Let us denote Ck = Γk + γk+1 − γk , which is a closed Jordan curve delimiting a domain Dk . The image of Ck by B is the closed curve obtained by adding to the unit circle the lower border and subtracting the upper border of a cut alongside η ending in wk = B(bk ), where bk is the common end of γk and γk+1 . Let ∆k be the domain in the w-plane bounded by that curve. The conformal correspondence theorem (see [8], p. 154) shows that B |Dk is a conformal map of Dk on ∆k with homeomorphic correspondence S S of their boundaries. Moreover, if Ak = Dk Lk lk , where Lk and lk represent the set of points of Γk , respectively γk , then B |Ak is a bijective and continuous map of Ak on the closed unit disc. ck , the symmetric of We can extend analytically B |Ak by symmetry to A S ck is Ak with respect to the unit circle. Obviously, the image of Ωk = Ak A the whole Riemann sphere. In fact B realizes a conformal map of the interior of Ωk on the Riemann sphere with a slit alongside η from wk to w = 1 and alongside the symmetric of η with respect to the unit circle from w = 1 to 1/wk . The sets Ωk form the partition we were referring to in the theorem.

1.5

1

0.5

–1

–0.5

0.5

–0.5

–1

–1.5

Fig. 1.

1

1.5

2

2.5

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2

1.5

1

0.5

–1

–0.5

0

0.5

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1.5

–0.5

–1

–1.5

Fig. 2.

The figures 1 and 2 exhibit computer generated fundamental domains for two Blaschke products.The first is of the form B1 (z) = b31 (z)b22 (z) where bk are of the form (2) with a1 = 0.4 + i 0.3 and a2 = 0.5 − i 0.6. The second is of the form B2 (z) = B1 (z)b33 (z), where a3 = 0.9 − i 0.2.We notice that the addition of b33 (z) changed in an obvious way only the fundamental domain containing a3 . Also, since a3 is closed to the unit circle, some of the corresponding domains are very small. We can reduce the infinite case to the finite one by using an exhaustion b \ E. Let us denote by Fm and by Hm the finite sets of sequence (Um ) of C branch points of B, respectively of roots of the equation B(z) = 1, situated in (Um ). Let us show that (Um ) can be chosen in such a way that every continuation over ηm (the polygonal line constructed with Fm as in the finite case) starting from a point of Hm lands in a point of Fm . Indeed, let ηm stretch from w = 1 to w = 0 and have other vertices exactly at the branch points bj ∈ Fm . Let ε > 0 be small enough such that every circle Cj of radius ε centered at bj intersects only the sides of ηm ending in bj . Let hj be the point of intersection of Cj and ηm situated between bj and 1. For ε small enough, the equation B(z) = hj has exactly qj distinct solutions zk where qj is the multiplicity of bj as solution of the equation B ′ (z) = 0. The continuation over ηm from every zk towards bj are all arcs connecting zk to bj . We complete these continuations up to w = 1. They are Jordan arcs γn,j connecting bj with the points Hm on the unit circle. The arcs γn,j cannot intersect each other, since then the image of the domain they are bounding would have no interior point, which is absurd. An easy analysis shows that we can arrange the exhaustion (Um ) in such a way that Fm and Hm are for every m the vertices of a connected bilateral graph with edges γn,j .

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Switching from (Um ) to (Um+1 ) implies just adding a finite number of vertices and edges to that graph, and hence obtaining a new one of the ∞ same type. Finally, every element of ∪∞ k=1 Fk and ∪k=1 Hk will appear in these graphs starting with an m. As in the finite case, the curves Cn,j = Γn,j − γn,j+1 + γn,j contain domains which are mapped by B conformally b \ E into fundamental domains follow on slit unit discs. The partition of C the same steps as in the finite case. Later on in this section we will find more about such a partition in the infinite case. Corollary 3.1. If B is a Blaschke product of order m, then ZZ ZZ 1 1 |B ′ (z)|2 dxdy = |B ′ (z)|2 dxdy = m, π π |z|1 Z 1 |B ′ (z)| | dz | ≥ m. 2π |z|=1

(8) (9)

Proof. Indeed, it is known (see [8], page 155) RR that if f represents conformally a domain D on D ∗ , then area(D ∗ ) = D |f ′ (z)|2 dxdy and that if f is regular on the closed contour Γ and in the domain enclosed by Γ, then R the length of f (Γ) is Γ |f ′ (z)| | dz |. With Dk , previously defined, instead of D and Z ZB(z) instead of f (z), we 1 ∗ have that D is the slit unit disc, therefore |B ′ (z)|2 dxdy = 1 for π Dk every k, consequently m=

ZZ ZZ m X 1 1 |B ′ (z)|2 dxdy = |B ′ (z)|2 dxdy. π π Dk |z|1 true. A trivial consequence for every infinite Blaschke product B, RR is that, RR ′ 2 ′ 2 |B (z)| dxdy = ∞. |B (z)| dxdy = |z|>1 |z| 0. We denote by N (n, l) the set of all multi-indexes α ≡ (α1 , . . . , αn ) ∈ Nn with |α| ≡ α1 + · · · + αn ≤ l. We say that a is a vector of coefficients of order l if a ≡ (aα )α∈N (n,l) is a real valued function on N (n, l). We denote by R(n, l) space of all vectors of coefficients of order l. Then, by ordering N (n, l) on arbitrary way, we identify R(n, l) with a finite dimension real vector space and we endow R(n, l) with the corresponding Euclidean norm | · |. For each a ∈ R(n, l) we denote by

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P P [a](ξ) = P [a](ξ1 , . . . , ξn ) the polynomial α∈N (n,l) aα ξ α and we denote P by Pj [a](ξ) the homogeneous polynomial |α|=j aα ξ α , for all j = 0, . . . , l. Then, we set L[a] ≡ P [a](∂x1 , . . . , ∂xn ), so that L[a] is a partial differential operator with constant coefficients. We say that L[a] is an elliptic operator of order l if Pl [a](ξ) 6= 0 for all ξ ∈ ∂Bn , where Bn ≡ {ξ ∈ Rn : |ξ| < 1} is the unit ball in Rn . We have the following. Theorem 2.1. Let n, k ∈ N, n ≥ 2, k ≥ 1. Let U be a bounded open subset of R(n, 2k) such that L[a] is an elliptic operator of order 2k for all vector of coefficients a in the closure of U. Then, there exist a real analytic function A(·, ·, ·) of U × ∂Bn × R to R and real analytic functions B(·, ·), C(·, ·) of U × Rn to R, such that the following statements hold. (i) There exists a sequence {fj (·, ·)}j∈N of real analytic functions of U × ∂Bn to R such that fj (a, −θ) = (−1)j fj (a, θ),

∀ (a, θ) ∈ U × ∂Bn

and A(a, θ, r) =

∞ X

fj (a, θ)rj ,

j=0

∀ (a, θ, r) ∈ U × ∂Bn × R,

where the series converges absolutely and uniformly in all compact subsets of U × ∂Bn × R. (ii) There exists a family {bα (·) : α ∈ Nn , |α| ≥ 2k − n} of real analytic functions of U to R such that X bα (a)z α , ∀ (a, z) ∈ U × Rn , B(a, z) = |α|≥2k−n

where the series converges absolutely and uniformly in all compact subsets of U × Rn . Furthermore, the function B can be chosen to be identically 0 if n is odd. (iii) The function C can be chosen to be identically 0 if n is odd. (iv) For all a ∈ U the function S(a, ·) of Rn \ {0} to R defined by S(a, z) ≡ |z|2k−n A(a, z/|z|, |z|) + B(a, z) log |z| + C(a, z) for z ∈ Rn \ {0} is a fundamental solution of L[a]. For the proof of Theorem 2.1 we refer to [4]. Here we only remark that the argument of the proof heavily exploits the construction of a fundamental solution given by John in [5]. We also observe that Theorem 2.1 resembles the results obtained by Mantlik [6,7] with much more general assumptions

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on the operator (see also Tr`eves [8].) Nevertheless it is not a corollary. Indeed, by Theorem 2.1 we obtain a suitably detailed expression for the fundamental solution, which cannot be deduced by Mantlik’s results.

3. The Support of the Single Layer Potential Our next step is to introduce the support of the single layer potential To do so we recall some technical facts of Lanza de Cristoforis and Rossi [2,3]. Let n, m ∈ N, n ≥ 2, m ≥ 1. Let 0 < λ < 1. We fix an open connected and bounded subset Ω of the euclidean space Rn , whose exterior Rn \ clΩ is also connected. Moreover we assume that the boundary ∂Ω of Ω is a compact sub-manifold of Rn of H¨ older class C m,λ . Then we introduce the class A∂Ω of the admissible functions on ∂Ω by the following definition. Definition 3.1. A∂Ω is the set of all functions φ ∈ C 1 (∂Ω, Rn ) which are injective and whose differential dφ(x) is injective for all x ∈ ∂Ω. One can verify that A∂Ω ∩C m,λ (∂Ω, Rn ) is an open subset of the Banach space C m,λ (∂Ω, Rn ). Moreover, if φ ∈ A∂Ω ∩ C m,λ (∂Ω, Rn ), then φ(∂Ω) splits Rn into two connected components. We denote by I[φ] the bounded one. Then I[φ] is an open and bounded subset of Rn , connected and with connected exterior, and whose boundary ∂I[φ] = φ(∂Ω) is a compact submanifold of Rn of H¨ older class C m,λ parametrized by the function φ. The set φ(∂Ω) will be the support of our single layer potential.

4. The Single Layer Potential Now we introduce our particular single layer potential. Let n, k ∈ N, n ≥ 2, k ≥ 1. Let U be an open bounded subset of R(n, 2k) such that L[a] is an elliptic operator of order 2k for all a in the closure of U. Then U satisfies the assumptions of Theorem 2.1 and we can consider the corresponding function S(a, ·) for all a ∈ U. Let m ∈ N, m ≥ 1, and λ ∈]0, 1[. We fix an open bounded and connected subset Ω of Rn with connected exterior and with boundary of class C m,λ . If a ∈ U, φ ∈ A∂Ω ∩ C m,λ (∂Ω, Rn ), µ ∈ C m−1,λ (∂Ω) and β ∈ N (n, 2k −1), we denote by vβ [a, φ, µ] the function of Rn to R defined by vβ [a, φ, µ](x) ≡

Z

φ(∂Ω)

(∂zβ S)(a, x − y) µ ◦ φ(−1) (y) dσy ,

∀ x ∈ Rn ,

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where the integral is understood in the sense of singular integrals if |β| = 2k − 1 and x ∈ φ(∂Ω), namely Z vβ [a, φ, µ](x) ≡ lim+ (∂zβ S)(a, x − y) µ ◦ φ(−1) (y) dσy . ǫ→0

φ(∂Ω)\(x+ǫBn )

Clearly, v(0,...,0) [a, φ, µ] is the single layer potential with support φ(∂Ω) and with density µ ◦ φ(−1) of the operator L[a] with respect to the fundamental solution S(a, ·). For |β| ≤ 2k − 2, vβ [a, φ, µ] coincides with the β-derivative of the single layer potential v(0,...,0) [a, φ, µ], namely vβ [a, φ, µ](x) = ∂xβ v(0,...,0) [a, φ, µ](x) for all x ∈ Rn . For |β| = 2k − 1, vβ [a, φ, µ] coincides with the β-derivative of v(0,...,0) [a, φ, µ] on Rn \ φ(∂Ω), but not on φ(∂Ω). Indeed, for |β| = 2k−1, the β derivative of v(0,...,0) [a, φ, µ] displays a jump across the boundary φ(∂Ω) and its value at x ∈ φ(∂Ω) is not defined. Nevertheless, one can verify that the restriction of ∂xβ v(0,...,0) [a, φ, µ] to the domain I[φ] admits a continuous extension to the closure of I[φ]. If we denote by v¯β [a, φ, µ] such an extension, then v¯β [a, φ, µ](x) = J(a, νφ )µ ◦ φ(−1) (x) + vβ [a, φ, µ](x),

∀ x ∈ φ(∂Ω),

where J(·, ·) is a real analytic function of U ×(Rn \{0}) to R and νφ denotes the outer unit normal of I[φ] (cf. [4].) 5. A Real Analyticity Theorem We state in this section our main Theorem 5.1, which is in some sense a natural extension of Theorem 3.23 of Lanza de Cristoforis and Preciso [1], where the Cauchy integral has been considered, of Theorem 3.25 of Lanza de Cristoforis and Rossi [2], where the Laplace operator ∆ has been considered, and of Theorem 3.45 of Lanza de Cristoforis and Rossi [3], where the Helmholtz operator has been considered. Here we consider more general operators, the constant coefficient elliptic operators of order 2k, with k ∈ N \ {0}, which can be factorized with operators of order 2. Let n, k ∈ N, n ≥ 2, k ≥ 1. Let U1 , . . . , Uk be bounded open subsets of R(n, 2) such that L[ai ] is an elliptic operator of order 2 for all ai in the closure of Ui and for all i = 1, . . . , k. One easily verifies that, for each ¯ of k-tuple (a1 , . . . , ak ) of U1 × · · · × Uk , there exists a unique element a R(n, 2k) such that P [¯ a](ξ) = P [a1 ](ξ) · P [a2 ](ξ) · · · · · P [ak ](ξ),

∀ ξ ∈ Rn .

(3)

Moreover, the map of U1 × · · · × Uk to R(n, 2k) which takes (a1 , . . . , ak ) to ¯ is real analytic. So, the set V of all the vectors of coefficients a ¯ ∈ R(n, 2k) a

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which satisfy Eq. (3), for some (a1 , . . . , ak ) ∈ U1 × · · · × Uk , is a bounded ¯ in subset of R(n, 2k) and L[¯ a] is an elliptic operator of order 2k for all a V. By means of a simple topological argument, it follows that there exists a bounded open neighborhood U of V in R(n, 2k) such that L[a] is an elliptic operator of order 2k for all a in the closure of U. Thus, U satisfies the assumption of Theorem 2.1 and we can introduce the corresponding function S(a, ·) of Rn \ {0} to R for all a ∈ U. Now, let m ∈ N \ {0}, λ ∈]0, 1[. Let Ω be a bounded open connected subset of Rn with connected exterior and with boundary of class C m,λ . Let vβ [a, φ, µ] be the function of Rn to R defined by Eq. (3), for all a ∈ U, φ ∈ A∂Ω ∩C m,λ (∂Ω, Rn ), µ ∈ C m−1,λ (∂Ω) and β ∈ N (n, 2k−1). We denote by Vβ [a, φ, µ] the function defined on the boundary of the fixed domain Ω by the following equation, Vβ [a, φ, µ](x) ≡ vβ [a, φ, µ] ◦ φ(x),

∀ x ∈ ∂Ω,

(4)

for all (a, φ, µ) ∈ U × (A∂Ω ∩ C m,λ (∂Ω, Rn )) × C m−1,λ (∂Ω) and for all β ∈ N (n, 2k − 1). Then we have the following real analyticity theorem. Theorem 5.1. The following statements hold. (i) If |β| ≤ 2k−2, then the map of (U1 ×· · ·×Uk )×(A∂Ω ∩C m,λ (∂Ω, Rn ))× C m−1,λ (∂Ω) to C m,λ (∂Ω), which takes (a1 , . . . , ak , φ, µ) to Vβ [¯ a, φ, µ], is real analytic. (ii) If |β| = 2k − 1, then the map of (U1 ×· · ·×Uk )×(A∂Ω ∩C m,λ (∂Ω, Rn ))× C m−1,λ (∂Ω) to C m−1,λ (∂Ω), which takes (a1 , . . . , ak , φ, µ) to Vβ [¯ a, φ, µ], is real analytic. For a proof of Theorem 5.1 we refer to [4]. The main idea stems from the papers of Lanza de Cristoforis and Preciso [1] and Lanza de Cristoforis and Rossi [2,3] and exploits the Implicit Mapping Theorem for analytic functions. Let β be a multi-index with |β| ≤ 2k − 2. Let (a1 , . . . , ak , φ, µ) belong to (U1 × · · · × Uk ) × (A∂Ω ∩ C m,λ (∂Ω, Rn )) × C m−1,λ (∂Ω). We introduce a system of coupled boundary value problems defined in a suitable neighborhood of ∂Ω. Such a system has a unique solution which uniquely identifies the function Vβ [¯ a, φ, µ]. By means of the Implicit Mapping Theorem, we show that the solution of the system depends real analytically on (a1 , . . . , ak , φ, µ). We deduce that Vβ [¯ a, φ, µ] depends real analytically on (a1 , . . . , ak , φ, µ). To prove the statement for |β| = 2k − 1, we exploit the jumping properties of the single layer potential.

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6. Some Applications 6.1. The Bi-Helmholtz Operator We denote by H2 [b1 , b2 ] the operator H2 [b1 , b2 ] ≡ (∆ + b1 ) (∆ + b2 ), where b1 and b2 are real coefficients. We fix two open and bounded subsets B1 and B2 of R. By a straightforward corollary of Theorem 2.1, we introduce a particular fundamental solution SH 2 (b1 , b2 , ·) of H2 [b1 , b2 ], for all (b1 , b2 ) ∈ B1 × B2 . Then, we fix two constants m ∈ N \ {0} and λ ∈]0, 1[. We fix a bounded open connected subset Ω of Rn with connected exterior and with boundary of class C m,λ . We denote by (vH 2 )β [b1 , b2 , φ, µ] the function of Rn to R defined by (vH 2 )β [b1 , b2 , φ, µ](x) Z (∂zβ SH 2 )(b1 , b2 , x − y) µ ◦ φ(−1) (y) dσy , ≡ φ(∂Ω)

∀ x ∈ Rn ,

for all (b1 , b2 , φ, µ) ∈ B1 × B2 × (A∂Ω ∩ C m,λ (∂Ω, Rn )) × C m−1,λ (∂Ω) and for all β ∈ N (n, 3), where the integral is understood in the sense of singular integrals if |β| = 3 and x ∈ φ(∂Ω). Then, we set (VH 2 )β [b1 , b2 , φ, µ](x) ≡ (vH 2 )β [b1 , b2 , φ, µ] ◦ φ(x),

∀ x ∈ ∂Ω,

for all (b1 , b2 , φ, µ) ∈ B1 × B2 × (A∂Ω ∩ C m,λ (∂Ω, Rn )) × C m−1,λ (∂Ω) and for all β ∈ N (n, 3). By Theorem 5.1, we immediately deduce the following. Proposition 6.1. If |β| ≤ 2, the map (VH 2 )β [·, ·, ·, ·] is real analytic from B1 × B2 × (A∂Ω ∩ C m,λ (∂Ω, Rn )) × C m−1,λ (∂Ω) to C m,λ (∂Ω). If |β| = 3, the map (VH 2 )β [·, ·, ·, ·] is real analytic from B1 × B2 × (A∂Ω ∩ C m,λ (∂Ω, Rn )) × C m−1,λ (∂Ω) to C m−1,λ (∂Ω). 6.2. The Lam´ e Equations We denote by L[b1 , b2 ] the vector valued operator L[b1 , b2 ] ≡ ∆ + b1 ∇div + b2 , where b1 and b2 are real coefficients. We fix two open and bounded subsets B1 and B2 of R and we assume that B1 does not contain the point −1. A fundamental solution SL (b1 , b2 , ·) of the operator L[b1 , b2 ] is given by the matrix-valued function defined by (SL (b1 , b2 , z))ij   ≡ δij ∆z +

b2 b1 + 1



2

−

b1 ∂ b1 + 1 ∂zi ∂zj



(5)   SH 2 b2 , b2 /(b1 + 1), z ,

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for all z ∈ Rn \ {0}, and for all i, j = 1, . . . , n, and for all (b1 , b2 ) ∈ B1 × B2 , where δij denotes the Kronecker delta symbol and SH 2 is the fundamental solution of the bi-Helmholtz operator H2 introduced in the previous Subsection 6.1 (cf. Kupradze, Gegelia, Bashele˘ıshvili and Burchuladze [9].) Now, let m ∈ N \ {0} and λ ∈]0, 1[. Let Ω be a bounded open connected subset of Rn with connected exterior and with boundary of class C m,λ . We set (vL )β [b1 , b2 , φ, µ](x) Z ≡ (∂zβ SL )(b1 , b2 , x − y) µ ◦ φ(−1) (y) dσy , φ(∂Ω)

∀ x ∈ Rn ,

where the integral is understood in the sense of singular integrals if |β| = 1 and x ∈ φ(∂Ω), and (VL )β [b1 , b2 , φ, µ](x) ≡ (vL )β [b1 , b2 , φ, µ] ◦ φ(x),

∀ x ∈ ∂Ω,

for all (b1 , b2 , φ, µ) ∈ B1 × B2 × (A∂Ω ∩ C m,λ (∂Ω, Rn )) × C m−1,λ (∂Ω, Rn ) and for all β ∈ N (n, 1). Then, by Theorem 5.1, we immediately have the following. Proposition 6.2. If |β| = 0, the map (VL )β [·, ·, ·, ·] is real analytic from B1 × B2 × (A∂Ω ∩ C m,λ (∂Ω, Rn )) × C m−1,λ (∂Ω, Rn ) to C m,λ (∂Ω, Rn ). If |β| = 1, the map (VL )β [·, ·, ·, ·] is real analytic from B1 × B2 × (A∂Ω ∩ C m,λ (∂Ω, Rn )) × C m−1,λ (∂Ω, Rn ) to C m−1,λ (∂Ω, Rn ). We now consider the double layer potential wL [b1 , b2 , φ, µ] of L[b1 , b2 ]. (i) We denote by SL the vector valued function given by the i-th column of SL , for all i = 1, . . . , n. We denote by T (b, A) the matrix (b − 1)(trA)1n + (A + At ), for all n × n real matrix A and all scalar b ∈ R, where 1n is the n × n unit matrix. We denote by νφ the outward unit normal to φ(∂Ω), for all φ ∈ A∂Ω ∩ C m,λ (∂Ω, Rn ). Then we set wL [b1 , b2 , φ, µ](x) Z  i h
(i) T b1 , Dz SL (b1 , b2 , x − y) νφ (y) · µ ◦ φ(−1) (y) ≡− φ(∂Ω)

dσy

i=1,...,n

for all x ∈ Rn and all (b1 , b2 , φ, µ) ∈ B1 × B2 × (A∂Ω ∩ C m,λ (∂Ω, Rn )) × C m,λ (∂Ω, Rn ), where the integral is understood in the sense of singular integrals if x ∈ φ(∂Ω). We denote by WL [b1 , b2 , φ, µ] the composition wL [b1 , b2 , φ, µ] ◦ φ. Then we have the following real analyticity Proposition 6.3 for WL [b1 , b2 , φ, µ]. We refer to [4] for a proof.

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Proposition 6.3. The map WL [·, ·, ·, ·] of B1 ×B2 ×(A∂Ω ∩C m,λ (∂Ω, Rn ))× C m,λ (∂Ω, Rn ) to C m,λ (∂Ω, Rn ) is real analytic. 6.3. The Stokes System We say that SS ≡ (SV , SP ) is a fundamental solution for the Stokes system in Rn if SV is a real analytic matrix-valued function of Rn \ {0}, SP is a real analytic vector valued function of Rn \ {0} and ∆SV (z) − ∇SP (z) = δ(z)1n ,

div SV (z) = 0,

∀ z ∈ Rn \ {0},

(cf. Ladyzhenskaya [10].) One can verify that a suitable choice of the functions SV and SP is given by the following equalities,   ∂ ∂2 S∆2 (z), (SP )i (z) ≡ − S∆ (z), (6) (SV (z))ij ≡ δij ∆ − ∂zi ∂zj ∂zi

for all z ∈ Rn \{0}, where we understand S∆2 (z) ≡ SH 2 (0, 0, z) and S∆ (z) ≡ ∆S∆2 (z), in accordance with the notation of subsection 6.1. Now, let m ∈ N \ {0} and λ ∈]0, 1[. Let Ω be a bounded open connected subset of Rn with connected exterior and with boundary of class C m,λ . We introduce the single layer potentials vV [φ, µ] and vP [φ, µ] by the equalities Z SV (x − y) µ ◦ φ(−1) (y) dσy , ∀ x ∈ Rn , (7) vV [φ, µ](x) ≡ φ(∂Ω) Z SP (x − y) · µ ◦ φ(−1) (y) dσy , ∀ x ∈ Rn , (8) vP [φ, µ](x) ≡ φ(∂Ω)

for all (φ, µ) ∈ (A∂Ω ∩ C m,λ (∂Ω, Rn )) × C m−1,λ (∂Ω, Rn ), where the integral in equation (8) is understood in the sense of singular integrals if x ∈ φ(∂Ω). As usual, we set VV [φ, µ] ≡ vV [φ, µ] ◦ φ and VP [φ, µ] ≡ vP [φ, µ] ◦ φ. Then, by equation (6) and by Proposition 6.1, we have the following. Proposition 6.4. The maps VV [·, ·] and VP [·, ·] of (A∂Ω ∩C m,λ (∂Ω, Rn ))× C m−1,λ (∂Ω, Rn ) to C m,λ (∂Ω, Rn ) and to C m−1,λ (∂Ω, Rn ), respectively, are real analytic. Now for each scalar b ∈ R and each n× n real matrix A we set T (b, A) ≡ (i) −b1n + (A + At ). Then we denote by SV the vector valued function given by the i-th column of SV for each i = 1, . . . , n. We define the double layer potential wV [φ, µ] by Z h   (i) T (SP )i (x − y), DSV (x − y) wV [φ, µ](x) ≡ − φ(∂Ω)

 i ·νφ (y) · µ ◦ φ(−1) (y)

i=1,...,n

dσy ,

∀ x ∈ Rn ,

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217

for all (φ, µ) ∈ (A∂Ω ∩ C m,λ (∂Ω, Rn )) × C m−1,λ (∂Ω, Rn ). As usual, we set WV [φ, µ] ≡ wV [φ, µ] ◦ φ and we have the following (see [4] for the proof.) Proposition 6.5. The map WV [·, ·] defined from (A∂Ω ∩ C m,λ (∂Ω, Rn )) × C m,λ (∂Ω, Rn ) to C m,λ (∂Ω, Rn ) is real analytic. We can also define the double layer potential wP [φ, µ] for the pressure and we can prove a real analyticity proposition for it similar to Proposition 6.5. To do so, we first need to obtain a suitable expression for wP [φ, µ]. For the sake of brevity, we omit to present here more details. We refer to [4] for a complete analysis. References 1. M. Lanza de Cristoforis and L. Preciso, On the analyticity of the Cauchy integral in Schauder spaces, J. Integral Equations Appl. 11, 363 (1999). 2. M. Lanza de Cristoforis and L. Rossi, Real analytic dependence of simple and double layer potentials upon perturbation of the support and of the density, J. Integral Equations Appl. 16, 137 (2004). 3. M. Lanza de Cristoforis and L. Rossi, Real analytic dependence of simple and double layer potentials for the Helmholtz equation upon perturbation of the support and of the density, to appear (2005). 4. M. Dalla Riva, Potential theoretic methods for the analysis of singularly perturbed problems in linearized elasticity, Doctoral dissertation, Advisor M. Lanza de Cristoforis, Universit` a degli Studi di Padova, 2008. 5. F. John, Plane waves and spherical means applied to partial differential equations, Interscience Publishers, New York-London, 1955. 6. F. Mantlik, Partial differential operators depending analytically on a parameter, Ann. Inst. Fourier (Grenoble) 41, 577 (1991). 7. F. Mantlik, Fundamental solutions for hypoelliptic differential operators depending analytically on a parameter, Trans. Amer. Math. Soc. 334, 245 (1992). 8. F. Tr`eves, Fundamental solutions of linear partial differential equations with constant coefficients depending on parameters, Amer. J. Math. 84, 561 (1962). 9. V. D. Kupradze, T. G. Gegelia, M. O. Bashele˘ıshvili and T. V. Burchuladze, Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity, North-Holland Series in Applied Mathematics and Mechanics, Vol. 25, Russian ed., North-Holland Publishing Co., Amsterdam, 1979, Edited by V. D. Kupradze. 10. O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Gordon and Breach Science Publishers, New York, 1963. Revised English edition, translated from the Russian by R. A. Silverman.

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DIFFERENTIAL PROPERTIES OF (α, Q)-HOMEOMORPHISMS ANATOLY GOLBERG Department of Applied Mathematics, Holon Institute of Technology, 52 Golomb St., P.O.Box 305, Holon 58102, Israel E-mail: [email protected]

Dedicated to Professor Promarz M. Tamrazov on the occasion of his 75th birthday. Quasi-invariance of an n-module is a characteristic property of quasiconformal mappings. We give inequalities that generalize quasi-invariance. Topological mappings satisfying these inequalities are called (α, Q)-homeomorphisms. We establish the ACL property and differentiability a.e. of (α, Q)homeomorphisms, distortions of α-modules and other properties. Keywords: Quasiconformal mappings, Q-homeomorphisms, inner and outer dilatations, α-moduli of families of k-dimensional surfaces.

1. Introduction In this paper, we continue to investigate the classes of mappings with finite integral dilatations, which play an important role in geometric function theory in multi-dimensional spaces. They include the well-known classes of mappings that are quasiconformal, quasiconformal in mean, etc. One of the characteristic properties of quasiconformal mappings is the quasi-invariance of an n-module of families of joining curves (or, equivalently, conformal capacity). It states that the module of a curve family can be changed by a K-quasiconformal mapping only up to a factor at most K. All other properties of K-quasiconformal mappings can be derived from this condition (see, e.g., [9]). In this paper, we give inequalities that generalize quasi-invariance. Our approach involves p-moduli of k-dimensional surfaces. This type of inequalities have been recently considered in [10,11]. The corresponding mappings are called Q-homeomorphisms. The authors stud-

Differential Properties of (α, Q)-Homeomorphisms

219

ied their various properties, like distortion, removability, boundary behavior under some conditions provided that Q belongs to BMO, FMO, etc. In this paper, we introduce more general (α, Q)-homeomorphisms that include all the above classes and investigate their differential and geometric properties: absolute continuity on lines (ACL), belonging to Sobolev’s class 1,1 Wloc , differentiability almost everywhere, distortion of moduli, etc. 2. Quasiconformality in Rn Let A : Rn → Rn be a linear bijection. The numbers HI (A) =

|detA| , ln (A)

HO (A) =

Ln (A) , |detA|

H(A) =

L(A) , l(A)

are called the inner, outer and linear dilatations of A, respectively. Here l(A) = min |Ah|, L(A) = max |Ah|, and detA is the determinant of A (see, |h|=1

|h|=1

e.g., [20]). Obviously, all three dilatations are not less than 1. They have the following geometric interpretation. The image of the unit ball B n under A is an ellipsoid E(A). Let BI (A) and BO (A) be the inscribed and the circumscribed balls of E(A), respectively. Then HI (A) =

mE(A) , mBI (A)

HO (A) =

mBO (A) , mE(A)

and H(A) is the ratio of the greatest and the smallest semi-axis of E(A). Here mA = mn A denotes the n-dimensional Lebesgue measure of a set A. Let λ1 ≥ λ2 ≥ . . . ≥ λn be the semi-axes of E(A). More precisely, the numbers λi are positive square root of the proper values of A∗ A where A∗ is the adjoint of A. Then L(A) = λ1 ,

l(A) = λn ,

|detA| = λ1 · . . . · λn ,

and we can also write HI (A) =

λ1 · . . . · λn−1 , λnn−1

HO (A) =

λ1n−1 , λ2 · . . . · λn

H(A) =

λ1 . λn

If n = 2 then HI (A) = HO (A) = H(A). In the general case, we have the relations: H(A) ≤ min(HI (A), HO (A)) ≤ H n/2 (A)

≤ max(HI (A), HO (A)) ≤ H n−1 (A).

(1)

Let G and G∗ be two bounded domains in Rn , n ≥ 2, and let a mapping f : G → G∗ be differentiable at a point x ∈ G. This means there exists

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a linear mapping f ′ (x) : Rn → Rn , called the (strong) derivative of the mapping f at x, such that f (x + h) = f (x) + f ′ (x)h + ω(x, h)|h|, where ω(x, h) → 0 as h → 0. We denote HI (x, f ) = HI (f ′ (x)),

HO (x, f ) = HO (f ′ (x)),

and L(x, f ) = L(f ′ (x)),

l(x, f ) = l(f ′ (x)),

J(x, f ) = J(f ′ (x)).

Proposition 2.1. Let f : G → G∗ be a K-quasiconformal homeomorphism, 1 ≤ K < ∞. Then (i) f is ACL (absolutely continuous on lines); 1,n (ii) f ∈ Wloc (G) (Sobolev class); (iii) for almost every x ∈ G, HI (x, f ) ≤ K,

HO (x, f ) ≤ K.

3. α-module of the families of k-dimensional surfaces Now we define the quasiconformality of a homeomorphism in other terms (geometric or modular). Let Sk be a family of k-dimensional surfaces S in Rn , 1 ≤ k ≤ n − 1 (curves for k = 1). S is a k-dimensional surface if S : Ds → Rn is a homeomorphic image of the closed domain Ds ⊂ Rk . The p-module of Sk is defined as Z ρp dx, p ≥ k, Mp (Sk ) = inf Rn

where the infimum is taken over all Borel measurable functions ρ ≥ 0 and such that Z ρk dσk ≥ 1 S

for every S ∈ Sk . We call each such ρ an admissible function for Sk . The following proposition characterizes a K-quasiconformality in the terms of n-moduli of Sk , (see, e. g., [17], cf. [1]).

Proposition 3.1. A homeomorphism f of a domain G ⊂ Rn is Kquasiconformal, 1 ≤ K < ∞, if for any family Sk , 1 ≤ k ≤ n − 1, of k-dimensional surfaces in G the double inequality k−n

n−k

K n−1 Mn (Sk ) ≤ Mn (f (Sk )) ≤ K n−1 Mn (Sk ), holds.

(2)

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221

It suffices to use even one of these double inequalities to get the basic properties of quasiconformal mappings. Now we will give an extension of (2) to a class of mappings which is essentially wider than the quasiconformal mappings. For linear bijections A : Rn → Rn , we consider the following quantities HI,α (A) =

|J(A)| , lα (A)

HO,α (A) =

Lα (A) |J(A)|

assuming that α ≥ 1. Note that these dilatations range between 0 and ∞, while the classical dilatations are greater than or equal to 1. In the case α = n, the quantities HI,α and HO,α coincide with the inner and outer dilatations, respectively. Now we consider the homeomorphisms f which are differentiable almost everywhere in G, and fix the real numbers α, β satisfying 1 ≤ α < β < ∞. Define Z Z β α β−α β−α HO,β (x, f ) dx, HOα,β (f ) = (x, f ) dx, HI,α HIα,β (f ) = G

G

where HI,α (x, f ) = HI,α (f ′ (x)), HO,β (x, f ) = HO,β (f ′ (x)). We call these quantities the inner and outer mean dilatations of a mapping f : G → Rn . Define for the fixed real numbers α, β, γ, δ such that 1 ≤ α < β < ∞, 1 ≤ γ < δ < ∞, the class of mappings with finite mean dilatations of such homeomorphisms f : G → G∗ which satisfy (iv) f and f −1 are ACL-homeomorphisms, (v) f and f −1 are differentiable, and their Jacobians J(x, f ) 6= 0 and J(y, f −1 ) 6= 0 a.e. in G and G∗ , respectively, (vi) the inner and the outer mean dilatations HIα,β (f ) and HOγ,δ (f ) are finite. On properties of mappings with finite mean dilatations, we refer to [4]. The relations (1) show that in the classical case of quasiconformal mappings, their inner and outer dilatations are finite or infinity simultaneously. However, this does not be true for the mean dilatations. The following example shows that the unboundedness of one from these dilatations does not depend on the value of another mean dilatation. Example 3.1. Consider the unit cube G = {x = (x1 , . . . , xn ) : 0 < xk < 1, k = 1, . . . , n} and let

  xn1−c , f (x) = x1 , . . . , xn−1 , 1−c

0 < c < 1.

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An easy computation shows that f belongs to the class of mappings with finite mean dilatations if and only if the following inequalities 0 < c < 1 − α/β

and 0 < c < 1 − (γ − 1)δ/(δ − 1)γ

are fulfilled. The cases 1 − α/β ≤ c < 1 and 1 − (γ − 1)δ/(δ − 1)γ ≤ c < 1 imply HIα,β (f ) = ∞ and HOγ,δ (f ) = ∞, respectively. Thus, by suitable choice of the parameters c, α, β, γ, δ, one obtains the desired relations between HIα,β (f ) and HOγ,δ (f ). The class of mappings with finite mean dilatations can be generalized, in turn, so that the conditions (iv)-(v) are preserved, but instead of (vi), one has a more general inequality. Theorem 3.1. Let f : G → G∗ be a homeomorphism satisfying (iv)-(v) with HI,α ∈ L1loc . Then for any family Sk , 1 ≤ k ≤ n − 1, of k-dimensional surfaces in G, and for each admissible ρ for Sk , we have the inequality Z Mα (f (Sk )) ≤ ρα HI,α (x, f )dx. (3) G

The proof goes in the line to the proof of Theorem 2 [4], cf. [13]. 4. (α, Q)-homeomorphisms Let Q : G → [1, ∞] be a measurable function. Due to [10], [11], a homeomorphism f : G → Rn is a Q-homeomorphism if Z M (f (Γ)) ≤ Q(x)ρn (x) dx G

for every family Γ of curves in G and for every admissible function ρ for Γ (see also [15]). Given a function Q : G → [1, ∞], we say that a sense preserving 1,n homeomorphism f : G → Rn is Q(x)-quasiconformal if f ∈ Wloc (G) and max{HI (x, f ), HO (x, f )} ≤ Q(x)

a.e.

Proposition 4.1. ([11]) Let f : G → Rn be a Q(x)-quasiconformal mapping. Then (a) f is differentiable a.e., (b) f satisfies N -property, (c) J(x, f ) ≥ 0 a.e. 1,n n−1 If, in addition, Q ∈ Lloc , then f −1 ∈ Wloc (G∗ ), and −1 (d) f is differentiable a.e.,

Differential Properties of (α, Q)-Homeomorphisms

223

(e) f −1 satisfies N -property, (f ) J(x, f ) > 0 a.e. The proof of Proposition 4.1 is based on the results of [14], [5] and [12]. Let Q : G → [0, ∞] be a measurable function. For a given integer k, 1 ≤ k ≤ n − 1, a homeomorphism f : G → Rn is called (α, Q)homeomorphism, if Mα (f (Sk )) ≤

Z

Q(x)ρα (x) dx

(4)

G

for every family of k-dimensional surfaces Sk in G and for every admissible function ρ for Sk . This inequality is a natural generalization of the right-hand side in (2). Note also that the integral in (4) can be interpreted as a weighted module (cf. [1], [19]). We now restrict ourselves by two cases: k = 1 and k = n − 1, which correspond to curve families and (n − 1)-dimensional surface families, respectively, and for similarity adapt them to ring domains. A ring domain D ⊂ Rn is a finite domain whose complement consists of two components C0 and C1 . Setting F0 = ∂C0 and F1 = ∂C1 . We obtain two boundary components of D. For definiteness, let us assume that ∞ ∈ C1 . We say that a curve γ joins the boundary components in D if γ lies in D, except for its endpoints, one of which lies on F0 and the second on F1 . A compact set Σ is said to separate the boundary components of D if Σ ⊂ D and if C0 and C1 are located in different components of CΣ. Denote by ΓD the family of all locally rectifiable curves γ which join the boundary components of D and by ΣD the family of all compact piecewise smooth (n − 1)-dimensional surfaces Σ which separate the boundary components of D. For each quantity V associated with D such as subset of D or a family of sets contained in D, we let V ∗ denote its image under f . We consider also certain set functions. Let Φ be a finite nonnegative function in G defined for open subsets E of G satisfying m X

Φ(Ek ) 6 Φ(E)

k=1

for any finite collection {Ek }m k=1 of nonintersecting open sets Ek ⊂ E. We denote the class of all such set functions Φ by F. The upper and lower derivatives of a set function Φ ∈ F at a point

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A. Golberg

x ∈ G are defined by Φ′ (x) = lim sup

h→0 d(Q) n − 1, we have the low estimate  n−α−1  n−1 α dα (C0 ) α−1 n−1 n−1 (5) ωn−2 Mα (ΓD ) ≥
α−n+1 , α−n+1 mA

which is given in [7] for the α-capacities of condensers; here ωn−2 denotes the (n − 2)-dimensional Lebesgue measure of the unit sphere S n−2 . The equivalence between the α-module of ΓD and the α-capacity of the condenser (C0 , A) was obtained in [6]. Now the proof of Theorem 4.1 will be accomplished in two steps following the construction applied in [9] (see, also [7]). First, let us show that f is ACL. Denote Z Q(x) dx. (6) Θ(V ) = mV ∗ and Ψ(V ) = V

It is easy to verify that Θ, Ψ ∈ F. Let Q be an open n-dimensional interval (parallelepiped) in G. Write Q = Q0 × J, where Q0 is (n − 1)-dimensional interval in Rn−1 and J is an open segment of xn -axis. Using the notation of [12], we have Θ(T, Q) = Θ(T × J) and Ψ(T, Q) = Ψ(T × J). The functions Θ(T, Q) and Ψ(T, Q) also belong to the class F for Borel set T ⊂ Q0 . ′ ′ Fix z ∈ Q0 such that Θ (z, Q) < ∞ and Ψ (z, Q) < ∞. Let ∆1 , . . . , ∆k be the disjoint closed subintervals of the segment Jz = {z} × J. Set C0,i = n n ∆i + rB and Ai = ∆i + 2rB n , where B is the closure of B n = B n (x, 1).

Differential Properties of (α, Q)-Homeomorphisms

225

The positive number r is chosen so that the domains Di are disjoint and Di ⊂ Q. Now from (4) and the admissibility of ρ = 1/dist(F0,i , F1,i ) for ΓDi one obtains Z 1 Q(x)dx, (7) Mα (Γ∗Di ) ≤ α r Ai where dist(F0,i , F1,i ) denotes the distance between F0,i and F1,i . Using (5), we have that Mα (Γ∗Di ) is estimated by
α ∗ d(C0,i ) n−1 1 ∗ Mα (ΓDi ) ≥ C n−1 (8)
α−n+1 . mA∗i n−1 Substituting (8) into (7) together with the notations from (6) gives   n−1   α−n+1 α α ∗ Ψ(Ai ) r1−n , d(C0,i ) ≤ C1 Θ(Ai )

where C1 depends only on α and n. We summarize such inequalities for 1 ≤ i ≤ k and apply H¨ older’s inequality. Then we obtain  n1 X X  n−1 k k k n X ∗ Θ(Ai ) d(C0,i ) ≤ C1 Ψ(Ai ) r1−n . i=1

i=1

i=1

Applying the inequalities k X i=1

Θ(Ai ) ≤ Θ(B n−1 (z, r), Q),

k X i=1

Ψ(Ai ) ≤ Ψ(B n−1 (z, r), Q),

and letting r → 0, we get the estimate   α−n+1  n−1  k α α X ′ ′ ∗ d(C0,i ) ≤ C2 Θ (z, Q) Ψ (z, Q) , i=1

where the constant C2 depends only on q and n. Thus f is ACL. 1,1 Let us now show that a given homeomorphism f belongs to Wloc . For x ˜, x ∈ G, x˜ 6= x, we get k(x) = lim sup x ˜→x

|f (˜ x) − f (x)| . |˜ x − x|

Consider for a point x ∈ G the spherical ring Dr (x) = {y : r < |x−y| < 2r}, choosing r > 0 so that Dr (x) ⊂ G. Estimating the integral in the right-hand side of (4) we have Z 1 ∗ Mα (ΓDr ) ≤ α Q(x)dx. (9) r Ai

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A. Golberg

Substituting (5) and (6) into (9) yields
n−1
n
α−n+1 ∗ Ψ(Ar ) α r1−n , d(C0,r ) ≤ C3 Θ(Ar ) α

where C3 depends only on q, n and K. Since k(x) ≤ lim sup r→0

∗
α−n+1
n−1 d(C0,r ) ′ ′ α Ψ (x) α , ≤ C4 Θ (x) r

it follows that k(x) < ∞ for a.e. x ∈ G. Here C4 also depends only on n and α. Applying now the Stepanov theorem [18], we conclude that f is differentiable a.e. in G. Moreover, for an arbitrary Borel set V ⊂ G, using H¨ older’s inequality we have Z
n−1
α−n+1 Ψ(V ) α < ∞. k(x) dx ≤ C4 Θ(V ) α V

In the case of Q-homeomorphisms a similar result was obtained in [16]. The following result shows that dilatation HI,α R is dominated in G by the upper derivative of the set function Ψ(V ) = V Q(x) dx; here V is an open subset of G. Theorem 4.2. Let f : G → G∗ be an (α, Q)-homeomorphism with Q ∈ L1loc (G) and α > n − 1. Then HI,α ∈ L1loc (G). Sketch of the proof. Let a ∈ G be an arbitrary point at which f is differentiable, and with J(a, f ) 6= 0. Preceding f , if necessary, by a rotation and a reflection, one reduces the proof to the case when f (a) = a = 0 and |f ′ (0)ei | = λi , i = 1, . . . , n; here eν denotes the νth unit basis vector. Let D be a ring domain obtained from n-dimensional interval {x : |xi | < r(tλi + 1), i = 1, . . . , n − 1, |xn | < rtλn }, t > 0, by deleting the points of (n − 1)-dimensional interval Qn−1 (0, r) = {x : |xi | ≤ r, i = 1, . . . , n − 1, xn = 0}. We choose r > 0 so that D ⊂ G and will show that λ1 · . . . · λn ′ ≤ Ψ (0). α λn ′

Using that a is arbitrary, one can conclude that HI,α (x, f ) ≤ Ψ (x) for almost all x ∈ G. Applying (ix) completes the proof of Theorem 4.2. There are the well-known distortion estimates of the module of planar ring domains under quasiconformal mappings obtained by Gr¨ otzsch and

Differential Properties of (α, Q)-Homeomorphisms

227

Belinskii (see, [2], [8]). One result of such type in multidimensional case is due to [3]. The following theorem presents the inequalities of Gr¨ otzsch and Belinskii types for homeomorphisms with locally integrable dilatation HI,α and more general for (α, Q)-homeomorphisms. Theorem 4.3. Let f : Rn → Rn , f (0) = 0, n ≥ 2, be a homeomorphism with HI,α ∈ L1loc . Then for every spherical ring R(a, b) = {x ∈ Rn : 0 < a < |x| < b < ∞} the double inequality 

n−α bn−α − an−α

α Z  n−1

R(a,b)

α HI, α−n+1 (x, f ) dx

|x|α

α 1−n

≤ Mα (f (ΣR )) ≤ ωn−1

Z

 n−1−α n−1

R(a,b)

HI,α (x, f ) dx |x|α

holds. The proof of Theorem 4.3 is based on applying Theorem 3.1 for suitable admissible functions ρ and on using the relationship between Mp (f (ΓR )) and Mα (f (ΣR )) 1−p Mp (ΓR ) = M p(n−1) (ΣR ) p−1

found in [21]. References 1. C. Andreian Cazacu, Some formulae on the extremal length in n-dimensional case, Proceedings of the Romanian-Finnish Seminar on Teichm¨ uller Spaces and Quasiconformal Mappings (Bra¸sov, 1969), 87–102. Publ. House of the Acad. of the Socialist Republic of Romania, Bucharest, 1971. 2. P. P. Belinskii, General properties of quasiconformal mappings, Nauka, Novosibirsk, 1974 (Russian). 3. C. Bishop, V. Y. Gutlyanski˘ı, O. Martio, M. Vuorinen, On conformal dilatation in space, Int. J. Math. Math. Sci. 22 (2003), 1397–1420. 4. A. Golberg, Homeomorphism with finite mean dilatations, Contemp. Math. 382 (2005), 177–186. 5. J. Heinonen, P. Koskela, Sobolev mappings with integrable dilatation, Arch. Rational Mech. Anal. 125 (1993), 81–97. 6. J. Hesse, A p-extremal length and p-capacity equality, Ark. Mat. 13 (1975), 131–144. 7. V. I. Kruglikov, Capacities of condensors and quasiconformal in the mean mappings in space, Mat. Sb. 130 (1986), no. 2, 185–206.

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A. Golberg

8. S. L. Krushkal, Quasiconformal mappings and Riemann surfaces, V. H. Winston & Sons, Washington, D.C.; John Wiley & Sons, New York-Toronto, Ont.-London, 1979. 9. O. Martio, S. Rickman, and J. V¨ ais¨ al¨ a, Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I 448 (1969), 1–40. 10. O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, Q-homeomorphisms, Contemp. Math., 364 (2004), 193–203. 11. O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, On Q-homeomorphisms, Ann. Acad. Sci. Fenn. Math. 30 (2005), no. 1, 49–69. 12. T. Rado, P. Reichelderfer, Continuous Transformations in Analysis, Springer-Verlag, 1955. 13. K. Rajala, Surface families and boundary behavior of quasiregular mappings, Illinois J. Math. 49 (2005), no. 4, 1145–1153. 14. Y. G. Reshetnyak, Space Mappings with Bounded Distortion, Transl. Math. Monogr., Amer. Math. Soc., vol. 73, 1989. 15. V. Ryazanov, R. Salimov, Weakly flat spaces and boundaries in the mapping theory, Ukrainian Math. Bull., 4 (2007), no. 2, 199–234. 16. R. Salimov, Absolute continuity and differentiability of Q-homeomorphisms, Ann. Acad. Sci. Fenn. Math., 2007, to appear. 17. B. V. Shabat, The modulus method in space, Soviet Math. Dokl. 130 (1960), no. 6, 1210–1213. 18. V. V. Stepanov, Sur les conditions de l’existence de la differentielle totale, Mat. Sb. 30 (1924), 487–489. 19. P. M. Tamrazov, Moduli and extremal metrics in nonorientable and twisted Riemannian manifolds, Ukrainian Math. J. 50 (1998), no. 10, 1586–1598. 20. J. V¨ ais¨ al¨ a, Lectures on n-dimensional Quasiconformal Mappings, SpringerVerlag, 1971. 21. W. P. Ziemer, Extremal length and p-capacity, Michigan Math. J. 16 (1969), 43–51.

229

ON CHARACTERIZATION OF THE EXTENSION PROPERTY A. P. GONCHAROV Department of Mathematics, Bilkent University, Ankara, 06800, Turkey E-mail: [email protected] The geometric characterization of the extension property for Cantor-type sets, found in [3], is related to the rate of growth of the values of the discrete logarithmic energies of compact sets that locally form the set. Keywords: Extension property, Cantor-type sets, discrete logarithmic energies.

1. Extension Problem Let K be a compact set in Rd . Then E(K) is the space of Whitney jets on K, that is the space of traces on K of C ∞ functions. Topology in the space E(K) can be given by the system of seminorms || f || q = inf | F | q , q ∈ N,

where the infimum is taken for all possible extensions of f to F and | F | q denotes the qth norm of F in C ∞ (Rd ). The Extension Problem is to characterize when there exists a linear continuous extension operator L : E(K) −→ C ∞ (Rd ). We say that the compact set K has the extension property if there exists a such operator. Tidten in [6] applied Vogt’s condition for a splitting of exact sequences of Fr´echet spaces and gave a topological characterization of the extension property: a compact set K has the extension property if and only if the space E(K) has a dominating norm (see for instance [2] for the definition of a dominating norm and for a recent account of the theory). Nevertheless, the problem of a geometric characterization of the extension property that goes back to the work [5] of Mityagin, is still open even for the one-dimensional case, in spite of the presence of numerous particular results. Here we consider the geometric characterization of the extension property for Cantor-type sets found in [3].

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A. P. Goncharov

2. Cantor-type Sets Given l1 with 0 < l1 < 1/2 and a sequence (αs )∞ s=2 with αs > 1 let us define α2 the sequence (ls )∞ in the following way: l 0 = 1, l1 , l2 = l1 , · · · , ls = s=0 α2 α3 ···αs (α n ) , · · · . Then by K we denote the symmetric Cantor-type set l1 T ∞ S 2s s=0 j=1 I j, s , where |I j, s | = l s for all j. Here the closed intervals I j, s we call basic intervals. Let x be an endpoint of some basic interval. Then there exists the minimal number q ( the type of x) such that x is the endpoint of some Ij, m for every m ≥ q. As in [3] we suppose that αs ≥ 1 + ε0 , s ≥ s0 for some positive ε0 and ls ≥ 4 ls+1 for all s. Let hs = ls − 2ls+1 be the gap between two adjacent intervals. We follow the notations used in [3]: πn, 0 = 1, and for n ≥ 1, s ≥ 1, we let s X πn, s = 2−s αn+1 αn+2 · · · αn+s , σn, s = πn, k . k=0

Theorem [3]. The following are equivalent: (i) The set K (αn ) has the extension property. s 2s−1 M · · · ln+s , ∀n ∀s ≥ sM . > ln2 ln+1 (ii) ∀M > 0 ∃ sM : ln+s (iii) σn, s+1 / σn, s ⇒ 1, as s → ∞ uniformly with respect to n. We see that the condition (ii) is purely geometrical, whereas the condition (iii) is related to the theory of logarithmic potential. In what follows log denotes the natural logarithm. 3. Discrete Logarithmic Energies Let K be a compact set in C and for given N points z1 , · · · , zN ⊂ K let µN = µN (z1 , · · · , zN ) denote the discrete measure that associates the mass 1/N to any point zk , 1 ≤ k ≤ N. The logarithmic potential of measure µN is given by U µN (z) =

N 1 1 X . log N |z − zk | k=1

Any discrete measure has infinite energy, but if we use the truncated kernels (see e.g.[1]), then we can define the corresponding logarithmic energy as in [4]: I(µN ) =

1 N2

N X

k,j=1,k6=j

log

1 . |zj − zk |

On Characterization of the Extension Property

231

Clearly, I(µN ) =

−2 log | V (z1 , · · · , zN )|, N2

where for the corresponding Vandermonde determinant we have Y V (z1 , · · · , zN ) = (zk − zj ). 1≤j (ln+s h2n+s−1 · · · h2n )2 = λ2 · a

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A. P. Goncharov

with log a = 2s (2 log

hn+s−1 ln+s−1

+ · · · + 2s log

> 21 . From this log hlkk > 1 − 2 lk+1 lk P n+s−1 n+s−k+2 ε0 −2s k=n 2 lk > −22s+3 lnε0 ,

−4

hn ln ).

lk+1 lk

By condition,

≥

−4 lkε0

hk lk

=

and log a >

which completes the proof.

Corollary 3.1. 1 1 1 1 < I(µN ) < σn, s log + 4 lnε0 . σn, s log 2 ln 2 ln

(1)

Theorem 3.1. If the set K (αn ) has the extension property then In (µ2s+1 )/In (µ2s ) → 1, as s → ∞, n → ∞. Here In (µ2s+1 ) stands for the discrete logarithmic energy defined by all endpoints of the type ≤ s + n on any basic interval of the length l n . Proof. Write γn = 8 lnε0 log−1 ln−1 . From (1) we have In (µ2s+1 ) σn, s σn, s + γn < + γn , < In (µ2s ) σn, s−1 σn, s−1 since σn, s−1 > 1. Now the result follows on the condition (iii) and decrease of the sequence γn . One can conjecture that the existence of a linear continuous extension operator for the space E(K) (at least for Cantor-type sets) is characterized by a regularity of growth of the minimal discrete logarithmic energies corresponding to compact sets that locally form the set K. The points (zk )N k=1 considered in Lemma give rather rough approximation of the minimal energy for the set K (αs ) ∩ [0, ln ]. The exact position of the Fekete points is not known even for rarefied Cantor-type sets. References 1. V. V. Andrievskii, H. P. Blatt, Discrepancy of Signed Measures and Polynomial Approximation, Springer, 2002. 2. L. Frerick, Extension operators for spaces of infinite differentiable Whitney jets, J. Reine Angew. Math. 602 (2007), 123–154. 3. A. Goncharov, On the geometric characterization of the extension property, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), 513–520. 4. J. Korevaar, Fekete extreme points and related problems, in Approximation Theory and Function Series, Budapest, 1996, 35–62. 5. B. S. Mitiagin, Approximative dimension and bases in nuclear spaces, Russian Math. Surveys 16 (1961), 59–127. 6. M. Tidten, Fortsetzungen von C ∞ -Funktionen, welche auf einer abgeschlossenen Menge in Rn definiert sind, Manuscripta Math. 27 (1979), 291–312.

233

ON SOME BOUNDARY PROPERTIES OF CONFORMAL MAPPING OLENA W. KARUPU Department of Higher and Numerical Mathematics, National Aviation University, 1 Komarov ave, Kyiv, Ukraine E-mail: [email protected] We consider some new estimates for the uniform curvilinear moduli of smoothness of arbitrary order of the function realizing a conformal mapping of the unit disk and a domain bounded by a smooth Jordan curve. Keywords: Conformal mapping; modulus of smoothness; finite difference smoothnesses.

1. Introduction Let G be a simply connected domain in the complex plane bounded by a smooth Jordan curve Γ, τ = τ (s) be the angle between the tangent to Γ and the positive real axis, s = s(w) be the arc length on Γ. Let w = ϕ(z) be a homeomorphism of the closed unit disk D = {z : |z| ≤ 1} onto the closure G of the domain G, conformal in the open unit disk D. Let z = ψ(w) be the function inverse to the function w = ϕ(z). The aim of this paper is to receive information about connection between the function τ = τ (s) and the functions w = ϕ(z) and z = ψ(w) in terms of the uniform curvilinear moduli of smoothness of arbitrary order. Kellogg in 1912 proved the theorem in which it had been established that if τ = τ (s) satisfies Holder condition with index α, 0 < α < 1, then the derivative ϕ′ (eiθ ) of the function ϕ(z) on ∂D satisfies Holder condition with the same index α. Afterwards this result was generalized in works by several authors: S. E. Warshawski, J. L. Geronimus, S. J. Alper, R. N. Kovalchuk, L. I. Kolesnik. P. M. Tamrazov [1] obtained solid reinforcement for the modulus of continuity for the derivative ϕ′ (z) of the function ϕ(z) on D. E. P. Dolzenko [2–4] received estimates for the modulus of continuity for the function re-

234

O. W. Karupu

alizing conformal mapping of Jordan domains in the case when boundaries of domains were characterized by moduli of oscilation or moduli of rectifiability. P. M. Tamrazov [5] solved the problem of estimating of finite difference smoothnesses for composite function. These results gave posibility to receive generalizations and inversations of Kellogg’s type theorems for general moduli of smoothness of arbitrary order. In particular, results in the terms of the arithmetic and uniform curvilinear moduli of smoothness of arbitrary order were received by author in [6–8] (for more details, see [1,6,9]). 2. Estimates for moduli of smoothness for the derivative of a function realizing a conformal mapping of the unit disk onto a Jordan domain Let the finite function w = f (z) be defined on a curve γ ⊂ C. Let (z0 , ..., zk ) be the collection of the points on the curve γ and [z0 , ..., zk ; f, z0 ] be the finite difference of order k for the function w = f (z). P. M. Tamrazov [1] defined on rectifiable curves the uniform curvilinear moduli of smoothness of order k for the function w = f (z) as ω e k,N,γ ((z), δ)p = sup sup |[z0 , ..., zk ; f, z0 ]| , where γw,δ (N ) is the set of collecw∈γ (z0 ,...,zk )∈γw,δ (N )

tions (z0 , ..., zk ) such that curvilinear (with respect to the curve γ) distances between points z0 , ..., zk ∈ γ satisfy the condition ρ (zi , zi+1 )/ρ (zj , zj+1 ) ≤ N , (N ∈ [1, ∞)), and ρ (zi , w) ≤ δ (i, j = 1, ..., k). In partial case when the function w = f (z) is defined on the real axis and points x0 , ..., xk form the arithmetic progression we receive the real arithmetic moduli of smoothness ωk (f (x), δ). The following results for the arithmetic and for the uniform curvilinear moduli of smoothness of arbitrary order for the the derivative ϕ′ (z) of the function ϕ(z) on ∂D generalizing Kellogg’s theorem were earlier obtained by the author [6]. Theorem 2.1. ([6]) Let modulus of smoothness ωk (τ (s), δ) of order k (k ∈ N ) for the function τ (s) satisfy the condition ωk (τ (s), δ) = O [ω (δ)] (δ → 0), where ω(δ) is normal majorant satisfying the condition Z l ω (t) dt < +∞ (1) t 0 Then the nonzero continuous on D derivative ϕ′ (z) of the function ϕ(z)

On Some Boundary Properties of Conformal Mapping

235

exists satisfying on ∂D the conditions ωk (arg ϕ′ (eiθ ), δ) = O (µ (δ)) (δ → 0) ,

ωk (log ϕ′ (eiθ ), δ) = O (µ∗ (δ)) (δ → 0) ,

ωk (ϕ′ (eiθ ), δ) = O (µ∗ (δ)) (δ → 0) ,

where µ (δ) = ωk (τ (s), δ) +

j−1 k−1 XX

...

rj−1 −1

Z

0

l

...

Z

0

∗

µ (δ) =

Z

0

+

l

k−r1 k

ri−1 −ri Qj k [ωk (τ (s) , xj )] i=2 [ωk (τ (s) , xi−1 )]    rp  dx1 ...dxj , Qj xp/ p=1 xp 1 + xp−1 rj k

l

[ωk (τ (s) , δ)]

rj =1

j=1 r1 =1

×

X

ω (x1 )  dx1  k x1 1 + (x1/δ )

j−1 k−1 XX

j=1 r1 =1

...

rj−1 −1

X

rj =1

δ k−r1

Z

0

l

...

Z

l 0

rj−1 xj+1

1+

Z

l

ω(y)

   j r Z l xj+1 j+11 Y ω (tp ) × 1+ 1+ dtp r p−1 −rp +1 xj x p tp p=1 rp−1 −1   xp × 1+ xrpp−1 −rp −1 dx1 ...dxj+1 . xp−1

r +1 dy

yj j !

xj+1

!

Corollary 2.1. If modulus of smoothness ωk (τ (s), δ) of order k for the function τ (s) satisfies Holder condition ωk (τ (s), δ) = O (δ α ) (δ → 0), 0 < α < k, then the modulus of smoothness ωk (ϕ′ (eiθ ), δ) of the same order k for the derivative ϕ′ (eiθ ) of the function ϕ(z) on ∂D satisfies the condition ωk (ϕ′ (eiθ ), δ) = O (δ α ) (δ → 0) with the same index α. Theorem 2.2. ([6]) Let modulus of smoothness ωk (τ (s), δ) of order k (k ∈ N ) for the function τ (s) satisfy the condition ωk (τ (s), δ) = O [ω (δ)] (δ → 0), where ω(δ) is normal majorant satisfying the condition (1). Then the uniform curvilinear modulus of smoothness ω e k,1,∂D (ϕ′ , δ) of ′ the same order k for the derivative ϕ (z) of the function ϕ(z) on ∂D satisfies the conditions ω ek,1,∂D (arg ϕ′ , δ) = O (e µ (δ)) (δ → 0) ,

ω ek,1,∂D (log ϕ′ , δ) = O (e µ∗ (δ)) (δ → 0) ,

ω ek,1,∂D (ϕ′ , δ) = O (e µ∗ (δ)) (δ → 0) ,

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O. W. Karupu

where µ e (δ) = µ (δ) + δ

k

Z

δ

l

µ (t) dt, tk+1

∗

∗

µ e (δ) = µ (δ) + δ

k

Z

δ

l

µ∗ (t) dt. tk+1

Corollary 2.2. If modulus of smoothness ωk (τ (s), δ) of order k for the function τ (s) satisfies Holder condition ωk (τ (s), δ) = O (δ α ) (δ → 0), 0 < α < k, then the modulus of smoothness ω e k,1,∂D (ϕ′ , δ) of the same order ′ k for the derivative ϕ (z) of the function ϕ(z) on ∂D satisfies the condition ω ek,1,∂D (ϕ′ , δ) = O (δ α ) (δ → 0) with the same index α. 3. Estimates for moduli of smoothness for a function realizing a conformal mapping of the unit disk onto a Jordan domain

Let us consider estimates for the modulus of smoothness for the function ϕ(z). Theorem 3.1. Let modulus of smoothness ωk (τ (s), δ) of order k for the function τ (s) satisfy the condition ωk (τ (s), δ) = O [ω (δ)] (δ → 0), where ω(δ) is normal majorant satisfying the condition [1]. Then the uniform curvilinear modulus of smoothness ω e k,1,∂D (ϕ, δ) of the function ϕ(z) on ∂D satisfies the condition   Z 2π µ ∗ (t) dt (δ → 0) . (2) ω ek,1,∂D (ϕ, δ) = O δ k t δ


Proof. Let us introduce the function χ of real variable θ as χ (θ) = ϕ eiθ . Then on ∂D the identity ϕ(z) = χ ◦ θ (z) takes place. Applying inequalities for finite difference smoothnesses of composite function obtained by P. M. Tamrazov ([5], p. 17), we receive the estimates of the modulus of smoothness ω ek,1,∂D (ϕ, δ) = ω ek,1,∂D (ϕ ◦ ϑ, δ) via the moduli of smoothness ωj (χ, δ), j = 1, ..., k. We have ω ek,1,∂D (ϕ, δ) = ω ek,1,∂D (ϕ ◦ ϑ, δ) ≤ ωk (χ, δ) + c1

k−1 X

ωj (χ, δ)δ k−j ,

j=1

where constant c1 does not depend on δ. We will also use the Marchaud inequalities:   Z 2π ωk (χ, t) j ωj (χ, δ) ≤ δ c2 + c3 dt , j = 1, ..., k − 1, tj+1 δ

(3)

On Some Boundary Properties of Conformal Mapping

237

where the constants c2 and c3 do not depend on δ. So, as the estimates Z 2π Z 2π ωk (χ, t) ωk (χ, t) dt ≤ aj dt, j = 1, ..., k − 1, j+1 t tk δ δ where constants a1 , ..., ak−1 do not depend on δ, are evident, then we obtain the estimates R 2π dt ω ek,1,∂D (ϕ, δ) ≤ ωk (χ, δ) + c4 δ k + c5 δ k δ ωk t(χ,t) k R (4) k 2π ωk (χ,t) ≤ c6 ωk (χ, δ) + c7 δ δ dt, tk

where constants c4 , c5 , c6 and c7 do not depend on δ.
dϕ(eiθ ) But χ′ (θ) = dθ = ieiθ ϕ eiθ , so

ωn (χ′ , δ) = ωn (ieiθ ϕ′ (eiθ ), δ), n = 1, 2, ...

We have to estimate the modulus of smoothness for the the product of functions ieiθ and ϕ′ (eiθ ). As k    ′ iθ
  iθ ′ iθ
X k k ϕ e ∆jh ieiθ ∆k−j = ∆h ie · ϕ e h j j=1

and ωr (ieiθ ), δ) ≤ br δ r , r = 0, ..., k, where constants b0 , ..., bk do not depend on δ, then ωn (χ′ , δ) ≤ c8

n X j=0


ωj (ϕ′ eiθ , δ)δ k−j , n = 1, 2, ...,

where constant c8 does not depend on δ. For the real arithmetic moduli of smoothness of the function χ(θ) the Marchaud inequalities take the form ωn (χ, δ) ≤ δωn−1 (χ′ , δ), n = 1, 2, ... So ωk (χ, δ) ≤ c9 δ

k−1 X j=0

k−1 X

ωj (ϕ′ eiθ , δ)δ k−j , ωj (ϕ′ eiθ , δ)δ k−1−j = c9 j=0

where constant c9 does not depend on δ. Taking into account the Marchaud inequalities (3) we will rewrite last estimate as
Z 2π ωk (ϕ′ eiθ , t) k ωk (χ, δ) ≤ c10 δ dt, (5) tk δ

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where constant c10 does not depend on δ. But by Theorem 2.1 the estimate ωk (ϕ′ (eiθ ), δ) ≤ c11 µ ∗ (δ) ,

(6)

where constant c11 does not depend on δ, is true. So, from (5) and (6) we have Z 2π
µ ∗ (t) ωk (ϕ eiθ , δ) = ωk (χ, δ) ≤ c12 δ k dt, (7) tk δ where constant c12 does not depend on δ. Then we obtain Z 2π Z 2π ∗ Z 2π µ (y) ωk (χ, t) dt ≤ c13 dy dt k t yk δ t δ Z 2π ∗ Z 2π ∗ µ (y) µ (y) ≤ c14 dy ≤ c15 dy, k−1 y yk t t

(8)

where constants c13 , c14 and c15 do not depend on δ. Let us substitute the estimates (7) and (8) to the estimate (4) and obtain Z 2π ∗ µ (t) k dt, ω ek,1,∂D (ϕ, δ) ≤ c16 δ t δ where constant c16 does not depend on δ. Theorem 3.1 is proved.

Corollary 3.1. If modulus of smoothness ωk (τ (s), δ) of order k for the function τ (s) satisfies Holder condition ωk (τ (s), δ) = O (δ α ) (δ → 0) , 0 < α < k, then the uniform curvilinear modulus of smoothness ω e k,1,∂D (ϕ, δ) of the function ϕ(z) on ∂D satisfies the condition
 α+1 (δ
→ 0) if 0 < α < k − 1, O δ 1 ω ek,1,∂D (ϕ, δ) = α = k − 1, O δ k log δ (δ → 0) if
 O δ k (δ → 0) if k − 1 < α < k.

4. Estimates for moduli of smoothness for a function realizing a conformal mapping of a Jordan domain onto the unit disk

The following result for the uniform curvilinear modulus of smoothness of arbitrary order for the the derivative ψ ′ (w) of the function ψ(w) on the curve Γ was earlier obtained by author [6]. Theorem 4.1. ([6]) Let modulus of smoothness ωk (τ (s), δ) of order k for the function τ (s) satisfy the condition ωk (τ (s), δ) = O [ω (δ)] (δ → 0), where ω(δ) is normal majorant satisfying the condition (1).

On Some Boundary Properties of Conformal Mapping

239

Then the nonzero continuous on G derivative ψ ′ (w) of the function ψ(w) exists satisfying on Γ the conditions ω ek,1,Γ (arg ψ ′ , δ) = O (e η (δ)) (δ → 0) , where

ω ek,1,Γ (log ψ ′ , δ) = O (e η ∗ (δ)) (δ → 0) ,

ω ek,1,Γ (ψ ′ , δ) = O (e η ∗ (δ)) (δ → 0) ,

ηe (δ) = µ e (δ) + δ 1−k(k−1)/2

Z

ηe∗ (δ) = µ e∗ (δ) + δ 1−k(k−1)/2

l

δ

Z

µ e (y) dy δ k y k+1

l δ

Z

l

µ (t) dt tk

δ

µ e∗ (y) dy δ k y k+1

Z

δ

l

! k(k+1)/2−1 k

µ e (t) dt tk

,

! k(k+1)/2−1 k

.

Proof of this theorem is similar to the proof of Theorem 3.1. Corollary 4.1. If modulus of smoothness ωk (τ (s), δ) of order k for the function τ (s) satisfies Holder condition ωk (τ (s), δ) = O (δ α ) (δ → 0) , 0 < α < k, then the uniform curvilinear modulus of smoothness ω e k,1,Γ (ψ ′ , δ) of the ′ derivative ψ (w) of the function ψ(w) satisfies the condition ω e k,1,Γ (ψ ′ , δ) = O(δ α )(δ → 0) with the same index α. Let us consider estimates for the modulus of smoothness for the function ψ(w).

Theorem 4.2. Let modulus of smoothness ωk (τ (s), δ) of order k(k ∈ N ) for the function τ (s) satisfy Holder condition ωk (τ (s), δ) = O (δ α ) (δ → 0), 0 < α < k, then the uniform curvilinear modulus of smoothness ω e k,1,Γ (ψ, δ) of the function ψ(w) satisfies the condition
  O δ α+1 (δ
→ 0) if 0 < α < k − 1, 1 ω ek,1,Γ (ψ, δ) = O δ k log
δ (δ → 0) if α = k − 1,  k O δ (δ → 0) if k − 1 < α < k.

Proof of this theorem is based on Theorems 2.2 and 4.1 and is similar to the proof of Theorem 3.1.

Theorem 4.3. Let G1 and G2 be the simply connected domains in the complex plane bounded by the smooth Jordan curves Γ1 and Γ2 . Let τ1 (s1 ) be the angle between the tangent to Γ1 and the positive real axis, s1 (ζ) be

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the arc length on Γ1 . Let τ2 (s2 ) be the angle between the tangent to Γ2 and the positive real axis, s2 (w) be the arc length on Γ2 . Let w = f (ζ) be a homeomorphism of the closure G1 of the domain G1 onto the closure G2 of the domain G2 , conformal in the domain G1 . Let moduli of smoothness ωk (τ1 (s1 ), δ) and ωk (τ2 (s2 ), δ) of order k (k ∈ N ) for the functions τ1 (s1 ) and τ2 (s2 ) satisfy Holder condition with the same index α, 0 < α < k. Then the uniform curvilinear modulus of smoothness ω e k,1,Γ1 (f, δ) of the function f (ζ) on Γ1 satisfies the condition
 α+1 (δ
→ 0) if 0 < α < k − 1, O δ 1 ω ek,1,Γ1 (f, δ) = if α = k − 1, O δ k log δ (δ → 0)
 k O δ (δ → 0) if k − 1 < α < k.

Proof of this theorem is based on Theorems 3.1 and 4.2 and on estimates of finite difference smoothnesses for composite function. References

1. P. M. Tamrazov, Smoothnesses and polynomial appoximations, Kiev: Naukova dumka, 1975 [in Russian]. 2. E. P. Dolzenko, Smoothness of harmonic and analytic functions at boundary points of domain, Dokl., Acad. Nauk SSSR, 29 (1965), no 5, 1069-1084 [in Russian]. 3. E. P. Dolzenko, Notes on modulus of continuity of the conformal mapping of the unit disk onto the Jordan domain, Mat. Zametki, 60 (1996), no 2, 176-184 [in Russian]. 4. E. P. Dolzenko, On boundary smoothness of conformal mappings for domains with nonsmooth boundary, Russian Acad. Nauk Dokl., 415 (2007), no 2, 155159 [in Russian]. 5. P. M. Tamrazov, Finite difference identities and estimates for moduli of smoothness of composite functions, Kiev: Institute of mathematics of Ukrainian Academy of sciences Publishers, 1977 [in Russian]. 6. O. W. Karupu, On moduli of smoothness of conformal mappings, Ukr. Math. J., 30 (1978), no 4, 540-545 [in Russian]. 7. O. W. Karupu, On some chracteristics of conformal mapping, in: Aviation in the XXI century, Proceedings of the second world congress, Ukraine, Kiev, 19-21 September, 2005, 5, (2005), NAU Publishers, 586-591. 8. O. W. Karupu, Finite difference properties of conformal mappings, Acad. Nauk Ukrain. Inst. Mat. Works, 3 (2006), no 4, 175-180 [in Ukranian]. 9. O. W. Karupu, On properties of moduli of smoothness of conformal mappings, in: Complex Analysis and Potential Theory, Proceedings of the Conference Satellite to ICM 2006, 231-238, World Scientific, Singapore, 2007.

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THE JUMP PROBLEM ON NONRECTIFIABLE CURVES AND METRIC DIMENSION B. A. KATS∗ Chair of Mathematics, Kazan State Architecture and Building University, Zelenaya Street, 1, Kazan, Tatarstan, 430043, Russian Federation E-mail: [email protected] We introduce a new version of metric dimension for plane compact sets and apply it to solving the so-called jump problem for holomorphic functions on open nonrectifiable arcs. Keywords: Jump problem, Nonrectifiable curve, Metric dimension.

Introduction The jump problem is a well-known boundary value problem for holomorphic functions. We cite its formulation for the case of an open arc. Let Γ be an arc on the complex plane C with beginning at a point a1 and end point a2 . Let a function f (t) be defined on this arc. We seek a holomorphic in C \ Γ function Φ(z), which has limit values Φ+ (t) and Φ− (t) from the left and and from the right correspondingly at any point t ∈ Γ◦ := Γ \ {a1 , a2 } and satisfies relation Φ+ (t) − Φ− (t) = f (t), t ∈ Γ◦ ;

(1)

in addition, we assume that Φ(∞) = 0. Let function f satisfy the H¨ older condition sup{

|f (t′ ) − f (t′′ )| ′ ′′ : t , t ∈ Γ, t′ 6= t′′ } := hν (f, Γ) < ∞ ν |t′ − t′′ |

(2)

with certain exponent ν ∈ (0, 1]. We denote by Hν (Γ) the set of all defined on Γ functions satisfying (2). ∗ The

author was supported in part by RFBR Grant #07-01-00166-a

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It is well known (see [1,2]), that the jump problem on a piecewise-smooth arc Γ has solution Z f (ζ)dζ 1 . (3) Φ(z) = 2πi Γ ζ − z If the jump f (t) satisfies the H¨ older condition (2), then the Cauchy integral (3) over piecewise - smooth arc has continuous boundary values on Γ◦ from both sides, and difference of these values equals to f . Then the restrictions on the arc Γ were weakened. Thus, E. M. Dyn’kin [3] and T. Salimov [4] proved independently that for closed non-smooth rectifiable arc the boundary values of the Cauchy integral (3) exist and are continuous if f ∈ Hν (Γ) for ν > 1/2. E. M. Dyn’kin [3] proved also that this result cannot be improved. For any ν ∈ (0, 1/2] he constructs a rectifiable curve Γ and a function f (t) ∈ Hν (Γ) such that the Cauchy integral (3) has not boundary values at a point of integration path. Thus, the jump problem is resolvable for rectifiable arcs under restriction ν > 1/2, and this restriction cannot be weakened on the whole class of rectifiable curves. R. K. Seifullaev [5] obtained important results concerning the jump problem on open non-smooth rectifiable arcs. Then the author [6] proved that for any (in general, nonrectifiable) simple closed curve Γ and for any function f ∈ Hν (Γ) such that ν>

1 Dm Γ 2

(4)

there exists a holomorphic in C \ Γ function Φ(z) satisfying relation (1) on the whole curve Γ. Generally speaking, it cannot be written as the Cauchy integral (3). Here Dm Γ is well known [7] upper metric dimension, i.e. Dm Γ = lim sup ε→0

log N (ε; Γ) , − log ε

where N (ε; Γ) is the least number of disks of diameter ε covering Γ. The upper metric dimension of rectifiable curve equals to 1, i.e. for rectifiable curves the condition (4) turns into the Dyn’kin - Salimov condition ν > 1/2. Then the author [8] obtained analogs of these results for open arcs. The restriction (4) cannot be weakened, too. If 0 < ν ≤ d/2 < 1, then there exists a curve Γ a function f ∈ Hν (Γ) such that Dm Γ = d and the jump problem (1) has not solutions [6]. But the conditions for solvability of the jump problem can be improved by means of another metric characteristics of Γ.

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243

The author [9] has introduced new characteristics (so called refined metric dimension) for closed curves and has proved that this dimension allows us to weaken the solvability conditions for the jump problem on that curves. In the present paper we define the refined metric dimension of arbitrary compact set on the complex plane. Then we use this dimension for improvement of conditions of solvability of the jump problem on open arcs. 1. Refined metric dimension of plane compact sets We use a concept of decomposition from the paper [9]. Let δ0 , δ1 , δ2 , . . . be finite simply connected domains, and let s0 , s1 , s2 , . . . be integer numbers such that s0 = +1 and sj for j ≥ 1 equals either to +1 or to −1. We denote by ∆ a sequence of pairs {δj , sj }, j = 0, 1, 2, . . . , and define partial sums ∆n of this sequence as follows: ∆0 = δ0 ; if n ≥ 1, then we put ∆n equaling to interior of union ∆n−1 ∪ δn for sn = +1 and to difference ∆n−1 \ δn for sn = −1. Everywhere below we suppose that δn ∩ ∆n−1 = ∅ for sn = +1 and δn ⊂ ∆n−1 for sn = −1. If the sequence ∆ is infinite, then we denote P∞ by ∆∞ = j=0 sj δj a set of all points z ∈ C such that z ∈ ∆j for all j beginning from certain j = j(z). We say that a sequence ∆ is decomposition of a domain D if D = ∆m , where m is number of pairs in this sequence (it is either finite integer or infinity). We call a decomposition rectifiable if all domains δj have rectifiable boundaries. Let λ(δ) stand for a length of rectifiable boundary of domain δ, and w(δ) is the most diameter of open disk which can be embedded in δ. Definition 1.1. Let Γ be a compact set on the complex plane. Let Q be a domain with rectifiable boundary (for instance, a square) such that Γ ⊂ Q and the set Q\Γ is not connected. Let Q1 , Q2 , . . . be connected components of Q \ Γ. The set e(Γ) consists of all real numbers p ≥ 1 such that each of components Q1 , Q2 , . . . has rectifiable decomposition ∆ = {{δj , sj }, 0 ≤ Pm j ≤ m} such that sum σ(∆) := j=0 λ(δj )wp−1 (δj ) is finite. Then the value rdm Γ := inf e(Γ) is called refined metric dimension of the compact Γ. Theorem 1.1. If Γ is a continuum, then 1 ≤ rdm Γ ≤ Dm Γ ≤ 2.

(5)

In addition, for any d ∈ (1, 2) there exists an open arc G such that rdm G < d = Dm G. If Γ is closed curve, then definition 1.1 coincides with Definition 1 from the paper [9]. The theorem 1.1 is a generalization of concerning closed curves

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Theorem 1 from the same paper. The proof of relation (5) is analogous. Here we have to construct an arc G, because the corresponding curve in [9] is closed. 2β = d. We put Kn = 2[nβ], where Let a value β be defined by equality β+1 square brackets mean entire part, and divide the segment [2−n , 2−n+1 ] of real axis into Kn equal parts of length αn = 2−n /Kn each one. We denote by xn,j the ends of these parts, i. e. xn,j = 2−n + jαn , j = 0, 1, . . . , 2[nβ] − 1, and consider vertical segments In,j = [xn,j , xn,j + i2−n ]. As shown in [6], 2[nβ] −1 the upper metric dimension of the union A = ∪∞ In,j equals to n=1 ∪j=0 2β β+1 , i. e. Dm A = d. Then we put εn = αµn , where µ > 1, and consider mutually disjoint rectangles δn,j = {z = x+iy : xn,j < x < xn,j +εn , 0 < y < 2−n }. Let δ0 be 2[nβ] −1 square {z = x+iy : 0 < x < 1, 0 < y < 1} and Q1 ≡ δ0 \(∪∞ δn,j ), n=1 ∪j=0 i. e. Q1 is unit square with countable set of rectangular cuts condensing to origin. The boundary of domain Q1 consists of three unit segments [0, i], [i, 1 + i], [1, 1 + i], and broken line connecting the points 0 and 1. We denote this line by G. Obviously, A ⊂ G. Easy estimation shows that Dm G = Dm A = d. Let us obtain an upper bound for rdm G. We put Q = {z = x + iy; 0 < x < 1, −1 < y < 1}. Then the set Q \ G consists of two components Q1 and Q2 = Q \ Q1 . The pairs {δ0 , +1}, {δn,j , −1}, n = 1, 2, . . . , j = 0, 1, . . . , 2[nβ] − 1 compose rectifiable decomposition ∆ of domain Q1 . We have λ(δn,j ) ∼ 2−n and w(δn,j ) = εn , hence, the seP∞ [nβ]−n p−1 εn = ries σp (∆) converges simultaneously with the series n=1 2 P∞ [nβ]−n−µ(p−1)([nβ]+n) (µ+1)β+(µ−1) . The domain Q2 , i. e. for p > n=1 2 µ(β+1) has decomposition ∆′ consisting of pairs {Q, +1}, {δ0, −1}, {δn,j , +1}, n = 1, 2, . . . , j = 0, 1, . . . , 2[nβ] − 1 with the same parameters. Consequently, rdm G ≤

(µ + 1)β + (µ − 1) . µ(β + 1)

(6)

The right side of (6) is less than d = Dm G for β > 1 and µ > 1. Thus, the arc G proves the last proposition of the theorem. 2. Solvability of the jump problem on open arcs We consider first a case where end points of an arc Γ belongs to piecewisesmooth boundary of a domain Q such that Γ ⊂ Q. Then Γ divides Q into two connected components Q1 and Q2 . If that domain Q exists, then we say that Γ is separating arc.

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245

Let f w (ζ) be the Whitney continuation (see, for instance, [10]) of a jump f ∈ Hν (Γ) on the whole complex plane. The boundary of component Q1 consists of nonrectifiable arc Γ and piecewise-smooth arc γ, which is a part of ∂Q. Let ∆ = {{δj , sj }, 0 ≤ j < ∞} be rectifiable decomposition of Q1 . We shall seek a solution of the jump problem (1) in the form Z Z w ∞ X f w (ζ)dζ f (ζ)dζ sj 1 − (7) Φ(z) = 2πi ∂δj ζ − z 2πi γ ζ − z j=0 We represent the Cauchy integrals over curves ∂δj in the form Z Z Z 1 f w (ζ)dζ ∂f w dζ ∧ dζ 1 = f w (z)χj (z) − . 2πi ∂δj ζ − z 2πi ζ −z δj ∂ζ Here χj (z) is characteristic function of the domain δj . By means of the Whitney decomposition [10], we can prove the following. Lemma 2.1. If δ is finite domain with rectifiable boundary, f ∈ Hν (γ) 1 , then and p < 1−ν ZZ |∇f w |p dxdy ≤ Chpν (f, γ)λ(δ)w1−p(1−ν) (δ), δ

where C is absolute constant.

By virtue of the lemma the series (7) converges and gives a solution of the boundary value problem (1) if 1 rdm Γ. (8) 2 As the arc γ is piecewise-smooth, since near the points aj , j = 1, 2, we have Φ(z) = O(log |z − aj |), j = 1, 2. Thus, the following is valid. ν>

Lemma 2.2. If Γ is a separating arc and f ∈ Hν (Γ), then under restriction (8) the jump problem (1) has a solution such that its singularities at the end points of the arc are logarithmic. Now we consider arbitrary nonrectifiable arc. In general, it is not separating. Let Γ1 ⊂ Γ be an arc with beginning and end at points b1 and b2 , b1 6= b2 . Boundary of its closed convex hull contains at least one interior point of the arc Γ1 . It is end point of a small segment of a straight line without another common points with Γ. We say that this point is attainable. As we has seen, any non-trivial arc of Γ contains at least one attainable point. Thus, the set of attainable points is everywhere dense on Γ. Hence, S Γ is representable as union Γj of countable family of arcs with attainable

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end points. All these arcs are separating. We denote by Φj (z) a solution of the jump problem on the separating arc Γj ; it exists by virtue of lemma 2.2. One can show easily, that if Γm and Γn are two neighboring arcs with common end point, then singularities of functions Φm and Φn at this point annihilate, i.e. the sum Φm + Φn is solution of the jump problem on arc P∞ Γm ∪ Γn . Hence, if series j=1 Φj (z) converges, then its sum is solution of the jump problem on the whole arc Γ. In general this series diverges, but we can regularize it. Indeed, there exists a sequence of rational functions Rj (z) P∞ with poles at points a1 and a2 such that the series j=1 (Φj (z) − Rj (z)) uniformly converges in domain C\Γ. These these rational functions exist by virtue of considerations analogous to the proof of Mittag-Leffler theorem. Thus, the following is valid. Theorem 2.1. The jump problem (1) on nonrectifiable arc Γ under assumption (8) has a solution. Unlike lemma 2.2, here we do not obtain any information on behavior of solution near the end points. That behavior is of importance for applications of this boundary value problem (see [1]). In this connection we imply certain estimates for solution near the end points. We consider a problem on unit jump Φ+ (t) − Φ− (t) = 1, t ∈ Γ◦ ; its solution is function kΓ (z) :=

z − a2 1 , ln 2πi z − a1

where the branch of logarithm is selected by means of cut along Γ and condition kΓ (∞) = 0. At the ends of arc Γ imaginary part of this function has logarithmic singularities. Its real part can be bounded (for instance, if the arc Γ is smooth), but it can have singularities of arbitrary high order if this arc curls into spiral. We shall show that under certain restrictions the jump problem (1) has a solution with the same order of singularities at the ends as kΓ . We fix r > 0 such that the arc Γ is situated in the disk |z| < r, and denote by ω(z) a smooth function equaling unit for |z| ≤ r and zero for |z| ≥ 2r. Let f w (z) stand for the Whitney extension (see, for instance, [10]) of the function f onto the whole complex plane. Then the product ψ(z) := kΓ (z)ω(z)f w (z) is continuous in C\ Γ and has there partial derivatives with respect to Re z and Im z of all orders. The function ψ has jump f on the

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247

arc Γ. In addition, this function has compact support. We seek solution of the problem (1) in the form ZZ 1 ∂ψ dζ ∧ dζ . (9) Ψ(z) := ψ(z) − 2πi C ∂ζ ζ − z w

is locally integrable with any According to lemma 2.1, the derivative ∂f ∂ζ degree lesser than (2 − rdm Γ)/(1 − ν) Thus, under restriction (8) it is integrable with certain degree p > 2. Hence, under assumption kΓ (z) = O(|z − aj |−α ), α < 1, j = 1, 2,

(10)

the function with compact support ψζ := ∂ψ is integrable, what make our ∂ζ definition of Ψ(z) correct. Moreover, under the same assumptions the function ψζ is locally integrable with certain degree p > 2 outside of arbitrary neighborhoods of ends of the arc Γ. Consequently, (see, for instance, [11]) the integral term of (9) is continuous in the whole plane with possible exception of end points of the arc Γ. In addition, the well known properties of integral operator ZZ ϕ(ζ)dζ ∧ dζ 1 ϕ(z) 7→ 2πi C ζ −z (see, for instance, [11]) imply that the function Ψ(z) is holomorphic in C \ Γ and vanishes at infinity point. Thus, under assumptions (8) and (10) the function (9) is a solution of the jump problem (1). It remains to estimate its singularities at the end points of Γ. The first term ψ(z) has at the points aj , j = 1, 2 singularities of the same order as kΓ (z), i.e., ψ(z) = O(|z − aj |−α ). Let p stand for degree of w . As above, 2 < p < (2 − rdm Γ)/(1 − ν). Then local integrability of ∂f ∂ζ near the point aj we have ZZ 1 ∂ψ dζ ∧ dζ 2πi C ∂ζ ζ − z 0) each time are determined with respect to some fixed collection of parameters and correspond to the inequalities C1 B ≤ A ≤ C2 B and A ≤ C3 B, where Ck > 0 (k = 1, 2, 3) are the constants independent of the mentioned collection of parameters.

Jackson-Bernstein Theorem in Lp on Closed Curves in the Complex Plane

263

Notice that for the cited generalized smoothness modulus the relations: (2)

(2)

ωp (f, η) ≤ δ −2 (δ + η)2 ωp (f, δ) (2) (2) ˜ p (f, δ) ω ˜ p (f, η) ≤ δ −2 (δ + η)2 ω

∀ δ > 0, ∀ η > 0 , ∀ δ > 0, ∀ η > 0

and (2)

(2)

η −2 ωp (f, η) ≤ 4 δ −2 ωp (f, δ) (2) (2) η −2 ω ˜ p (f, η) ≤ 4 δ −2 ω ˜ p (f, δ)

for for

0 n ≥ 1, e2n+2m + (−1)n e2m−2n 2   1 e2n+1 e2m+1 = ∀n ≥ m ≥ 1, e2n+2m+1 + (−1)m e2n−2m+1 2   1 e2n e2m = −e2n+2m+1 + (−1)m e2n−2m+1 ∀n ≥ m ≥ 1. 2 It is evident that here e1 , e2 , e3 form a harmonic triad of vectors. Theorem 3.1. In order that a function Φ : Qξ −→ F of the form Φ(xe1 + ye2 + ze3 ) =

∞ X

Uk (x, y, z) ek ,

(6)

k=1

where Uk : Q −→ R, be monogenic in the domain Qξ ⊂ E3 , it is necessary and sufficient that the functions Uk , k = 1, 2, . . . , be differentiable in the domain Q, that the conditions 1 ∂U2 (x, y, z) ∂U1 (x, y, z) =− , ∂y 2 ∂x ∂U1 (x, y, z) 1 ∂U5 (x, y, z) ∂U2 (x, y, z) = + , ∂y ∂x 2 ∂x 1 ∂U4 (x, y, z) ∂U3 (x, y, z) =− , ∂y 2 ∂x 1 ∂U2k−1 (x, y, z) 1 ∂U2k+3 (x, y, z) ∂U2k (x, y, z) = + , k = 2, 3, . . . , ∂y 2 ∂x 2 ∂x ∂U2k+1 (x, y, z) 1 ∂U2k−2 (x, y, z) 1 ∂U2k+2 (x, y, z) =− − , k = 2, 3, . . . , ∂y 2 ∂x 2 ∂x

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S. A. Plaksa

∂U1 (x, y, z) 1 ∂U3 (x, y, z) ∂U2 (x, y, z) 1 ∂U4 (x, y, z) =− , =− , (7) ∂z 2 ∂x ∂z 2 ∂x ∂U1 (x, y, z) 1 ∂U5 (x, y, z) ∂U3 (x, y, z) = − , ∂z ∂x 2 ∂x ∂Uk (x, y, z) 1 ∂Uk−2 (x, y, z) 1 ∂Uk+2 (x, y, z) = − , k = 4, 5, . . . , ∂z 2 ∂x 2 ∂x be satisfied in Q, and that the following relations be fulfilled: ∞ X ∂Uk (x, y) < ∞, (8) ∂x k=1 ∞ X Uk (x+εh1 , y+εh2, z +εh3 ) − Uk (x, y, z) − ∂Uk (x, y, z) εh1 lim ε→0+0 ∂x k=1 ∂Uk (x, y, z) ∂Uk (x, y, z) εh2 − εh3 ε−1 = 0 ∀ h1 , h2 , h3 ∈ R . (9) − ∂y ∂z

Proof. Necessity. If the function (6) is monogenic in the domain Qζ , then for h = e1 the equality (4) turns into the equality Φ′ (ζ) =

∞ X ∂Uk (x, y, z) ek , ∂x k=1

and the relation (8) is fulfilled. Now, setting in series h = e2 and h = e3 in the equality (4), we obtain the conditions (7) for components of monogenic function (6). At last, for h1 , h2 , h3 ∈ R and ε > 0, denoting h := h1 e1 + h2 e2 + h3 e3 and taking into account the equalities (7), we have
Φ(ζ + εh) − Φ(ζ) ε−1 − h Φ′ (ζ) X ∞   = Uk (x + εh1 , y + εh2 , z + εh3 ) − Uk (x, y, z) ek k=1

 ∞ X ∂Uk (x, y, z) ek ε−1 −ε (h1 e1 + h2 e2 + h3 e3 ) ∂x k=1 ∞  X = ε−1 Uk (x + εh1 , y + εh2 , z + εh3 ) − Uk (x, y, z) k=1

 ∂Uk (x, y, z) ∂Uk (x, y, z) ∂Uk (x, y, z) εh1 − εh2 − εh3 ek . − ∂x ∂y ∂z

(10)

Therefore, in as much as the function (6) is monogenic in the domain Qζ , the relation (9) is fulfilled in Qζ .

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Sufficiency. Let ε > 0 and h := h1 e1 + h2 e2 + h3 e3 , where h1 , h2 , h3 ∈ R. Then under the conditions (7) and (8) the equality (10) is true. Now, it follows from the equality (10) and the relation (9) that the function (6) is monogenic in the domain Qζ . The theorem is proved. Note that conditions (7) are similar by nature to the Cauchy-Riemann conditions for monogenic functions of complex variables, and the relations (8), (9) are conditioned by the infinite dimensionality of the algebra F. It is clear that if the Gateaux derivative Φ′ of monogenic function Φ : Qξ −→ F, in turn, is monogenic function in the domain Qξ , then all components Uk of expansion (6) satisfy Eq. (3) in Q in consequence of condition (5). At the same time, the following statement is true even independently of relation between solutions of the system of equations (7) and monogenic functions. Theorem 3.2. If the functions Uk : Q −→ R have continuous second-order partial derivatives in the domain Q and satisfy the conditions (7), then they satisfy Eq. (3) in Q. Proof. It is easy to show that if the functions Uk are doubly continuously differentiable in the domain Q, then the equalities ∆Uk (x, y, z) = 0 for k = 1, 2, . . . are corollaries of the system (7). Note that the algebra F is isomorphic to the algebra F of absolutely convergent trigonometric Fourier series g(τ ) = a0 +

∞  X

ak ik cos kτ + bk ik sin kτ

k=1



with real coefficients a0 , ak , bk and the norm kgkF := |a0 | +

∞  P

k=1

 |ak | + |bk | .

In this case, we have the isomorphism e2k−1 ←→ ik−1 cos (k − 1)τ , e2k ←→ ik sin kτ between basic elements. Let us write the expansion of a power function of the variable ξ = xe1 + ye2 + ze3 in the basis {ek }∞ k=1 , using spherical coordinates ρ, θ, φ which have the following relations with x, y, z: x = ρ cos θ,

y = ρ sin θ sin φ,

z = ρ sin θ cos φ .

(11)

In view of the isomorphism of the algebras F and F, the construction of expansions of this sort is reduced to the determination of relevant Fourier

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coefficients. So we have  ξ = ρ Pn (cos θ) e1 n

+2

n


n! Pnm (cos θ) sin mφ e2m + cos mφ e2m+1 , (n + m)! m=1 n X

(12)

where n is a positive integer, Pn and Pnm are Legendre polynomials and associated Legendre polynomials, respectively, namely: Pn (t) :=

1 2n

dn 2 (t − 1)n , n! dtn

Pnm (t) := (1 − t2 )m/2

dm Pn (t) . dtm

We obtain in exactly the same way the following expansion of the exponential function: ζ

e =e

ρ cos θ

∞  X
 J0 (ρ sin θ)e1 + 2 Jm (ρ sin θ) sin mφ e2m +cos mφ e2m+1 , m=1

where Jm are Bessel functions, namely Z (−1)m π it cos τ e cos mτ dτ . Jm (t) := π 0 Thus, 2n + 1 linearly independent spherical functions of the n-th power are components of the expansion (12) of the function ξ n . Using the expansion (12) and rules of multiplication for basic elements of the algebra F, it is easy to prove the following statement. Theorem 3.3. Every spherical function   n   X m an,m cos mφ + bn,m sin mφ Pn (cos θ) ρ an,0 Pn (cos θ) + n

m=1

where an,0 , an,m , bn,m ∈ R, is the first component of expansion of the monogenic function  n  X (n + m)!  an,0 e1 + (−1)m bn,m e2m + an,m e2m+1 ξ n n! m=1 in the basis {ek }∞ k=1 , where ξ := xe1 + ye2 + ze3 , and x, y, z have the relations (11) with the spherical coordinates ρ, θ, φ.

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4. Relation between monogenic functions and harmonic vectors Let us establish some important properties of solutions of the system (7). Theorem 4.1. Every solution of the system (7) in a domain Q ⊂ R3 generates a harmonic vector V := (U1 , − 12 U2 , − 21 U3 ) in Q. Proof. The equalities (2) for a vector V := (U1 , − 21 U2 , − 21 U3 ) are corollaries of the system (7). Really, the system (7) contains the third equation and the fourth equation from (2). The second equation from (2) is a corollary of the third condition and the seventh condition of the system (7). Finally, the first equation from (2) is a corollary of the second condition and the eighth condition of the system (7). Theorem 4.2. For any function U1 : Q −→ R harmonic in a simply connected domain Q ⊂ R3 there exist harmonic functions Uk : Q −→ R, k = 2, 3, . . . , such that the conditions (7) are fulfilled in Q. Proof. Let us to add and to subtract the second condition and the eighth condition as well as the third condition and the seventh condition of the system (7). Let us also to add and to subtract the fourth condition for k = m and the ninth condition of the system (7) for k = 2m + 1, where m = 2, 3, . . . . Let us else to add and to subtract the fifth condition for k = m and the ninth condition of system (7) for k = 2m, where m = 2, 3, . . . . Thus, we rewrite the system (7) in the following equivalent form: 1 ∂U2 1 ∂U3 ∂U3 ∂U2 ∂U1 − − = 0, − = 0, ∂x 2 ∂y 2 ∂z ∂y ∂z 1 ∂U2 ∂U1 1 ∂U3 ∂U1 + = 0, + = 0, ∂y 2 ∂x ∂z 2 ∂x ∂U2k ∂U2k−2 ∂U2k−1 ∂U2k+1 ∂U2k−2 ∂U2k−1 =− − , = − ∂x ∂z ∂y ∂x ∂y ∂z ∂U2k ∂U2k+1 ∂U2k−2 ∂U2k ∂U2k+1 ∂U2k−1 − = , + = , ∂z ∂y ∂x ∂y ∂z ∂x

(13)

where k = 2, 3, . . .. First of all, note that there exists a harmonic vector V0 := (U1 , v20 , v30 ) in the domain Q. Moreover, for any vector V := (U1 , v2 , v3 ) harmonic in Q, the components v2 , v3 are determined accurate within the real part and the imaginary part of any function f1 (t) holomorphic in the domain

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{t = z + iy : (x, y, z) ∈ Q} of the complex plane, i.e. the equalities

v2 (x, y, z) = v20 (x, y, z)+Re f1 (z+iy), v3 (x, y, z) = v30 (x, y, z)+Im f1 (z+iy) are true for all (x, y, z) ∈ Q. Then, using Theorem 4.1, we find the functions U2 and U3 , namely: U2 := −2 v2 , U3 := −2 v3 . Now, let us show that the last four conditions of the system (13) allow to determine the functions U2k , U2k+1 , if the functions U2 , U3 , . . . , U2k−1 are already determined. Really, integrating the fifth equation and the sixth equation of the system (13), we obtain the expressions  Z x ∂U2k−2 (τ, y, z) ∂U2k−1 (τ, y, z) dτ + u e2k (z, y), + U2k (x, y, z) = − ∂z ∂y x0  Z x ∂U2k−2 (τ, y, z) ∂U2k−1 (τ, y, z) dτ + u e2k+1 (z, y) − U2k+1 (x, y, z) = ∂y ∂z x0

for (x, y, z) belonging to a certain neighborhood N ⊂ Q of any point (x0 , y0 , z0 ) ∈ Q. Substituting these expressions into the seventh equation and eighth equation of the system (13) and taking into account that U2k−2 , U2k−1 are harmonic functions in the domain N , we obtain the following inhomogeneous Cauchy–Riemann system (see, for example, [5]) for finding the functions u e2k , u e2k+1 : ∂e u2k (z, y) ∂e u2k+1 (z, y) ∂U2k−2 (x, y, z) , − = ∂z ∂y ∂x x=x0 (14) u2k+1 (z, y) ∂U2k−1 (x, y, z) ∂e u2k (z, y) ∂e + = . ∂y ∂z ∂x x=x0

Solutions of the system (14) are determined accurate within the real part and the imaginary part of any function holomorphic in the domain {t = z + iy : (x, y, z) ∈ N } of the complex plane. Therefore, in as much as the domain Q is simply connected, taking into account the unicity theorem for spatial harmonic functions, it is easy to continue the functions U2k , U2k+1 defined in the neighborhood N into the domain Q.

Let us note that the functions U2m , U2m+1 satisfying the last four conditions of the system (13) for k = m ≥ 2 are determined accurate within the real part and the imaginary part of any function fm (t) holomorphic in the domain {t = z + iy : (x, y, z) ∈ Q} of the complex plane, i.e. the equalities U2m (x, y, z)

0 = U2m (x, y, z) + Re fm (z + iy) ,

0 U2m+1 (x, y, z) = U2m+1 (x, y, z) + Im fm (z + iy)

An Infinite-Dimensional Banach Algebra and Spatial Potential Fields

277

0 0 are true for all (x, y, z) ∈ Q, where U2m , U2m+1 are functions forming together with the functions U1 , U2 , . . . , U2m−1 a particular solution of the system (13), in which k = 2, 3, . . . , m.

References 1. M. A. Lavrentyev and B. V. Shabat, Problems of hydrodynamics and their mathematical models, Moscow: Nauka, 1977 [in Russian]. 2. F. Klein, Vorlesungen u ¨ber die entwicklung der mathematik im 19 jahrhundert, V.1, Berlin: Verlag von Julius Springer, 1926. 3. I. P. Mel’nichenko, On expression of harmonic mappings by monogenic functions, Ukr. Math. J., 27 (1975), no. 5, 606–613. 4. I. P. Mel’nichenko, Algebras of functionally-invariant solutions of the threedimensional Laplace equation, Ukr. Math. J., 55 (2003), no. 9, 1284–1290. 5. I. N. Vekua, Generalized analytic functions, London: Pergamon Press, 1962.

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Session 04

Complex Analytic Methods for Applied Sciences

SESSION EDITORS V. V. Mityushev S. V. Rogosin

Pedagogical University, Krak´ ow, Poland Belarus State University, Minsk, Belarus

281

ON THE CURVATURE OF AN INNER CURVE IN A SCHWARZ-CHRISTOFFEL MAPPING A. ANDERSSON International Centre for Mathematical Modelling, V¨ axj¨ o University, Sweden and School of Engineering, J¨ onk¨ oping University, Sweden In the so-called outer polygon method, an approximate conformal mapping for a given simply connected region Ω is constructed using a Schwarz–Christoffel mapping for an outer polygon, a polygonal region of which Ω is a subset. The resulting region is then bounded by a C ∞ -curve, which among other things means that its curvature is bounded. In this work, we study the curvature of an inner curve in a polygon, i.e., the image under the Schwarz–Christoffel mapping of R, the unit disk or upper halfplane, to a polygonal region P of a curve inside R. From the Schwarz–Christoffel formula, explicit expressions for the curvature are derived, and for boundary curves appearing in the outer polygon method, asymptotic boundaries for the curvature are given. Keywords: Curvature, Schwarz-Christoffel mapping, inner curve, outer polygon.

1. Introduction The Schwarz–Christoffel mapping from the upper half-plane or the unit disk to a polygonal region in the complex plane is a well-known and often used conformal mapping for which also efficient numerical methods exist by the work of Trefethen and Driscoll [1–3]. In a recent paper [4] by the author, this mapping is used for different non-polygonal regions by seeing the region under consideration as an inner region in a polygon. The motivation for the method is that in some applications, see for example [5], it is necessary that the approximate conformal mapping maintains a good control of the boundary curve direction, especially towards infinity when concerning unbounded regions. Furthermore, if the boundary curve in the region has bounded curvature, this property should be maintained in the

282

A. Andersson

approximate region produced by the conformal mapping. Hence, an important issue is to find bounds for the variation of the curvature. Curvature in the context of conformal mappings has been studied in for example [6] where mappings from the disk to convex domains is treated. When bounded regions are handled in [4], the approximate regions have boundaries that are level curves of Green’s functions. Curvature for such curves is dealt with in for example [7–9]. The purpose of this paper is to investigate the curvature for some inner curves in polygons, both bounded and unbounded, convex and non-convex, curves that typically appear as boundary curves in the regions produced by the methods in [4]. From the Schwarz–Christoffel formula, explicit expressions for the curvature of inner curves are given, and furthermore, asymptotic bounds for the curvature are given. The plan of the paper is as follows. In Sec. 2, a short description of the method and the ideas in [4] is given, and in Sec. 3 and Sec. 4, the curvature for inner curves in Schwarz–Christoffel polygons in general and in the special cases relevant for the methods in [4] is calculated. Finally, in Sec. 5, bounds for the curvature are given. 2. The outer polygon method

ǫ

Rǫ ǫi

Rǫ

Rǫ (ǫ − β)π

βπ a

Fig. 1.

Inner regions in polygons used in [4].

◦ In [4], methods for constructing an outer polygon Pǫ,Ω to a region Ω

On the Curvature of an Inner Curve in a Schwarz-Christoffel Mapping

283

in the z-plane, such that a Schwarz–Christoffel mapping from R, the up◦ per half-plane or unit disk in the w-plane, to Pǫ,Ω maps a region Rǫ ∈ R approximately on Ω. In the examples in the paper, three different cases are treated: (1) Ω is a bounded region, R is the unit disk and Rǫ the disk {w : |w| ≤ 1 − ǫ}. (2) Ω is an unbounded region with one infinite boundary point, R is the upper half-plane and Rǫ = {w : Im w ≥ ǫ}. (3) Ω is an infinite channel with parallel straight line walls at both ends, R is the upper half-plane and Rǫ the region above two rays in the upper half-plane going out from the point on the real axis which is mapped to infinity at one of the channel ends. An algorithm with tangent polygons and a quasi-Newton method is used to determine an outer polygon and an approximate conformal mapping for Ω. The resulting region has a boundary with bounded curvature in all finite points. However, to make the algorithm work, it is often necessary to use a small ǫ, and as will be apparent, the curvature might in that case vary in a wide range. 3. Curvature of an inner curve in a polygon Let s be a Schwarz–Christoffel mapping Z wY n (ω − wk )αk −1 dω + B s(w) = A

(1)

w0 k=1

from R, the upper half-plane or the unit disk in the w-plane, to a polygonal region P in the z-plane with inner angles α1 π, . . . , αn π, which means that for each k, αk ∈ [0, 2]. If s is a mapping from the unit disk, each wk is a complex number on the unit circle; if s is a mapping from the upper half–plane, each wk is a real number, and since there are three degrees of freedom in the Schwarz–Christoffel mapping, we can without loss of generality assume that they are all in the interval [−1, 1]. (This restriction is used in SC toolbox [3], which is the numerical software that [4] uses. Note also that this type of restriction affects the value of A.) Let w(t) with real parameter t be a curve in the interior of R. We will study the curvature of the curve s(w(t)), residing inside P . The angular direction of the curve s(w(t)) is given by X Φ(t) = arg(s′ (w(t))) = arg(A)+arg(w ′ (t))+ (αk−1) arg(w(t)−wk ). (2) k

284

A. Andersson

Setting w(t) − wk = xk (t) + yk (t)i, arg(w(t) − wk ) = arctan(yk (t)/xk (t)), and we get X (αk − 1)(xk (t)y ′ (t) − x′ (t)yk (t)) d k k . (3) arg(w′ (t)) + Φ′ (t) = dt (xk (t))2 + (yk (t))2 k

2

Since the denominator (xk (t))2 +(yk (t))2 = |w(t) − wk | , and by setting x (t) yk (t) , hk (t) = k′ xk (t) yk′ (t)

the curvature κ(t) of s(w(t)) is

X d arg(w′ (t)) + (αk − 1)hk (t)/ |w(t) − wk |2 ′ dt Φ (t) k Y . κ(t) = ′ = α −1 ′ |s (w(t))| |A| |w (t)| |w(t) − wk | k

(4)

k

If we separate the terms, Eq. (4) can be written Y |w(t) − wk |1−αk d arg(w′ (t)) k κ(t) = dt |A| |w′ (t)| +

X k

(αk − 1)hk (t)

Y

j6=k

1−αj

|w(t) − wj |

|A| |w′ (t)| |w(t)

. (5)

1+αk

− wk |

We will in the following mainly study curves passing close to a polygon vertex zm and their curvature near this vertex. Let for some prevertex wm , ǫ = min{|w(t) − wm |} and let t = tm be the point for which this minimum is reached. If furthermore Y 1−αk pm = |w(tm ) − wk | , (6) k6=m

κ(tm ) =

pm |A| |w′ (tm )| ǫαk

ǫ


(αm − 1)hm (tm ) d arg w′ (tm ) + dt ǫ +ǫ

X (αk − 1)hk (tm ) 2

k6=m

|w(tm ) − wk |

!

. (7)

4. Three special cases For the boundary curve of the inner region s(Rǫ ) in the examples mentioned in Sec. 2 and treated in [4], we apply the result from the previous section.

On the Curvature of an Inner Curve in a Schwarz-Christoffel Mapping

285

4.1. A bounded region Assume that P is a bounded region, i.e., with no infinite vertex, and that R is the unit disk. Let for some ǫ, 0 < ǫ < 1, w(t) = (1 − ǫ)eit ,

0 ≤ t < 2π.

(8)

Here, the prevertices w1 , . . . , wn are all numbers on the unit circle; set wk = eitk for k = 1, . . . , n. Furthermore, arg(w ′ (t)) = π/2 + t, xk (t) = (1 − ǫ) cos t − cos tk and yk (t) = (1 − ǫ) sin t − sin tk , and hence, 

hk (t) = (1 − ǫ) cos t (1 − ǫ) cos t − cos tk + sin t (1 − ǫ) sin t − sin tk
= (1 − ǫ) 1 − ǫ − cos(t − tk ) . Therefore, Y κ(t) =

k

|w(t) − wk |1−αk |A| (1 − ǫ) +

X k

(αk − 1)(1 − ǫ − cos(t − tk ))

Y

1−αj

|w(t) − wj |

j6=k 1+αk

|A| |w(t) − wk |

. (9)

Since ǫ ≤ |w(t) − wk | ≤ 2 − ǫ, the first term in Eq. (9) dominates when ǫ is close to 1, and the curvature is almost constant. This means that the inner curve is nearly circular. Note that in this case, the images under the Schwarz–Christoffel mapping of origin-centered circles with radius less than one are level curves of Green’s function for the polygonal region. The result harmonizes well with the results by Walsh [8], where the properties of Green’s function level curves for general bounded regions are developed thoroughly. On the other hand, for small ǫ, κ(t) is near the mth vertex in P dominated by the corresponding term in Eq. (9). Since w(tm ) = (1 − ǫ)wm , it follows that
! X (αk − 1) 1 − ǫ − cos(tm − tk ) ǫ pm , +1−αm +ǫ κ(tm ) = |A| ǫαm 1−ǫ (1−ǫ)2 −2(1−ǫ) cos(tm −tk )+1 k6=m

(10)

which, with an asymptotic expression, can be written
 ǫ (1 − αm )pm  2 1 + . + O ǫ κ(tm ) = |A| ǫαm 2

(11)

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4.2. Polygon with one infinite vertex Assume that P has one vertex at infinity and that R is the upper half-plane. Then, Z w n−1 Y s(w) = A (ω − wk )αk −1 dω + B, (12) w0 k=1

with real prevertices wk . Let for some ǫ > 0, w(t) = t + ǫi. It follows that xk (t) = t − wk and yk (t) = ǫ, so hk (t) = −ǫ and the curvature Y 1−αj ǫ(1 − αk ) |t + ǫi − wj | X j6=k . (13) κ(t) = 1+αk |A| |t + ǫi − wk | k

If ǫ is small, the mth term dominates in Eq. (13) when t ≈ wm . Setting tm = wm , we have ! X pm 1 − αk 2 (14) κ(tm ) = 1 − αm + ǫ 2 |A| ǫαm |wm − wk + ǫi| k6=m



(1 − αm )pm 1 + O ǫ2 . = α m |A| ǫ

(15)

4.3. Channel with parallel walls at the ends Finally, assume that P is a polygonal channel with parallel walls at both ends, and that the mapping s from the upper half-plane to P maps infinity to one infinite end of the channel, and that the real number a is mapped on the other infinite channel end. Then, Z w n−2 Y (ω − a)−1 (ω − wk )αk −1 dω + B. (16) s(w) = A w0

k=1

iϕ

Let for ϕ ∈ (0, π), w(t) = a + te , 0 ≤ t < ∞, i.e., w(t) is a ray from a in the upper half-plane making the angle ϕ with the positive real axis. It follows immediately that arg(w ′ (t)) = ϕ, and furthermore, we have xk (t) = a − wk + t cos ϕ x′k (t)

X k

yk′ (t) = sin ϕ,

= cos ϕ

so hk (t) = (a − wk ) sin ϕ. From Eq. (5), it follows that

κ(t) =

yk (t) = t sin ϕ

(αk − 1)(a − wk ) sin ϕ

Y a − wj + teiϕ 1−αj

j6=k

1+αk

|A| |a − wk + teiϕ |

.

(17)

On the Curvature of an Inner Curve in a Schwarz-Christoffel Mapping

287

Consider the curvature near a vertex s(wm ). A ray passing close to wm must have a small ϕ if wm > a and a ϕ slightly less than π if wm < a. In both cases, a − wm + teiϕ ≥ |a − wm | sin ϕ, with equality for t = (wm − a) cos ϕ. Hence, setting tm = (wm − a) cos ϕ, ǫ = |a − wm | sin ϕ and vk,m = (wk − a)/ |wm − a|, it follows from Eq. (7) that  X pm  (1 − αk )vk,m 2 ±(1 − α ) + ǫ (18) κ(tm ) = m |A| ǫαk (wm − wk )2 − ǫ2 (1 ± 2vk,m ) k6=m



(1 − αm )pm =± 1 + O ǫ2 , α m |A| ǫ

(19)

where ± is the sign of wm − a. 5. Conclusions

In all the considered cases, an estimation of the curvature results in similar expressions, see Eqs. (11), (15) and (19). For small ǫ, a term 1 − αm Y 1−αk |wm − wk + ǫc| (20) ± |A| ǫαm k6=m

with |c| = 1 dominates near the mth vertex, and the curvature at that point grows for αm 6= 1 infinitely when ǫ → 0. To give an asymptotic upper bound for the curvature, we start with a preliminary lemma. Lemma 5.1. Let f , g0 and g1 be functions, sufficiently smooth in a neighbourhood Ω of x˜, c a real and β a real non-zero constant, 0 < ǫ < 1, and let x∗ (ǫ) ∈ Ω be a local maximum of f . Assume that

β f (x) = (h(x)) g0 (x) + ǫg1 (x) + O ǫ2 , (21) where the function h(x) is positive on Ω and defined by either of h(x) = ǫ2 + (x − x˜)2 , 2

2

4

h(x) = ǫ + (1 − ǫ)(x − x ˜) + O (x − x ˜) and furthermore that lim x∗ = x ˜.

ǫ→0

Then, for x∗ ∈ Ω,


x∗ − x ˜ = O ǫ2 .




(22) ,

(23)

(24)

(25)

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A. Andersson

Proof. 

f ′ (x) = (h(x))β−1 βh′ (x) g0 (x) + ǫg1 (x) + O ǫ2 +

h(x) g0′ (x) + ǫg1′ (x) + O ǫ2



From Eq. (24), x∗ − x ˜ = Bǫ + O ǫ2 . Since h′ (x∗ ) = 2Bǫ + O ǫ2 , 0 = f ′ (x∗ ) = (h(x∗ ))

β−1

2Bβǫg0 (x) + O ǫ2
Hence, B = O (ǫ) and it follows that x∗ − x ˜ = O ǫ2 .







 ,

.

Let {wk } and w(t) be defined as before, and let Y 1−αk qm = |wm − wk | .

(26)

k6=m

Then pm =

Y

k6=m

1−αk

|wm −wk +ǫc|

 X = qm 1+ǫc

k6=m


1 − αk 2 +O ǫ |wm −wk |

(27)

and with tm defined as in any of Secs. 4.1, 4.2 or 4.3, it follows from Eqs. (11), (15) and (19) that |κ(tm )| = where

rm



|1 − αm | qm 2 , 1 + r ǫ + O ǫ m α |A| ǫ m

(28)

X |1 − αk | 1   + , for bounded regions, Sec. 4.1   |wm −wk | 2 k6=m = X |1 − α | (29) k   , for unbounded regions, Secs. 4.2, 4.3.   |wm − wk | k6=m

Theorem 5.1. Let t∗ be a local optimum point for the boundary curvature in any of the three cases considered in Sec. 4, and let Im be an interval containing t∗ but no other optimum point and tm , chosen such that |κ(t)| ≤ |κ(t∗ )| for all t ∈ Im . Then,

|1 − αm | qm 2 1 + r ǫ + O ǫ m α |A| ǫ m
on Im . Furthermore, |κ(t∗ ) − κ(tm )| = O ǫ4 . |κ(t)| ≤

(30)

On the Curvature of an Inner Curve in a Schwarz-Christoffel Mapping

289

Proof. The first statement follows from the second together with Eq. (28). From Eqs. (9), (13) and (17), it is seen that κ(t) can be written in the 2 form of Eq. (21) with x ˜ = wm , β = (1 + αm )/2 and h(t) = |w(t) − wm | . This means that h takes the form of Eq. (23) for bounded regions and Eq. (22) for the two types of unbounded regions considered. Since κ for ǫ > 0 is an analytic function with a local optimum at t∗ ,
1 κ(tm ) = κ(t∗ ) + κ′′ (t∗ )(tm − t∗ )2 + O (tm − t∗ )3 , 2
and it follows from Lemma 5.1 that κ(t∗ ) − κ(tm ) = O ǫ4 .

Assume that a curvature bounded by C is desired near the mth vertex. Then, if the vertex is convex, i.e., αm < 1, and if ǫ is small enough to leave out the higher order terms, Theorem 5.1 shows that ǫ should be chosen such that ǫαm >

|1 − αm | qm . |A| C

(31)

Similarly, if the vertex is concave and therefore αm > 1, ǫ should if possible be chosen such that ǫαm |1 − αm | qm > . 1 + rm ǫ |A| C

(32)

This can of course be difficult to achieve. However, in the method used in [4], tangent points can be put arbitrarily dense on the boundary curve of the considered region. This means that one can to some extent rule the size of αm in the outer polygon. By using many tangent points to achieve obtuse angles with αk close to 1, it is also possible to bring down the size of qm . Note also the different curvature in the curves we consider close to convex and concave vertices respectively. The parameter αm ∈ [0, 2], and is less than one if the vertex is convex and greater than one if it is concave. For example, a right-angled convex vertex means αm = 0.5, but if it is concave, αm = 1.5. This results in a significantly greater curvature in the latter when ǫ is small. This difference is even greater for acute angles. Acknowledgements I would like to thank professor B¨ orje Nilsson, V¨ axj¨ o University, Sweden, for many valuable comments, ideas and discussions on this work.

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A. Andersson

References 1. T. A. Driscoll and L. N. Trefethen, Schwarz-Christoffel mapping, Cambridge Monographs on Applied and Computational Mathematics, Vol. 8, Cambridge University Press, Cambridge, 2002. 2. L. N. Trefethen, SIAM J. Sci. Statist. Comput. 1, 82 (1980). 3. T. A. Driscoll, Schwarz–Christoffel toolbox for Matlab available from http://www.math.udel.edu/~driscoll/SC/. 4. A. Andersson, Schwarz–Christoffel mappings for non-polygonal regions, Report 07104, MSI, V¨ axj¨ o University, V¨ axj¨ o, 2007; submitted to SIAM J. Sci. Comput. 5. B. Nilsson, Proc. R. Soc. Lond. A 458, 1555 (2002). 6. V. Y. Gutlyanski˘ı and S. A. Kopanev, Ukra¨ın. Mat. Zh. 44, 1330 (1992). 7. J. L. Walsh, Amer. Math. Monthly 44, 202 (1937). 8. J. L. Walsh, Amer. Math. Monthly 60, 671 (1953). 9. D. B. Shaffer, Trans. Amer. Math. Soc. 158, 143 (1971).

291

AN R-LINEAR PROBLEM WITH DERIVATIVES FOR DOUBLY PERIODIC FUNCTIONS AND ITS APPLICATION ´ PIOTR DRYGAS Department of Mathematics, University of Rzeszow, ul. Rejtana 16a, 35-959 Rzeszow, Poland E-mail: [email protected] We discuss the R-linear conjugation problem for a disc in a class of doubly periodic functions with derivatives, i.e., the R-linear problem with derivatives on the torus. We prove that this problem can be solved by the method of successive approximations in a Hardy-type space under some natural conditions. Keywords: R-Linear problem, functional equations, boundary value problem, Hardy space.

1. Introduction Let D+ be a simply connected domains in the complex plane C bounded by smooth curves ∂D + , D be the complement of all closures of ∂D + to the extended complex plane C ∪ {∞}. Let ∂D + be orientated in counter clockwise direction. Let a(t), b(t) and c(t) be given H¨ older continuous function on ∂D+ ; a(t) 6= 0. The R-linear conjugation problem on C is states as follows. To find a function ϕ(z) analytic in D, D + , continuous in the closures of the considered domains with the following conjugation condition ϕ+ (t) = a(t)ϕ− (t) + b(t)ϕ− (t) + c(t),

t ∈ ∂D + .

(1)

The problem (1) in class of doubly periodic functions has application in composite. Mityushev in [13] shown that the condition (1) with approat coefficient a, b, c express the perfect contact between different materials. One can find constructive results for the effective conductivity tensor in [12]. The perfect contact condition is an idealization which is valid in a large variety of situations. However, imperfect contact is known to exist in composites. It is expressed by a discontinuity in the temperature distribution between matrix and inclusions. The effect of this interface phenomenon

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on the effective conductivity has been studied by Lipton ([9]), Torquato & Cheng ([2]), Benveniste & Miloh ([1]), Goncales & Kolodziej ([7]). The present paper is devoted to R-linear problem in a class of doubly periodic functions with derivatives ′

ϕ− (t) = a(t)ϕ+ (t)−b(t)ϕ+ (t)+c(t)(ϕ+ (t))′ +d(t)(ϕ+ (t)) ,

|t| = r, (2)

where a(t), b(t), c(t), d(t) be a given H¨ older continuous function on ∂D + . The condition (2) corresponds to the thermal resistance condition introduced by Kapitza ([1]). In contrast with the contributions, which were previously mentioned, we apply the method of functional equations ([10,12]). A simple analytical algorithm to obtain the coefficients is presented in Sec. 5. This paper is organized as follows. The problem is presented in Sec. 2. Complex potential and functional equations which govern the local field are introduced in Sec. 3. Corresponding spaces and operators is introduced in Sec. 4. Solution to functional equations is given in Sec. 5. 2. Statement of the problem It is convenient to use the complex variable z = z + iy, where i is the imaginary unit. Let Z[i] be a set of complex numbers with integer real and imaginary parts. Consider a square lattice Q which is defined by two fundamental translation vectors expressed in the form of the complex numbers 1 and i. Introduce the zero-th cell Q0 := {z = t1 + it2 : −1/2 < tj < 1/2 (j = 1, 2)}. The lattice Q consists of the cells Qm := {z ∈ C : z − m ∈ Q0 }, where m ∈ Z[i]. Let the disc D + = {z ∈ C : |z| < r} lies in the cell Q0 , D− be the complement of the closure of D + to Q0 . Let n = (n1 , n2 ) denote the outward unit normal vector to the circle ∂ the normal derivative to ∂D + . Then according to the direction ∂D+ , ∂n of the normal vector the signs “+” and “−” are assigned to the domains D+ and D− . We have to find a function u+ (z) harmonic in D + and u− (z) harmonic in D − , continuously differentiable in the closures of the considered domains with the following conjugation conditions ∂u+ ∂u− (t) = λ (t), ∂n ∂n λ

∂u+ (t) + γ(u+ (t) − u− (t)) = 0, ∂n

t ∈ ∂D + .

(3) (4)

Hereafter, the letter z is used for a variable in domains, t for a variable on curves. The relation (3) and (4) model the imperfect thermal contact between inclusion and matrix [4].

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293

Let the function u− (z) has the unit jump per a cell along the x–axis and is periodic along the y–axis u− (z + 1) − u− (z) = 1,

u− (z + i) − u− (z) = 0.

(5)

3. Complex potentials and boundary value problem Introduce the complex potentials ϕ± (z) = u± (z) + iv ± (z)

(6)

analytic in D + , D− , continuously differentiable in the closures of the considered domains. Two real relations (3) and (4) yield the complex relation (see for details [3,10])   r2 + λ+1 + ′ + ′ − + ϕ (t)−ρϕ (t)+µt(ϕ (t)) +µ (ϕ (t)) , |t| = r, (7) ϕ (t) = 2 t where

ρ=

λ−1 , λ+1

µ=

1+ρ . 2rγ

(8)

For µ = 0, (7) becomes the well-studied R–linear conjugation problem ([10,13]). The relations (5) in terms of the complex potentials become ([13]) ϕ− (z + 1) − ϕ− (z) = 1 + id1 ,

ϕ− (z + i) − ϕ− (z) = id2 ,

(9)

where d1 and d2 are undetermined real constants. Differentiating (7) along the curve ∂D + and using the relation ([10, p. 52])   2 ′  2  r 2 r r =− , (10) ϕ+ ψ+ z z z we arrive at the following condition   r 2 λ+1 − ψ (t) = (1 + µ)ψ + (t) + (ρ − µ) ψ + (t)+ 2 t #  2 2 r + ′ |t| = r. [ψ + (t)]′ , +µt[ψ (t)] − µt t

(11)

After differentiation the conditions (9) become ψ − (z + 1) − ψ − (z) = 0,

ψ − (z + i) − ψ − (z) = 0.

Lemma 3.1. The function ψ + (z) satisfies the relations ψ + (z) = ψ + (z)

and

ψ + (z) = ψ + (−z),

|z| ≤ r.

(12)

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The proof of Lemma 3.1 follows from the symmetry of the problem (11). It follows from lemma that the Taylor expansion of ψ + (z) has the form ψ + (z) =

∞ X

ψ2k z 2k ,

(13)

k=0

where the coefficients ψ2k are real. 4. Spaces and operators We introduce the space CA of function analytic on D + and continuous
in the closure of on D + , endowed with the norm kf kCA = sup f reiθ . Let 0≤θ≤2π

1 CA be the space of functions analytic in D + and continuously differentiable in the closure of D + , endowed with the norm kf kCA1 = kf kCA + kf ′ kCA . We define the Hilbert spaces L2 (∂D+ ) of functions f (t) which satisfy
iθ 2 the condition f re ∈ L (0, 2π) with respect to the variable θ ∈ (0, 2π), R 2π
f reiθ 2 dθ. endowed with the norm kf k2 2 + = L (∂D )

0

+ The space H2 is introducedR as the space
of analytic function on D , 2π iθ 2 satisfying the condition sup 0 |f r0 e | dθ < ∞, endowed with the 0 2, Q(z) is a given square matrix of order n of the following special quasidiagonal form: Every block Qr = (qik )r is a lower (upper) triangular matrix satisfying the conditions r r = qr , = · · · = qm q11 s ,ms r r qik = qi+s,k+s

(i + s ≤ n,

|q r | ≤ q0 < 1,

k + s ≤ n);

moreover Q(z) ∈ Wp1 (C), p > 2 and Q(z) ≡ 0 outside of some circle of the complex plane C. By a solution of system (1), we mean a so-called regular solution, i.e., w(z) ∈ L2 (D), wz , wz ∈ Lλ (D′ ), λ > 2, D ′ ⊂ D. System (1) is to be fulfilled almost everywhere on D. The following equation ∂z Ψ − ∂z (Q′ Ψ) − A′ (z) Ψ − B ′ (z) Ψ = 0 ∗ Corresponding

author.

(2)

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is called the conjugate equation of equation (1), where prime ′ denotes the transposition of the matrices. When A = B = 0, system (1) takes form (1′ )

∂z w − Q(z) ω(z) = 0.

The solutions of equation (1′ ) are called Q-holomorphic vectors. The theory of generalized analytic vectors was constructed by B. Bojarski [1]. These results and some further development of this theory are presented in [4]. 2. Boundary Value Problems for Elliptic System (1) on the Plane with Angular Points We introduce suitable classes of generalized analytic vectors. Let Γ be a piecewise smooth curve. Denote by Ep± (Γ, Q, A, B, ρ) the class of solutions of the system (1) representable by generalized Cauchy type integrals w(z) =

1 2πi

Z n N o X ck Wk (z), (3) Ω1 (z, t) dQ tϕ(t) − Ω2 (r, t) dQ tϕ(t) + Γ

k=1

where ck are arbitrary real constants, Wk (z) are continuous vectors on the whole plane vanishing at infinity, {Wk } (k = 1, N ) forms a complete system of linearly independent solutions of the Fredholm equation Z   1 v(t, z) A(t) w(t) + B(t) W (t) dσt = 0, Kw ≡ w(z) − (4) π D

the matrix v(t, z) is generalized Cauchy kernel for the equation (1′ ), the m Q |t−tk |νk , −1 < νk < p−1, p > 1, ϕ(t) ∈ Lp (Γ, ρ) weight function ρ(t) = k=1

and satisfies the condition Z
dQ t ϕ(t), Ψj = 0 (j = 1, N ), Im Γ


dQ t = I dt + Q dt ,

(5)

here Ψj form a similar system for the conjugate equation (2), Ω1 and Ω2 are the fundamental kernels of (1) representable by the resolvent Γ1 and Γ2 Z Γ1 (z, τ ) v(t, τ ) dστ , Ω1 (z, t) = v(t, z) + D Z (6) Γ2 (z, t) v(t, τ ) dστ . Ω2 (z, t) = D

Some Problems for Elliptic Systems on the Plane

305

Denote by Eq± (Γ, Q′ , −A′ , −B ′ , ρ1−q ), q = p/p − 1 the class of solutions of the equation (2) representable in the form Z n N o X 1 ck Ψk (z), (7) Ψ(z) = Ω1′ (t, z) dQ′ th(t) − Ωr′ (t, z) dQ′ th(t) + 2πi Γ k=1

1−q

where the density h(t) ∈ Lq (Γ, ρ ) satisfies the conditions Z
dQ′ t h(t), Wj (t) = 0 (j = 1, N ). Im

(8)

Γ

Our model problem is the following: Find a vector w(z) = (w1 , . . . , wn ) ∈ Ep± (Γ, Q, A, B, ρ) satisfying the boundary condition w+ (t) = a(t) w− (t) + b(t) w− (t) + c(t),

(9)

almost everywhere on Γ. Γ is piecewise smooth closed curve, the knot points of Γ (where Γ loosed the smoothness) are included in the set of {tk } points; a(t) and b(t) are given piecewise continuous n × n-matrices on Γ, inf | det a(t)| > 0 and c(t) is given vector of the class Lp (Γ, ρ). The boundary value problem (9) for holomorphic vectors was studied by A. Markushevich and is called the generalized Hilbert problem [8]. The boundary condition (9) contains the conjugate value of the desired vector. Therefore, the Noetherity condition and the index formula of this problem depend on the values of the angles at the knot points of the boundary curve Γ. Substituting the integral representation formula (3) into the boundary condition (9) for the unknown vector ϕ(t), we obtain the following singular integral equation (M ϕ)(t) = f (t), f (t) = 2c(t) + 2

(10) N X

k=1

ck [a(t) − I] Wk (t) + b(t) Wk (t)

and the solution is subject to conditions (5). It is easy to see that problem (9) is Noetherian in the class Ep± (Γ, Q, A, B, ρ) if and only if the singular integral operator (M ϕ)(t) is Noetherian in Lp (Γ, ρ). The necessary and sufficient Noetherity condition for the integral operator M ϕ is [2] (11) inf 1 det M (tk , ξ) > 0 (k = 1, m), t∈Γ, ξ∈R

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where the matrix M (tk , ξ) is defined in the following way: In the points different from the knot points M (tk , ξ) = (1 + sgn ξ) I + (1 − sgn ξ) a(t), and in the knot points   (1−s0 )I +(1+s0 )a(t−0) s1 (I −a(t−0)) , M (tk , ξ) = s2 (I −a(t+0)) (1+s0 )I +(1+s0 )a(t+0)

(12)

(13)

where s0 = cos thω, s1 = e(α−1)ω / sin hω, s2 = e(1−α)ω / sin hω, α = η/π, and η is the angle between the tangents at the knot points, 0 < η < 2π, 1+ν ωj = π(iβj + ξ), βj = p j . If the Noetherity condition (11) is fulfilled, then the necessary and sufficient solvability conditions are Z
f (t), dQ f Ψk (t) = 0, (14) Im Γ

where Ψk (z) form a complete system of linearly independent solutions of the homogeneous problem Ψ+ (t) = a′ (t) Ψ− (t) + b(t)[t′ ] + Q′ (t) [ t ′ ] Ψ− (t)

(15)

for the equation (2) in the class Eq± (Γ, Q′ , −A′ , −B ′ , ρ1−q ). Starting from the properties of Mellin transform and the hyperbolic trigonometric functions, we define the index of singular integral operator and therefore the index of our problem (9) in Eq± (Γ, Q, A, B, ρ). 3. One Class of Two-Dimensional Third-Kind Fredholm Integral Equation We have correct setting and complete analysis of boundary value problem for sufficiently wide class of equations. They are first order singular equations. In the works of L. Mikhailov and Z. Usmanov the integral equations with first order singularities are investigated. Further development of this theory is given in the works of A. Tungatarov, R. Akhmedov, M. Reissig, A. Timofeev, H. Begehr, D. Q. Dai and etc. Somewhat different equations were studied by L. Vinogradov and N. Bleiv. The equations of higher order singularities are undoubtedly of much more theoretical and practical interest. In the works of Kh. Najhmidinov and Z. Usmanov of such type equations are investigated, however the equations studied below cannot be reduced to them.

Some Problems for Elliptic Systems on the Plane

307

Let us consider the system m X

z νk Ak

k=0

∂kw =0 ∂z k

(16)

in some domain G of z plane, where m, ν are given natural numbers, Ak (0 ≤ k ≤ m) are given complex n × n-matrices; G is deleted neighborhood of z = 0. Assume that det λ0 6= 0,

det Am 6= 0,

Ak · Aj = Aj · Ak ,

0 ≤ j,

k ≤ m.

(17)

Construct all possible polynomials of the form τm ζ m + τm−1 ζ m−1 + · · · + τ1 ζ + τ0 = 0,

(18)

where the coefficient τk is some eigen-value of the matrix Ak (0 ≤ k ≤ n). Denote by ∆ the set of all complex roots of these polynomials and introduce a number δ0 = max |ζ|, obviously δ0 > 0. ζ∈∆

Along with the solution w(z) of the system (16) construct its characteristic function n m−1 X X ∂ p ωk iQ Tw (ρ) = max (ρ e ) , ρ > 0. (19) p 0≤ϕ≤2π ∂z p=0 k=1

Theorem 3.1. Let ν ≥ 2 and ψ(z) be some analytic function in G. Let the solution w(z) of the system (16) satisfy the condition    δ , z → 0, (20) Tw (z) = O |ψ(z)| exp |z|σ where δ is some number and σ < ν − 1. Then the solution w(z) is identically the zero vector function. Moreover, when condition (20) is fulfilled then w(n) is also trivial if   ν −3 . (21) σ = ν − 1, δ < δ0 cos πβ, β = max ν, 2ν − r Note that in particular, when ν = 2 for system (16) we are succeeded to state correct boundary value problem to take its complete analysis. Example 3.1. Consider the integral equation z ν w + pn w = f,

(22)

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where ν > 0 is given integer, n is natural number, f is a given function on bounded domain G containing the origin, ZZ a1 (ζ) w(ζ) 1 dG(ζ), p1 w(z) = − π ζ−z G .. (23) . ZZ 1 an (ζ) pn−1 (w(ζ)) pn w(z) = − dG(ζ), π ζ −z G is the sequence of singular integral operators, w is desired function; by ak = ak (z) (1 ≤ k ≤ n) are denoted the holomorphic in G and continuous in G functions. Here we mean that ν (24) m = ≥ 2. n Consider now the homogeneous integral equation z ν w + pn w = 0.

(25)

By a solution of this equation, we mean a function w ∈ Lp (G), p > 2, satisfying (25) almost everywhere in G. From the definition and on the basis of known properties of the operator p1 w [7] it follows that every solution of the equation (25) is continuous in every point of G, except maybe the origin. Moreover, the function w0 (z) ≡ z w(z),

z ∈ G,

(26)

has continuous derivatives of order n − 1 in G and the function ∂ n−1 w , z ∈ G, ∂z n−1 has generalized derivative with respect to z and w1 (z) ≡ z ν

(27)

(w1 )z + R(z) w = 0, where R(z) =

n Q

k=1

(28)

ak (z), z ∈ G.

Assume that there exists a holomorphic function b(z) in domain G, such that  n b(z) = −R(z), z ∈ G and b(0) 6= 0. (29) Denote by

αk = cos

2π(k − 1) 2π(k − 1) + i sin , n n

1≤k≤n

Some Problems for Elliptic Systems on the Plane

309

all possible roots from 1 of order n. Taking into account (24), (27), (28) and (29) we obtain that every solution of the homogeneous equation (25) is representable in the form w(z) =

m X

k=1

Qk (z),

z ∈ G \ {0},

(30)

where Qk (z) = Φk (z) exp{αk c(z) z}, c(z) = b(z) z m and Φk (z) (k = 1, n) are holomorphic in G \ {0} functions. It follows from (30) the validity of the representation n

∂q w X q = αk c(z)q Qk (z) (1 ≤ q ≤ n − 1), ∂z q

(31)

A · Ψ(z) = Ω(z),

(32)

k=1

which simultaneously with (30) admit the matrix form


Ψ(z) = column Q1 (z), Q2 (z), . . . , Qn (z) ,   ∂ n−1 w 1 1 ∂w Ω(z) = column w, , ,..., c(z) ∂z c(z)n−1 ∂z n−1   1 1 ··· 1  α1 α2 · · · αn    2 2 2  A=  α1 α2 · · · αn  .  . . . . . . . . . . . . . . . . . . .

(33)

α1n−1 α2n−1 · · · αnn−1

From (32) we have that every function Qk (k = 1, n) is a linear combination of the components of the vector (33). Therefore, in the neighborhood of z = 0, the following estimation holds  1  Q(z) = O ν , z → 0. (34) z Consequently, we get the next result. Theorem 3.2. The homogeneous integral equation (25) has no nontrivial solutions in Lp (G), p > 2. Let us consider the inhomogeneous case (22), when the right hand side function is s-analytic in G f (z) = F0 (z) + z F1 (z) + · · · + z s−1 Fs−1 (z),

(35)

here s is a natural number, Fk (0 ≤ k ≤ s − 1) are arbitrary holomorphic functions in G. The following theorem is valid.

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Theorem 3.3. The inhomogeneous integral equation (22) is unsolvable in the class Lp (G), p > 2, if the right hand side function has the form (35), where at least one function Fk (0 ≤ k ≤ s − 1) is not identically zero and s ≤ n. Remark 3.1. We have established that the homogeneous integral equation (25) and the inhomogeneous equation (22) with sufficiently general righthand sides have no solutions. Note that the exact description of the image of our integral operator for which equation (22) will have solutions is very complicated. References 1. B. V. Bojarski˘ı, Theory of generalized analytic vectors. (Russian) Ann. Polon. Math. 17 (1966), 281–320. 2. R. Duduchava, On general singular integral operators of the plane theory of elasticity. Rend. Sem. Mat. Univ. Politec. Torino 42 (1984), no. 3, 15–41. 3. R. V. Duduchava, General singular integral equations and fundamental problems of the plane theory of elasticity. (Russian) Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 82 (1986), 45–89. 4. R. P. Gilbert, J. L. Buchanan, First order elliptic systems: A function theoretic approach. Mathematics in Science and Engineering, 163. Academic Press, Orlando, FL, 1983. 5. G. Makatsaria, Correct boundary value problems for some classes of singular elliptic differential equations on a plane. Mem. Differential Equations Math. Phys. 34 (2005), 115–134. 6. G. Manjavidze, Boundary value problems of the theory of generalized analytic vectors. Functional analytic methods in complex analysis and applications to partial differential equations (Trieste, 1993), 13–51, World Sci. Publ., River Edge, NJ, 1995. 7. I. Vekua, Generalized analytic functions, Pergamon, Oxford, 1962. 8. N. P. Vekua, Systems of singular integral equations. Noordhoff, Groningen 1967.

311

NEUMANN PROBLEM FOR GENERALIZED POISSON AND BI-POISSON EQUATIONS ¨ AKSOY U. Department of Mathematics, Atılım University, Ankara, Turkey E-mail: [email protected] A. O. C ¸ ELEBI˙ ˙ Department of Mathematics, Yeditepe University, Istanbul, Turkey E-mail: [email protected] Using the harmonic and biharmonic Neumann functions N1 (z, ζ) and N2 (z, ζ), and operators having these functions as kernels, the Neumann problem for second and fourth order complex partial differential equations with main parts as the harmonic and biharmonic operators, are discussed. Solvability conditions are obtained. Keywords: Neumann Problem; Neumann Functions; Poisson Equation; BiPoisson Equation; Singular Integral Operators.

1. Introduction The Neumann problem is investigated for several complex model equations. These equation include the inhomogeneous Cauchy-Riemann, Bitsadze, Poisson and bi-Poisson equations; see, e.g, [3–8]. The main subject of this paper is to investigate the solvability of the Neumann problem for linear general second and fourth order complex partial differential equations with harmonic and biharmonic operators as main parts, which will be called as generalized Poisson and generalized bi-Poisson equations, respectively. In Section 2 we give a review of the harmonic Neumann and bi-harmonic Neumann functions and the Neumann problems defined for Poisson and bi-Poisson equations. In Section 3, the operators related to these problems are introduced and some particular properties are discussed. In the last section, we consider the Neumann problems for generalized Poisson and bi-Poisson equations and we give the conditions for solvability of the problems.

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¨ Aksoy & A. O. C U. ¸ elebi

2. Neumann Functions and Neumann Problems for Poisson and Bi-Poisson Equations The harmonic Neumann function for the unit disc D is ¯ |2 . N1 (z, ζ) = log | (ζ − z)(1 − z ζ)

(1)

Its outward normal derivative is ∂νz N1 (z, ζ) = (z∂z + z¯∂z¯)N1 (z, ζ) = 2

(2)

for z ∈ ∂D, ζ ∈ D, and it satisfies Z 1 dz =0 N1 (z, ζ) 2πi ∂D z

for ζ ∈ D [3,5]. Neumann function is defined slightly different in [11]. Biharmonic Neumann function N2 (z, ζ) is attained by convolution of the harmonic Neumann function as ZZ 1 ˜ 1 (ζ, ˜ ζ)dξd˜ ˜ η N1 (z, ζ)N (3) N2 (z, ζ) = π D

¯ and also called the Neumann function of second order [5]. For z and ζ in D with z 6= ζ ¯ |2 −4] − (1− | z |2 )(1− | ζ |2 ) N2 (z, ζ) = | ζ − z |2 [log | (ζ − z)(1 − z ζ) ˜2 (z, ζ) + 2(2 − z ζ¯ − z¯ζ) − N where Z dζ˜ ¯ ˜ ˜2 (z, ζ) = − 2 [(2−z ζ˜− z¯ζ) log | ζ˜−z |2 −(1+ | z |2 )] log | ζ˜−ζ |2 N 2πi ∂D ζ˜  2 ZZ  1 1− | z | 1 ¯˜ |2 dξd˜ ˜ η . (4) − 1 log | (ζ˜ − ζ)(1 − ζ ζ) + − ¯ ˜ π ˜ 1 − z¯ζ D 1 − zζ

Evaluating the integrals, another form of the biharmonic Neumann function is given by Begehr in [6,7] as ∞ X   1 ¯ |2 − 4 ¯ k + (¯ N2 (z, ζ) =| ζ − z |2 4 − log | (ζ − z)(1 − z ζ) [(z ζ) z ζ)k ] k2 k=2   ¯ ¯ 2 − (1 + |z|2 )(1 + |ζ|2 ) log(1−z ζ) + log(1− z¯ζ) . −2(z ζ¯ + z¯ζ) log |1−z ζ| z¯ζ z ζ¯

The biharmonic Neumann function satisfies ∂z ∂z¯N2 (z, ζ) = N1 (z, ζ) in D,

∂νz N2 (z, ζ) = 2(| ζ |2 −1) on ∂D

(5)

Neumann Problem for Generalized Poisson and Bi-Poisson Equations

313

for any ζ ∈ D. Also, the normalization condition Z 1 dz =0 N2 (z, ζ) 2πi ∂D z

holds for ζ ∈ D. The following problems for the Poisson and bi-Poisson equations are posed and their unique solutions are obtained by Begehr [6,7]. Neumann problem. Find the solution to ∂z ∂z¯w = f in D, ∂ν w = γ1 on ∂D,

1 2πi

Z

w(z)

∂D

dz = c, z

for f ∈ L (D), γ1 ∈ C(∂D), c ∈ C. The solution to Neumann problem is Z ZZ 1 dζ 1 − γ1 (ζ) log |ζ − z|2 N1 (z, ζ)f (ζ)dξdη w(z) = c − 2πi ∂D ζ π D 1

(6)

if and only if 1 2πi

Z

2 dζ = γ1 (ζ) ζ π ∂D

ZZ

f (ζ)dξdη.

D

Neumann-2 problem. Find the solution to (∂z ∂z¯)2 w = f in D , f ∈ Lp (D), 1 < p < ∞ , ∂ν w = γ1 , ∂ν ∂z ∂z¯w = γ2 on ∂D , γ1 , γ2 ∈ C(∂D). 1 2πi

Z

∂D

w(ζ)

1 dζ = c1 , ζ 2πi

Z

∂D

wζ ζ¯(ζ)

dζ = c 2 , c1 , c2 ∈ C . ζ

The unique solution is given as Z 1 dζ w(z) = c1 − (1− | z |2 )c2 − {N1 (z, ζ)γ1 (ζ) + N2 (z, ζ)γ2 (ζ)} 4πi ∂D ζ ZZ 1 N2 (z, ζ)f (ζ)dξdη + π D if and only if Z ZZ 1 2 dζ = 2c2 + γ1 (ζ) (| ζ |2 −1)f (ζ)dξdη , 2πi ∂D ζ π D Z ZZ 2 dζ 1 = γ2 (ζ) f (ζ)dξdη . 2πi ∂D ζ π D

(7)

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3. Integral Operators Related to Harmonic and Biharmonic Neumann Problems In this section, we introduce the operators Pk,l related to Poisson and biPoisson equations. Uniform boundedness and uniform continuity of them are obtained in case k + l < 4. For the case k + l = 4, Lp boundedness of Pk,l is proved. Using the second order Neumann functions defined in the previous section, we introduce the following operators. Definition 3.1. For k, l ∈ N0 with (k, l) 6= (2, 2) and k + l ≤ 4 ZZ 1 ∂zk ∂z¯l N2 (z, ζ)f (ζ)dξdη Pk,l f (z) = π D

where f ∈ L1 (D). If k + l < 4, then the operators Pk,l are weakly singular linear operators and if k + l = 4, then they are strongly singular operators. Using this definition, we can write ZZ ZZ 1 1 ¯ 2 f (ζ)dξdη, N1 (z, ζ)f (ζ)dξdη = log |(ζ − z)(1 − z ζ)| P1,1 f (z) = π D π D  ZZ ZZ  1 1 ζ¯ 1 P2,1 f (z) = + ∂z N1 f (ζ)dξdη = − ¯ f (ζ)dξdη, π D π D (ζ −z) (1−z ζ) ZZ 1 P3,1 f (z) = ∂ 2 N1 f (ζ)dξdη π D z  ZZ  ζ¯2 1 1 + =− ¯ 2 f (ζ)dξdη . π D (ζ − z)2 (1 − z ζ)

ˆ 0, Π ˆ 1 and P3,1 is the P1,1 and P2,1 are modified forms of the operators Π ˆ operator Π2 given by Vinogradov [12] which are also stated by Vekua [11] in his book. Lemma 3.1. Let f ∈ Lp (D), p > 2 and k + l < 4. Then, for z ∈ D.

|Pk,l f (z)| ≤ Ckf kLp (D)

ζ −z ≤ 1 for |z| ≤ 1, |ζ| < 1, it can be Proof. Using the fact that 1 − z ζ¯ shown that k∂zk ∂z¯l N2 (z, ζ)kLq (D) < C

Neumann Problem for Generalized Poisson and Bi-Poisson Equations

315

for q < 2 and for some constant C, where ¯ 2 − 4) ∂z N2 = −(ζ − z)(log |(ζ − z)(1 − z ζ)|   ζ¯ 1 + − |ζ − z|2 + z¯(1 − |ζ|2 ) − 2ζ¯ − ∂z N˜2 , ζ − z 1 − z ζ¯   ζ¯2 2(ζ − z) 2(ζ − z)ζ¯ 1 2 2 ˜ + + ∂z2 N2 = − |ζ − z| ¯ 2 − ∂z N2 , (ζ − z) (ζ − z)2 1 − z ζ¯ (1 − z ζ)   1 ζ¯3 3(ζ − z) 3(ζ − z)ζ¯2 3 ˜ 2 3 ∂z N2 = + ¯ 2 − 2|ζ − z| (ζ −z)3 + (1−z ζ) ¯ 3 − ∂z N2 (ζ − z)2 (1 − z ζ) and ∂zk ∂z¯l N2 = ∂z¯k ∂zl N2 for k = 1, 2, 3 where N˜2 is given by (4). The required result follows from the use of H¨ older’s inequality. The following lemma proves the uniform continuity of the operators Pk,l for k + l < 4. Lemma 3.2. Let f ∈ Lp (D), p > 2 and k + l < 4. Then for z1 , z2 ∈ D,  |z1 − z2 |(p−2)/p , if k + l = 3, |Pk,l f (z1 ) − Pk,l f (z2 )| ≤ Ckf kLp(D) |z1 − z2 |, otherwise. Proof. The result is achived for the case k + l 6= 3 by the use of the mean value theorem since we have ∂z Pk,l = Pk+1,l ,

∂z¯Pk,l = Pk,l+1

which are known to be bounded. To prove the result for the remaining operators, we use the identity in [2, p. 680] and H¨ older’s inequality. The strongly singular operators Pk,l for k + l = 4 are shown to be bounded on Lp (D) in the next lemma. Lemma 3.3. For f ∈ Lp (D) and k + l = 4, we have Pk,l f ∈ Lp (D) and kPk,l f kLp (D) ≤ Ap kf kLp(D)

(p > 1)

with kP1,3 kL2 (D) = kP3,1 kL2 (D) = 1 .

(8)

ˆ 2 is bounded in Lp (D) and kΠ ˆ 2 kL2 (D) = 1. Proof. It is proved in [11] that Π p Thus the L boundedness of the operators P3,1 and P1,3 and equality (8) follows.

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P4,0 has the kernel ∂z4 N2 given by   ζ¯4 8(ζ − z) 8(ζ − z)ζ¯3 1 4 ˜ 2 + = ¯ 3 − 6|ζ − z| (ζ − z)4 + (1 − z ζ) ¯ 4 − ∂z N2 (ζ − z)3 (1 − z ζ)

∂z4 N2

and P0,4 has the kernel ∂z4 N2 = ∂z¯4 N2 . They can be decomposed as P4.0 f = T−2,2 f + S4,0 f and P0,4 f = T2,−2 f + S0,4 f where the operators T−2,2 and T2,−2 are known to be bounded in Lp and unitary in L2 space [2] and the operators S4,0 and S0,4 are shown to be bounded in Lp by the use of Schur’s Theorem and Forelli-Rudin type estimates, see [9]. 4. Neumann Problems for Generalized Poisson and Bi-Poisson Equations We start with the problem for a generalized Poisson equation with the Neumann conditions. Problem N1. Find w ∈ W 2,p (D) as a solution to the second order complex differential equation  X  (1) ∂2w ∂2w ∂2w (2) qkl (z) l k + qkl (z) l k + ∂z∂ z¯ k+l=2 ∂ z¯ ∂z ∂z ∂ z¯ k6=l

+

X 

akl (z)

k+l 1. As we mentioned above the characterization of the weight pairs (υ, ω) providing the boundedness of the operators Rα and Wα from Lpω (0, ∞) to Lqυ (0, ∞) under the given conditions are not new, but there are many proof techniques which the authors used in previous studies. In this context our approach to the problem can be considered different. We denote the weighted Lebesgue space by Lpω (0, ∞), where 1 < p < ∞. It consists of all measurable functions f (x) on (0, ∞) such that kf kp,ω = kf kLpω =

Z

∞ 0

1/p |f (x)| ω (x) dx < ∞. p

(1)

The measurable functions υ, ω : (0, ∞) → (0, ∞) are called weight functions. We use the notation Z ∞ Z x ′ υ (t) dt (2) ω 1−p (t) dt , V (x) = W (x) = (1−α)q t x 0 and assume that W and V satisfy x V ∈ D : ∃ η > 0, V ≤ 2η V (x) for x > 0, 2 x ≤ 2−δ W (x) for x > 0. W ∈ RD : ∃ δ > 0, W 2

(3) (4)

Moreover, we use

W (x) =

Z

∞

′

ω 1−p (t) dt , V (x) =

x

Z

x 0

υ (t) t(1−α)q

dt

(5)

and assume that W and V satisfy V ∈ RD : ∃δ > 0,

W ∈ D : ∃η > 0,

V W

x 2


x 2




≤ 2−δ V (x) for η

≤ 2 W (x)

x > 0,

(6)

for x > 0.

(7)

Solution of Weighted Problems for Riemann-Liouville and Weyl Operators

323

2. Main results Theorem 2.1. Let 1 < p ≤ q < ∞, 1/q < α < 1 or α > 1. The inequality kRα f kLqυ ≤ A kf kLpω , where A > 0 does not depend on f , is fulfilled if and only if  1/q Z t 1/p′  Z ∞ p υ (x) 1−p′ ′ . dx ω (x) dx < ∞, p = B := sup p−1 x(1−α)q t>0 0 t

Morever, if A is the best constant, then A ≈ B.

Proof. Sufficiency.  Let f ∈ Lp and Ωxn = 2−n−1 x, 2−n x , n ∈ N. Let’s first write Z 2−n x Z 2−m−1 x m P α−1 α−1 (Rα f ) (x) = (x − t) f (t) dt + (x − t) f (t) dt. n=0

2−n−1 x

0

Now, applying norm we obtain kRα f kLqυ ≤

Z m X

∞

n=0

0

+

Z

υ (x)

Z

2−n x

2−n−1 x

∞

υ (x)

0

Z

(x − t)

α−1

2−m−1 x

(x − t)

0

f (t) dt

α−1

!q

f (t) dt

!1/q

dx

!q

!1/q

dx

= Jn + Jm . First, let us show that Jn is finite. If we take (x − t)α−1 out of the inner integral, then we have !q q(α−1) Z ∞  Z 2−n x υ (x) 1 q f (t) dt dx. Jn ≤ 1 − n 2 x(1−α)q 2−n−1 x 0 Let h (t) = ω (t) Jnq ≤ C1 = C1

Z

∞

0

Z

∞

0

υ (x) x(1−α)q υ (x) x(1−α)q

q/p′

f (t) and apply H¨ older’s inequality to get !q/p Z −n Z −n 2

x

2−n−1 x

Z

n Ωx

2

p

|h (t)| dW (t) p

|h (t)| dW (t)

!q/p

x

!q/p′

dW (t)

2−n−1 x

dx

′

W (Ωxn )q/p dx,

1 q(α−1) ) . But from (4) we have 2n
′ ′ q/p′ q/p′ ≤ (W 2−n x )q/p ≤ 2−nδq/p W (x) ≤ 2−n(α−1)/q W (x)

where C1 = (1 − W (Ωxn )

p′ −1

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M. Avcı, S. O˘ gra¸s & R. A. Mashiev

since δ > 0 is an arbitrary number and δq/p′ ≥ (α − 1) /q. So !q/p Z ∞ Z ′ dV (x) q/p p Jnq ≤ −C1 2−n(α−1)/q V (x) W (x) |h (t)| dW (t) . V (x) n 0 Ωx (8)
1/p R∞ p p ≤ C and Now, let us assume that kf kLω = 0 |f (t)| ω (t) dt 1/p R p q/p′ ≤ C. that B1 = V (x) W (x) < ∞ hold. Then Ω n |f (t)| ω (t) dt x R q/p R Moreover, we can write Ω n |f (t)|p ω (t) dt ≤ C q−p Ω n |f (t)|p ω (t) dt. x

x

Now, the inequality (8) together with the differentials of V (x) and Z ∞ Z 2−n x dV (x) p q , where W (x), imply that Jn ≤ −C2 |f (t)| ω (t) dt V (x) 0 2−n−1 x C2 = C1 C q−p 2−n(α−1)/q . Applying Fubini theorem, we have Z ∞ Z 2n+1 t dV (x) p q Jn ≤ −C2 dt. Since V (x) satisfies (3), we |f (t)| ω (t) V (x) n 0 2 t Z 2n+1 t dV (x) V (2n t) can write − = log ≤ η log 2, t > 0, n ∈ N. V (x) V (2n+1 t) 2n t Eventually we obtain Z ∞ 1/q Jnq ≤ η log 2C2 |f (t)|p ω (t) dt, Jn ≤ (η log 2)1/q C2 C p/q = C3 , (9) 0

1/q

where C3 = C(1 − 2−n )(α−1) 2−n(α−1) (η log 2) . Now let us show Jm is finite. From Lebesgue’s convergence theoR 2−m−1 x R∞ α−1 α−1 rem, since we have 0 (x − t) f (t) dt ≤ 0 (x − t) f (t) dt ∈ L1 (0, ∞), as m → ∞, we obtain immediately !q !1/q Z Z −m−1 ∞

Jm =

2

x

0

α−1

(x − t)

υ (x)

0

f (t) dt

dx

→ 0.

(10)

As a result, (9) together with (10) imply that kRα f kLqυ < ∞. Necessity. 1/q Assume that kRα f kLqυ < ∞ holds. Let B2 = B and choose test func′  R 1/p t 1−p′ ∈ Lp . Then the right-hand dx tions ft (x) = χ(0,t) (x) 0 ω (x) q

side will be smaller than kRα f kLqυ . So we have q kRα f kLqυ

≥ =

Z

∞

υ (x)

t

Z

t

Z

t

0

∞

υ (x)

Z

0

t

α−1

(x − y)

α−1

(x − y)

Z

t

1−p′

ω (x)

0

dy

q Z

0

1/p′ !q dx dy dx

t

ω (x)

1−p′

q/p′ dx dx

Solution of Weighted Problems for Riemann-Liouville and Weyl Operators

1 = q α

Z

∞

t

325

 Z t q/p′ 1−p′ υ (x) (x − (x − t) ) dx ω (x) dx . α

α q

0

α

For some positive k, it holds that (xα − (x − t) ) ≤ kxα−1 . Therefore Z ∞ 1/q Z t 1/p′ υ (x) 1−p′ dx kRα f kLqυ ≥ C4 ω (x) dx = C4 B. x(1−α)q 0 t The proof is complete. Remark 2.1. It is easy to see that, for the convergence of the series as m → ∞, the conditions p > 1 and α > 1 are necessary.

m P

C3

n=0

Theorem 2.2. Let 1 < p ≤ q < ∞, and 1/q < α < 1 or α > 1. The inequality kWα f kLqυ ≤ A kf kLpω , where A > 0 does not depend on f , is fulfilled if and only if 1/q Z ∞ 1/p′ Z t υ (x) 1−p′ dx < ∞. B := sup ω (x) dx (1−α)q t>0 t 0 x Morever, if A is the best constant, then A ≈ B. Proof. The proof is similar to the proof of Theorem 2.1 and we omit it. References 1. J. S. Bradley, “Hardy inequality with mixed norms”, Canad. Math. Bull., 21 (1978), 405-408 2. D. E. Edmunds, W. D. Evans and D. J. Harris, “Approximation numbers of certain Volterra integral operators”, J. London Math. Soc., (2), 37 (1988), 471-489. 3. I. Genebashvili, A. Gogatishvili and V. Kokilashvili, “Solution of two weigth problems for integral transforms with positive kernels”, Georgian Math. J., 3 (1996), No. 1, 319-342. 4. H. P. Heinig, “Weigted inequalities in Fourier analysis”, Nonlinear Analysis, Function Spaces and Appl. 4, Proc. Spring School, 1990, 42-85. 5. K. F. Andersen and H. P. Heinig, “Weighted norm inequalities for certain integral operators”, SIAM J. Math. Anal., 14 (1983), No. 4, 834-844. 6. V. M. Kokilashvili, “On Hardy’s inequalities in weighted spaces”(in Russian), Soobshch. Akad. Nauk Gruzin. SSR, 96 (1979), 37-40. 7. V. G. Maz’ya, Sobolev spaces, Springer, 1985. 8. A. Meskhi, “Solution of some weighted problems for Riemann-Liouville and Weyl operators”, Georgian Math. J., 5 (1998), 319-342.

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9. B. Muckhenhoupt, “Hardy’s inequality with weights”, Studia Math., 44 (1972), 31-38. 10. V. Stepanov, “Two weighted estimates for Riemann-Liouville integrals”, Report No. 39, Math. Inst., Czechoslovak Acad. Sci., 1988. 11. V. Stepanov, “Weighted inequalities of Hardy type for higher derivatives and their applications”, Soviet Math. Dokl., 38 (1989), 389-393. 12. G. Talenti, “Osservazioni spora una classe di disuguaglianze”, Rend. Sem. Mat. Fiz. Milano, 39 (1996), 171-185.

327

A POLYHARMONIC DIRICHLET PROBLEM OF ARBITRARY ORDER FOR COMPLEX PLANE DOMAINS H. BEGEHR Math. Institut, Freie Universit¨ at Berlin 14195, Berlin, Germany E-mail: [email protected] T. S. VAITEKHOVICH Belarusian State University, 220030, Minsk, Belarus E-mail: [email protected]

Dedicated to Professor Dr. Ilppo Simo Louhivaara on the occasion of his 80th birthday. The consecutive convolution of the harmonic Green function with itself provides a polyharmonic Green function of any order. Convoluting such a polyharmonic Green function with a polyharmonic Green-Almansi function gives a polyharmonic Green function of new type. It leads to a certain Dirichlet problem for a Poisson equation of certain order. This problem is well-posed. Its solution is given here. Keywords: Polyharmonic equation, Dirichlet boundary value problems, Green functions, Poisson equation.

1. Introduction The harmonic Green function G1 (z, ζ) for a regular complex domain D satisfies for any ζ ∈ D the following properties. • • • •

G1 ( · , ζ) is harmonic in D \ {ζ} and continuous in D \ {ζ}, G1 (z, ζ) + log |ζ − z| is harmonic in z in the neighborhood of ζ, G1 (z, ζ) = 0 for z ∈ ∂D, G1 (z, ζ) = G1 (ζ, z) for z, ζ ∈ D, z 6= ζ.

328

H. Begehr & T. S. Vaitekhovich

Any function w ∈ C 2 (D; C) ∩ C 1 (D; C) can be represented through Z Z 1 1 ∂νζ G1 (z, ζ)w(ζ)dsζ − G1 (z, ζ)wζ ζ (ζ)dξdη. w(z) = − 4π ∂D π D

(1)

This formula provides the unique solution to the Dirichlet problem for the Poisson equation wzz = f in D, w = γ on ∂D,

(2)

for f ∈ Lp (D; C), 2 < p, γ ∈ C α (∂D; C), 0 < α < 1. The iteration process Z 1 e 1 (ζ, e ζ)dξde e η , 2 ≤ n, Gn (z, ζ) = − Gn−1 (z, ζ)G π D

(3)

leads to a hierarchy of polyharmonic Green functions, which with respect to the Dirichlet problem and formula (1) satisfy for ζ ∈ D • Gn ( · , ζ) is polyharmonic of order n in D \ {ζ}, |ζ − z|2(n−1) log |ζ − z|2 is polyharmonic of order n in z in the • Gn (z, ζ) + (n − 1)!2 neighborhood of ζ, • (∂z ∂z )µ Gn (z, ζ) = 0 for z ∈ ∂D, 0 ≤ µ ≤ n − 1, • Gn (z, ζ) = Gn (ζ, z) for z, ζ ∈ D, z 6= ζ. This polyharmonic Green function is related to the Dirichlet problem (∂z ∂z )n w = f in D, (∂z ∂z )µ w = γµ on ∂D, 0 ≤ µ ≤ n − 1,

(4)

for f ∈ Lp (D; C), 2 < p, γµ ∈ C 0,α (∂D; C), 0 < α < 1. Its unique solution is given via the representation formula for w ∈ C 2n (D; C) ∩ C 2n−1 (D; C), see [1], Z n X 1 w(z) = − ∂νζ Gµ (z, ζ)(∂ζ ∂ζ )µ−1 w(ζ)dsζ 4π ∂D µ=1 Z 1 Gn (z, ζ)(∂ζ ∂ζ )n w(ζ)dξdη. − π D Theorem 1.1. For f ∈ Lp (D; C), 2 < p, γµ ∈ C 0,α (∂D; C), 0 ≤ µ ≤ n− 1, the unique solution to the Dirichlet problem (4) is given by 1 w(z) = − 4π

Z

n−1 X

1 ∂νζ Gµ+1 (z, ζ)γµ (ζ)dsζ − π ∂D µ=0

Z

D

Gn (z, ζ)f (ζ)dξdη.

A Polyharmonic Dirichlet Problem for Complex Plane Domains

329

Besides, this polyharmonic Green function Gn (z, ζ) there is the GreenAlmansi function Gm (z, ζ). It satisfies for ζ ∈ D the following properties (see [1,2]). • Gm ( · , ζ) is polyharmonic of order m in D \ {ζ}, |ζ − z|2(m−1) log |ζ − z|2 is polyharmonic of order m in z in • Gm (z, ζ) + (m − 1)!2 the neighborhood of ζ, • (∂z ∂z )µ Gm (z, ζ) = 0, 0 ≤ 2µ ≤ m − 1, ∂νz (∂z ∂z )µ Gm (z, ζ) = 0, 0 ≤ 2µ ≤ m − 2, for z ∈ ∂D, • Gm (z, ζ) = Gm (ζ, z) for z, ζ ∈ D, z 6= ζ. The related representation formula for w ∈ C 2m (D; C) ∩ C 2m−1 (D; C) is X 1 Z ∂νζ (∂ζ ∂ζ )m−µ−1 Gm (z, ζ)(∂ζ ∂ζ )µ w(ζ)dsζ w(z) = − 4π ∂D µ=0 [ m−1 2 ]

[m 2 ]−1

X

+

µ=0

1 − π

Z

D

1 4π

Z

∂D

(∂ζ ∂ζ )m−µ−1 Gm (z, ζ)∂νζ (∂ζ ∂ζ )µ w(ζ)dsζ

Gm (z, ζ)(∂ζ ∂ζ )m w(ζ)dξdη,

see [1]. Its provides the unique solution to the Dirichlet problem (∂z ∂z )m w = f in D, (∂z ∂z )µ w = γµ , 0 ≤ 2µ ≤ m − 1,

(5)

bµ , 0 ≤ 2µ ≤ m − 2, on ∂D. ∂νζ (∂z ∂z )µ w = γ

Theorem 1.2. For f ∈ Lp (D; C), 2 < p, γµ ∈ C m−2µ,α (∂D; C), 0 ≤ 2µ ≤ m − 1, γ bµ ∈ C m−2µ−1,α (∂D; C), 0 ≤ 2µ ≤ m − 2, 0 < α < 1, the unique solution to the Dirichlet problem (5) is given as 1 w(z) = − 4π +

1 4π

Z

Z

[ m−1 2 ]

X

∂D µ=0

[m 2 ]−1

X

∂D µ=0

∂νζ (∂ζ ∂ζ )m−µ−1 Gm (z, ζ)γµ (ζ)dsζ

(∂ζ ∂ζ )m−µ−1 Gm (z, ζ)b γµ (ζ)dsζ −

1 π

Z

D

Gm (z, ζ)f (ζ)dξdη.

The explicit solution to this problem in the case of the unit disk is given in [3,4] and for the upper half plane in [5]. For the explicit expressions of the polyharmonic Green–Almansi functions for the unit disk see [1–4] and for the upper half plane see [5,7,8]. The function Gn (z, ζ) is evaluated only

330

H. Begehr & T. S. Vaitekhovich

for n = 2 for the unit disk, half planes, quarter plane, circular ring, see [6,8–10]. In particular for the unit disk D and the upper half plane H0 1 − zζ 2 (n − 1)!2 Gn (z, ζ) = |ζ − z|2(n−1) log ζ −z n−1 X1 − |ζ − z|2(n−1−µ) (1 − |z|2 )µ (1 − |ζ|2 )µ , µ µ=1 ζ − z 2 (n − 1)!2 Gn (z, ζ) = |ζ − z|2(n−1) log ζ − z +

n−1 X µ=1

1 |ζ − z|2(n−1−µ) (z − z)µ (ζ − ζ)µ , µ

respectively. Moreover, for D, H0 , the right upper quarter plane Q1 and the circular ring R = {0 < r < |z| < 1} 1 − zζ 2 G2 (z, ζ) = |ζ − z|2 log ζ −z   log(1 − zζ) log(1 − zζ) + (1 − |z|2 )(1 − |ζ|2 ) for D, + zζ zζ ζ − z 2 − (ζ − ζ)(z − z) log |ζ − z|2 for H0 , G2 (z, ζ) = |ζ − z|2 log ζ − z 2 ζ − z 2 2 ζ + z 2 2 + (ζ − G2 (z, ζ) = |ζ − z| log 2 ζ)(z − z) log ζ − z ζ − z2 ζ + z 2 for Q1 , + (ζ + ζ)(z + z) log ζ + z and for R

G2 (z, ζ) = |ζ −z| +2Re

− +



2

"

∞ ζ −z Y log |z|2 log |ζ|2 (z − r2k ζ)(ζ − r2k z) 2 − log 2 log r 1−zζ k=1 (zζ −r2k )(1−r2k zζ)

(1 − |ζ|2 ) log |z|2 r2 (1 − |z|2 ) ζ (1 − |z|2 ) log |ζ|2 + − log |ζ|2 log r2 log r2 1 − r2 z

(1 − r2 ) log |z|2 log |ζ|2 zζ log |z|2 log |ζ|2 r2 (1 − |ζ|2 ) z 2 − log |z| + 1 − r2 ζ log r2 (log r2 )2

h  r2 2 (1 − |z|2 )(1 − |ζ|2 ) log r2 2 2 1 − |z| + (1 − |ζ| ) log(1 − zζ) 1 − r2 zζ zζ

#

A Polyharmonic Dirichlet Problem for Complex Plane Domains

− (1 − r2 )

331

∞   i X 1 z r2n ζ  2n − log 1 − log(1 − r zζ) r2n ζ z r2n zζ n=1

∞ X (r2n |ζ|2 − 1) (1 − |z|2 ) log(1 − r2n zζ) 2n r zζ n=1  2  ∞   X z r2n  |z| − r2 r2n z  − (1 − r2 ) log 1 − (r2n − |ζ|2 ) − log 1 − ζ ζ zζ zζ n=1   ∞  X (|z|2 − r2 )ζ r2n ζ  2n 2 2n − (1 − r )zζ log(1 − r zζ) − (1 − r ) log 1 − z z n=1

−

∞ ∞ X ∞  X X r2n z  (1 − r2n ) (1 − |z|2 )ζ 2 + (1 − r ) log 1 − r2n z ζ n=1 n=1 k=1  2n 2    2(n+k)  r |ζ| − 1 1 z r ζ 2(n+k) × zζ) − 2k log(1 − r log 1 − r2n r2k ζ z r zζ  2k    r2(n+k)  r2k z r r2(n+k) z  log 1 − − − (r2n − |ζ|2 ) log 1 − ζ ζ zζ zζ   2k  2(n+k)  r ζ r ζ 2(n+k) 2k 2n zζ) − r zζ log(1 − r log 1 − − (1 − r ) z z     r2(n+k)  r2(n+k) z  1 − r2n zζ ζ . log 1 − log 1 − + 2n − r r2k r2k z ζ zζ

−

2. A New Polyharmonic Dirichlet Problem For a regular domain D on the complex plane the convolutions of the polyharmonic Green and Green–Almansi functions Z 1 e n (ζ, e ζ)dξde e η, Gm (z, ζ)G Hm,n (z, ζ) = − π D Z b n,m (z, ζ) = − 1 e m (ζ, e ζ)dξde eη H Gn (z, ζ)G π D

provide two polyharmonic Green functions of order m + n. They differ from one another, but are related by b n,m (ζ, z), Hm,n (z, ζ) = H

as Gn (z, ζ) and Gm (z, ζ) are symmetric functions of their two variables. The properties of the hybrid Green function Hm,n (z, ζ) follow from the fact that as well Gn (z, ζ) as Gm (z, ζ) are fundamental solutions to the operators (∂z ∂z )n and (∂z ∂z )m , respectively, and from their related Dirichlet

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problems. As a function of z it satisfies for any ζ ∈ D the Dirichlet problem (∂z ∂z )m Hm,n (z, ζ) = Gn (z, ζ) in D, (∂z ∂z )µ Hm,n (z, ζ) = 0, 0 ≤ 2µ ≤ m − 1, ∂νζ (∂z ∂z )µ Hm,n (z, ζ) = 0, 0 ≤ 2µ ≤ m − 2, on ∂D. Thus Hm,n ( · , ζ) has for any ζ ∈ D the following properties. • Hm,n ( · , ζ) is polyharmonic of order m + n in D \ {ζ}, |ζ − z|2(m+n−1) log |ζ − z|2 is polyharmonic of order m + n • Hm,n (z, ζ) + (m + n − 1)!2 in z in the neighborhood of ζ, • (∂z ∂z )µ Hm,n (z, ζ) = 0, 0 ≤ 2µ ≤ m − 1, m ≤ µ ≤ m + n − 1, ∂νz (∂z ∂z )µ Hm,n (z, ζ) = 0, 0 ≤ 2µ ≤ m − 2, for z ∈ ∂D. As a function of ζ it satisfies for any z ∈ D the Dirichlet problem (∂ζ ∂ζ )n Hm,n (z, ζ) = Gm (z, ζ) in D, (∂ζ ∂ζ )µ Hm,n (z, ζ) = 0, 0 ≤ µ ≤ n − 1, on ∂D. Thus Hm,n (z, ·) has for any z ∈ D the following properties. • Hm,n (z, ·) is polyharmonic of order m + n in D \ {z}, |ζ − z|2(m+n−1) log |ζ − z|2 is polyharmonic of order m + n • Hm,n (z, ζ) + (m + n − 1)!2 in ζ in the neighborhood of z,   • (∂ζ ∂ζ )µ Hm,n (z, ζ) = 0, 0 ≤ µ ≤ n + m−1 2 ,  ∂νζ (∂ζ ∂ζ )µ Hm,n (z, ζ) = 0, n ≤ µ ≤ n + m−2 , for ζ ∈ ∂D. 2

Theorem 2.1. Any w ∈ C 2(m+n) (D; C) ∩ C 2(m+n)−1 (D; C) can be represented as −1 w(z) = 4π

Z

∂D

m−1 2 ] m+n−1 h[ X X i + ∂νζ (∂ζ ∂ζ )m+n−µ−1 Hm,n (z, ζ)(∂ζ ∂ζ )µ w(ζ)dsζ

µ=m

µ=0

Z [X 1 (∂ζ ∂ζ )m+n−µ−1 Hm,n (z, ζ)∂νζ (∂ζ ∂ζ )µ w(ζ)dsζ 4π ∂D µ=0 Z 1 Hm,n (z, ζ)(∂ζ ∂ζ )m+n w(ζ)dξdη. − π D m 2

+

]−1

A Polyharmonic Dirichlet Problem for Complex Plane Domains

333

Proof. From the Gauss theorem it follows that Z 1 Hm,n (z, ζ)(∂ζ ∂ζ )m+n w(ζ)dξdη π D Z  h i 1 ∂ζ Hm,n (z, ζ)∂ζm+n ∂ζm+n−1 w(ζ) = 2π D h i h i + ∂ζ Hm,n (z, ζ)∂ζm+n−1 ∂ζm+n w(ζ) − ∂ζ ∂ζ Hm,n (z, ζ)(∂ζ ∂ζ )m+n−1 w(ζ) h i − ∂ζ ∂ζ Hm,n (z, ζ)(∂ζ ∂ζ )m+n−1 w(ζ)  + 2∂ζ ∂ζ Hm,n (z, ζ)(∂ζ ∂ζ )m+n−1 w(ζ) dξdη Z  h i 1 Hm,n (z, ζ) ∂ζm+n ∂ζm+n−1 w(ζ)dζ − ∂ζm+n−1 ∂ζm+n w(ζ)dζ = 4πi ∂D i h − (∂ζ ∂ζ )m+n−1 w(ζ) ∂ζ Hm,n (z, ζ)dζ − ∂ζ Hm,n (z, ζ)dζ Z 1 ∂ζ ∂ζ Hm,n (z, ζ)(∂ζ ∂ζ )m+n−1 w(ζ)dξdη + π D Z  1 Hm,n (z, ζ)∂νζ (∂ζ ∂ζ )m+n−1 w(ζ) = 4π ∂D  − ∂νζ Hm,n (z, ζ)(∂ζ ∂ζ )m+n−1 w(ζ) dsζ Z 1 + ∂ζ ∂ζ Hm,n (z, ζ)(∂ζ ∂ζ )m+n−1 w(ζ)dξdη. π D Repeating this argumentation leads to Z 1 Hm,n (z, ζ)(∂ζ ∂ζ )m+n w(ζ)dξdη π D m+n−2 X 1 Z  = (∂ζ ∂ζ )µ Hm,n (z, ζ)∂νζ (∂ζ ∂ζ )m+n−µ−1 w(ζ) 4π ∂D µ=0  − ∂νζ (∂ζ ∂ζ )µ Hm,n (z, ζ)(∂ζ ∂ζ )m+n−µ−1 w(ζ) dsζ Z 1 (∂ζ ∂ζ )m+n−1 Hm,n (z, ζ)∂ζ ∂ζ w(ζ)dξdη. + π D Because (∂ζ ∂ζ )m+n−1 Hm,n (z, ζ) = (∂ζ ∂ζ )m−1 Gm (z, ζ) = G1 (z, ζ), and, see e.g. [1], Z 1 G1 (z, ζ)∂ζ ∂ζ w(ζ)dξdη π D

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=−

1 4π

finally w(z) = −

m+n−1 X µ=0

Z

1 4π

∂D

Z

  ∂νζ G1 (z, ζ)w(ζ) − G1 (z, ζ)∂νζ w(ζ) dsζ − w(z),

∂D

 ∂νζ (∂ζ ∂ζ )µ Hm,n (z, ζ)(∂ζ ∂ζ )m+n−µ−1 w(ζ)

 − (∂ζ ∂ζ )µ Hm,n (z, ζ)∂νζ (∂ζ ∂ζ )m+n−µ−1 w(ζ) dsζ Z 1 − Hm,n (z, ζ)(∂ζ ∂ζ )m+n w(ζ)dξdη π D m+n−1 X 1 Z  ∂νζ (∂ζ ∂ζ )m+n−µ−1 Hm,n (z, ζ)(∂ζ ∂ζ )µ w(ζ) =− 4π ∂D µ=0  − (∂ζ ∂ζ )m+n−µ−1 Hm,n (z, ζ)∂νζ (∂ζ ∂ζ )µ w(ζ) dsζ Z 1 − Hm,n (z, ζ)(∂ζ ∂ζ )m+n w(ζ)dξdη. π D

Observing the boundary behavior of Hm,n (z, ζ) for z ∈ D gives the representation formula. Dirichlet problem. Find a solution to the (m + n)-Poisson equation (∂z ∂z )m+n w = f in the regular domain D, satisfying (∂z ∂z )µ w = γµ , 0 ≤ 2µ ≤ m − 1, m ≤ µ ≤ m + n − 1, ∂νz (∂z ∂z )µ w = γ bµ , 0 ≤ 2µ ≤ m − 2 on ∂D.

Theorem 2.2. For f ∈ Lp (D; C), 2 < p, γµ ∈ C 0,α (∂D; C), 0 < α < 1, for m ≤ µ ≤ m + n − 1, γµ ∈ C m−2µ,α (∂D; C) for 0 ≤ 2µ ≤ m − 1, γ bµ ∈ C m−2µ−1,α (∂D; C) for 0 ≤ 2µ ≤ m − 2, the unique weak solution to the Dirichlet problem is 1 w(z) = − 4π

Z

∂D

2 ] [ X m−1

µ=0

+

m+n−1 X 

∂νζ (∂ζ ∂ζ )m+n−µ−1 Hm,n (z, ζ)γµ (ζ)dsζ

µ=m

Z [X 1 γµ (ζ)dsζ (∂ζ ∂ζ )m+n−µ−1 Hm,n (z, ζ)b 4π ∂D µ=0 Z 1 Hm,n (z, ζ)f (ζ)dξdη. − π D m 2

+

]−1

A Polyharmonic Dirichlet Problem for Complex Plane Domains

335

Proof. To verify the boundary behavior one rewrites 1 w(z) = − 4π −

1 4π

[ m−1 2 ] X

Z

∂D µ=0

Z

n−1 X

∂D µ=0

∂νζ (∂ζ ∂ζ )m−µ−1 Gm (z, ζ)γµ (ζ)dsζ

∂νζ (∂ζ ∂ζ )n−µ−1 Hm,n (z, ζ)γµ+m (ζ)dsζ

Z [m 2 ]−1 X 1 (∂ζ ∂ζ )m−µ−1 Gm (z, ζ)b γµ dsζ + 4π ∂D µ=0 Z 1 Hm,n (z, ζ)f (ζ)dξdη. − π D

Here the first three terms are relevant for the boundary behavior. According to Theorem 1.2 the first and the third terms together take care of the conditions (∂z ∂z )µ w = γµ , 0 ≤ 2µ ≤ m − 1, ∂νz (∂z ∂z )µ w = γ bµ , 0 ≤ 2µ ≤ m − 2,

on ∂D because for 0 ≤ 2ρ ≤ m − 1 Z ρ ∂νζ (∂ζ ∂ζ )n−µ−1 Hm,n (z, ζ)γµ (ζ)dsζ (∂z ∂z ) ∂D

1 =− π

Z

n−µ−1

∂D

∂νζ (∂ζ ∂ζ )

Z

D

e n (ζ, e ζ)dξde e η γµ (ζ)dsζ (∂z ∂z )ρ Gm (z, ζ)G

vanishes for z ∈ ∂D. This is similarly true for the ∂νz (∂z ∂z )ρ derivative of these terms for 0 ≤ 2ρ ≤ m − 2. According to Theorem 1.1 the second term gives for m ≤ ρ ≤ m + n − 1 Z ρ ∂νζ (∂ζ ∂ζ )n−µ−1 Hm,n (z, ζ)γµ (ζ)dsζ (∂z ∂z ) ∂D Z ρ−m ∂νζ (∂ζ ∂ζ )n−µ−1 Gn (z, ζ)γµ (ζ)dsζ = (∂z ∂z ) Z∂D ρ−m ∂νζ Gµ+1 (z, ζ)γµ (ζ)dsζ , = (∂z ∂z ) ∂D

which is γρ−m (z) on ∂D up to the factor −4π for µ = ρ − m and 0 for µ 6= ρ − m. As for m ≤ ρ ≤ m + n − 1, ζ ∈ ∂D, (∂z ∂z )ρ Gm (z, ζ) = 0, z ∈ D, there are no contributions from the first and third terms.

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References 1. H. Begehr, T. Vaitekhovich. Iterated Dirichlet problem for higher order Poisson equation. Preprint, FU Berlin, 2006. 2. H. Begehr. Orthogonal decomposition of the function space L2 (D; C). J. Reine Angew. Math., 549, 2002. 3. H. Begehr, T. N. H. Vu, Z.-X. Zhang. Polyharmonic Dirichlet problems. Proc. Steklov Inst. Math., 255, 2006. 4. H. Begehr. A particular polyharmonic Dirichlet problem. Complex Analysis and Potential Theory, eds. T. Aliyev Azero˘ glu, P. M. Tamrazov, World Sci., Singapore, 2007. 5. H. Begehr, E. Gaertner. A Dirichlet problem for the inhomogeneous polyharmonic equation in the upper half plane. Georgian Math. J., 14, 2007. 6. H. Begehr, T. Vaitekhovich. Green functions in complex plane domains. Preprint, FU Berlin, 2007. 7. E. Gaertner. Basic complex boundary value problems in the upper half plane. Ph.D. thesis, FU Berlin, 2006. http://www.diss.fu-berlin.de/2006/320 8. H. Begehr, T. Vaitekhovich. Polyharmonic Green functions for particular plane domains. Preprint, FU Berlin, 2007. 9. T. S. Vaitekhovich. Boundary value problems to second order complex partial ˇ differential equations in a ring domain. Siauliai Mathematical Seminar, 2(10), 2007. 10. T. S. Vaitekhovich. Biharmonic Green function of a ring domain. Preprint, FU Berlin, 2007.

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THE FOURIER TRANSFORM METHOD IN CONTROLLABILITY PROBLEMS FOR THE FINITE STRING EQUATION WITH A BOUNDARY CONTROL BOUNDED BY A HARD CONSTANT L. V. FARDIGOLA Mathematical Division, Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine Kharkiv, 61103, Ukraine E-mail: [email protected] Necessary and sufficient conditions for (approximate) null-controllability are obtained for the control system wtt = wxx − q 2 w, wx (0, t) = u(t), wx (π, t) = 0, x ∈ (0, π), t ∈ (0, T ), where q ≥ 0, T ∈ (0, π], u is a control bounded by a hard constant. These problems are considered in Sobolev spaces. Controls solving them are found explicitly. Continuous controls solving the approximate nullcontrollability problem are constructed with the aid of the Ces` aro means of a Fourier series generated by the data of the control system. Keywords: Wave equation, half-axis, controllability problem, control bounded by a hard constant, Fourier transform, Sobolev space, Ces` aro mean.

1. Introduction Controllability problems for the wave equation on a segment were investigated in a number of papers [1–7] (and many others). We should note that most of the papers investigating controllability of the wave equation consider L2 -controllability or, more generally, Lp -controllability (2 ≤ p < +∞). But only L∞ -controls can be realized practically. Moreover, such controls should be bounded by a hard constant for practical purposes. Sometimes for a system arising in Nature and Technology we need to control the system letting it fluctuating when trying to drive it to a desired state without forcing it too much. Therefore we construct continuous controls solving the approximate null-controllability problem and bounded by a given constant. Consider the control system wtt = wxx −q 2 w, wx (0, t) = u(t), wx (π, t) = 0, x ∈ (0, π), t ∈ (0, T ), (1)

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where T ∈ (0, π], q ≥ 0. We assume that the control u satisfies the restriction u ∈ B U (0, T ) = v ∈ L2 (0, T ) | |v(t)| ≤ U a. e. on (0, T ) where U > 0 is given. The wave equation on a segment controlled by boundary conditions was studied for q = 0 in the Sobolev spaces [4,7]. In the present paper the most of these results are extended on the case of the Neumann control for q ≥ 0. If q = 0, the cases of the Dirichlet control and the Neumann control are rather similar. If q > 0, investigation of the null-controllability problem for equation (1) is essentially more complicated. In the present paper the operator Φ describing the influence of a control on a target state is introduced and studied in the Sobolev spaces H0s , s ∈ R. In fact, application of Φ is the most essential new point of the paper. In addition, continuous controls u ∈ B U (0, T ) solving the approximate null-controllability problem are constructed in the present paper. Continuous controls for a finite string were considered in the case q = 0 recently [2]. In the case q > 0 the wave equation controlled by the Dirichlet boundary conditions was investigated [6] in special classes of functions for the time T ≥ 2π (in contrast to the present paper where T ≤ π). It is the principal distinction because in the case T > π the considered boundary control conditions may be reduced to the ones wx (0, t) = u1 (t), wx (π, t) = u2 (t), t ∈ (0, π), e.g., u(t) = u1 (t) − u2 (t − π), if T = 2π. If we replace q by iq in (1) then replacing q by iq throughout this paper we obtain the results analogous to ones obtained in Sections 3–5. 2. Notation We define the spaces used in this work. Let S be the Schwartz space   n
l o < +∞ , S = ϕ ∈ C ∞ (R) | ∀m ∈ N, ∀l ∈ N, sup ϕ(m) (x) 1 + |x|2 x∈R

and let S ′ be the dual space [8], where | · | is the Euclidean norm. Denote by Hls (s, l ∈ R) the Sobolev spaces o n
s/2
l/2 ϕ ∈ L2 (R) , 1 + D2 Hls = ϕ ∈ S ′ | 1 + x2

2
1/2 R∞
s/2 with kϕksl = −∞ (1+x2 )l/2 1+D2 ϕ(x) dx where D = −i∂/∂x. Let F : S ′ → S ′ be Rthe Fourier transform operator. For ϕ ∈ S we have ∞ (Fϕ) (σ) = (2π)−1/2 −∞ e−ixσ ϕ(x) dx and (Ff, ψ) = (f, F −1 ψ) for f ∈ S ′ , s 0 ψ ∈ S. It is well known [9, Chapter 1] that FH0s = Hs0 and kϕk0 = kFϕks ,

Fourier Transform Method in Controllability Problems for the Finite String

339

if ϕ ∈ H0s . It is easy to see that for f ∈ H00 , supp f ⊂ [−π, π], we have

0

X

∞ 1 0 0

√ f (x + 2πn) kf k0 ≤ (2)

≤ 2 kf k0 . 2 1+π −1 n=−∞

Let Ω be the odd extension operator and Ξ be the even extension operator. Further, we use the spaces  Hls = ϕ ∈ S ′ × S ′ | supp ϕ ⊂ [0, π] and Ξϕ ∈ Hls × Hls−1 ,  e ls = ϕ ∈ Hls × H s−1 | ϕ is even 2π-periodic H l
1/2 with |||ϕ|||s = (kϕ0 ks0 )2 + (kϕ1 ks−1 )2 . l 3. Conditions for (approximate) null-controllability  0 w0 0 For w = ∈ H01 consider system (1) with the initial conditions w10 w(x, 0) = w00 (x),

wt (x, 0) = w10 (x),

x > 0. (3)   w(·, t) Let W 0 , W (·, t) be even 2π-periodic extensions for w 0 , ∂w(·, t)/∂t 0 1 1 e e respectively, t ∈ [0, T ]. Evidently, W ∈ H−1 , W (·, t) ∈ H−1 , t ∈ [0, T ], since the norm kf k1−1 is equivalent [9] to the norm kf k0−1 + kf ′ k0−1 . It is easy to see that control problem (1), (3) is equivalent to the following one !   0 1 dW 0  
= W− u, t ∈ (0, T ), (4) d 2 − q2 0 2δ(x) dt dx W (x, 0) = W 0

(5)

1 1 e −1 e −1 where W (·, t) ∈ H , t ∈ [0, T ], W 0 ∈ H , u ∈ B U (0, T ). Here δ is the Dirac distribution, δ = H ′ , H is the Heaviside function H(ξ) = 1, if ξ > 0, and H(ξ) = 0 otherwise. Consider for (4),(5) the steering condition

W (·, T ) = W T

(6)

1 e −1 where W T ∈ H . 0 For a given T > 0, w0 ∈ H01 denote by RU T (w ) the set of the states 1 e −1 WT ∈ H for which there exists a control u ∈ B U (0, T ) such that problem (4)–(6) has a unique solution.

Definition 3.1. A state w0 ∈ H01 is called null-controllable at a given time 0 T ∈ (0, π], if 0 belongs to RU T (w ) and approximately null-controllable at a 0 e1 given time T ∈ (0, π], if 0 belongs to the closure of RU T (w ) in H−1 .

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Let Φ : S ′ → S ′ with D(Φ) S ′ | g is odd andsupp g ⊂ [−T, T ]}  = {g ∈   p  −1 2 + q2 √ −i Fg such that (Φg) (x) = Fσ→x (x), g ∈ D(Φ). In σ 2 2 σ +q

the Appendix it is proved that Φ is invertible, Φ−1 : S ′ → S ′ , D(Φ−1 ) = R(Φ) = {g ∈ S ′ | g is even and supp g ⊂ [−T, T ]} (Lemma 5.3). Moreover, Φ is bounded from H0p−1 to H0p (Lemma 5.1) and Φ−1 is bounded from H0p to H0p−1 (Lemma 5.3), p ∈ R. Proposition 3.1. Let W 0 ∈ S ′ × S ′ , u ∈ B U (0, T ). Then " #   ∞ X ΩU 0 W (x, T ) = E(x, T ) ∗ W (x)− Φ (x + 2πn) , t ∈ [0, T ], (7) ΩU ′ n=−∞ where U(t) = u(t) (H(t) − H(t − T )), W is a unique solution of (4)–(5), ∗ is the convolution with respect to x,  p   2 + q2 σ sin t 1 ∂/∂t 1 −1 p Fx→σ (8) E(x, t) ≡ √ 2 (∂/∂t) ∂/∂t 2π σ2 + q2

e p , ΩU ∈ H p−1 and ΞU ∈ H p−2 then W (·, T ) ∈ In particular, if W 0 ∈ H −1 0 0 e p , p ∈ R. H −1

Proof. It is easy to see that  p  r   p sin t σ 2 + q 2
−1   = π J0 q t2 − x2 H t2 − x2 sgn t (9) p Fσ→x 2 σ2 + q2 where Jν (ξ) is the Bessel function (ν ∈ R). Applying the Fourier transform with respect to x to problem (4)–(6) and using (9) we conclude that (7) and (8) are true. With regard to Lemmas 5.1 and 5.7 we get that the last assertion of the proposition is also true. The proposition is proved.

Theorem 3.1. Let w0 ∈ H01 , T ∈ (0, π] and supp w00 ⊂ [0, T ]; −1 0
Φ Ξw0 (ξ) ≤ U a.e. on [0, T ];  

d 0 −1 0 Ξw1 = Φ . sgn ξ Φ Ξw0 dξ

(10) (11) (12)

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341

Then the state w0 is null-controllable at the time T . Moreover, the solution of the null-controllability problem (the control u) is unique and √
Z T qtI1 q x2 − t2 0 ′ √ w0 (x) dx u(t) = w00 ′ (t) + x2 − t2 t √
Z T qxI1 q x2 − t2 0 0 √ = w1 (t) + w1 (x) dx a.e. on (0, T ). (13) x2 − t2 t Here Iν (ξ) = i−ν Jν (iξ) is the modified Bessel function.

Proof. Put Ub = Φ−1 Ξw00 . It follows from Lemma 5.3 and (11) that Ub is odd and Ub ∈ L∞ (R). Put U(t) = H(t)Ub(t) and denote by u(t) its restriction on [0, T ]. Using Lemma 5.4 we conclude that u ∈ B U (0, T ) and the first part of assertion (13) is true for it. Taking into account (12), we get    

d d −1 0 =Φ sgn t Φ ΦΩU (sgn tΩU) = ΦΩU ′ . (14) Ξw1 = Φ dt dt With regard to (14) we obtain from (7) that w00 is null-controllable. It follows from (14) and Lemma 5.5 that (ΞU)′ = w e10 ′ where w e10 ∈ H00 is √ 2 2 R T qxI1 (q x −t ) 0 √ even, supp w e10 ⊂ [−T, T ] and w e10 (t) = Ξw10 (t) + |t| w1 (x) dx. x2 −t2 0 0 Since ΞU and w e1 have compact supports then ΞU = w e1 . Therefore the second part of (13) is true. The theorem is proved. Theorem 3.2. If a state w0 ∈ H01 is approximately null controllable at a given time T ∈ (0, π] then assertions (10)–(12) are valid.

0 Proof. For each n ∈ N there exists a state W n ∈ RU T (w ) such that n 1 n 1 |||W |||−1 < 1/n. According to Lemma 5.7 |||E(·, T ) ∗ W |||−1 → 0 as   P∞ ΩUn n → ∞. With regard to (7) we have Φ (x + 2πn) → n=−∞ ΩUn′ 1 e −1 W 0 (x) as n → ∞ in H for some un ∈ B U (0, T ), here Un (t) = P∞ (j) 0 un (t) (H(t) − H(t − T )). Hence n=−∞ ΦΩUn (x + 2πn) → Wj (x) as 0 n → ∞ in H−1 . It follows from (2) that

ΦΩUn(j) → Ξwj0

0 as n → ∞ in H−1 .

(15)

Put Ub = Φ−1 Ξw00 . With regard to Lemma 5.3 we get supp Ub ⊂ [−T, T ], ∞ Ub is odd and ΩUn → Ub as n → ∞ in H0−1 . Then the sequence
{ΩUn }n=1 ′ converges to Ub as n → ∞ in S ′ and consequently in L2 (R) because ∞ {ΩUn }n=1 is uniformly bounded on R and S is dense in L2 (R). By the b Riesz theorem Ub ∈ L2 (R). Since un ∈ B U (0, T ) then |U(t)| ≤ U a.e. on

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[−T, T ]. With regard to Lemma 5.1 we obtain from here that (10), (11) b are true. Setting U(t) = H(t)U(t) and taking into account (15) we obtain (j) −1 (j) −1 0 ΩUn → ΩU = Φ Ξwj as n → ∞ in H−1 . Hence (12) holds. The theorem is proved. Remark 3.1. According to Lemma 5.6 condition (12) is equivalent to Z ∞ I1 (q(ξ − x)) w00 (ξ) dξ, x > 0. (12′′ ) w10 (x) = w00 ′ (x) + q ξ − x x Remark 3.2. If q = 0 then (12) is of the form w10 = w00 ′ and (13) is of the form u(t) = w00 ′ (t) = w10 (t) a.e. on [0, T ].

Remark 3.3. According to Lemma 5.5 condition (11) is equivalent to √
Z T qxI1 q x2 − t2 0 0 √ w1 (x) dx ≤ U a.e. on [0, T ], (11′ ) w1 (t) + 2 2 x −t t if (10) and (12) hold. With regard to Lemma 5.5 the condition w10 (x) ≤ U ′ ) a.e. on [0, T ] is sufficient for (11 ) (and 11)) and the condition I0 (qT w10 (x) ≤ U (1 + qT ) a.e. on [0, T ] is necessary for (11′ ) (and 11)). If q = 0 we from here that (11′ ) and (11) are equivalent to the condition 0 obtain w1 (x) ≤ U a.e. on [0, T ]. 4. Continuous controls

In this section we construct continuous controls that solve the approximate null-controllability problem and are bounded by U . They are obtained with the help of the Ces` aro means for a Fourier series determined by the data of the considered system. These controls generate a continuous steering state, if an initial state is continuous. Consider control system (4), (5) and assume that for T =
π and e w0 ∈ H01 conditions (10)–(12) hold. Set U(t) = H(t) Φ−1 Ξw00 (t) and denote by u e theR restriction of Ue on [0, π]. By Theorem 3.1 u e ∈ B U (0, π). π e(t) sin(nt) Put νn = π2 0 u  p dt, n = 0, ∞. According to Lemma 5.3 R 2n π we get νn = π 0 cos x n2 − q 2 w00 (x) dx, n = 0, ∞. Put un (t) = Pn Pk Pn n+1−l 1 n = 0, ∞. Evidently, un k=0 l=0 n+1 νl sin(lt), l=0 νl sin(lt) = n+1 P∞ ν is a Ces` aro mean for the Fourier series l=0 l sin(lt), n = 0, ∞. Set P e)(t−2πk). It is well known [11, Section 3.3] that un (t) = ν(t) = ∞ (Ω U k=−∞  2 R sin((n+1)ξ) 1 1 π ν(t + ξ)F (ξ) dξ where F (ξ) = is the Fej´er kern n π −π 2(n+1) sin(ξ) R 1 π U nel and π −π Fn (ξ) dξ = 1. Therefore un ∈ B (0, π). It is easy to see that Un ∈ H0p , p < 3/2, n = 0, ∞, where Un (t) = un (t) (H(t) − H(t − π)).

Fourier Transform Method in Controllability Problems for the Finite String

Since u e ∈ B U (0, π) ⊂ L2 (0, π) then

π 2

343

 2 2 ν = ke u k . Taking 2 l=0 l L (0,π)

P∞

this into account we have from [11, Section 3.1] that the first and the second Ces` aro means tend to ke ukL2 (0,π) as n → ∞, i.e.

∞ n ∞ π X 2 X (n + 1 − l)(n + 2 − l) 2 πX 2 νl → νl , νl n+1 2 (n + 1)(n + 2) 2 l=0 l=0 l=0 l=0 Pn Pn l l2 2 as n → ∞. Therefore l=0 n+1 νl2 → 0 as n → ∞ hence l=0 (n+1) 2 νl → 0 as n → ∞. Summarizing we obtain !1/2 r ∞ n X X l2 π 2 2 ν νl + → 0 as n → ∞. ke u − un kL2 (0,π) = 2 (n + 1)2 l n X n+1−l

νl2 →

l=n+1

l=0

Taking into account Lemmas 5.7, 5.1 and Proposition 3.1 we get the following result.

Theorem 4.1. Let T = π, w0 ∈ H01 , s ≤ 1. Let also conditions (10)– (12) be fulfilled. Then un ∈ B U (0, π), n = 0, ∞, and |||W n (·, π)|||1−1 ≤ 2Mq Lq ke u − un kL2 (0,π) → 0 as n → ∞ where W n is the solution of (4), (5) corresponding to the control un , Lq is the norm of Φ acting from H00 to H01 . Moreover, Un ∈ H0p , p < 3/2, n = 0, ∞, where Un (t) = e p ⊂ C 1 (R) × C(R), un (t) (H(t) − H(t − π)). In addition, W n (·, π) ∈ H −1 if w0 ∈ H0p , 3/2 < p < 5/2. p
pπ FΞw00 (λn ) where λn = Denote ω ejn = n2 − q 2 , n ∈ N, j = q q 2 

e0n . 1, 2. Then νn = i π2 FΩ Ue (πn) = i π2 F Φ−1 Ξw00 (πn) = − 2n π ω Condition (12) is equivalent to Ωw10 = ΦΩ Ue′ . Therefore the following holds. Remark 4.1. (12) is equivalent to ω e 1n = 5. Appendix

2 T

P∞ ((−1)l+n −1)l2 ωe 0l l=0

l2 −n2

, n = 0, ∞.

We assume throughout this section that p ∈ R, T > 0. Denote ∂ : S ′ → S ′ with D(∂) = {f ∈ S ′ | f is even and supp f ⊂ [−T, T ]} such that ∂f = f ′ , f ∈ D(∂). Evidently, R(∂) = {g ∈ S ′ | f is odd and supp g ⊂ [−T, T ]}. Then ∂ is bounded linear operator from H0p to H0p−1 . Due to the Paley–Wiener theorem we obtain that ∂ is invertible. According to the inverse operator theorem we conclude that ∂ −1 is also bounded. Lemma 5.1. Let g ∈ D(Φ). Then supp Φg ⊂ [−T, T ], Φg is even, Φ is bounded from H0p−1 to H0p . If q = 0 then Φ = ∂ −1 .

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Proof. Due to the generalized Paley–Wiener theorem [10, Chapter 3] we conclude that 1) G = Fg is a regular functional; 2) G is of a polynomial growth on R; 3) G can be extended to an entire function of the order ≤ 1 and the type ≤ T . Evidently, √ G is odd. Applying again the generalized

Paley–Wiener theorem to

G(

√

σ2 +q2 )

we obtain that supp Φg ⊂ [−T, T ].

σ2 +q2

Obviously, Φg is even. We have p−1 k(Φg)′ k0

=

2

Z

q

∞

|G(µ)|2

1−p

(1 − q 2 + µ2 )

s

q2 1 − 2 dµ µ

!1/2

e pq kgkp−1 , ≤L 0

e pq > 0. Since ∂ −1 is bounded then Φ is bounded from H p−1 to H p . where L 0 0   g)(σ) If q = 0 then Φg = F −1 (F(iσ) = ∂ −1 g. The lemma is proved. Lemma 5.2. If g ∈ D(Φ) ∩ H00 , then Z ∞  p  (Φg)(x) = − J0 q t2 − x2 g(t) dt. |x|

Proof. With regard to the definition of Φ and (9) we conclude that the assertion of the lemma is true. Denote Ψ : S ′ → S ′ with D(Ψ) = {f ∈  S ′ | f is even  and supp g ⊂ p −1 2 2 (t), f ∈ D(Ψ). [−T, T ]} such that (Ψf ) (t) = Fµ→t iµ Ff µ −q Lemma 5.3. Let f ∈ D(Ψ). Then supp Ψf ⊂ [−T, T ], Ψf is odd, Ψ is bounded from H0p to H0p−1 . If q = 0 then Ψ = ∂. In addition, R(Ψ) = D(Φ), D(Ψ) = R(Φ) and Ψ = Φ−1 .

Proof. Reasoning as in the proof of Lemma 5.1 we conclude that supp Ψf ⊂ [−T, T ] and Ψf is odd. Hence R(Ψ) ⊂ D(Φ). We have −1 iµ (Fx→ξ (Φg) (x)) (ξ)|ξ=√µ2 −q2 = g, R(Φ) ⊂ D(Ψ) and ΨΦg = Fµ→t

g ∈ D(Φ), i.e. Φ : D(Φ) → R(Φ) is invertible and Φ−1 = Ψ. Denote by Φp the restriction of Φ on H0p−1 . According to Lemma 5.1 Φp is bounded from H0p−1 to H0p . Evidently, Φp is invertible and Φ−1 p = Ψp−1 where Ψp−1 is the restriction of Ψ on H0p . Applying the inverse operator theorem we conclude that Ψp−1 is bounded from H0p to H0p−1 . If q = 0 then d F −1 ((Ff ) (σ)) = ∂f . The lemma is proved. Φf = dt Lemma √5.4. If f ∈ D(Φ−1 ) ∩ H01 then (Φ−1 f )(t) R ∞ I1 (q x2 −t2 ) ′ √ qt |t| f (x) dx. x2 −t2

=

f ′ (t) +

Fourier Transform Method in Controllability Problems for the Finite String

345

p
−1 d √ −i (Ff ′ )( µ2 − q 2 ) . Since Proof. We have g(t) = dt Fµ→t 2 2 µ −q p
p sin t σ 2 + q 2 / σ 2 + q 2 is an entire function with respect to (σ, q) then replacing q by iq and reasoning as in the proof of Lemma 5.2 we conclude that the assertion of the lemma is true. Lemma 5.5. Let f ∈ D(Φ−1 ) ∩ H00 . Then Φ−1 f = ge′ where ge ∈ H00 is even, supp ge ⊂ [−T, T ] and Z ∞  p  (16) ge(t) = − f ′ (x)I0 q x2 − t2 dx, f (x) = −

|t| ∞

Z

|x|

 p  ge ′ (t)J0 q t2 − x2 dt.

(17)

Moreover, if |f (x)| ≤ F a.e. on [−T, T ] then |e g (t)| ≤ FI0 (qT ) a.e. on [−T, T ], if |e g (t)| ≤ G a.e. on [−T, T ] then |f (x)| ≤ G(1 + qT ) a.e. on [−T, T ]. √
R∞ Proof. It follows from Lemma 5.4 that − |t| I0 q x2 − t2 f ′ (x) dx = ge(t) + C. Evidently, supp ge ⊂ [−T, T ] iff C = 0. Hence (16) is true. It follows from Lemma also true. If |f (x)| ≤ F a.e. on [−T, T ] √ (17) is
R T ∂5.2 that 2 2 then |e g (t)| ≤ F |t| ∂x I0 q x − t dx ≤ FI0 (qT ). If |e g (t)| ≤ G a.e. on RT qt [−T, T ] then |f (x)| ≤ G + G |x| √t2 −x2 dt ≤ G(1 + qT ).
d sgn tΦ−1 f . Lemma 5.6. Let f ∈ H01 be even, supp f ⊂ [−T, T ], h = Φ dt R∞ Then h(x) = sgn x f ′ (x) + q |x| f (ξ) I1 (q(ξ−|x|)) dξ. ξ−|x|



1 d Proof. Put F = Ff . Denote G(ξ) = iξ Ft→ξ dt sgn t(Φ−1 f )(t) (ξ). p  R∞ 1 2 µ2 − q 2 dµ. Therefore h = Then G = π2 V.p. 0 ξ2 −µ 2µ F √   −q|x|  p  G σ2 +q2 −1 −1 −∆q Fσ→x σ2 +q2 G = − 12 ∆q e q ∗ Fσ→x where σ2 + q2
d 2 2 ∆q = dx − q . Taking into account (9) where q is replaced by iq and R 2 1 cosh(ξt) √ dt = I0 (ξ) we obtain π 0 1−t2  1 e−q|x| h(x) = ∆q (f ′ (x) ∗ (I0 (qx) sgn x))|x=0 2 q  + sgn x (f (x) ∗ (I0 (qx) sgn x)) − f (x) ∗ I0 (qx) Z ∞ = sgn x f ′ (x) − f (ξ + |x|) (∆q I0 (qξ)) dξ. 0

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Since ∆q I0 (qξ) = q 2 /2 (I2 (qξ) − I0 (qξ)) = −qI1 (qξ)/ξ, we conclude that the assertion of the lemma os true.   p f0 e p , Mq = 40(2 + π 2 ) (1 + q 2 ). Then ∈ H Lemma 5.7. Let f = −1 f1 |||E(x, t) ∗ f |||p−1 ≤ Mq |||f |||p−1 , t ∈ [−π, π]. Proof. Put fj0 = (1+D 2 )−

p−j 2

 i h p−j (1 + D2 ) 2 fj (H(x + π) − H(x − π)) , p−j

j = 0, 1. Evidently, fj0 ∈ H0p−j , supp(1 + D 2 ) 2 fj0 ⊂ [−π, π]. Let E = {emj }1m,j=0 . Therefore for t ∈ [−π, π] we have

0   X p−m

|||E(x, t) ∗ f |||p−1 ≤

emj (·, t) ∗ (1 + D 2 ) 2 fj0 (x + 2πn) . −1

m,j=0,1

Taking into account (8) and applying the Paley–Wiener theorem we

p−m conclude that supp emj (·, t) ∗ (1 + D2 ) 2 fj0 ⊂ [−2π, 2π]. We have

P∞

0 0

h(x + 2πn) ≤ 20 khk for h ∈ H 0 , supp h ⊂ [−2π, 2π]. With n=−∞

−1

0

0

regard to (2) and (8) it follows from here that X

emj (·, t) ∗ f 0 p−m ≤ Mq |||f |||p , t ∈ [−π, π], |||E(x, t) ∗ f |||p−1 ≤ 20 j 0 −1 m,j=0,1


sin(t√σ2 +q2 ) 2 since 1 + |σ|2 √ 2 2 ≤ t2 + 2. The lemma is proved. σ +q

References

1. W. Krabs and G. Leugering, Math. Methods Appl. Sci., 17, 77 (1994). 2. M. Gugat, ZAMM, 86, 134 (2006). 3. M. Negreanu and E. Zuazua, C. R. Math. Acad. Sci. Paris, 338, No. 5, 413 (2004). 4. M. Gugat, G. Leugering and G. M. Sklyar, SIAM J. Control Optim., 44, No. 1, 49 (2005). 5. H. O. Fattorini, Infinite dimensional optimization and control theory, Cambridge University Press, 1999. 6. V. A. Il’in and E. I. Moiseev, Dokl. Akad. Nauk, Ross. Acad. Nauk, 394, No. 2, 154 (2004). 7. L. V. Fardigola and K. S. Khalina, Ukr. Mat. Zh., 59, No. 7, 939 (2007). 8. L. Schwartz, Th´eorie des distributions, 1, 2, Hermann, Paris, 1950–1951. 9. S. G. Gindikin and L. R. Volevich, Distributions and convolution equations, Gordon and Breach Sci. Publ., Philadelphia, 1992. 10. I. M. Gelfand and G. E. Shilov, Generalized Functions, 3, Fismatgiz, Moscow, 1958. 11. A. Zygmund, Trigonometric series, 1, Cambridge University Press, 1959.

347

DENSITY PROBLEM OF MONODROMY REPRESENTATION OF FUCHSIAN SYSTEMS G. K. GIORGADZE Department of Mathematics, Tbilisi State University, Tbilisi, 0186, Georgia E-mail: [email protected] This article is dedicated to the investigation of the density problem of monodromy groups of Fuchsian systems on complex manifolds in linear groups. We consider the so-called inverse problem. We apply the Riemann-Hilbert monodromy problem and we show that there exists a Fuchsian system with dense monodromy subgroup in the special unitary group. Keywords: Fuchsian system; Monodromy representation; Dense subgroup; Fundamental group; Riemann surface.

1. Introduction and statement of the problem In this paper we investigate some properties of monodromy represenation of Fuchsian systems [12]. This problem arises in the theory of quantum computation in connection with the construction of a universal set of gates for a quantum computer [9]. Let X be an m-dimensional complex analytic manifold and suppose D = ∪ni=1 Dj is a divisor such that Dj are generic 1-codimensional submanifolds of X. Let df = ωf

(1)

be a completely integrable Pfaffian system on X, where ω is a d × d matrix valued holomorphic 1-form on X \ D. Complete integrability condition means that ω satisfies dω − ω ∧ ω = 0. The system (1) defines the monodromy representation ρ : π1 (X \ D) → GL(n, C).

(2)

The main problem is to investigate the properties of the system (1) when the represenation (2) is a dense subset of GL(n, C), i.e ρ(π1 (X \ d) = GL(n, C).

348

G. K. Giorgadze

In case, when π1 (X \D)-not free, for example, braid group, then problem is difficult. M. Freedman, M. Larsen and Z. Wang [16] constructed Fuchsian system with density monodromy representation in SU (n).When π1 (X \ D) is free group and dim X = 1, then existence of regular system (1) with density monodromy representation (2) follows from positivity solution of Riemann-Hilbert monodromy problem in class of regular systems [12]. In addition if system (1) is 2 × 2, then there exist the solution of problem in class of Fuchsian system [17]. 2. Fuchsian system and entangled operator Consider a system [17] dy = B(z)y (3) dz of p linear differential equations with rational coefficients on the Riemann sphere CP1 . Let D = {a1 , ..., an } be the set of singular points of (3), consisting of the poles of the matrix function B(z). Consider the matrix differential form ω = B(z)dz. In what follows we will rather write (3) in its invariant form dy = ωy

(4)

in terms of which D is a singular divisor of ω. Let Y denote a fundamental solution of (4), holomorphic in a neighbourhood of a given non-singular point z0 ∈ C. Analytic continuation of Y along a loop in CP1 \ D yields a new fundamental solution Y˜ = Y G for some matrix G ∈ GL(p, C). This defines the monodromy representation χ : π1 (CP1 \ D; z0 ) → GL(p, C)

(5)

of the system, with respect to Y . Since the fundamental group of CP1 \ D is generated by the homotopy classes of all elementary loops γi , where i, i = 1, ..., n, encloses the only singular point ai , the monodromy representation of (4) is defined by the local monodromy matrices Gi corresponding to these loops. These matrices satisfy a priori the only relation G1 · · · Gn = I. The Fuchsian differential equation is a linear differential equation whose singularities are all regular [17]. It frequently appears in a range of problems in mathematics and physics. For example, the famous Gauss hypergeometric differential equation is a canonical form of the second-order Fuchsian differential equation with three singularities on the Riemann sphere CP1 . Global properties of solutions, i.e., the monodromy, often play decisive roles

Density Problem of Monodromy Representation of Fuchsian Systems

349

in the applications of these equations in physics and other areas of mathematics. For example consider a Schr¨ odinger equation of the form ∂Ψ(t) = H(t)Ψ(t) (6) i ∂t with time dependent Hamiltonian H(t) = (Hij (t)), i, j = 1, ..., N , where H11 = ε(t), H12 = V2 , H13 = V3 , ..., H1N = VN , H21 = V2 , H31 = V3 , ..., HN 1 = VN , and Hij = 0 otherwise, Ψ(t) = (ψ1 (t), ..., ψN (t)) is a wave function, Vj , j = 2, . . . , n are constants, and the time dependent part ε has the form ε(t) = E1 tanh(t/T ) with constant E1 and T . Theorem 1. ([6,8]) Equation (6) is reducible to an N -dimensional Fuchsian system of Okubo type (zIN − B)

dΦ(z) = AΦ(z) dz

with B = diag(i, −i, ..., −i). In two-dimensional case (i.e., for N = 2) this result can be generalized by permitting Hamiltonians of more general form. Namely, consider a Schr¨ odinger equation with two-component phase function ∂f (t) = H(t)f (t), (7) i ∂t where f (t) = (f1 (t), f2 (t)), and time dependent Hamiltonian H(t) has the form   ε(t) V (t) (8) H(t) = V (t) −ε(t)

where

ε(t) =

V0 T dy E0 T + E1 T y dy , V (t) = p 2 1+y dt 1 + y 2 dt

with some monotonically increasing differentiable function y : R → R satisfying y(t) → ±∞ as t → ±∞. Theorem 2. ([8]) Equation (7) is reducible to a system of two hypergeometric equations of the form d2 g dg + (γ − (1 + α + β)) − αβg(z) = 0 2 dz dz with appropriate constants α, β, γ. z(z − 1)

(9)

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G. K. Giorgadze

For the proofs of Theorems 1 and 2, see [6–8]. 3. Fuchsian system and entangled operator One-qubit gates are elements of SU (2) and act on C2 . For realization of quantum algorithms one needs all operators from SU (2n ). This problem can be reduced to finding such elements Mi of SU (4) which together with SU (2) ⊗ SU (2) generate the whole SU (4). Definition 1. ([2]) A two-qubit gate A is universal for the quantum computation if A together with local unitary transformations (i.e transformation of one qubit states) generates all unitary transformations of the complex vector space of dimension 2n to itself. For example it is known that a controlled NOT [1] is a universal gate. Definition 2. A gate A is a said to be entangling if there is a vector |ab >= |a > ⊗|b >∈ C2 ⊗ C2 such that A|ab > is not decomposable as a tensor product of two qubits. Theorem 3. ([2]) A two-qubit gate A is universal if and only if it is entangling. Below we give, following Kauffman [3], a topological description of the entangling operator. Let R : V ⊗ V → V ⊗ V be a linear map. If R satisfies braided Yang-Baxter equation (R ⊗ I)(I ⊗ R)(R ⊗ I) = (I ⊗ R)(R ⊗ I)(I ⊗ R), then R is called braiding. Theorem 4. ([4]) All 4 × 4 unitary solutions to the Yang-Baxter are similar to one of  1     √ 0 0 √12 000 a000 2  0 √1 − √1 0     0 0 b 0   2 2 0 b 0 R1 =   , R2 =   0 c 0 0  , R3 =  0 0 c  0 √12 √12 0  d00 000d − √12 0 0 √12 where a, b, c, d are unit complex numbers.

equation  a 0 , 0 0

Operator R1 acts on the standard basis of C2 ⊗ C2 and transforms it into the Bell basis [1]. As the operators R2 and R3 they are described in the following proposition.

Density Problem of Monodromy Representation of Fuchsian Systems

351

Proposition 1. ([3]) The braidings R2 and R3 are universal gates when ad − bc 6= 0 for certain internal parameters a, b, c and d. From this proposition follows that nor every braiding is universal. For example, if a = b = c = d = 1, we obtain so called SW AP operator, which does not universal. By the SW AP operator, when by the definition act on the tensor product u ⊗ v ∈ V ⊗ V the following manner SW AP (u ⊗ v) = v ⊗ u, we define the operator R13 : V ⊗ V ⊗ V → V ⊗ V ⊗ V as R13 = (I ⊗ SW AP )(R ⊗ I)(I ⊗ SW AP ). Denote by R12 = R ⊗ I and R23 = I ⊗ R. Definition 3. The equation (with respect to R) R12 R13 R23 = R23 R13 R12 is called the algebraic Yang-Baxter equation. Proposition 2. R is a solution of algebraic Yang-Baxter equation if and only if R ◦ SW AP is solution of braided Yang-Baxter equation. In the paper [3], the relation between the topological and quantum entangling is investigated and it is proved that the unitary solution of YangBaxter equation detect topological linking if and only if the gates corresponding to these solutions can entangle quantum states. We will apply results of this section to a special divisor. Namely, suppose D = ∪i b−a , f (x, b) = 0 λ γ2 with the asymptotic behavior f (x, y) = 0[(d−y) ], γ2 > d−c . Then solution 2 of equation (1) of class Cxy (D) is representable in the form u(x, y) = K1 [ϕ1 (x), u(x, y) = K2 [ϕ1 (x), u(x, y) = K3 [ϕ2 (x), u(x, y) = K4 [ϕ2 (x),

ψ1 (y), ψ2 (y), ψ1 (y), ψ2 (y),

f (x, y)], f (x, y)], f (x, y)], f (x, y)],

when when when when

(x, y) ∈ D1 , (x, y) ∈ D2 , (x, y) ∈ D3 , (x, y) ∈ D4 ,

(2)

where ϕ1 (x), ψ1 (y) are arbitrary functions of points of Γ1 and Γ2 , ϕ2 (x), ψ2 (y) are related to ϕ1 (x), ψ1 (y) by  µ Z b t − a b−a f (t, y) ψ2 (y) + , c d−c 2 Then any solution of equation (1) of class Cxy (D) is representable in the form (2), where ϕ1 (x), ψ1 (y) are arbitrary functions of points of Γ1 and Γ2 , ϕ2 (x), ψ2 (y) and ϕ1 (x), ψ1 (y) are related by (3), (4), (5), C10 is an arbitrary constant, C2 and C10 are related by (6).

Remark 0.1. Statements similar to Theorems 1 and 2 are obtained also in the cases λ > 0, µ < 0 and λ < 0, µ > 0. Remark 0.2. By fulfillment of all conditions of Theorem 0.1, any solution 2 (D) as x → a and y → c is unbounded and of equation (1) of class Cxy

Hyperbolic Equations for Which All of the Boundary Consist of Singular Lines

359

its behavior is determined from the asymptotic formula U (x, y) = O[(x − µ λ a)− b−a ] and U (x, y) = O[(y − c)− d−c ]; as x → b and y → d, it vanishes and its behavior is determined from the asymptotic formula U (x, y) = o[(b − µ λ x) b−a ] and U (x, y) = o[(d − x) d−c ]. Remark 0.3. By fulfillment of conditions of Theorem 0.2, any solution of 2 equation (1) of class Cxy (D) as x → a, y → c vanishes and its behavior |µ|

|λ|

is determined from U (x, y) = o[(x − a) b−a ] and U (x, y) = o[(y − c) d−c ]; as x → b and y → d, it is unbounded and its behavior is determined from |λ| |µ| U (x, y) = O[(x − a) b−x ] and U (x, y) = O[(d − x) d−c ].

In the case δ 6= λµ that which has been proved, the problem of determination of solutions of (1) in domain D is equivalent to the problem of determination of solutions of following equations in domains Dj (1 ≤ j ≤ 4):  Z x Z y dt u(t, s)ds ω(x, y) u(x, y) − δ1 ω(t, s) (s − c)(d − s) a (t − a)(b − t) c = K1 [ϕ1 (x), ψ1 (y), f (x, y)], (x, y) ∈ D1 , u(x, y) + δ1

Z

b

x

dt (t − a)(b − t)

Z

y

c



ω(x, y) ω(t, s)



u(t, s)ds (s − c)(d − s)

= K2 [ϕ1 (x), ψ2 (y), f (x, y)], (x, y) ∈ D2 , u(x, y) + δ1

Z

x

a

dt (t − a)(b − t)

Z

y

d

ω(x, y) ω(t, s)



u(x, y) − δ1

b

x

dt (t − a)(b − t)

Z

d y



ω(x, y) ω(t, s)



(8)

u(t, s)ds (s − c)(d − s)

= K3 [ϕ2 (x), ψ1 (y), f (x, y)], (x, y) ∈ D3 , Z

(7)

(9)

u(t, s)ds (s − c)(d − s)

= K4 [ϕ2 (x), ψ2 (y), f (x, y)], (x, y) ∈ D4 , (10)  µ   λ  d−y d−c b−x b−a . The solution of equation where δ1 = λµ−δ, ω(x, y) = x−a y−c (7) is found in the class of function u(x, y) representable in D1 in the form u(x, y) = ω(x, y)

∞ X

(x − a)n+γ1 (b − x)u1n (y)

n=0

(11)

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N. Rajabov

or u(x, y) = ω(x, y)

∞ X

n=0

(y − c)n+γ2 (d − y)νn1 (x)

(12)

where Un1 (y), νn1 (x) are unknown functions, γ1 > 0, γ2 > 0. Correspondingly the solution of equation (8) is found in the class of functions u(x, y) representable in D2 in the form u(x, y) = ω(x, y)

∞ X

(b − x)n+γ3 (x − a)u2n (y)

(13)

n=0

or (12). The solution of equation (9) is found in the class of functions u(x, y) representable in D3 in the form (11) or u(x, y) = ω(x, y)

∞ X

n=0

(d − y)n+γ4 (y − c)νn2 (x), γ4 > 0.

(14)

Similarly the solution of the integral equation (10) is found in the class of functions representable in D4 in the form (13) or (14). In this connection assume that in (7)-(10) the right part, also is expandable in corresponding series. Then for determination of the unknown functions u1n (y), u2n (y), νn1 (x), νn2 (x) we obtained an infinite number of system of one-dimensional integral equations with boundary singular points, theory of which is developed in [3-8,9]. For example, when we seek the solution of (7) in the form (11), we assume that the function f (x, y) is represented in the form f (x, y) = ω(x, y)

∞ X

(x − a)n+γ (b − x)fn1 (y).

(f1 )

n=0

Then at ϕ1 (x) = 0, ψ(y) = 0, for determination of the unknown function u1n (y), we obtain the following infinite number of one-dimensional systems of integral equations with singular left boundary points: Z y Z y 1 u10 (s)ds f01 (s)ds δ1 1 = , (u10 ) u0 (y) − (b − a)γ c (s − c)(d − s) (b − a)γ c (s − c)(d − s) u1n (y)

δ1 − (n + γ)(b − a) ′

=

Z

y

c

(u1n−1 (y)) 1 + (b − a) (n + γ)(b − a)

n = 1, 2, 3, . . . , c < y < y0 < d.

u1n (s)ds (s − c)(d − s) Z

c

y

fn1 (s)ds , (s − c)(d − s)

(u1n )

Hyperbolic Equations for Which All of the Boundary Consist of Singular Lines

361

Similarly, when we seek the solution of (9) in the form (11) with unknown u3n (y), and with condition that f (x, y) repressented in the form (f1 ), provided that ϕ2 (x) = 0, ψ1 (y) = 0, then for determining the unknown function u3n (y), we obtain the following infinite number of one-dimensional system of integral equations with singular right boundary points: Z y Z y 1 δ1 u30 (s)ds f01 (s)ds = , (u30 ) u30 (y) + (b − a)γ c (s − c)(d − s) (b − a)γ c (s − c)(d − s) δ1 (b − a)(b − a)

u30 (y) +

Z

d

y

′

(u3n−1 (y)) 1 − (b − a) (n + γ)(b − a)

=

u3n (s)ds (s − c)(d − s) Z

d y

fn1 (s)ds . (s − c)(d − s)

(u3n )

In according to [4] and [9], when the solution the system of the integral equation (u1n )(n = 0, 1, 2, . . .) in D1 exist, then it is given by formulas δ1    y − c γ(b−a)(d−c) 1 u0 (y) = C01 d−y +

1 γ(b − a)

u1n (y) 

=

Z



c

y



y−c d−y

d−c s−c

δ1  γ(b−a)(d−c)

 f01 (s) ≡ T01 [C01 , f01 (y)], (s − c)(d − s)

δ1   (n+γ)γ(b−a)(d−c)

Cn1

+

Z

c

y



d−c s−c

δ1  (γ+n)(b−a)(d−c)

  ′ (u1n−1 (s)) fn1 (s) 1 ds ≡ Tn1 [C01 , fn1 (y)], + b−a (γ + n)(b − a) (s − c)(d − s)

n = 1, 2, 3, · · · . The solution of system integral equation (u3n )(n = 0, 1, 2, · · · ) in D3 is given by formulas δ1 δ1     Z d 1 d − c γ(b−a)(d−c) y − c γ(b−a)(d−c) C03 + u30 (y) = d−y γ(b − a) y s−c  f01 (s) ≡ T01 [C01 , f01 (y)], (s − c)(d − s) u3n (y)

=



y−c d−y

δ1  (n+γ)γ(b−a)(d−c)



Cn3 −

1 − (b − a)

Z

y

d



d−c s−c

δ1  (γ+n)(b−a)(d−c)

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N. Rajabov



′

(u3n−1 (s)) +

  fn1 (s) ds ≡ Tn3 [C03 , fn1 (y), u3n−1 (s)]. (γ + n)(s − c)(d − s)

Moreover from condition continuity solution equation (1) in lines y = y0 (u10 (y0 ) = u3n (y0 )), follows that δ1  Z d f01 (s)ds d − s γ(b−a)(d−c) 1 3 1 , (A10 ) C0 − C0 = γ(b − a) c s−c (s − c)(d − s) Cn3 −Cn1 = Z y0 c

d−s s−c

1 (γ +n)(b−a)

(γ+n)(bδ−1a)(d−c)

Z d c

d−s s−c ′

(γ+n)(bδ−1a)(d−c)

(u1n−1 (s)) ds +

Z d y0

d−s s−c

fn1 (s 1 ds+ (s−c)(d−s) b−a

(γ+n)(bδ−1a)(d−c)

 ′ (u3n−1 (s)) ds .

(A1n ) In the case, when we the solution of the Integral Equation (8) will be seek in the form (13) and will be assume that, function f (x, y) representable in the form ∞ X f (x, y) = ω(x, y) (b − x)n+γ (x − a)fn2 (y), (f2 ) n=0

then at ϕ2 (x) = 0, ψ2 (y) = 0 for determined unknown functions u2n (y), we obtained the following one dimensional Volterra type Integral Equation with left fixed singular point Z y Z y 1 u20 (s)ds f02 (s)ds δ1 2 = , (u20 ) u0 (y) + γ(b − a) c (s − c)(d − c) λ(b − a) c (s − c)(d − s) u2n (y) 1 = (b − a)

δ1 + (γ + n)(b − a)

 ′ (u2n−1 (y)) +

Z

y

c

1 (γ + n)

u2n (s)ds (s − c)(d − s)

Z

c

y

 fn2 (s)ds . (s − c)(d − s)

(u2n )

Similarly, when the solution of the Integral Equation (10) we will be seek in the form (13) with unknown functions u4n (y) and with condition that, function f (x, y) repressing in the form (f2 ), at provided that ϕ2 (x) = 0, ψ2 (y) = 0, then for determined unknown functions u4n (y) we obtained the following infinity number the one - dimensional Volterra type system Integral Equation with right boundary singular point Z d Z d δ1 1 u40 (s)ds f02 (s)ds 4 u0 (y) + = , (u40 ) γ(b − a) y (s − c)(d − c) γ(b − a) y (s − c)(d − s)

Hyperbolic Equations for Which All of the Boundary Consist of Singular Lines

u4n (y) "

δ1 + (γ + n)(b − a)

Z

d

y

′ 1 1 = (u4n−1 (y)) + (b − a) (γ + n)

363

u4n (s)ds (s − c)(d − s) Z

d y

# fn2 (s)ds . (s − c)(d − s)

(u4n )

The solution of the system integral Equation (u2n ) in D2 is given by formulas, n = 1, 2, 3, · · · , δ1 δ1     Z y s − c γ(b−a)(d−c) 1 d − y γ(b−a)(d−c) 2 2 C0 + u0 (y) = y−c (b − a) c d−c  2 f0 (s) ≡ T02 [C02 , f02 (y)] (s − c)(d − s) δ1 δ1     Z d d − y (n+γ)(b−a)(d−c) s − c (γ+n)(b−a)(d−c) 2 2 un (y) = Cn + y−c d−s y    2 ′ (un−1 (s)) 1 + ds ≡ Tn2 [Cn2 , fn2 (y), u2n−1 (y)]. b−a (γ +n)(b−a)(s−c)(d−s)

The solution of system Integral Equation (u4n ) in D4 is given by formulas, n = 1, 2, 3, · · · δ1 δ1     Z d 1 d − y γ(b−a)(d−c) s − c γ(b−a)(d−c) 4 4 C0 + u0 (y) = y−c (b − a) y d−c  2 f0 (s) ≡ T04 [C04 , f02 (y)] (s − c)(d − s) δ1 δ1     Z d d − y (n+γ)(b−a)(d−c) s − c (γ+n)(b−a)(d−c) 4 4 un (y) = Cn − y−c d−s y    4 ′ (un−1 (s)) 1 ds ≡ Tn4 [Cn4 , fn2 (y), u4n−1 (y)]. + b−a (γ +n)(b−a)(s−c)(d−s)

Moreover from condition continuity the solution hyperbolic equation equation in lines y = y0 , follows, that u2n (y0 ) = u4n (y0 ). From here it follows that the constants Cn4 and Cn2 (n = 0, 1, 2, . . .) among themselves connected in the following way, n = 1, 2, 3, · · · δ1  Z d f02 (s)ds s − c γ(b−a)(d−c) 1 4 2 C0 − C0 = (B01 ) γ(b − a) c d−s (s − c)(d − s), Cn4 −Cn2

1 = (γ +n)(b−a)

Z d c

s−c d−s

δ1  (γ+n)(b−a)(d−c)

fn2 (s) 1 ds+ (s−c)(d−s) (b−a)

364

Z

c

N. Rajabov y0

s−c d−s

(γ+n)(bδ−1a)(d−c)

′

2 (νn−1 (s)) ds+

Z d y0

s−c d−s

(γ+n)(bδ−1a)(d−c)

 ′ 4 (νn−1 (s)) ds .

(Bn1 )

Thus we have proved the following. Theorem 0.3. Let in equation (1) the coefficients λ, µ, δ be such that δ1 = λµ − δ > 0, f be a function in D1 , D3 representable in the form (f1 ) and in D2 , D4 representable in the form (f2 ), where fn1 (c) = 0 with 1 δ1 , n= the asymptotic behavior fn1 (y) = O[(y − c)γn ], γn1 > (γ+n)(b−a)(d−c) 2 0, 1, 2, 3, · · · , as y → c and fn (d) = 0, with the asymptotic behavior 2 δ1 fn2 = O[(d − c)γn ], γn2 > (γ+n)(b−a)(d−c) , n = 0, 1, 2, 3, · · · as y → d. Then any solution of equation (1) is in the class of functions representable in domains D1 , D3 in the form (11), and in domains D2 , D4 in the form (13), that is, u(x, y) = ω(x, y) u(x, y) = ω(x, y) u(x, y) = ω(x, y) u(x, y) = ω(x, y)

∞ X

n=0 ∞ X

n=0 ∞ X

n=0 ∞ X n=0

(x−a)n+γ (b−x)Tn1 [Cn1 , fn1 (y), u1n−1 (y)], (x, y) ∈ D1 , (b−x)n+γ (x−a)Tn2 [Cn2 , fn2 (y), u2n−1 (y)], (x, y) ∈ D2 , (x−a)n+γ (b−x)Tn3 [Cn3 , fn1 (y), u3n−1 (y)], (x, y) ∈ D3 , (b−x)n+γ (x−a)Tn4 [Cn4 , fn2 (y), u4n−1 (y)], (x, y) ∈ D4 ,

where constant Cn1 , Cn3 among themselves connected by formulas (A1n ) (n = 0, 1, 2, 3, · · · ) and constants Cn2 , Cn4 among themselves connected by formulas (Bn1 )(n = 0, 1, 2, 3, · · · ). Using the integral representation of the from (2), immediately testing, easily we can see, that if exist the solution of the equation (1), then it have the following characteristics    λ  µ   x − a b−a y y − c d−c u(x, y) Pλ (u) = ϕ1 (x), = ψ1 (y), d−c b−x y=c x=a   λ  ′ µ y − c d−c x Pµ (u) = ϕ1 (x) + ϕ1 (x), d−c (x − a)(b − x) y=c   λ  ′ µ y − c d−c x Pµ (u) = ϕ2 (x) + ϕ2 (x) d−c (x − a)(b − x) y=d

Hyperbolic Equations for Which All of the Boundary Consist of Singular Lines



y−c d−y

λ  d−c





 u(x, y)

x−a b−x x−a b−x

= ϕ2 (x), y=d

µ  b−a

µ  b−a

(Pλy )2 u



(Pλy )2 U



x−a b−x

µ  b−a

= ψ2 (y), x=b

= ψ11 (y) +

λψ1 (y) , (y − c)(d − y)

= ψ21 (y) +

λψ2 (y) , (y − c)(d − y)

x=a



 Pλy (u)

x=b

365

µ µ ∂ ∂ + (x−a)(b−x) , Pλy ≡ ∂y + (y−c)(d−y) . where Pµx ≡ ∂x Integral representation (2) and higher mentioned conditions the solution of the equation (2), give possibility for equation (1) place and investigate the following boundary value problems. 2 Problem R1 . Find the solution of equation (1) in class Cxy (D) with one of the following boundary conditions:

1)



y−c d−y

λ  d−c

 U (x, y)

= A1 (x), y=d



x−a b−x

µ  b−a

Pλy



= B1 (y); x=b

    λ  µ x − a b−a y y − c d−c = A2 (x), = B2 (y); U (x, y) Pλ 2) d−y b−x y=c x=a  λ  µ     y − c d−c x x − a b−a y 3) = B3 (y); Pµ (u) = A3 (x), Pλ d−y b−x x=a y=c      λ  µ x − a b−a y y − c d−c x Pµ (u) Pλ = A4 (x), = B4 (y); 4) d−y b−x y=d x=b 

where Aj (x), Bj (y), j = 1, 2, 3, 4− are given functions Γ1 and Γ2 . 2 Problem R2 . Find the solution of equation (1) in class Cxy (D) with the following boundary conditions:

1)

2)



x−a b−x

µ   b−a (Pλy )2 (u)



y−c d−y

λ  d−c



Pµx (u)

= B5 (y), x=a

= A6 (y), y=d





y−c d−y

x−a b−x

λ  d−c

µ  b−a

 Pµx (u) 

(Pλy )2 (u)

= A5 (x) y=c

(R21 )

= B6 (y)

x=b

(R22 ) The solutions of problems R1 and R2 can be found using the integral representation (2) and properties mentioned above.

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N. Rajabov

For example, the solution of problem R21 reduce to solution of the following linear differential equations with two boundary singular points: ψ11 (y) +

λ ψ1 (y) = B5 (y), (y − c)(d − y)

(15)

ϕ11 (x) +

µ ϕ1 (x) = A5 (x). (x − a)(b − x)

(16)

By solving differential equations (15) and (16) according to [3], finding  µ M1 , a < x ≤ x0 ϕ1 (x) = (17) M2µ , x0 ≤ x < b

 µ  b−x b−a x where M1µ [C1 , A5 (x)] = x−a [C1 + Ta,µ [A5 (x)]], M2µ [C2 , A5 (x)] =   µ b−x b−a x [C2 − Tb,µ [A5 (x)]], C1 is an arbitrary constant, C2 with C1 conx−a nected in the following way  µ Z b A5 (t)dt t − a b−a , (18) C2 = C1 + b−t (t − a)(b − t) a ψ1 (y) =



N1λ [C3 , B5 (y)], N2λ [C4 , B5 (y)],

c < y ≤ y0 y0 ≤ y < d

(19)

λ  d−c  where N1λ [C3 , B5 (y)] = d−s [C3 + Tc,y λ [B5 (y)]], N2λ [C4 , B5 (y)] = s−c  λ  y d−s d−c [C4 − Td, λ [B5 (y)]], C3 is an arbitrary constant, C4 with C3 cons−c nected in the following way  λ Z d s − c d−c B5 (s)ds (20) C4 = C3 + d − s (s − c)(d − s) c

Theorem 0.4. Let in equation (1), λ, µ, c, f (x, y) satisfy the conditions of Theorem 0.1, and in problem (R21 ), let A5 (x) ∈ C(Γ1 ), A5 (b) = 0 µ , B5 (y) ∈ with the asymptotic behavior A5 (x) = O[(b = x)γ5 ], γ5 > b−a C(Γ2 ), B5 (d) = 0 with the asymptotic behavior B5 (y) = O[(d − y)γ6 ], λ . Then problem (R21 ) is always solvable and its solution is given γ6 > d−c by formulas (2), (17), (19), (3), (4), (5), (18), (20), where C10 , C1 , C3 are arbitrary constants. Remark 0.4. Results similar to Theorem 0.2 are obtained about the solvability of problems R1 and R22 .

Hyperbolic Equations for Which All of the Boundary Consist of Singular Lines

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References 1. M. Holodnyok, A. Klich, M. Kubichek, M. Marec, Methods of analysis of dynamical models, Moscow, Mir, 1991, 365. 2. N. Rajabov, Integral representation and boundary value problem for certain differential equation with singular lines or singular surfaces, Part 4, TSU publisher, Dushanbe 1985, 147. 3. N. Rajabov, An introduction to the theory of partial differential equations with super-singular coefficients, Tehran 1997, 230. 4. N. Rajabov, Introduction to ordinary differential equations with singular and super-singular coefficients, Dushanbe 1998, 160. 5. N. Rajabov, System of linear integral equations of Volterra type with singular and super-singular kernels. Ill-posed and non-classical problems of mathematical physics and analysis, Proc. Intern. Cons. Samarqand, Uzbekistan, September 11-12, 2000, Kluwer, Utrecht-Boston, 103-124. 6. N. Rajabov, About one Volterra type integral equation, Russian Acad. Sci. Doklady, 2002, v. 383, No. 3, 314-317. 7. N. Rajabov, Volterra type integral equation with interior fixed singular and super-singular point in kernel, proceedings international conference “Differential Equation and its Supplement”, Samara 2002, 279-283. 8. N. Rajabov, About one class of Volterra type linear integral equations with interior fixed singular or super-singular point, Topics in analysis and applications, Kluwer, Dordrecht, Boston, London, 2004, 317-326. 9. L. N. Rajabov, About one class hyperbolic equation with singular lines, Bulletin National University, No. 5 (31), 2002, 44-51. 10. N. Rajabov, Volterra type Integral equation with boundary and interior singular and super-singular kernels its application, Dushanbe, 2007, 222.

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ASYMPTOTIC STUDY OF AN ANISOTROPIC PERIODIC ROTATING MHD SYSTEM R. SELMI Department of Mathematics and Informatics, Faculty of Sciences, University of Gabes, Gabes 6000, Tunisia E-mail: [email protected] Asymptotic behavior of three-dimensional anisotropic periodic rotating magnetohydrodynamic system is investigated as the Rossby number goes to zero. Since the system is mixed, the divergence free condition is crucial in the proofs. The main tools are the spectral properties of the penalization operator, the energy method, Schochet’s method and product laws in anisotropic Sobolev spaces. A filtering process is used to handle the singular perturbation, and elementary Fourier analysis to deal with the nonlinear part. Keywords: Anisotropic MHD system; asymptotic behavior; limit system; divergence free condition; singular perturbation.

1. Introduction We consider the following mixed periodic incompressible and perturbed magnetohydrodynamic system, denoted by (M HD ε ):  1 1   ∂t u − ν∆h u + u · ∇u − curl b × b + curl b × e2 + u × e3 = −∇p   ε ε   1 ∂t b − η∆h b + u · ∇b − b · ∇u + curl (u × e2 ) = 0 ε    div u = 0, div b = 0    (u, b)|t=0 = (u0 , b0 ),

where the velocity field u, the induced magnetic perturbation b and the pressure p are unknown functions of time t ∈ R+ and space variable x = (x1 , x2 , x3 ) ∈ T3 , e2 and e3 are the cartesian coordinate system, ν and η designate respectively the dynamic viscosity and the magnetic diffusivity, ε is the Rossby number destined to tend to zero and ∆h = ∂12 + ∂22 . About physical motivation, see [1] and references therein. Applying P, the L2 (T3 )orthogonal projection onto divergence free vector fields, to the first equation

Asymptotic Study of an Anisotropic Periodic Rotating MHD System

369

of (M HDε ), one sees that U := (u, b) satisfies  ε + 3  ∂t U + Q(U, U ) + a2 (D)U + L (U ) = 0 in R × T (S ε ) div u = 0, div b = 0 in R+ × T3  U |t=0 = U0 in T3 ,

where after simplification, the quadratic term Q, the viscous term a2 (D) and the linear perturbation Lε are given by  P(u · ∇u) − P(b · ∇b)  Q(U, U ) = u · ∇b − b · ∇u a2 (D)U = (−ν∆h u, −η∆h b) 1  ∂2 b + P(u × e3 )  1 . Lε (U ) = L(U ) = ∂2 u ε ε ′

As in [5], for (s, s′ ) ∈ R2 the anisotropic Sobolev space H s,s (T3 ) is the set of tempered distributions f satisfying, for any k = (k ′ , k3 ) = (k1 , k2 , k3 ) ∈ R3 , s

s′

kf kH s,s′ := k(1 + |k ′ |2 ) 2 (1 + |k3 |2 ) 2 F(f )kL2 < ∞. Following ideas in our previous work [1], we prove theorem below. Note that product law given by Theorem 1.4 in reference [6] still apply in the periodic case since that if any integral converges in+∞ its corresponding series do. Theorem 1.1. Let s > 12 be a real number and U0 = (u0 , b0 ) ∈ H 0,s (T3 ), such that div u0 = 0 and div b0 = 0. There exists a positive time T such that for all ε > 0, there exists a unique solution U ε of (S ε ), satisfying U ε ∈ CT0 (H 0,s (T3 )) and ∇h U ε ∈ L2T (H 0,s (T3 )). Moreover, U ε fulfils Z t ε 2 kU (t, .)kH 0,s (T3 ) +min(ν, η) k∇h U ε (τ, .)k2H 0,s (T3 ) dτ ≤ kU0 k2H 0,s (T3 ) . (1) 0

Furthermore, if a constant c exists, such that kU0 kH 0,s (T3 ) ≤ c min(ν, η), then the solution is global. The aim of this work is to focus on the asymptotic behavior of the unique solution, as the Rossby number ε goes to zero. However, since (∂t U ε ) is not bounded with respect to ε, then the classical methods, based on taking directly such limit in the system, no longer work. Lε being skew symmetric, we overcome this difficulty by using the method introduced in [10]. This consists in filtering the singular system by the associated group L(t), to obtain a filtered one, where the penalized perturbation disappears. Thus, it will be possible to look, in the sense of distribution, for the limit system

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R. Selmi

satisfied by the limit V of the filtered solution V ε := L(− εt )U ε . Namely, we establish the following convergence result: Theorem 1.2. Let s > 12 and U0 = (u0 , b0 ) ∈ H s (T3 ), such that div u0 = div b0 = 0. Let U ε = (uε , bε ) be the family of solutions of (S ε ) given by Theorem 1. Then, for all s′ < s, the family V ε = (v ε , cε ) converges strongly ′ in CT0 (H 0,s ) to the solution V of system (LS) defined by  0 3  ∂t V + Q (V, V ) + a2 (D)V = 0 in [0, T ] × T 3 (LS) div v = div c = 0 in [0, T ] × T  V |t=0 = U0 = (u0 , b0 ) in T3 where we set, in the sense of distributions, Q0 (V, V ) := lim Qε (V, V ). ε→0

The proof uses the divergence free condition, anisotropic product law [6] and properties of P. In fact, ∂3 U3 = −divh Uh implies that U is more regular with respect to x3 then expected. We motion here references [7,8], where it was proved global in time existence result of anisotropic Navier-Stokes equation, when ε is sufficiently small but for initial data associated with horizontal mean equal to zero in H 0,s . Such condition is purely technic and dictated by the use of some Gagliardo-Nirenberg inequality needed in the framework of Littlewood-Paley computation for the critical periodic case s = 21 . Here, since s > 21 , we can extend existence and uniqueness results proved in [1] for the case R3 without any additional physically meaningless condition on the initial data. Moreover, although it is local in time, our convergence result presents the advantage to use classical techniques, such as commutator inequalities, to handle the nonlinear part in an elegent and rapid manner in opposition to tedious computation involved by the anisotropic littelwood-Paley theory used in [8]. 2. Convergence result 2.1. Study of the limit system Following ideas in [9], we prove the lemma below. Lemma 2.1. The “wave equation” given by the following system  3  ∂t U + L(U ) = 0 in R × T div u0 = div b0 = 0 in R × T3  U (0) = U0 in T3

(2)

has a global solution U (t) = L(t)U0 , such that for all (s, s′ ) ∈ R2 and U0 ′ in H s,s (T3 ), kL(t)U0 kH s,s′ (T3 ) = kU0 kH s,s′ (T3 ) = kt L(t)U0 kH s,s′ (T3 ) .

Asymptotic Study of an Anisotropic Periodic Rotating MHD System

Proof. System (2) is equivalent to  3   ∂t u + P(u × e3 ) + ∂2 b = 0 3 in R × T  ∂t b + ∂2 u = 0 in R × T  div u0 = div b0 = 0 in R × T3   U (0) = U0 in T3 .

371

(3)

Applying “curl” to the first equation, then by Fourier Transform we get   ∂t iM (k)ˆ u(t, k) + ik3 u ˆ(t, k) − k2 M (k)ˆb(t, k) = 0 in R × T3    ˆ ∂t b(t, k) + ik2 uˆ(t, k) in R × T3 (4)  k · uˆ0 = k · ˆb0 = 0 in R × T3    U(0) ˆ ˆ0 in T3 . =U The eigenvalues of M (k) are ±i|k|, 0 and the corresponding eigenvectors k which is not divergence free. Vectors ρ(k)± are given by are ρ(k)± and |k| 
 1, − i |kk33 | , 0 if k12 + k22 = 0, 
α(k), β(k), γ(k) ρ(k)+ = 
1 if k12 + k22 6= 0,  |α(k)|2 +|β(k)|2 +|γ(k)|2 2

α(k), β(k), γ(k) = − k1 k3 − i k2 |k|, −k2 k3 + i k1 |k|, k12 + k22 , and ρ(k)− = ρ(k)+ . In this eigenbase, u and b can be rewritten such that u ˆ(t, k) = uˆ+ (t, k)ρ(k)+ + uˆ− (t, k)ρ(k)− ˆb(t, k) = ˆb+ (t, k)ρ(k)+ + ˆb− (t, k)ρ(k)− . In term of variables u ˆ+ , u ˆ− , ˆb+ and ˆb− , system ( 4) is summarized in ( ˆ ± (t, k) + A± (k)U ˆ ± (t, k) = 0 in R × T3 ∂t U k·u ˆ = k · ˆb = 0 in R × T3 , ˆ ± = (ˆ where U u± , ˆb± ) and  k3 ik2  ∓i . A± (k) =  |k| ik2 0 

Let A = diag(A+ , A− ), its eigenvalues satisfy ω3 = ω2 , ω4 = ω1 and q i (−k3 + k32 + 4k22 |k|2 ) ω1 (k) = 2|k| q i (−k3 − k32 + 4k22 |k|2 ). ω2 (k) = 2|k|

(5)

372

R. Selmi

If we denote by ̺j , 1 ≤ j ≤ 4, the corresponding eigenvectors, then X
exp ωj (k)t < F(U )(0, k)|̺j (k) > ̺j (k). F(U )(t, k) = j∈{1,2,3,4}

  Define V ε by V ε (t, x) := L − εt U ε (t, x) = (v ε , cε ). Then,


F V ε (t, k) =

X

j∈{1,2,3,4}


exp ωj (k)t < F(V ε )(0, k)|̺j (k) > ̺j (k)

and V ε satisfies the following system  ε ε ε ε ε  ∂t V + Q (V , V ) + a2 (D)V = 0 ε ε ε (S˜ ) div v = div c = 0 in R × T3  ε V |t=0 = U0 = (u0 , b0 ) in T3

(6)

in R × T3

where the “filtered” quadratic form Qε is given by  t  t t  Qε (V ε , V ε ) = L − PQε L V ε, L Vε . ε ε ε   In Fourier variables 2F Qε (V1ε , V2ε ) (n) = (a11 , a21 ), where a11 = − a21 = −

X

X

1≤j,j ′ ,j ′′ ≤4

k+m=n

1≤j,j ′ ,j ′′ ≤4

k+m=n

1≤j,j ′ ,j ′′ ≤4

k+m=n

1≤j,j ′ ,j ′′ ≤4

k+m=n

X

X

X

X

X

X

t

j,j ′ ,j ′′

′

t

j,j ′ ,j ′′

′

t

j,j ′ ,j ′′

′

′′

t

j,j ′ ,j ′′

′

′′

′′

e−i ε ωk,m,n P (n) < v j (k)mv j (m)|̺j (n) > ̺j (n) ′′

e−i ε ωk,m,n P (n) < cj (k)mcj (m)|̺j (n) > ̺j (n) e−i ε ωk,m,n < v j (k)mcj (m)|̺j (n) > ̺j (n) e−i ε ωk,m,n < cj (k)mv j (m)|̺j (n) > ̺j (n). ′

′′

′

′′

j,j ,j In the expression above, ωk,m,n := ω j (k) + ω j (m) − ω j (n),   2 n1 n1 n2 n1 n3 1  n2 n1 n22 n2 n3  P (n) := Id − 2 n1 + n22 + n23 n3 n1 n3 n2 n23

and for any vector X, xj :=< F(X)(n)|̺j (n) > ̺j (n). The filtering procedure presents the advantage that the singular perturbation disappears. So, contrarily to ∂t U ε in system (S ε ), ∂t V ε in system (S˜ε ) is already bonded in L∞ ([0, T ], H −N (T3 )), for some sufficiently large integer N . Consequently, ′ classical compactness argument leads to U ε → U in C([0, T ], H s,s (T3 )), for all s < 0 and s′ < 21 . As in [4], one shoes that in D ′ the stationary phase

Asymptotic Study of an Anisotropic Periodic Rotating MHD System

373

theorem implies that lim Qε (V ε , V ε ) = Q0 (V, V ). So, when ε goes to 0, one ε→0

obtains formally the following limit system:  0 3  ∂t V + Q (V, V ) + a2 (D)V = 0 in [0, T ] × T 3 (LS) div v = div c = 0 in [0, T ] × T  V |t=0 = U0 = (u0 , b0 ) in T3 .

Theorem 2.1. Under assumptions of Theorem 1.1, there exists T > 0 and 0,s a unique V ∈ L∞ (T3 )) ∩ L2T (H 1,s (T3 )) solution of (LS) satisfying T (H Z t kV (t, .)k2H 0,s (T3 ) + ν k∇h v(τ, .)k2H 0,s (T3 ) dτ 0 Z t (7) +η k∇h c(τ, .)k2H 0,s (T3 ) dτ ≤ kU0 k2H 0,s (T3 ) . 0

Furthermore, If a constant c exists, such that kU0 kH 0,s (T3 ) ≤ c min(ν, η), then the solution is global. The proof follows the lines of the one of Theorem 1.1. Remark 2.1. Note that the time T appearing in Theorem 2.1 can be different from the one of Theorem 1.1. Here we are not dealing with the possible relation between the life span of the solution of the system (S ε ) and the life span of the solution of the corresponding limit system (LS). But since we are proving a local in time convergence result, any uniform time of existence of both systems, denoted also T , is suitable for our purpose. We mention that the method used in [7] can be followed to prove global well posedeness of (LS) and to deduce the same for (M HD ε ), as ε is small enough. 2.2. Proof of Theorem 1.2 Let W ε = V ε − V = (v ε − v, cε − c) = (W1ε , W2ε ). Then, W ε satisfies  ε ε ε ε ε ε  ∂t W + Q (W , W + 2V ) + a2 (D)W = Rosc ε ε div W1 = div W2 = 0  W ε |t=0 = (0, 0),

(8)

ε where Rosc = Q0 (V, V ) − Qε (V, V ). The right hand side of system (8) is an oscillating term which converges weakly to zero but not strongly. The ε method we use to deal with Rosc is inspired from ideas introduced in [10]. ε ε,N It consists to divide Rosc into high frequency term Rosc and low frequency

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ε term Rosc,N , defined respectively, for any arbitrary cut-off integer N ≥ 1, ε,N ε ε by Rosc = Rosc − Rosc,N and

  ε ε (t, n) = 1{|n|≤N } F(Rosc;|k|,|m|≤N )(t, n), F Rosc,N

ε ε where Rosc;|k|,|m|≤N is the part of Rosc for which k and m are such that |k|, |m| ≤ N . To absorb the low frequency term, we adopt the change ˜ε ˜ε of function: ϕεN = W ε + εR osc,N , where Rosc,N is obtained by dividing ′

′′

′

′′

j,j ,j j,j ,j ε exp(−i εt ωk,m,n ) by iωk,m,n in the expression Rosc,N . Function ϕεN satisfies: ε ε ε,N ˜ osc,N + εrosc,N , (9) ∂t ϕεN + Qε (ϕεN , ϕεN − 2εR + 2V ) + a2 (D)ϕεN = Rosc ε ˜ε ˜ε ˜ ε,t ˜ε where εrosc,N = −Qε (εR osc,N , εRosc,N − 2V ) − εa2 (D)(Rosc,N ) + εRosc,N ˜ ε,t is the expression of R ˜ε and R osc,N where we have changed, the nonlinear osc,N factor by its time derivative. Note that equation (9) presents the advantage that the low frequency terms have disappeared up to an ε. For any real 1 number x, denote by hxi the quantity (1 + |x|2 ) 2 and by Λ3 the operator 1 defined by Λ3 = (1 + ∂32 ) 2 , that is the operator of multiplication by hk3 i in the frequency space. Clearly, for all real numbers s and s′ , Λ3 is an isometry ′ ′ from H s,s to H s,s −1 . 1 Taking the scalar product in H 0,− 2 (T3 ), equation (9) gives

kϕεN k20,− 1 2

+ 2ν

Z

= −2 −2 Let

Z

Q

ε

T3

Z

(ϕεN , ϕεN

t

k∇h ϕεN,1 k20,− 1 dτ 2

Z0 t Z Z0

0

t

ZT3

T3

+ 2η

Z

0

t

k∇h ϕεN,2 k20,− 1 dτ 2

−1 ε ˜ε Qε (ϕεN , ϕεN − 2εR osc,N + 2V )Λ3 ϕN dxdτ

(10)


−1 ε ε,N ε Λ3 ϕN dxdτ. Rosc + εrosc,N

−1 ε ˜ε − 2εR osc,N + 2V )Λ3 ϕN dxdτ =

4 X

Ii , where

i=1

 t
i  −t
h t
ε ε ˜ε L P L (ϕN −2εR +2V ) ϕεN Λ−1 ∇ h osc,N 3 ϕN dx ε ε ε h ZT3   t
i −t
h t
ε ε ε ˜ osc,N P L (ϕN −2εR +2V ) ∂3 L ϕεN Λ−1 I2 = L 3 ϕN dx ε h ε ε 3 ZT3  t
i t
ε  −t
ε ˜ε P L ϕN ∇h L (ϕεN −2εR Λ−1 L I3 = osc,N +2V ) 3 ϕN dx ε ε ε h 3 ZT i −t
h t
ε   t
ε ε ˜ε P L ϕN ∂3 L (ϕN −2εR +2V ) Λ−1 L I4 = osc,N 3 ϕN dx. ε ε ε 3 T3 I1 =

L

Asymptotic Study of an Anisotropic Periodic Rotating MHD System

375

Ideas in [1] lead to 1

3

ε 2 ε 2 ˜ε |I1 | ≤ kϕεN − 2εR osc,N + 2V k 12 ,s kϕN k0,− 1 kϕN k1,− 1 2 1

2 3

ε 2 ε 2 ˜ε |I3 | ≤ Ck(ϕεN − 2εR osc,N + 2V )k 21 ,s kϕN k0,− 1 kϕN k1,− 1 2 2 ˜ε |I4 | ≤ CkϕεN − 2εR osc,N + 2V k 21 ,s 3 1 1 3
ε 2 ε 2 2 2 × kϕεN k0,− . + kϕεN k0,− 1 kϕN k 1 kϕN k 1,− 1 1,− 1 2

2

2

2

To estimate I2 , one uses Parseval’s formula and property of Λ3 to obtain X  −t

t
ε ˜ε P L (ϕN − 2εR I2 = (2π)−3 F L osc,N + 2V ) 3 ε ε k∈Z  

 1 × ∂3 L εt ϕεN F ϕεN (−k). (k) hk3 i

Since one has F(Pu) = F(u)− < F(u),

k |k|

>

k |k| ,

then by (6), it comes that

X XX n3 j,j ′ ,j ′′ t I2 = C exp(ωk−n,n,k )F(ϕεN )(−k) ε hk3 i n j,j ′ ,j ′′ n∈Z k∈Z ′ ′′ ˜ε < (ϕε − 2εR + 2V )j (k − n)(ϕε )j (n)|̺j (k) > ̺j (k)− N

osc,N

N

′

′′

j ε j j j ˜ε ̺ (k), k k o . > |k| |k|

Using the change of variable (k, n) ↔ (−n, −k), we obtain  n X XX k3  3 j,j ′ ,j ′′ t I2 = C )F(ϕεN )(−k) − exp(ωk−n,n,k ε hk3 i hn3 i n j,j ′ ,j ′′ n∈Z k∈Z j′ ε j ′′ j j ˜ε < (ϕεN − 2εR osc,N + 2V ) (k − n)(ϕN ) (n)|̺ (k) > ̺ (k)− ′′ ′ j j j ε j ˜ε ̺ (k), o k k > . |k| |k| We recall the following elementary equality due to [6]: n  1 k3 n3 −k3 (n3 −k3 )k3 (n3 + k3 ) 1  3 . − + + = ≤ |n3 −k3 | hk3 i hn3 i hk3 i hn3 ihk3 i(hn3 i+hk3 i) hn3 i hk3 i Reapply change of variables (k, n) ↔ (−n, −k), this inequality leads to |I2 | ≤ C

X X |n3 − k3 |

n∈Z k∈Z

hk3 i

ε ˜ε |F((ϕεN − 2εR osc,N + 2V )3 )(k − n)||F(ϕN )(n)|

|F(ϕεN )(−k)|.

376

R. Selmi

As W , V and ϕεN are divergence free, then |I2 | ≤ C

2 ˜ε XXX |nj − kj ||F((ϕεN − 2εR osc,N + 2V )j )| |F(ϕεN )(n)| hk i 3 j=1

n∈Z k∈Z

|F(ϕεN )(−k)|.

Let V ′ such that F(Vj′ ) = |F(Vj )|. So, kVj′ ks,s′ = kVj ks,s′ . Denote |Dj | the operator of multiplication by |ξj |. Use Parseval’s formula to obtain |I2 | ≤ C Since

|Dj |Vj′

2 XX k∈Z j=1

′ ′ε ′ε ˜ε |Dj |(ϕεN − 2εR osc,N + 2V )j ϕ N Λ3 ϕ N . ′

and ∂j Vj′ has the same H s,s norm, then as for I3 , one has 3

1

ε 2 ε 2 ˜ osc,N . + 2V k 21 ,s kϕεN k0,− |I2 | ≤ CkϕεN − 2εR 1 kϕN k 1,− 1 2

2

4 Using estimations of |Ii |, 1 ≤ i ≤ 4 and ab ≤ 14 a4 + 43 a 3 , (10) becomes Z t Z t kϕεN k20,− 1 + 2ν k∇h ϕεN,1 k20,− 1 dτ + 2η k∇h ϕεN,2 k20,− 1 dτ 2 2 2 0 Z t Z t 0 ε,N ε 2 ε 2 (11) kRosc + εrosc,N k−1,− 1 dτ ≤ 2min(ν, η) kϕN k1,− 1 dτ + C 2 2 0 0Z t 4 ˜ε +C kϕεN k20,− 1 (kϕεN − 2εR osc,N + 2V k 12 ,s + 1)dτ. 2 0

About the low-frequency terms, we have the following lemma:

Lemma 2.2. A constant CN (T ) = C(N, T ) exists, such that ε krosc,N k

1

≤ CN (T )

L2T (H −1,− 2 ) ˜ε ∞ kR k osc,N LT (H 0,s )∩L2T (H 1,s )

≤ CN (T ).

To prove this lemma, recall that all considered functions are truncated j,j ′ ,j ′′ in low frequencies. Moreover, ωk,m,n 6= 0 implies that j,j1′ ,j′′ ≤ C(N ). |ωk,m,n |

ε,N Since ϕεN = V ε − V + Rosc , equation (11) leads to Z t Z t k∇h ϕεN,1 k20,− 1 dτ + 2η k∇h ϕεN,2 k20,− 1 dτ kϕεN k20,− 1 + 2ν 2 2 2 0 0 Z t Z t ε,N ε kRosc + εrosc,N k20,− 1 dτ ≤ 2min(ν, η) kϕεN k21,− 1 dτ + C 2 2 0 0 Z t ε 2 ε 4 ε 4 4 +C kϕN k0,− 1 (kV k 1 ,s + kV k 1 ,s + ε C(N ) + 1)dτ. 2

0

2

2

It follows that

ε,N 2 k 2 kϕεN k20,− 1 ≤ CkRosc 2

LT

1 (H 0,− 2

)

+ ε2 CN (T ) + C

Z

0

t

kϕεN k20,− 1 h(τ )dτ, (12) 2

Asymptotic Study of an Anisotropic Periodic Rotating MHD System

377

where h(τ ) = kV ε k41 ,s + kV ε k41 ,s + ε4 C(N ) + 1. An interpolation gives 2

Z

0

T

2

kV ε k41 ,s (τ )dτ ≤ kV ε k2L∞ (H 0,s ) kV ε k2L2 (H 1,s ) . T

2

T

The same holds for V . Thus, h belongs to L1T . Gronwall’s Lemma implies
ε,N 2 kϕεN k20,− 1 ≤ CkRosc k 2 0,− 1 + ε2 CN (T ) 2 2) L (H T h
ε 2 0,s ) kV k 2 × exp C + ε4 C(N ) T + CkV ε kL∞ (13) LT (H 1,s ) T (H i 2 0,s ) kV k 2 + CkV kL∞ . L (H 1,s ) T (H T

About the high-frequency term

ε,N Rosc ,

we introduce the following lemma:

0 0,s Lemma 2.3. For  any functionf ∈ CT (H ) with s ∈ R, the high frequency N −1 1[N,+∞[ F(f ) goes to zero in CT0 (H 0,s ), when N → ∞. term f = F

Its proof is based on ideas from [2] and [3]; since kf N (t)k2H s = 2s 2 N 2 |k|≥N |k| |F(f )(t, k)| , Dini’s Theorem implies that kf (t)kH s → 0 uniformly in t. This lemma implies, in a straightforward way, the following: P

1

ε,N Proposition 2.1. Rosc → 0 in CT0 (H 0,s ) ∩ L2T (H 0,− 2 ), uniformly in ε, as N → ∞.

Using Proposition 2.1, letting ε → 0 and N → +∞, we get W ε → 0 in 1 C 0 ([0, T ], H 0,− 2 (T3 )). An interpolation argument completes the proof. Remark 2.2. Depending on which is the minimum of ν and η, either 1 1 W1ε → 0 in L2 ([0, T ], H 1,− 2 (T3 )) or W2ε → 0 in L2 ([0, T ], H 1,− 2 (T3 )). References 1. J. Benameur and R. Selmi: Study of Anisotropic MHD System in Anisotropic Sobolev Spaces, Annales Math´ematique de la Facult´e des Sciences de Toulouse, to appear. 2. I. Gallagher: Applications of Schochet’s Methods to Parabolic Equation, Journal de Math´ematiques Pures et Appliqu´ees 77, 989-1054, 1998. 3. I. Gallagher: Asymptotics of the Solutions of Hyperbolic Equations with a Skew-Symmetric Perturbation, Journal of Differential Equations 150, 363384, 1998. 4. E. Grenier: Oscillatory Perturbations of the Navier-Stokes Equations, Journal de Math´ematiques Pures et Appliqu´ees 76, 477-498, 1997. 5. D. Iftimie: Resolution of the Navier-Stokes Equations in Anisotropic Spaces, Revista Matem´ atica Iberoamericana 15, 1-36, 1999.

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6. D. Iftimie: A Uniqueness Result for the Navier-Stokes Equations with Vanishing Vertical Viscosity, SIAM Journal of Mathematical Analysis 33, 14831493, 2002. 7. M. Paicu: Etude Asymptotique pour les Fluides Anisotropes en Rotation Rapide dans le Cas Periodique, Journal de Math´ematiques Pures et Appliqu´ees 83, 163-242, 2004. 8. M. Paicu: Equation Periodique de Navier-Stokes sans Viscosit´e dans une Direction, Communications in Partial Differential Equations 30, 1107-1140, 2005. 9. R. Selmi: Convergence Results for MHD System, International Journal of Mathematics and Mathematical Sciences 2006, Article 28704, 19 pp., 2006. 10. S. Schochet: Fast Singular Limits of Hyperbolic PDEs, Journal of Differential Equations 114, 476-512, 1994.

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HILBERT-TYPE BOUNDARY VALUE PROBLEM FOR POLYANALYTIC FUNCTIONS ∗ YUFENG WANG School of Mathematics and Statistics, Wuhan University, Wuhan 430072, P. R. China E-mail: wh [email protected] JINPING WANG Department of Mathematics, Ningbo University, Ningbo 315211, P. R. China We discuss Hilbert-type boundary value problems of polyanalytic functions with the same factors. Using the decomposition of polyanalytic functions and the classical regularization method, we obtain explicit expressions of solutions and conditions of solvability. Keywords: Polyanalytic function, Hilbert-type boundary value problem, order, poly-Schwarz operator.

1. Introduction Three kinds of basic boundary value problems (BVPs) of complex equations of order one in complex analysis have been investigated in details by H. Begehr, and the explicit expression of solution and the condition of solvability have been obtained [5]. Those BVPs include the Schwartz problem, the Dirichlet problem and the Newmann problem. By Cauchy-Pompeiu formula and the method of iteration, BVPs of two-order complex equations, including the Poisson equation and the nonhomogeneous Bitsadze equation, have been systematically studied, and the explicit expression of solution and the condition of solvability have also been obtained. In addition, the so-called mixed BVPs of the Poisson equation and the nonhomogeneous ∗ This

first author is supported by Tianyuan Fund of Mathematics (10626039), NNSF of China (10471107) and RFDP of Higher Eduction of China (20060486001). The second author is supported by Natural Science Foundation of Zhejiang Province (Y606093) and Professor Foundation of Ningbo University.

380

Y. Wang & J. Wang

Bitsadze equation have been solved, which are the Dirichlet-Newmann type problem, the Newmann-Dirichlet type problem, and so on [4]. Further, BVPs of the model complex equation of higher order, i.e. polyanalytic equation and polyharmonic equation, have also been investigated, and, by the iteration, the explicit expression of solution and the condition of solvability have been obtained [1,5,7,14]. Because of the similar properties of polyanalytic function with analytic function, Riemann and Hilbert type BVPs of polyanalytic function have been considered, and the explicit expression of solution and the condition of solvability have also been obtained [2,8,12,13]. In this article we will discuss Hilbert boundary value problems of polyanalytic function with the same factors, which is different from Hilbert type BVPs discussed in [13]. Using the regularization method, we finally obtain explicit expressions of solutions and conditions of solvability. Let Hn (G) denote the set of polyanalytic functions on the open set G as in [8]. In the sequel we need the following decomposition. Proposition 1.1. Let D be the unit disc. Then Hn (D\{0}) = H1(D\{0})⊕(¯ z +z)H1(D\{0})⊕· · ·⊕(¯ z +z)n−1H1(D\{0}), (1)

where (¯ z + z)j H1 (D \ {0}) = {(¯ z + z)j h(z) : h ∈ H1 (D \ {0})} for j = 0, 1, · · · , n − 1. By Proposition 1.1, if V ∈ Hn (D \ {0}) then

V(z) = V0 (z)+(¯ z +z)V1 (z)+· · ·+(¯ z +z)n−1fn−1 (z) with Vj ∈ H1(D\{0}) (2) for j = 0, 1, · · · , n − 1. 2. Preliminary Results Let D(z0 , r) = {z ∈ C : |z − z0 | < r} denote the neighborhood of z0 on the ◦

complex plane C, and D(z0 , r) = D(z0 , r) \ {z0 } the deleted neighborhood. Similarly to [2] the growth order of a finite point for polyanalytic functions is introduced as follows. ◦

Definition 2.1. Suppose f ∈ Hn (D(z0 , r)). If there is an integer m such that +∞ > lim supz→z0 |(z − z0 )m f (z)| > 0, then we say f possesses order m at z0 , denoted by Ord(f, z0 ) = m. If lim supz→z0 |(z − z0 )m f (z)| = +∞ for an arbitrary integer m, then we also call f possesses order +∞ at z 0 , denoted by Ord(f, z0 ) = +∞. We assume Ord(f, z0 ) = −∞ if and only if f ≡ 0.

Hilbert-Type Boundary Value Problem for Polyanalytic Functions

381

◦

Let f ∈ Hn (D(z0 , r)), and one has [8] f (z) = f0 (z) + (¯ z − z¯0 )f1 (z) + · · · + (¯ z − z¯0 )n−1 fn−1 (z)

(3)

◦

where fj ∈ H1 (D(z0 , r)), j = 0, 1, · · · , n − 1. Observe fk (z) =

+∞ X

j=−∞

ak,j (z − z0 )j with ak,j ∈ C

for k = 0, 1, · · · , n − 1, and hence f (z) =

+∞ X

pj (z, z¯)(z − z0 )j

(4)

ak,j−k (¯ z − z¯0 )k (z − z0 )−k .

(5)

j=−∞

with pj (z, z¯) =

n−1 X k=0

Similar to [2] we have the following conclusions. ◦

Theorem 2.1. Suppose f ∈ Hn (D(z0 , r)). Ord(f, z0 ) = m if and only if f (z) =

+∞ X

j=−m

pj (z, z¯)(z − z0 )j with

n−1 X k=0

|ak,j−k | = 6 0.

(6)

In this case the point z0 is called a pole of f . ◦

Corollary 2.1. Suppose f ∈ Hn (D(z0 , r)). Then Ord(f, z0 ) = max{Ord(fk , z0 ) + k, k = 0, 1, · · · , n − 1}, where fk is determined by (3). Corollary 2.2. Suppose V ∈ Hn (D\{0}). Then Ord(V, 0) = max{Ord(Vk , 0) + k, k = 0, 1, · · · , n − 1}, where Vk is determined by (2). 3. Hilbert BVP of Polyanalytic Functions In this section we investigate the following Hilbert type BVP: Find a V (z) ∈ k Hn (D) such that ∂∂ z¯Vk , k = 0, 1, · · · , n − 1 are continuous on D, satisfying the Hilbert type boundary conditions (  j + ) ∂ V (t) = γj (t), t ∈ ∂D, j = 0, 1, · · · , n − 1, (7) Re [a(t) + ib(t)] ∂ z¯j

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where a(t), b(t), γj (t) ∈ C(∂D), j = 0, 1, 2, · · · , n − 1 and a2 (t) + b2 (t) = 1, t ∈ ∂D. When a(t) = 1, b(t) = 0 the problem (7) is just the Schwarz problem for polyanalytic functions [1]. Let X(z) = iz κ exp{S[α](z)},

(8)

where S is the Schwartz operator defined as Z 1 t + z dt S[α](z) = with α(t) = arg{t−κ [a(t) − ib(t)]} α(t) 2πi ∂D t−z t

(9)

and the index κ is given by

1 {arg[a(t) − ib(t)]}∂D . 2π Then X has the following properties [8]. κ=

(1) (2) (3) (4)

Re{[a(t) + ib(t)]X + (t)} = 0, t ∈ ∂D. X(z) is analytic in D\{0} and continuous up to ∂D. X + (z) 6= 0, z ∈ D\{0}, X +(t) ∈ C(∂D). X(z) possesses order −κ at z = 0, that is, limz→0 z −κ X(z) = iS[α](0). Our first problem is to find a V ∈ Hn (D\{0}) ∩ C n−1 (D\{0}) such that (  + ) j ∂ V   Re (t) = 0, t ∈ ∂D, j = 0, 1, · · · , n − 1, ∂ z¯j (10)   Ord(V, 0) ≤ m.

This is the simplest Hilbert BVP. Introduce the class of symmetric polynomials ( ) k X j SΠk = qk (z) = cj z : Rec0 = 0, cj = −¯ c−j , j = 1, 2, · · · , k j=−k

as in [13]. If k < 0, we assume SΠk = {0}. Theorem 3.1. The simplest Hilbert BVP (10) is solvable and the set of solutions is SΠm ⊕ (¯ z + z)SΠm−1 ⊕ · · · ⊕ (¯ z + z)n−1 SΠm−n+1 where (¯ z + z)j SΠm−j = {(¯ z + z)j qm−j : qm−j ∈ SΠm−j }, j = 0, 1, · · · , n − 1.

Proof. Let V (z) = V0 (z) + (¯ z + z)V1 (z) + · · · + (¯ z + z)n−1 Vn−1 (z) with Vj ∈ H1 (D \ {0}) for j = 0, 1, · · · , n − 1. By Corollary 2.2, Ord(V, 0) ≤ m is equivalent to Ord(Vj , 0) ≤ m − j, j = 0, 1, · · · , n − 1.

(11)

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On the other hand, the boundary conditions in (10) are equivalent to ReVj+ (t) = 0, t ∈ ∂D j = 0, 1, · · · , n − 1.

(12)

By [8] we know (11) and (12) imply Vj ∈ SΠm−j , j = 0, 1, · · · , n − 1. Next we consider the homogeneous Hilbert type BVP of (7) as follows (  j + ) ∂ V Re [a(t) + ib(t)] (t) = 0, t ∈ ∂D, j = 0, 1, · · · , n − 1. (13) ∂ z¯j For convenience we introduce the symbol Sm = SΠm ⊕ (¯ z + z)SΠm−1 ⊕ · · · ⊕ (¯ z + z)n−1 SΠm−n+1 .

(14)

Theorem 3.2. The homogeneous Hilbert type BVP (13) is solvable and its solution is V (z) = iX(z)qκ(z),

(15)

where X is defined by (8) and qκ ∈ Sκ . Proof. By Property 1, 2 and 3 of X defined by (8), the boundary conditions (13) are equivalent to (  + ) V ∂j Re (t) = 0, t ∈ ∂D, j = 0, 1, · · · , n − 1. (16) ∂ z¯j iX V , 0) ≤ Ord( X1 , 0) = κ. And hence By Property 4 of X we know Ord( iX Theorem 3.1 implies

V = qκ ∈ S κ , iX which leads to (15). Introduce the poly-Schwarz operator n−1 X (−1)k 1 Z t+z dt γk (t)(t−z + t−z)k S[γ0 , · · · , γn−1 ](z) = k! 2πi ∂D t−z t

(17)

k=0

for z ∈ D, where the kernel functions γk ∈ C(∂D), k = 0, 1, · · · , n − 1. According to Theorem 3.4 in [1], +  j ∂ S[γ0 , · · · , γn−1 ] (t) = γj (t), t ∈ ∂D, j = 0, 1, · · · , n − 1. (18) Re ∂ z¯j

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Now the boundary conditions (7) are equivalent to (  + ) ∂j V γj (t) Re , (t) = j ∂ z¯ iX i[a(t) + ib(t)]X + (t)

t ∈ ∂D,

where j = 0, 1, · · · , n − 1. And hence   γn−1 γ0 e (z), z ∈ D ,··· , V (z) = iX(z) · S i(a + ib)X + i(a + ib)X +

is the special solution of BVP (7). Therefore (19) and (20) give !#+ ) (" V − Ve ∂j (t) = 0, t ∈ ∂D, j = 0, 1, · · · , n − 1. Re ∂ z¯j iX

(19)

(20)

(21)

When κ ≥ 0, by Theorem 3.2, the solution of BVP (7) may be written as V (z) = Ve (z) + iX(z)qκ(z), qκ ∈ Sκ .

(22)

When κ < 0, if the Hilbert type problem (7) is solvable then Theorem 3.2 imply that its solution may be represented as (20) or say n−1 X (−1)k X(z) Z γk (t)(t − z + t − z)k t + z dt (23) V (z) = k! 2πi ∂D [a(t) + ib(t)]X + (t) t − z t k=0

for z ∈ D. Obviously V (z) defined by (23) is the solution of (7) if and only if limz→0 V (z) exists. If limz→0 V (z) exists, then one has Ord(Vj , 0) ≤ −κ − j, j = 0, 1, · · · , n − 1 with Vj (z) =

n−1 X k=j

(−1)k Ckj X(z) · k! 2πi

Z

∂D

γk (t)(t¯ + t)k−j t + z dt [a(t) + ib(t)]X + (t) t − z t

(24)

(25)

for j = 0, 1, · · · , n − 1. Observe, at the neighborhood of z = 0,

−κ−j−1 +∞ X X z z z −κ−j tκ+j+1 t+z =1+2 ( )ℓ = 1 + 2 , ( )ℓ + t−z t t t−z ℓ=1

one has

n−1 X k=j

ℓ=1

(−1)k Ckj 1 · k! 2πi

Z

∂D

dt γk (t)(t¯ + t)k−j =0 [a(t) + ib(t)]X + (t) tℓ+1

(26)

for ℓ = 0, 1, · · · , −κ − j − 1 and j = 0, 1, · · · , n − 1. Under the conditions (26), (23) may be rewritten as V (z) =

n−1 X k=1

(¯ z + z)j (−1)j Vj (z)

(27)

Hilbert-Type Boundary Value Problem for Polyanalytic Functions

385

with Vj (z) =z −j

n−1 X

Z

k=j

∂D

(−1)k Ckj exp{iS[α](z)} · k! 2πi

tj dt γk (t)(t¯ + t)k−j , + [a(t) + ib(t)] exp{iS [α](t)} t − z

(28)

for j = 0, 1, · · · , n − 1. Let z = reiθ , θ ∈ [0, 2π]. By (27) and (28), one gets lim V (z) =

r→0

n−1 X

cj (1 + e−2iθ )j

(29)

j=1

with j

cj = (−1)

n−1 X k=j

(−1)k Ckj exp{iS[α](0)} k! 2πi

Z

∂D

γk (t)(t¯ + t)k−j tj−1 dt (30) [a(t)+ib(t)] exp{iS + [α](t)}

for j = 0, 1, · · · , n − 1. Lemma 3.1. Let cj (j = 0, 1, · · · , n − 1) and c be complex numbers. Then

n−1 P k=j

cj (1 + e−2iθ )j ≡ c for all θ ∈ [0, 2π] if and only if c0 = c, cj = 0 for

j = 0, 1, · · · , n − 1.

Pn−1 Proof. If k=1 cj (1 + e−2iθ )j ≡ c for ∀θ ∈ [0, 2π], let θ = π2 , then c0 = c, Pn−1 Pn−1 and hence k=1 cj (1 + e−2iθ )j ≡ 0, which implies k=1 cj (1 + e−2iθ )j−1 ≡ Pn−1 0. Therefore c1 = 0 similarly. If c1 = · · · = ck = 0 then j=k+1 cj (1 + Pn−1 −2iθ j −2iθ j−k−1 e ) ≡ 0, ∀θ which leads to j=k+1 cj (1 + e ) ≡ 0, ∀θ. Hence ck+1 = 0. By the principle of induction, we know cj = 0 (j = 0, 1, · · · , n−1). The converse is obvious. By Lemma 3.1, if V (z) is the solution of BVP (7) then cj = 0 (j = 0, 1, · · · , n − 1) or say n−1 X k=j

(−1)k Ckj 1 · k! 2πi

Z

∂D

γk (t)(t¯ + t)k−j tj−1 dt = 0 [a(t) + ib(t)] exp{iS + [α](t)}

(31)

for j = 0, 1, · · · , n − 1. Combining (26) and (31) we have n−1 X k=j

(−1)k Ckj 1 · k! 2πi

Z

∂D

dt γk (t)(t¯ + t)k−j =0 [a(t) + ib(t)]X + (t) tℓ+1

(32)

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for ℓ = 0, 1, · · · , −κ − j and j = 0, 1, · · · , n − 1. Conversely, when (32) is fulfilled, BVP (7) is solvable and its solution is (23). In general we obtain the following result. Theorem 3.3. The Hilbert-type BVP (7) is solvable if and only if the conditions (32) are fulfilled, and its solution may be written as (22). References 1. H. Begehr and D. Schmersau, The Schwarz problem for polyanalytic functions, ZAA, 2005, 24(2), 341–351. 2. J. Du and Y. Wang, On boundary value problem of polyanalytic function on the real axis, Complex Variables, 2003, 48(6), 527–542. 3. M. B. Balk, Polyanalytic Functions, Akademie Verlag, Berlin, 1991. 4. H. Begehr, Complex Analytic Methods for Partial Differential Equations: An Introductory Text, World Scientific, Singapore, 1994. 5. H. Begehr, Boundary value problems in complex analysis I. Bol. Asoc. Mat. Venez., 2005, 12, 65–85; II. Bol. Asoc. Mat. Venez,. 2005, 12, 217–250. 6. H. Begehr and G. N. Hile, A hierarchy of integral operators, Rocky Mountain J. Math., 1997, 27, 669–706. 7. H. Begehr and A. Kumar, Boundary value problem for inhomogeneous polyanalytic equation, Analysis, 2005, 25, 55–71. 8. J. Du and Y. Wang, Riemann boundary value problem of polyanalytic function and metaanalytic function on the closed curve, Complex Variables, 2005, 50, 521–533. 9. B. F. Fatulaev. The main Haseman type boundary value problem for metaanalytic function in the case of circular domains, Math. Model. Anal., 2001, 6, 68–76. 10. J. Lu, Boundary Value Problems for Analytic Functions, World Scientific, Singapore, 1993. 11. N. I. Muskhelishvili, Singular Integral Equations, 2nd ed., Noordhoff, Groningen, 1968. 12. Y. Wang and J. Du, On Riemann boundary value problem of polyanalytic function on the real axis, Acta. Math. Sci., 2004, 24B, 663–671. 13. Y. Wang and J. Du, Hilbert boundary value problem of polyanalytic function on the unit circumference, Complex Var. Elliptic Equ., 2006, 51, 923–943. 14. H. Begehr, J. Du and Y. Wang, A Dirichlet problem for polyharmonic functions, Ann. Mat. Pura Appl., to appear.

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EXISTENCE AND UNIQUENESS OF THE SOLUTION OF A FREE BOUNDARY PROBLEM FOR A PARABOLIC COMPLEX EQUATION YONGZHI XU Department of Mathematics, University of Louisville, Louisville, KY 40292 E-mail: [email protected] We present a proof of the existence and uniqueness of the solution of a free boundary problem for a nonlinear parabolic complex equation of second order in a simply connected region in the complex plane with Dirichlet boundary condition. A more general problem is discussed in [15], but the uniqueness is not proved. The problem is originally motivated by a free boundary problem model of breast cancer. In this paper we convert the free boundary problem to a sequence of boundary value problems of nonlinear parabolic complex equations of second order. Then we use iteration method to construct its solution and prove its existence and uniqueness. Keywords: Free boundary problem, complex parabolic equation, existence and uniqueness.

1. The free boundary problem In [15] we considered a free boundary value problem for a parabolic complex equation. Let Dt be a simply connected bounded domain in the z = x + iy plane C with the boundary Γt for each t ≥ 0. D0 = {z||z| < 1} with boundary Γ0 = {z||z| = 1}. Denote D = {(z, t)|z ∈ Dt , 0 < t ≤ T } for some T > 0. ∂D = ∂D 1 ∪ ∂D2 is the parabolic boundary of D, where ∂D 1 is the bottom of D and ∂D 2 is the side boundary of D, respectively. Let ζ = ϕ(z, t) be the bijective comformal map that maps the bounded simply connected region Dt onto the closed unit disk D0 = {ζ||ζ| ≤ 1} such that ϕ(0, t) = 0, ϕ′ (0, t) = 1, and z = ψ(ζ, t) is its inverse function. Then Γt = {z|z = ψ(eiθ , t), 0 ≤ θ ≤ 2π}. Let u = u(z, t) be a function definded in D, and satisfy the nonlinear parabolic complex equation and initial and free boundary value conditions ut − A0 uzz¯ + Re[A1 uz ] + A2 u = A3 ,

(1)

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Y. Xu

u(z, 0) = g(z), z ∈ D0 , ∂u + a2 (z, t)u = a3 (z, t), z ∈ Γt , t > 0, ∂ν and the free boundary Γt satisfies the condition Z  ∂Re{ψ(eiθ , t)} = Re µ(u(z, t), t) − uˆ)dz , t > 0, ∂t Ct a1 (z, t)

(2) (3)

(4)

where Ct = {z|z = ρψ(eiθ , t), 0 ≤ ρ ≤ 1} is the line segment connecting z = 0 and z = ψ(eiθ , t). Since Γ0 = {z||z| = 1}, we have ψ(ζ, 0) = ζ. (3) may be written in the complex form 2Re[a1 (z, t)λ(z, t)uz ] + a2 (z, t)u = a3 (z, t), z ∈ Γt , t > 0.

(5)

Here z = x + iy, A0 = A0 (z, t, u, uz , uzz¯), A1 = A1 (z, t, u, uz ), A2 = A2 (z, t, u), and A3 = A3 (z, t); µ and uˆ are known real constants; ν is the unit vector at every point on Γt , and g(z), aj (z, t) (j = 1, 2, 3), λ(z, t) = cos(ν, x) − i cos(ν, y) are known functions. 2. Preliminary results 2.1. Initial mixed boundary value problem In [10] the following conditions are introduced: (a) A0 (z, t, u, uz , uzz¯), A1 (z, t, u, uz ), A2 (z, t, u), and A3 (z, t), are measurable for any continuously differentiable function u(z, t) and measurable functions uzz¯ ∈ L2 (D) and satisfy the conditions ¯ ≤ k1 , p > 4. 0 < δ ≤ A0 ≤ δ −1 , |Aj | ≤ k0 , j = 1, 2, Lp [A3 , D]

(6)

(b) The above functions with respect to u ∈ R, uz ∈ C are continuous for almost every point (z, t) ∈ D and uzz¯ ∈ R. (c) Let F (z, t, u, uz , uzz¯) = A0 uzz¯ − Re[A1 uz ] − A2 u − A3 . For almost every point (z, t) ∈ D and u ∈ R, uz ∈ C, V ∈ R, there is F (z, t, u, uz , V 1 ) − F (z, t, u, uz , V 2 ) = A˜0 (V 1 − V 2 ), δ < A˜0 ≤ δ −1 .

(7) (8)

(d) For any u ∈ R, uz ∈ C, V ∈ R, there is

F (z, t, u1 , u1z , V ) − F (z, t, u2 , u2z , V ) ¯ = Re[A˜1 (u1 − u2 )z ] + A˜2 (u1 − u2 ) − (u1 − u2 )t on D,

(9)

Solution of a Free Boundary Problem for a Parabolic Complex Equation

389

where A˜j (j = 1, 2) satisfy |A˜j | ≤ k0 , in D. Here δ > 0, q0 < 1, q1 > 0, k0 , k1 are nonnegative constants. Following Wen [10], the conditions (a)-(c) are called condition C, and the conditions (a)-(d) the condition C ′ . We also assume that g(z), aj (z, t) (j = 1, 2, 3), λ(z, t) = cos(ν, x) − i cos(ν, y) satisfy the conditions ∂g + a2 (z, 0)g = a3 (z, 0), z ∈ Γ0 , ∂ν 2,1 2,1 Cα,α/2 [η, ∂D2 ] ≤ k0 , η = (λ, a1 , a2 ), Cα,α/2 [a3 , ∂D2 ] ≤ k2 , Cα2 [g, D0 ] ≤ k2 , a1 (z, 0)

aj (z, t) ≥ 0, j = 1, 2, a1 + a2 ≥ 1, cos(ν, n) ≥ η > 0 on ∂D 2 .

(10) (11) (12)

If u = u(z, t) is a function definded in the unit cylinder D0 × [0, T ] and satisfies the nonlinear parabolic complex equation and mixed boundary value conditions ut − A0 uzz¯ + Re[A1 uz ] + A2 u = A3 , in ∂D0 × [0, T ], u(z, 0) = g(z), z ∈ D0 ,

2Re[a1 (z, t)λ(z, t)uz ] + a2 (z, t)u = a3 (z, t), z ∈ ∂D0 × [0, T ].

(13) (14) (15)

We call this problem M . This is a special case of the complex parabolic equation considered by Wen in [10]. More precisely in this paper we need only the results in a simple connected domain, and for the case that Re{Quzz } = 0, while [10] presented results in a multiple-connected region with the term Re{Quzz }. The conditions C and C ′ presented here are stronger than that in [10]. The following results are special cases of that in [10]. Lemma 2.1. Let equation (1) satisfy condition C. Then any continuous solution u(z, t) of problem M for (1) satisfies the estimate C[u, D] ≤ M1 = M1 (δ, α, k, p, D),

(16)

where k = k(k0 , k1 , k2 ), M1 is a nonnegative constant only dependent on δ, α, k, p, D. Lemma 2.2. Under the same condition as in Theorem 2.1, any solution u(z, t) of problem M for (1) satisfies the estimates 1,0 Cβ,β/2 [u, D∗ ] ≤ M2 , kukW 2,1 (D∗ ) ≤ M3 , 2

(17)

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Y. Xu

where D ∗ = D ∩ {∩(z∗ ,t∗ )∈∂D2 [|z − z ∗ |2 + |t − t∗ |] ≥ ǫ > 0}, ǫ is a small positive number, and k = (k0 , k1 , k2 ), Mj = Mj (δ, α, k, p, D), (j = 2, 3) are nonnegative constants. Lemma 2.3. Under condition C, problem M for (1) has a solution u(z, t). Lemma 2.4. Under condition C ′ , the solution u(z, t) of problem M for (1) is unique. 2.2. The equivalent coupled boundary value problem In [15] we proved that the free boundary value problem (1) (2) (4) (5) is equivalent to an initial-boundary value problem of coupled nonlinear differential and integral equations vt − A0 |ϕz |2 vζ ζ¯ + Re[(A1 ϕz + ϕt )vζ ] + A2 v = A3 , v(ζ, 0) = g(ψ(ζ, 0)), ζ ∈ D0 ,

(18) (19)

2Re[a1 (ψ(ζ, t), t)λ(ψ(ζ, t), t)ϕz vζ ] + a2 (ψ(ζ, t), t)v = a3 (ψ(ζ, t), t), Re[ψ(eiθ , t)] = Re[ψ(eiθ , 0)]e

ζ ∈ Γ0 , t > 0,

RtR 0

1 0

iθ

µ[v(ρe ,τ )−ˆ u]dρdτ

where v(ζ, t) = u(ψ(ζ, t), t), ζ ∈ D0 , t ≥ 0. The folowing lemmas are proved in [15].

(20) , t > 0, 0 ≤ θ ≤ 2π, (21)

Lemma 2.5. Under condition C ′ , and if ψ(eiθ , 0) = eiθ , then (i) There is M1 > 0 such that 0 ≤ u(z, t) ≤ M1 , for (z, t) ∈ D. (ii) e−µ˜uT ≤ |Re[ψ(eiθ , t)]| ≤ eµ(M1 −˜u)T , 0 ≤ θ ≤ 2π, 0 ≤ t ≤ T . ∂Re[ψ(eiθ ,t)] iθ (iii) −µ˜ u|Re[ψ(e , t)]| ≤ ≤ µ(M1 − u˜)|Re[ψ(eiθ , t)]|, 0 ≤ ∂t θ ≤ 2π, 0 ≤ t ≤ T .

Lemma 2.6. For given Re[ψ(eiθ , t)] and ψ(0, t) = 0 for 0 ≤ θ ≤ 2π and t ≥ 0, ψ(ζ, t) is determined uniquely in the unit disk D0 , Z 2π eiθ + ζ 1 ψ(ζ, t) = dθ, for ζ ∈ D0 , t ≥ 0. (22) Re[ψ(eiθ , t)] iθ 2π 0 e −ζ Moreover, if Z 2π 1 eiθ + eiθ0 dθ, for 0 ≤ θ0 ≤ 2π, t ≥ 0, ψ(eiθ0 , t) = P V Re[ψ(eiθ , t)] iθ 2π 0 e − eiθ0 (23) where P V stands for Cauchy principal value, then limζ→eiθ0 ψ(ζ, t) = ψ(eiθ0 , t), and for some constant T > 0, M = M (T ) > 0, |ψ(eiθ0 , t)| ≤ M (T )|Re[ψ(eiθ0 , t)]|, for 0 ≤ t ≤ T.

(24)

Solution of a Free Boundary Problem for a Parabolic Complex Equation

391

Lemma 2.7. Let the mixed problem (1), (2) and (4) satisfy the condition C ′ and the coefficients in problem (18)-(20) are corresponding coefficients of the Mixed problem (1), (2) and (4). If ψ(ζ, t) is known, then the Mixed problem (18)-(20) has a unique solution. 2.3. The iterated boundary value problem We use an iterative algorithm to construct a solution of the (18)-(21). Let ψ 0 (ζ, t) = ζ for ζ ∈ D0 = {ζ||ζ| < 1}. v n+1 , (n = 0, 1, 2, ...) is the solution of the mixed problem on D0 × [0, T ] + Re[(An1 ϕnz + ϕnt )vζn+1 ] + An2 v n+1 = An3 , vtn+1 − An0 |ϕnz |2 vζn+1 ζ¯ 2Re[a1 (ψ

n

v n+1 (ζ, 0) = g(ψ n (ζ, 0)), ζ ∈ D0 ,

(ζ, t), t)λ(ψ n (ζ, t), t)ϕnz vζn+1 ] n

= a3 (ψ (ζ, t), t),

n

(25) (26)

+ a2 (ψ (ζ, t), t)v

n+1

ζ ∈ Γ0 , t > 0,

(27)

where An0 = A0 (z, t, un , unz , unzz¯), An1 = A1 (z, t, un , unz ), An2 = A2 (z, t, un ), and An3 = A3 (z, t); ψ n (ζ, t), (n = 1, 2, ...) are analytic functions in the unit disk for t ≥ 0, satisfying the boundary condition Re[ψ n (eiθ , t)] = Re[ψ n (eiθ , 0)]e

RtR1 0

0

µ[v n (ρeiθ ,τ )−ˆ u]dρdτ

t > 0, 0 ≤ θ ≤ 2π, n = 1, 2, 3, · · ·.

, (28)

In view of Lemmas 2.1–2.4, we have the following. Theorem 2.1. Under the condition C ′ , for each interger n ≥ 0, there exists a unique solution (v n+1 , ψ n+1 ) to the problem (25)-(28). Moreover, we have the estimates C[v n , D] ≤ M1 = M1 (δ, α, k, p, D),

(29)

where k = k(k0 , k1 , k2 ), M1 is a nonnegative constant only dependent on δ, α, k, p, D. 1,0 Cα,α/2 [v n , D∗ ] ≤ M2 , kv n kW 2,1 (D∗ ) ≤ M3 , 2

(30)

where D ∗ = D ∩ {∩(z∗ ,t∗ )∈∂D2 [|z − z ∗ |2 + |t − t∗ |] ≥ ǫ > 0}, ǫ is a small positive number, and k = (k0 , k1 , k2 ), Mj = Mj (δ, α, k, p, D), (j = 2, 3) are nonnegative constants.

392

Y. Xu

3. Existence and uniqueness In this section we outline the proof of the existence and uniquenesse of the solution of an equivalent coupled boundary value problem for the case that a1 (z, t) = 0, a2 (z, t) = 1 and a3 (z, t) = a3 = constant. The proof is inspired by [9, Chapter 6]. The general mixed boundary condition case is still an open problem. n n First we show that the sequence a limit {v, ψ},
{v , ψ } converges to3/2−δ 1/2−δ 1 where v ∈ C [0, T ], C (D) for δ > 0, and ψ ∈ C ([0, T ]) for δ > 0. In view of Sobolev embedding theorem, Theorem 2.1 and Lemmas 2.5, 2.6 imply the following. Theorem 3.1. Under the assumptions of Theorem 2.1, we
have (i) {v n : n = 0, 1, ...} is compact in C 1/2−δ [0, T ], C 1 (D) , for δ > 0; (ii) {ψ n : n = 0, 1, ...} is compact in C 3/2−δ ([0, T ]), for δ > 0. Therefore {v n : n = 0, 1, ...} and {ψ n : n = 0, 1, ...} have limit points v and ψ. We need to show that v, s are unique and satisfy the free boundary problem. If (v1n , ψ1n ) and (v2n , ψ2n ) are two sequences, then for i = 1, 2, n+1 n+1 n+1 vi,t −Ani,0 |ϕni,z |2 vi,ζ +Re[(Ani,1 ϕni,z +ϕni,t )vi,ζ ]+Ani,2 vin+1 = Ani,3 , ζ¯

(31)

vin+1 (ζ, 0) = g(ψin (ζ, 0)), ζ ∈ D0 , vin+1 = a3 , ζ ∈ Γ0 , t > 0, RtR1 n iθ Re[ψin (eiθ , t)] = Re[ψin (eiθ , 0)]e 0 0 µ[vi (ρe ,τ )−ˆu]dρdτ ,

(33)

t > 0, 0 ≤ θ ≤ 2π, n = 1, 2, 3, · · · .

(34)

(32)

Let v n+1 = v1n+1 − v2n+1 and ψ n+1 = ψ1n+1 − ψ2n+1 . Then vtn+1 −An1,0 |ϕn1,z |2 vζn+1 +Re[(An1,1 ϕn1,z +ϕn1,t )vζn+1 ]+An1,2 v n+1 = F n , ζ¯ v n+1 (ζ, 0) = 0, ζ ∈ D0 ,

v n

iθ

n+1

= 0, ζ ∈ Γ0 , t > 0, n

iθ

RtR1

(36) (37)

µ[v1n (ρeiθ ,τ )−ˆ u]dρdτ

Re[ψ (e , t)] = Re[ψ (e , 0)]e 0 0 i RtR1 n iθ t > 0, 0 ≤ θ ≤ 2π, n = 1, 2, 3, · · ·, · 1 − e− 0 0 µ[v (ρe ,τ )]dρdτ h

(35)

(38)

 n+1  where F n = [An2,3 − An1,3 ] + An2,0 |ϕn2,z |2 − An1,0 |ϕn1,z |2 v2,ζ ζ¯ n o   n+1 +Re [(An2,1 ϕn2,z +ϕn2,t )−(An1,1 ϕn1,z +ϕn1,t )]v2,ζ + An2,2 −An1,2 v2n+1 . (39)

Solution of a Free Boundary Problem for a Parabolic Complex Equation

393

In view of Lemma 2.5 (i) for vin+1 , we know that there is a constant C > 0, such that kvin+1 k∞ ≤ C and kv n+1 k∞ = kv1n+1 − v2n+1 k∞ ≤ C. (We will use C for a general constant that may take different values without mention each time.) From (38) and (24), |ψ n+1 (ζ, t)| ≤ Ctkv n+1 kL∞ (D×[0,t]) , 0 ≤ t ≤ T.

(40)

It follows, in view of condition C ′ , that |F n | ≤ |An2,3 − An1,3 | + An2,0 |ϕn2,z |2 − An1,0 |ϕn1,z |2 kv2n+1 kH 2

+|Re[(An2,1 ϕn2,z +ϕn2,t )−(An1,1 ϕn1,z +ϕn1,t )]|kv2n+1 kH 1 + |An2,2 −An1,2 |kv2n+1 kL∞ ≤ |An2,3 − An1,3 | + An2,0 |ϕn2,z |2 − An1,0 |ϕn1,z |2 kv2n+1 kH 2 +|(An2,1 ϕn2,z + ϕn2,t ) − (An1,1 ϕn1,z + ϕn1,t )|kv2n+1 kH 1 + |An2,2 − An1,2 |kv2n+1 kL∞ ≤ C1 |ψ(ζ, t)| ≤ Ctkv n+1 kL∞ .

(41)

Note that v n+1 , as a solution of the linear parabolic equation (35), has integral representation Z th
i
α(t,τ )L eα(t,τ )L Re[(An1,1 ϕn1,z + ϕn1,t ) + ∂x eN An1,2 v n+1 (·, t) = − 0 Z t v n+1 (·, τ )dτ + eα(t,τ )L F n+1 dτ, (42) 0

where e

α(t,τ )L

is the solution operator of the heat operator Lv n+1 = vtn+1 − An1,0 |ϕn1,z |2 vζn+1 , ζ¯ α(t,τ )

is the solution with vanishing Dirichlet boundary condition, and eN operator for the vanishing Neumann boundary condition. So Z t r kv n+1 (t)kL∞ (D) ≤ M1 τ − 2 kv n+1 kL∞ (D×[0,t]) dτ 0 Z t r + M2 τ − 2 τ kv n+1 kL∞ (D×[0,t]) dτ (43) 0

For r < 2 and t > 0 small enough, kv n+1 kL∞ (D×[0,t]) ≤ Ct

2−r 2

kv n+1 kL∞ (D×[0,t]) < kv n+1 kL∞ (D×[0,t]) .

Since the estimates (40)-(43) are true for 0 ≤ t ≤ T , we can repeat the above argument and conclude that v1 (·, t) = v2 (·, t) in [0, T ]. This finishes the uniqueness proof. It must be that v n+1 (·, t) → 0 and ψ n+1 (t) → 0 in [0, t]. Therefore, the limit points v, ψ are unique and satisfy the equivalent coupled boundary value problem (18)-(21). It proves the local existence. But the a priori

394

Y. Xu

estimates in Lemma 2.5 hold for [0, T ], so by repeating the process we obtain the global existence. References 1. H. Begehr and R. Gilbert, Transformations, Transmutations and Kernel Functions, Longman Scientific & Technical, New York, 1992. 2. H. M. Byrne and M. A. J. Chaplain, Growth of nonnecrotic tumours in the presence and absence of inhibitors, Math. Biosciences, 130 (1995), 151-181. 3. J. R. Cannon and J. van der Hoek, Diffusion subject to the specification of mass, J. Math. Anal. Appl., 115, no. 2 (1986), 517-529. 4. Y. S. Choi and K.-Y. Chan, A parabolic equation with nonlocal boundary conditions arising from electrochemistry, Nonlinear Anal. TMA, 18, no. 4 (1992), 317-331. 5. A. Friedman and F. Reitich, Analysis of a mathematical model for the growth of tumours, J. Math. Biol., 38 (1999), 262-284. 6. R. Gilbert, Function Theoretic Methods in Partial Differential Equations, Academic Press, New York, 1969. 7. R. Gilbert and Y. Xu, A function theory for thin elastic shells, J. Elasticity, 22 (1989), 81-93. 8. H. P. Greenspan, Models for the growth of a solid tumour by diffusion, Studies in Appl. Math., 52 (1972), 317-340. 9. M. Taylor, Partial Differential Equations III: Nonlinear Equations, Springer, New York, 1996. 10. G. Wen, Initial-mixed boundary value problems for nonlinear parabolic complex equations of second order with measurable coefficients, Acta Scientiarum Naturalium, Universitatis Pekinensis, 31, no. 5 (1995), 511-519. 11. G. Wen, Linear and Nonlinear Elliptic Complex Equations, Shanghai Science and Technology Publishers, 1986. 12. G. Wen and H. Begehr, Boundary value problems for elliptic equations and systems, Longman Scientific & Technical, New York, Wiley, 1990 13. Y. Xu, Dirichlet problem for a class of second order nonlinear elliptic systems, Complex Variables, 15, no. 4 (1990), 241-258. 14. Y. Xu, A free boundary problem model of ductal carcinoma in situ, Discrete and Continous dynamical systems, Series B, 4, no. 1 (2004), 337-348. 15. Y. Xu, A free boundary problem of parabolic complex equation, Complex Var. Elliptic Equ., 51, Nos. 8-11, (2006), 945-951. 16. N. I. Yurchuk, A mixed problem with an integral condition for some parabolic equations, Differentsial’nye Uravneniya, 22, no. 12 (1986), 2117-2126.

395

FIRST-ORDER DIFFERENTIAL OPERATORS ASSOCIATED TO THE CAUCHY-RIEMANN OPERATOR OF CLIFFORD ANALYSIS ¨ U. YUKSEL Department of Mathematics, Atılım University, ˙ 06836 Incek, Ankara, Turkey E-mail: [email protected] Consider initial value problems of type ∂u(t, x) = Lu(t, x), ∂t

u(0, x) = u0 (x),

(1)

where L is a linear first-order differential operator and the desired solution is sought in a function space defined as the kernel of a linear differential operator D. Mainly two assumptions are required for such initial value problems to be solvable. Firstly, the operators have to be associated, i.e., L must ransform solutions of Du = 0 again into solutions of this equation. Secondly, an interior estimate constant kukΩ kLukΩ′ ≤ δ must be true, where Ω ′ is a subdomain of Ω having the positive distance δ from the boundary of Ω, and k·k denotes the norm of a suitably chosen function space (with respect to Ω ′ and Ω). This paper formulates sufficient conditions on the real-valued coefficients of the operator L having the form Lu :=

X

A,B,i

P

(A)

CB,i

∂uB eA , ∂xi

uB eB is a Clifford-algebra-valued function B uB under which L turns out be associated to

where u =

(2)

with real-valued com-

ponents the Cauchy-Riemann operator D of Clifford analysis, i.e., L transforms monogenic functions into monogenic functions again. The operator L is constructed in the special case of n = 2 using this criterion. See [2] for the construction of L for n = 1. In [2] the initial value problem (1) is also solved in Banach spaces with Lp -norm. Keywords: Monogenic Functions; Initial Value Problems; Cauchy-Kovalevsky Theorem; Interior Estimates; Associated Operators.

396

U. Y¨ uksel

1. A Criterion for the Construction of the Operator L Let A be the Clifford algebra generated by e0 , e1 , . . . , en , where e0 = 1, e2j = −1, ei ej + ej ei = −2δij for i, j = 1, ..., n, and Ω be a bounded domain in Rn+1 whose points are denoted by x = (x0 , x1 , ..., xn ). A Clifford-algebraP uA eA satisfying the differential equation Du = 0 valued function u = A

is called a (left-)monogenic function, where D =

∂ ∂x0

+

n P

j=1

∂ ej ∂x is the j

Cauchy-Riemann operator of Clifford analysis (see [1]). Define the constants kj,A,B for j = 1, ..., n and arbitrary A and B by the relation ej eA =

X

kj,A,B eB .

(3)

B

Then the following statement is true provided the coefficients of the operator L are continuously differentiable. Theorem 1.1. L is associated to D if the coefficients satisfy the systems P

   P (A) (A) kl,F,D CD,j − kj,D,B CB,0

−

E

D

P



B

  P (E) (E) =0 kl,E,A CF,j − kj,F,B CB,0

(4)

B

for all A, F ∈ S and j, l = 1, 2, ..., n with j ≤ l, ∂ ∂x0 +



(A) CD,j

n P P

E m=1



−

P B

(A) kj,D,B CB,0

∂ km,E,A ∂xm



  P (E) (E) =0 CD,j − kj,D,B CB,0

(5)

B

for all A, D ∈ S and j = 1, 2, ..., n. Proof. See [2]. Remark 1.1. In Theorem 1.1 we assume that all coefficients of L are continuously differentiable. However, there are also operators L associated to D with a number of continuous coefficients (cf. [2]).

Differential Operators Associated to C-R Operator of Clifford Analysis

397

2. Construction of the Operator L 2.1. The special case of n = 2 2.1.1. A system of linear algebraic equations If n = 2, then we have 4 basis elements 1, e1 , e2 , e12 and, therefore, 2 · 4 = 8 (A) equations (3) in the Clifford algebra for the 3 · 4 · 4 = 48 coefficients CB,i . These equations lead to k1,0,1 = −k1,1,0 = k1,2,12 = −k1,12,2 = 1, while all other k1AB = 0. Analogously k2,0,2 = −k2,1,12 = −k2,2,0 = k2,12,1 = 1, and k2AB = 0 otherwise. Substituting these values into (4), one obtains a system of 48 linear algebraic equations. 32 of the 48 equations coincide pairwise and therefore only the following 48 − 16 = 32 equations remain: (0)

(0)

(1)

(1)

C0,0 + C1,1 + C0,1 − C1,0 (2) (0) (0) (1) (1) (2) C1,2 + C2,1 + C0,2 − C2,0 + C0,1 − C1,0 (2) (2) (0) (0) C0,0 + C2,2 + C0,2 − C2,0 (0) (0) (1) (1) C0,1 − C1,0 − C0,0 − C1,1 (0) (0) (1) (1) (2) (2) C0,2 + C12,1 − C1,2 − C12,0 − C0,0 − C1,1 (0) (0) (2) (2) C12,2 − C1,0 − C1,2 − C12,0 (0) (0) (1) (1) C2,0 + C12,1 + C2,1 − C12,0 (0) (0) (1) (1) (2) (2) C0,1 − C12,2 − C0,0 − C2,2 − C2,1 + C12,0 (2) (2) (0) (0) C0,2 − C2,0 − C0,0 − C2,2 (0) (0) (1) (1) C2,1 − C12,0 − C2,0 − C12,1 (0) (0) (1) (1) (2) (2) C1,1 − C2,2 + C12,2 − C1,0 + C2,0 + C12,1 (0) (0) (2) (2) C1,2 + C12,0 + C12,2 − C1,0 (1) (1) (0) (0) (12) (12) −C1,2 − C2,1 + C0,2 − C2,0 + C0,1 − C1,0 (1) (1) (12) (12) −C0,0 − C2,2 + C0,2 − C2,0 (1) (1) (0) (0) (12) (12) C0,2 + C12,1 + C1,2 + C12,0 + C0,0 + C1,1 (1) (1) (12) (12) C12,2 − C1,0 + C1,2 + C12,0 (1) (1) (0) (0) (12) (12) C0,1 − C12,2 + C0,0 + C2,2 + C2,1 − C12,0 (1) (1) (12) (12) C0,2 − C2,0 + C0,0 + C2,2 (1) (1) (0) (0) (12) (12) −C1,1 + C2,2 + C12,2 − C1,0 + C2,0 + C12,1 (1) (1) (12) (12) −C1,2 − C12,0 + C12,2 − C1,0 (2) (2) (12) (12) C0,0 + C1,1 + C0,1 − C1,0 (0) (0) (2) (2) (12) (12) C1,2 + C2,1 + C0,2 − C2,0 − C0,1 + C1,0

=0 =0 =0 =0 =0 =0 =0 =0 =0 =0 =0 =0 =0 =0 =0 =0 =0 =0 =0 =0 =0 =0

(6)

398

U. Y¨ uksel (2)

(12)

(2)

(12)

C0,1 − C1,0 − C0,0 − C1,1 (0) (0) (2) (2) (12) (12) C0,2 + C12,1 − C1,2 − C12,0 + C0,0 + C1,1 (2) (2) (12) (12) C2,0 + C12,1 + C2,1 − C12,0 (12) (12) (0) (0) (2) (2) C0,1 − C12,2 − C0,0 − C2,2 + C2,1 − C12,0 (12) (12) (2) (2) C2,1 − C12,0 − C2,0 − C12,1 (2) (2) (12) (12) (0) (0) −C1,1 + C2,2 − C12,2 + C1,0 + C2,0 + C12,1 (12) (12) (2) (2) (1) (1) −C1,2 − C2,1 + C0,2 − C2,0 − C0,1 + C1,0 (12) (12) (2) (2) (1) (1) C0,2 + C12,1 + C1,2 + C12,0 − C0,0 − C1,1 (12) (12) (2) (2) (1) (1) C0,1 − C12,2 + C0,0 + C2,2 − C2,1 + C12,0 (12) (12) (2) (2) (1) (1) −C1,1 + C2,2 + C12,2 − C1,0 − C2,0 − C12,1

=0 =0 =0 =0 =0 =0 =0 =0 =0 =0

(7)

This system can be written in matrix form as M · K = 0, where the 48 × 1 column matrix K is  (0) (1) (2) (12) (0) (1) (2) (12) K = C0,0 C0,0 C0,0 C0,0 C0,1 C0,1 C0,1 C0,1 (0)

(1)

(2)

(12)

(0)

(1)

(2)

(12)

(0)

(1)

(2)

(12)

(0)

(1)

(2)

(12)

(0)

(1)

(2)

(12)

(0)

(1)

(2)

(12)

(0)

(1)

(2)

(12)

(0)

C0,2 C0,2 C0,2 C0,2 C1,0 C1,0 C1,0 C1,0

C1,1 C1,1 C1,1 C1,1 C1,2 C1,2 C1,2 C1,2

C2,0 C2,0 C2,0 C2,0 C2,1 C2,1 C2,1 C2,1 (1)

(2)

(12)

C2,2 C2,2 C2,2 C2,2 C12,0 C12,0 C12,0 C12,0 (0)

(1)

(2)

(12)

(0)

(1)

(2)

(12)

C12,1 C12,1 C12,1 C12,1 C12,2 C12,2 C12,2 C12,2

T

and the 32 × 48 coefficient matrix M is given in the Appendix. It is easy to show using Mathematica that the rank of the matrix M is 20 and the solution of the system (6)-(7) can be written as (0)

C0,0 = −p6 + p9 − p16 − p17 − p26 (1) C0,0 = p5 + p10 − p15 − p18 + p25 (2) C0,0 = p8 − p11 − p14 + p19 + p28 (12) C0,0 = −p7 − p12 − p13 + p20 − p27 (0) C0,1 = p4 + p5 + p25 (1) C0,1 = p3 + p6 + p26 (2) C0,1 = p2 − p7 − p27 (12) C0,1 = p1 − p8 − p28 (0) C0,2 = p28 (1) C0,2 = p27 (2) C0,2 = p26 (12) C0,2 = p25

(0)

C2,0 = −p8 + p11 − p19 (1) C2,0 = −p7 − p12 + p20 (2) C2,0 = −p6 + p9 − p17 (12) C2,0 = −p5 − p10 + p18 (0) C2,1 = p20 (1) C2,1 = p19 (2) C2,1 = p18 (12) C2,1 = p17 (0) C2,2 = p16 (1) C2,2 = p15 (2) C2,2 = p14 (12) C2,2 = p13

(8)

Differential Operators Associated to C-R Operator of Clifford Analysis (0)

C1,0 = p4 − p10 − p22 (1) C1,0 = p3 + p9 + p21 (2) C1,0 = p2 + p12 + p24 (12) C1,0 = p1 − p11 − p23 (0) C1,1 = p16 + p17 + p21 (1) C1,1 = p15 + p18 + p22 (2) C1,1 = p14 − p19 − p23 (12) C1,1 = p13 − p20 − p24 (0) C1,2 = p24 (1) C1,2 = p23 (2) C1,2 = p22 (12) C1,2 = p21

399

(0)

C12,0 = p12 (1) C12,0 = p11 (2) C12,0 = p10 (12) C12,0 = p9 (0) C12,1 = p8 (1) C12,1 = p7 (2) C12,1 = p6 (12) C12,1 = p5 (0) C12,2 = p4 (1) C12,2 = p3 (2) C12,2 = p2 (12) C12,2 = p1

(9)

where the pk = pk (t, x), k = 1, 2, ..., 28, are real-valued functions of t and x (see the Appendix). 2.1.2. A system of partial differential equations Analogously, substituting the values of kj,A,B into (5), one obtains the system of 32 linear first order partial differential equations (0)

(1)

(2)

∂x0 K0,1 − ∂x1 K0,1 − ∂x2 K0,1 (2) (1) (0) ∂x0 K0,2 − ∂x1 K0,2 − ∂x2 K0,2 (2) (1) (0) ∂x0 K1,1 − ∂x1 K1,1 − ∂x2 K1,1 (2) (1) (0) ∂x0 K1,2 − ∂x1 K1,2 − ∂x2 K1,2 (2) (1) (0) ∂x0 K2,1 − ∂x1 K2,1 − ∂x2 K2,1 (0) (2) (1) ∂x0 K2,2 − ∂x1 K2,2 − ∂x2 K2,2 (2) (1) (0) ∂x0 K12,1 −∂x1 K12,1 −∂x2 K12,1 (2) (1) (0) ∂x0 K12,2 −∂x1 K12,2 −∂x2 K12,2 (0) (12) (2) ∂x0 K0,1 − ∂x1 K0,1 + ∂x2 K0,1 (0) (12) (2) ∂x0 K0,2 − ∂x1 K0,2 + ∂x2 K0,2 (0) (12) (2) ∂x0 K1,1 − ∂x1 K1,1 + ∂x2 K1,1 (0) (12) (2) ∂x0 K1,2 − ∂x1 K1,2 + ∂x2 K1,2 (0) (12) (2) ∂x0 K2,1 − ∂x1 K2,1 + ∂x2 K2,1 (0) (12) (2) ∂x0 K2,2 − ∂x1 K2,2 + ∂x2 K2,2 (0) (12) (2) ∂x0 K12,1 −∂x1 K12,1 +∂x2 K12,1 (0) (12) (2) ∂x0 K12,2 −∂x1 K12,2 +∂x2 K12,2

=0 =0 =0 =0 =0 =0 =0 =0 =0 =0 =0 =0 =0 =0 =0 =0

(1)

(0)

(12)

∂x0 K0,1 + ∂x1 K0,1 + ∂x2 K0,1 = 0 (12) (0) (1) ∂x0 K0,2 + ∂x1 K0,2 + ∂x2 K0,2 = 0 (12) (0) (1) ∂x0 K1,1 + ∂x1 K1,1 + ∂x2 K1,1 = 0 (0) (1) (12) ∂x0 K1,2 + ∂x1 K1,2 + ∂x2 K1,2 = 0 (12) (0) (1) ∂x0 K2,1 + ∂x1 K2,1 + ∂x2 K2,1 = 0 (12) (0) (1) ∂x0 K2,2 + ∂x1 K2,2 + ∂x2 K2,2 = 0 (12) (0) (1) ∂x0 K12,1 +∂x1 K12,1 +∂x2 K12,1 = 0 (12) (0) (1) ∂x0 K12,2 +∂x1 K12,2 +∂x2 K12,2 = 0 (12) (1) (2) ∂x0 K0,1 + ∂x1 K0,1 − ∂x2 K0,1 = 0 (1) (2) (12) ∂x0 K0,2 + ∂x1 K0,2 − ∂x2 K0,2 = 0 (1) (2) (12) ∂x0 K1,1 + ∂x1 K1,1 − ∂x2 K1,1 = 0 (1) (2) (12) ∂x0 K1,2 + ∂x1 K1,2 − ∂x2 K1,2 = 0 (1) (2) (12) ∂x0 K2,1 + ∂x1 K2,1 − ∂x2 K2,1 = 0 (1) (2) (12) ∂x0 K2,2 + ∂x1 K2,2 − ∂x2 K2,2 = 0 (1) (2) (12) ∂x0 K12,1 +∂x1 K12,1 −∂x2 K12,1 = 0 (1) (2) (12) ∂x0 K12,2 +∂x1 K12,2 −∂x2 K12,2 = 0, (10)

400

U. Y¨ uksel

where (A)

K0,1 (A) K0,2 (A) K1,1 (A) K1,2

(A)

(A)

:= C0,1 − C1,0 (A) (A) := C0,2 − C2,0 (A) (A) := C0,0 + C1,1 (A) (A) := C1,2 + C12,0

(A)

K2,1 (A) K2,2 (A) K12,1 (A) K12,2

:= := := :=

(A)

(A)

C2,1 − C12,0 (A) (A) C0,0 + C2,2 (A) (A) C2,0 + C12,1 (A) (A) C12,2 − C1,0

(11)

Taking into account (11) and (8), (9), the system (10) can be written in terms of the pk as follows: ∂x0 (p5 +p10 +p22 +p25 )−∂x1 (p6−p9−p21 +p26 )−∂x2 (−p7−p12−p24−p27 ) = 0 ∂x0 (p8−p11 +p19 +p28 )−∂x1 (p7 +p12−p20 +p27 )−∂x2 (p6−p9 +p17 +p26 ) = 0 ∂x0 (−p6 +p9 +p21−p26 )−∂x1 (p5 +p10 +p22 +p25 )−∂x2 (p8−p11−p23 +p28 ) = 0 ∂x0 (p12 +p24 )−∂x1 (p11 +p23 )−∂x2 (p10 +p22 ) = 0 ∂x0 (−p12 +p20 )−∂x1 (−p11 +p19 )−∂x2 (−p10 +p18 ) = 0 ∂x0 (−p6 +p9−p17−p26 )−∂x1 (p5 +p10−p18 +p25 )−∂x2 (p8−p11 +p19 +p28 ) = 0 ∂x0 (p11−p19 )x0−∂x1 (−p12 +p20 )−∂x2 (p9−p17 ) = 0 ∂x0 (p10 +p22 )x0−∂x1 (−p9−p21 )−∂x2 (−p12−p24 ) = 0 ∂x0 (p6−p9−p21+p26 )+∂x1 (p5+p10+p22+p25 )+∂x2 (−p8+p11+p23−p28 ) = 0 ∂x0 (p7+p12−p20+p27 )x0 +∂x1 (p8−p11+p19+p28 )+∂x2 (p5+p10−p18+p25 ) = 0 ∂x0 (p5+p10+p22+p25 )x0 +∂x1 (−p6+p9+p21−p26 )+∂x2 (−p7−p12−p24−p27 ) = 0 ∂x0 (p11 +p23 )x0 +∂x1 (p12 +p24 )+∂x2 (p9 +p21 ) = 0 ∂x0 (−p11 +p19 )x0 +∂x1 (−p12 +p20 )+∂x2 (−p9 +p17 ) = 0 ∂x0 (p5+p10−p18+p25 )x0 +∂x1 (−p6+p9−p17−p26 )+∂x2 (−p7−p12+p20−p27 ) = 0 ∂x0 (−p12 +p20 )x0 +∂x1 (p11−p19 )+∂x2 (−p10 +p18 ) = 0 ∂x0 (−p9−p21 )x0 +∂x1 (p10 +p22 )+∂x2 (p11 +p23 ) = 0 ∂x0 (−p7−p12−p24−p27 )−∂x1 (−p8+p11+p23−p28 )+∂x2 (p5+p10+p22+p25 ) = 0 ∂x0 (p6−p9 +p17 +p26 )−∂x1 (p5 +p10−p18 +p25 )+∂x2 (p8−p11 +p19 +p28 ) = 0 ∂x0 (p8−p11−p23 +p28 )−∂x1 (−p7−p12−p24−p27 )+∂x2 (−p6 +p9 +p21−p26 ) = 0 ∂x0 (p10 +p22 )−∂x1 (p9 +p21 )+∂x2 (p12 +p24 ) = 0 ∂x0 (−p10 +p18 )−∂x1 (−p9 +p17 )+∂x2 (−p12 +p20 ) = 0 ∂x0 (p8−p11 +p19 +p28 )−∂x1 (−p7−p12 +p20−p27 )+∂x2 (−p6 +p9−p17−p26 ) = 0 ∂x0 (p9−p17 )−∂x1 (−p10 +p18 )+∂x2 (p11−p19 ) = 0 ∂x0 (−p12−p24 )−∂x1 (p11 +p23 )+∂x2 (p10 +p22 ) = 0 ∂x0 (−p8 +p11 +p23−p28 )+∂x1 (−p7−p12−p24−p27 )−∂x2 (p6−p9−p21 +p26 ) = 0 ∂x0 (p5 +p10−p18 +p25 )+∂x1 (p6−p9 +p17 +p26 )−∂x2 (p7 +p12−p20 +p27 ) = 0 ∂x0 (−p7−p12−p24−p27 )+∂x1 (p8−p11−p23 +p28 )−∂x2 (p5 +p10 +p22 +p25 ) = 0 ∂x0 (p9 +p21 )+∂x1 (p10 +p22 )−∂x2 (p11 +p23 ) = 0 (12)

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∂x0 (−p9 +p17 )+∂x1 (−p10 +p18 )−∂x2 (−p11 +p19 ) = 0 ∂x0 (−p7−p12 +p20−p27 )+∂x1 (p8−p11 +p19 +p28 )−∂x2 (p5 +p10−p18 +p25 ) = 0 ∂x0 (−p10 +p18 )+∂x1 (p9−p17 )−∂x2 (−p12 +p20 ) = 0 ∂x0 (p11 +p23 )+∂x1 (−p12−p24 )−∂x2 (−p9−p21 ) = 0 (13) 2.1.3. Solution of system (12)-(13) in special cases depending on p k Case 1. Let  continuously differentiable functions, if 1 ≤ k ≤ 4 or 13 ≤ k ≤ 16 pk = constants, if 9 ≤ k ≤ 12 or 17 ≤ k ≤ 28 Then it can easily be obtained from the system (12)-(13) that pk , 5 ≤ k ≤ 8, are constants. Case 2. Assume that  continuously differentiable functions, if 1 ≤ k ≤ 4 or 13 ≤ k ≤ 16 pk = constants, if 17 ≤ k ≤ 24. Then it follows from the system (12)-(13) that pk , 9 ≤ k ≤ 12, are constants, pk , 5 ≤ k ≤ 8, and pk , 25 ≤ k ≤ 28 are continuously differentiable functions such that p25+k + p5+k , 0 ≤ k ≤ 3, are constants. Considering the above cases and taking into account (8) and (9) yield the following. (A)

(A)

Lemma 2.1. Assume that C2,2 and C12,2 are continuously differentiable (A)

(A)

(A)

(A)

(A)

functions, and C0,2 , C1,2 , C2,1 , C12,0 , and C12,1 are constants. Assume (A)

(A)

(A)

(A)

(A)

further that C0,0 , C0,1 , C1,0 , C1,1 , and C2,0 are given as (0)

C0,0 (1) C0,0 (2) C0,0 (12) C0,0

= = = =

(0)

(12)

(2)

(12)

(0)

(2)

−C12,1 +C12,0 −C2,2 −C2,1 −C0,2 (12) (2) (1) (2) (12) C12,1 +C12,0 −C2,2 −C2,1 +C0,2 (0) (1) (2) (1) (0) C12,1 −C12,0 −C2,2 +C2,1 +C0,2 (1) (0) (12) (0) (1) −C12,1 −C12,0 −C2,2 +C2,1 −C0,2

(0)

C1,0 (1) C1,0 (2) C1,0 (12) C1,0

= = = =

(0)

(2)

(2)

C12,2 − C12,0 − C1,2 (1) (12) (12) C12,2 +C12,0 + C1,2 (2) (0) (0) C12,2 + C12,0 + C1,2 (1) (1) (12) C12,2 − C12,0 − C1,2

C0,1 (1) C0,1 (2) C0,1 (12) C0,1 (0)

C1,1 (1) C1,1 (2) C1,1 (12) C1,1

= = = =

= = = =

(0)

(0)

(12)

(12)

C12,2 +C12,1 +C0,2 (1) (2) (2) C12,2 +C12,1 +C0,2 (2) (1) (1) C12,2 −C12,1 −C0,2 (12) (0) (0) C12,2 −C12,1 −C0,2 (12)

(12)

C2,2 + C2,1 + C1,2 (1) (2) (2) C2,2 + C2,1 + C1,2 (2) (1) (1) C2,2 − C2,1 − C1,2 (0) (0) (12) C2,2 − C2,1 − C1,2

402

U. Y¨ uksel (0)

C2,0 (1) C2,0 (2) C2,0 (12) C2,0

= = = =

(0)

(1)

(1)

−C12,1 + C12,0 − C2,1 (1) (0) (0) −C12,1 − C12,0 + C2,1 (2) (12) (12) −C12,1 + C12,0 − C2,1 (2) (12) (2) C2,1 − C12,1 − C12,0 .

Then L is associated to D. (A)

(A)

Lemma 2.2. Let C2,2 and C12,2 be continuously differentiable functions, (A)

(A)

(A)

(A)

(A)

and C1,2 , C2,1 , and C12,0 be constants. Suppose that C0,2 and C12,1 are continuously differentiable functions such that their sum (for each A) is a (A) (A) (A) (A) (A) constant. Suppose further that C0,0 , C0,1 , C1,0 , C1,1 , and C2,0 are given as in Lemma 2.1. Then L is associated to D. 3. Appendix We give the Mathematica code used to solve the system (6)-(7). ClearAll; M=Table[0,{32},{48}]; M[[1,1]]=1;M[[1,6]]=1;M[[1,14]]=-1;M[[1,17]]=1;M[[2,7]]=1; M[[2,10]]=1;M[[2,15]]=-1;M[[2,21]]=1;M[[2,26]]=-1;M[[2,29]]=1; M[[3,1]]=1;M[[3,11]]=1;M[[3,27]]=-1;M[[3,33]]=1;M[[4,2]]=-1; M[[4,5]]=1;M[[4,13]]=-1;M[[4,18]]=-1;M[[5,3]]=-1;M[[5,9]]=1; M[[5,19]]=-1;M[[5,22]]=-1;M[[5,38]]=-1;M[[5,41]]=1;M[[6,13]]=-1; M[[6,23]]=-1;M[[6,39]]=-1;M[[6,45]]=1;M[[7,25]]=1;M[[7,30]]=1; M[[7,38]]=-1;M[[7,41]]=1;M[[8,2]]=-1;M[[8,5]]=1;M[[8,31]]=-1; M[[8,34]]=-1;M[[8,39]]=1;M[[8,45]]=-1;M[[9,3]]=-1;M[[9,9]]=1; M[[9,25]]=-1;M[[9,35]]=-1;M[[10,29]]=1;M[[10,37]]=-1;M[[10,26]]=-1; M[[10,42]]=-1;M[[11,14]]=-1;M[[11,17]]=1;M[[11,27]]=1; M[[11,33]]=-1;M[[11,43]]=1;M[[11,46]]=1;M[[12,15]]=-1; M[[12,21]]=1;M[[12,37]]=1;M[[12,47]]=1;M[[13,8]]=1;M[[13,9]]=1; M[[13,16]]=-1;M[[13,22]]=-1;M[[13,25]]=-1;M[[13,30]]=-1; M[[14,2]]=-1;M[[14,28]]=-1;M[[14,34]]=-1;M[[15,4]]=1;M[[15,10]]=1; M[[15,20]]=1;M[[15,21]]=1;M[[15,37]]=1;M[[15,42]]=1;M[[16,14]]=-1; M[[16,24]]=1;M[[16,40]]=1;M[[16,46]]=1;M[[17,1]]=1;M[[17,6]]=1; M[[17,32]]=1;M[[17,33]]=1;M[[17,40]]=-1;M[[17,46]]=-1;M[[18,4]]=1; M[[18,10]]=1;M[[18,26]]=-1;M[[18,36]]=1;M[[19,13]]=-1;M[[19,18]]=-1; M[[19,28]]=1;M[[19,34]]=1;M[[19,44]]=1;M[[19,45]]=1;M[[20,16]]=-1; M[[20,22]]=-1;M[[20,38]]=-1;M[[20,48]]=1;M[[21,3]]=1;M[[21,8]]=1; M[[21,16]]=-1;M[[21,19]]=1;M[[22,5]]=-1;M[[22,12]]=1;M[[22,13]]=1; M[[22,23]]=1;M[[22,28]]=-1;M[[22,31]]=1;M[[23,4]]=-1;M[[23,7]]=1; M[[23,15]]=-1;M[[23,20]]=-1;M[[24,1]]=1;M[[24,11]]=1;M[[24,17]]=1; M[[24,24]]=-1;M[[24,40]]=-1;M[[24,43]]=1;M[[25,27]]=1;M[[25,32]]=1; M[[25,40]]=-1;M[[25,43]]=1;M[[26,4]]=-1;M[[26,7]]=1;M[[26,29]]=1; M[[26,36]]=-1;M[[26,37]]=-1;M[[26,47]]=-1;M[[27,28]]=-1;M[[27,31]]=1;

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403

M[[27,39]]=-1;M[[27,44]]=-1;M[[28,16]]=1;M[[28,19]]=-1;M[[28,25]]=1; M[[28,35]]=1;M[[28,41]]=1;M[[28,48]]=-1;M[[29,6]]=-1;M[[29,11]]=1; M[[29,14]]=1;M[[29,24]]=-1;M[[29,27]]=-1;M[[29,32]]=-1;M[[30,2]]=-1; M[[30,12]]=1;M[[30,18]]=-1;M[[30,23]]=1;M[[30,39]]=1;M[[30,44]]=1; M[[31,3]]=1;M[[31,8]]=1;M[[31,30]]=-1;M[[31,35]]=1;M[[31,38]]=1; M[[31,48]]=-1;M[[32,15]]=-1;M[[32,20]]=-1;M[[32,26]]=-1;M[[32,36]]=1; M[[32,42]]=-1;M[[32,47]]=1; Print["Coefficients Matrix M = ", MatrixForm[M]] Print["Rank of M = ", MatrixRank[M]] V = NullSpace[M]; Print["MatrixForm[V] = Vectors of NullSpace[M] in Matrix Form = "] MatrixForm[V] L1 = Length[V] Print["Number of Vectors in NullSpace[M] = ", L1] L2 = Length[V[[1]]] Print["Number of Components of a Vector in NullSpace[M] = ", L2] For[i = 1, i 41 , . (|ˆ u0 (ξ)| + |ˆ u1 (ξ)|) (1 + t)ρ , m2 ∈ (0, 41 ).

Zone Z3 (N ): Finally, in the zone Z3 (N ) we obtain |E1 (t, 0, ξ)| . C, |E2 (t, 0, ξ)| . |ξ|−1 ,

|E1,t (t, 0, ξ)| . |ξ|, |E2,t (t, 0, ξ)| . C. Consequently, |ˆ ut (t, ξ)| . |ξ||ˆ u0 (ξ)| + |ˆ u1 (ξ)|,

|ξ||ˆ u(t, ξ)| . |ξ||ˆ u0 (ξ)| + |ˆ u1 (ξ)|

and

|ˆ u(t, ξ)| . |ˆ u0 (ξ)| + |ˆ u1 (ξ)|.

Summarizing we have shown for all (t, ξ) in the case m2 6= 1/4 that   1 2 2 2 2 kut (t, ·)k + k∇x u(t, ·)k + p(t) ku(t, ·)k EKG (u)(t) = 2

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.

where p(t) =

 

  1 ku1 (·)k2 + k∇x u0 (·)k2 + ku0 (·)k2 = CEKG (u)(0), 2

1 , (1+t)1/2 1 √  , (1+t)(1+ 1−4m2 )/2

2.2. The case m2 =

m2 > 41 , m2 ∈ (0, 14 ).

1 4

We consider the transformed problem (6) with ρ = 1/2, the logarithmic case. For this problem the fundamental system of solutions is given by w ˆ1 (z) = Ψ(1/2, 1; z) and w ˆ2 (z) = ez Ψ(1/2, 1; −z). In general, we need some properties of the function Ψ = Ψ(α, β; z) represented in the following lemma (see [1]). Lemma 2.2. (i) Ψ is as a single-valued function holomorphic in C \ {0}. 1 (ln z − ψ(α) + 2γ) + O(|z ln z|) ∼ ln z for small z, (ii) Ψ(α, 1; z) = − Γ(α) ′

(α) where ψ(α) = ΓΓ(α) and γ is Euler’s constant. (iii) |Ψ(α, β; z)| ≤ Cαβ |z|−Reα for large |z| with −(3/2)π < arg z < (3/2)π. (iv) Ψ(α − 1, β; z) = (α − β + z) Ψ(α, β; z) − z dz Ψ(α, β; z).

Transforming back the fundamental solutions yields 1 1 u ˜1 (t, ξ) = ((1 + t)|ξ|) 2 e−i(1+t)|ξ| Ψ( , 1; 2i(1 + t)|ξ|), 2 1 1 i(1+t)|ξ| u ˜2 (t, ξ) = ((1 + t)|ξ|) 2 e Ψ( , 1; −2i(1 + t)|ξ|) 2

and their first derivatives 1

u ˜1,t (t, ξ) = ((1 + t)|ξ|)− 2 |ξ|e−i(1+t)|ξ|   1 1 × −Ψ(− , 1; 2i(1 + t)|ξ|) + i(1 + t)|ξ|Ψ( , 1; 2i(1 + t)|ξ|) ; 2 2 1 u ˜2,t (t, ξ) = ((1 + t)|ξ|)− 2 |ξ|ei(1+t)|ξ|   1 1 × −Ψ(− , 1; −2i(1+t)|ξ|) − i(1+t)|ξ|Ψ( , 1; −2i(1+t)|ξ|) . 2 2

We want to estimate the solution u ˆ(t, ξ) = E1 (t, s, ξ)ˆ u(s, ξ) + E2 (t, s, ξ)ˆ ut (s, ξ) and the first derivative uˆt (t, ξ) = E1,t (t, s, ξ)ˆ u(s, ξ) + E2,t (t, s, ξ)ˆ ut (s, ξ), s, t ≥ 0. First, we have to determine the denominator u ˜ 2,t (s, ξ)˜ u1 (s, ξ) − u ˜1,t (s, ξ)˜ u2 (s, ξ). Thus we set u ˜1 (t, ξ) = f (t, ξ)w ˆ1 (2i(1+t)|ξ|) and u˜2 (t, ξ) =

Generalized Energy Conservation

421

f (t, ξ)w ˆ2 (2i(1 + t)|ξ|), where f (t, ξ) = ((1 + t)|ξ|)(1/2) e−i(1+t)|ξ| . We need the Wronskian given by 1

w ˆ1 (z)w ˆ2,z (z) − w ˆ2 (z)w ˆ1,z (z) = eπi 2 z −1 ez . Furthermore, we have u ˜2,t (t, ξ)˜ u1 (t, ξ) − u ˜1,t (t, ξ)˜ u2 (t, ξ)

= fw ˆ1 (ft w ˆ2 + f w ˆ2,t ) − f w ˆ2 (ft w ˆ1 + f w ˆ1,t )

= f 2 (wˆ1 w ˆ2,t − w ˆ1,t w ˆ2 ) = f 2 2i|ξ|(w ˆ1 w ˆ2,z − w ˆ1,z w ˆ2 ) 1

1

= f 2 (2i|ξ|)eǫπi 2 (2i(1 + t)|ξ|)−1 e2i(1+t)|ξ|) = eπi 2 |ξ| = i|ξ|. We want to investigate the multipliers in the three zones Z1 (N ), Z2 (N ) and Z3 (N ) given in Definition 2.1. The estimates are based on Lemma 2.2. Zone Z1 (N ): We start with E1 (t, 0, ξ) by applying Lemma 2.2(ii): 1 1 |E1 (t, 0, ξ)| . |ξ|−1 |ξ| 2 ((1 + t)|ξ|) 2 × ln(2i(1 + t)|ξ|) ln(−2i|ξ|) e−it|ξ| − ln(−2i(1 + t)|ξ|) ln(2i|ξ|) eit|ξ| .

The natural logarithm of a complex number z = aeib is given by ln z = ln a + ib. Hence, we obtain

π2 π − i ln(1 + t), 4 2 π π2 + i ln(1 + t). ln(−2i(1 + t)|ξ|) ln(2i|ξ|) = ln(2(1 + t)|ξ|) ln(2|ξ|) + 4 2 We have ln(2i(1 + t)|ξ|) ln(−2i|ξ|) e−it|ξ| − ln(−2i(1 + t)|ξ|) ln(2i|ξ|) eit|ξ|    π 2  −it|ξ| e − eit|ξ| = ln(2(1 + t)|ξ|) ln(2|ξ|) + 4   π −it|ξ| it|ξ| +e − i ln(1 + t) e 2   2 π = ln(2(1 + t)|ξ|) ln(2|ξ|) + 2 sin(−t|ξ|) − π ln(1 + t) cos(−t|ξ|) 4   2 π ≤ ln(2(1 + t)|ξ|) ln(2|ξ|) + sin(−t|ξ|) + π ln(1 + t) 4 . 1 + ln(1 + t). ln(2i(1 + t)|ξ|) ln(−2i|ξ|) = ln(2(1 + t)|ξ|) ln(2|ξ|) +

We continue the computations for E1 and obtain in the same way for E2 , E1,t and E2,t : 1

|E1 (t, 0, ξ)|, |E2 (t, 0, ξ)| . (1 + t) 2 (1 + ln(1 + t))

and

422

C. B¨ ohme & M. Reissig 1

|E1,t (t, 0, ξ)|, |E2,t (t, 0, ξ)| . (1 + t)− 2 (1 + ln(1 + t)). Finally, we get in the zone Z1 (N ) for all times t: |ˆ ut (t, ξ)| . |ˆ u0 (ξ)| + |ˆ u1 (ξ)|,

|ξ||ˆ u(t, ξ)| . |ˆ u0 (ξ)| + |ˆ u1 (ξ)|

and

1 2

|ˆ u(t, ξ)| . (1 + t) (1 + ln(1 + t)) (|ˆ u0 (ξ)| + |ˆ u1 (ξ)|).

Zone Z2 (N ): Similar to the considerations for m2 6= 1/4 we estimate the energy terms in Z2 (N ) by |ˆ ut (t, ξ)| . |ˆ u0 (ξ)| + |ˆ u1 (ξ)|,

|ξ||ˆ u(t, ξ)| . |ˆ u0 (ξ)| + |ˆ u1 (ξ)|

and

1 2

u0 (ξ)| + |ˆ u1 (ξ)|). |ˆ u(t, ξ)| . (1 + t) (1 + ln(1 + t))(|ˆ

Zone Z3 (N ): In the third zone Z3 (N ) we obtain for m2 = 1/4 exactly the same results as for m2 6= 1/4. Altogether, we can estimate the energy in the case m2 = 1/4 by EKG (u)(t)   1 1 2 2 2 ku(t, ·)k kut (t, ·)k + k∇x u(t, ·)k + = 2 (1 + t)(1 + ln(1 + t))2   1 . ku1 (·)k2 + k∇x u0 (·)k2 + ku0 (·)k2 = CEKG (u)(0). 2 Conclusively, the statements of Theorem 2.1 are shown. Remark 2.1. For the moment we have energy estimate only from above by the result of Theorem 2.1. Further investigations should bring lower bounds for the energy EKG (u)(t). 3. Generalized energy conservation for wave equations In contrast to the energy estimates of the scale-invariant Klein-Gordon equation, suitably defined energies of wave equations with time-dependent speed of propagation can be estimated from below by a constant depending on the initial energy. Thus, we introduce the notion generalized energy conservation. That is, the estimates C −1 E(u)(0) ≤ E(u)(t) ≤ CE(u)(0)

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423

hold for a suitable energy E for a Cauchy problem with a suitable positive constant C. Let us investigate the energy   1 2 2 2 (7) EW (u)(t) := kut (t, x)k + a(t) k∇x u(t, x)k 2 for the Cauchy problem of the wave equation

utt − a(t)2 △ u = 0, u(0, x) = u0 (x), ut (0, x) = u1 (x).

(8)

The application of WKB representations leads to the first generalized energy conservation result given in [4]. Theorem 3.1. (Reissig-Smith, [4]) Consider the Cauchy problem (8) with u0 ∈ H 1 , u1 ∈ L2 . If a = a(t) satisfies   a(t) ∈ C 2 (R+ ),   0 < a1 ≤ a(t) ≤ a2 < ∞ for all t, (9) k     a(k) (t) ≤ C 1 , k = 1, 2, k

1+t

then generalized energy conservation holds referring to the energy E W (u)(t) given in (7) for all times t and a positive constant C.

We can show that a higher regularity in (9) has no influence on the proof of Theorem 3.1. However, in [3] it is proved that higher regularity is useful under an additional stabilization condition on the propagation speed a and, consequently, changed conditions on the derivatives of a up to order M ≥ 2. We summarize this result in the following theorem. Theorem 3.2. (Hirosawa, [3]) Consider the Cauchy problem (8) with u 0 ∈ H 1 , u1 ∈ L2 . If a = a(t) satisfies  M   a(t) ∈ C (R+ ), M ≥ 2,   0 < a1 ≤ a(t) ≤ a2 < ∞ for all t, Rt p   0 |a(s) − a∞ |ds ≤ Ct for all t ≥ 1 and p ∈ [0, 1),     qk   a(k) (t) ≤ Ck 1 for k = 1, ..., M and q < 1, 1+t and q ≥ p + 1−p M , then generalized energy conservation holds referring to the energy EW (u)(t) given in (7) for all times t and a positive constant C.

The question of generalized energy conservation for wave equations with low regularity propagation speed appears. We can give the following statement.

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C. B¨ ohme & M. Reissig

Theorem 3.3. Consider the Cauchy problem (8) with u 0 ∈ H 1 , u1 ∈ L2 . If a = a(t) satisfies ( 0 < a0 ≤ a(t) ≤ a1 < ∞ for all t, a′ (t) ∈ L1 (R+ ),

then generalized energy conservation holds referring to the energy E W (u)(t) given in (7) for all times t and a positive constant C. Sketch of the proof. First, we rewrite the Fourier-transformed Cauchy problem by the Liouville transformation for a damped wave equation with time-dependent dissipation term b = b(t) ∈ L1 (R+ ). Due to [6], we have a scattering result to this problem. That means, the energy to the damped wave problem behaves asymptotically like the energy (3) of the classical wave equation (1). This implies the statements. Remark 3.1. We are also interested in statements about generalized energy conservation for Klein-Gordon equations with more general timedependent mass m = m(t) under certain conditions of the coefficient. Moreover, additional time-depending speed of propagation will play an essential role. References 1. H. Bateman and A. Erdelyi, Higher Transcendental Functions, vol. 1, McGraw-Hill, New York, 1953. 2. D. Del Santo, T. Kinoshita and M. Reissig, Rendiconti dell’Istituto di Matematica dell’Universita di Trieste 39 (2007), 1–35. 3. F. Hirosawa, Mathematische Annalen 339 (2007), 819–838. 4. M. Reissig and J. Smith, Hokkaido Mathematical Journal 34 (2005), 541–586. 5. J. Wirth, Mathematical Methods in the Applied Sciences 27 (2004), 101–124. 6. J. Wirth, Advances in Differential Equations 12 (2007), 1115–1133.

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THE CAUCHY PROBLEM FOR A HYPERBOLIC OPERATOR WITH LOG-ZYGMUND COEFFICIENTS Daniele Del Santo Dipartimento di Matematica e Informatica, Universit` a di Trieste Via A. Valerio 12/1, 34127 Trieste, Italy E-mail: [email protected] We use the Littlewood-Paley dyadic decomposition to obtain a well-posedness result for a hyperbolic equation with coefficients log-Zygmund-continuous in t and C ∞ in x. Keywords: Hyperbolic operators, Littlewood-Paley decomposition.

Cauchy

problem,

energy

estimates,

1. Introduction Consider the operator L = ∂t2 − ∂x (a(t, x)∂x ),

(1)

and suppose that L is strictly hyperbolic with bounded coefficients, i. e. Λ0 ≥ a(t, x) ≥ λ0 > 0

(2)

for all (t, x) ∈ [0, T ] × R. If the coefficient a is Lipschitz–continuous with respect to t, uniformly with respect to x, and C ∞ , bounded with bounded derivatives with respect to x, uniformly with respect to t, then the Cauchy T problem for (1) is well–posed in H∞ (R) = s Hs (R). This conclusion can be deduced from an energy estimate “without loss of derivatives” i.e. from an estimate of the type sup {ku(t, ·)kHm+1 + k∂t u(t, ·)kHm }

0≤t≤T

≤ Cm (ku(0, ·)kHm+1 + k∂t u(0, ·)kHm + 2

∞

Z

0

(3)

T

kLu(t, ·)kHm dt),

which is valid for all u ∈ C ([0, T ], H (R)). If the coefficient a is not Lipschitz–continuous with respect to t then the estimate (3) is no more true in general; nevertheless the H ∞ –well–posedness

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D. Del Santo

may be recovered from an energy estimate “with loss of derivatives” (see e. g. the estimate (8) below) and it may be proved under weaker regularity assumption on a. A result of this type was obtained in the well–known paper of Colombini, De Giorgi and Spagnolo [3], where the case of the coefficient a depending only on t is considered. The regularity condition under which the H ∞ –well– posedness is proved is the following one: there exists C > 0 such that Z T −ε 1 (4) |a(t + ε) − a(t)| dt ≤ Cε log( + 1) ε 0 for all ε ∈ (0, T ]. The energy estimate is obtained by using the Fourier transform with respect to x of the equation together with an approximation of the coefficient which is different in different zones of the phase space (the so called approximate energy technique, see [4]). The case of the coefficient a depending on t and x was considered by Colombini and Lerner in [6], where the regularity condition was: there exists C > 0 such that 1 (5) sup |a(t + ε, x) − a(t, x)| dt ≤ Cε log( + 1) ε (t,x)∈[0,T −ε]×R

for all ε ∈ (0, T ]. In this situation the use of the Littlewood-Paley dyadic decomposition replaces the Fourier transform with respect to x, while the approximate energy technique remains a crucial point. Recently, in [10], Tarama considered again the case of the coefficient a depending only on t, proving the H∞ –well–posedness under the condition: there exists C > 0 such that Z T −ε 1 (6) |a(t + ε) + a(t − ε) − 2a(t)| dt ≤ Cε log( + 1) ε ε for all ε ∈ (0, T /2]. The improvement with respect to [3] is obtained introducing a new type of approximate energy which involves the second derivatives of the approximating coefficient (see section 3.2 below). In the present communication the case of the coefficient depending on t and x, under a regularity condition inspired by (6) and (5), will be studied. The dyadic decomposition technique will be applied as in [6] (see also [2] and [7]) together with Tarama’s approximate energy. Acknowledgments I want to thank J.–M. Bony, F. Colombini, G. M´etivier and M. Reissig for the useful discussions on the topics of this communication and the anonymous referee for the corrections and the improvements suggested.

The Cauchy Problem for a Hyperbolic Operator with Log-Zygmund Coefficients

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2. Results Consider the operator (1) under the condition (2). Suppose that there exists C > 0 such that 1 sup |a(t + ε, x) + a(t − ε, x) − 2a(t, x)| ≤ Cε log( + 1) (7) ε (t,x)∈[ε,T −ε]×R for all ε ∈ (0, T /2]. Suppose moreover that for all t ∈ [0, T ] the function x 7→ a(t, x) is C ∞ (R) and for all α ∈ N the functions (t, x) 7→ ∂xα a(t, x) are L∞ ([0, T ] × R) (we resume all these requirements on the regularity of the coefficient a saying that a is log–Zygmund–continuous in t, uniformly with respect to x, and B ∞ in x, uniformly with respect to t). The main result of this communication is the following one. Theorem 2.1. The Cauchy problem for (1) is well–posed in H ∞ . This result is a consequence of the energy estimate “with loss of derivatives” contained in the following statement. Theorem 2.2. There exists β > 0 and, for all m ∈ R, there exists Cm > 0 such that sup {ku(t, ·)kHm+1−βt + k∂t u(t, ·)kHm−βt } 0≤t≤T Z T (8) ≤ Cm (ku(0, ·)kHm+1 + k∂t u(0, ·)kHm + kLu(t, ·)kHm−βt dt) 0

for all u ∈ C 2 ([0, T ], H∞ (R)).

3. Proof of Theorem 2.2 3.1. Dyadic decomposition We collect here some well–known facts on the Littlewood-Paley decomposition, referring to [1] and [6] for the details. Let ϕ0 ∈ C0∞ (Rξ ), 0 ≤ ϕ0 (ξ) ≤ 1, ϕ0 (ξ) = 1 if |ξ| ≤ 1, ϕ0 (ξ) = 0 if |ξ| ≥ 2, ϕ0 even and ϕ0 decreasing on [0, +∞). We set ϕ(ξ) = ϕ0 (ξ/2) − ϕ0 (ξ) and, if ν is an integer greater or equal than 1, ϕν (ξ) = ϕ(2−ν ξ). Let w be a tempered distribution in H−∞ (R); we define Z Z 1 1 ixξ wν (x) = ϕν (Dx )w(x) = e ϕν (ξ)w(ξ) ˆ dξ = ϕˆν (y)w(x − y) dy. 2π 2π

For all ν, wν is an entire analytic function belonging to L2 and for all m ∈ R there exists Km > 0 such that ∞ ∞ X 1 X kwν k2L2 22mν ≤ kwk2Hm ≤ Km kwν k2L2 22mν . Km ν=0 ν=0

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Moreover we have 2ν−1 kwν kL2 ≤ k∂x wν kL2 ≤ 2ν+1 kwν kL2 ,

(9)

where the inequality on the right–hand side holds for all ν while the other one holds only for all ν ≥ 1. 3.2. Approximation of the coefficient We extend the value of a to R2 setting a(t, x) = a(0, x) if t < 0 and a(t, x) = a(T, x) if t > T . The condition (7) remains valid with the supremum evaluated on R2 . We set Z aε (t, x) = ρε (t − s)a(s, x) ds, R

s 1 ε ρ( ε )

with ρ ∈ C ∞ (R), 0 ≤ ρ ≤ 1, supp ρ ⊆ [−1, 1] and where ρε (s) = R ρ(s) ds = 1. We obtain that there exists C > 0 such that 1 sup |aε (t, x) − a(t, x)| ≤ Cε log( + 1) ε (t,x)∈R2

(10)

and 1 1 sup |∂t2 aε (t, x)| + |∂t aε (t, x)|2 ≤ C log( + 1), ε ε

(11)

(t,x)∈R2

for all ε > 0. In particular (10) is obtained from (7) remarking that Z 1 ρε (s)(a(t − s, x) + a(t + s, x) − 2a(t, x))ds, aε (t, x) − a(t, x) = 2 |s|≤ε where we have used the fact that ρ is an even function. Similarly Z 1 ρ′′ (s)(a(t − s, x) + a(t + s, x) − 2a(t, x))ds, ∂t2 aε (t, x) = 2 |s|≤ε ε and the estimate for |∂t2 aε (t, x)| follows as before using (7), while that one for |∂t aε (t, x)|2 is a consequence of a standard argument similar to that one used to prove the Glaeser inequality (see [8] or [9]) using the bound on aε and ∂t2 aε . √ Finally, since 0 < λ0 ≤ aε (t, x), a(t, x) ≤ Λ0 , if we set a(t, x) = √ (a(t, x))1/2 and aε (t, x) = (aε (t, x))1/2 , it is easy to deduce from (11) that there exists C > 0 such that √ √ 1 1 (12) sup |∂t2 aε (t, x)| + |∂t aε (t, x)|2 ≤ C log( + 1), ε ε (t,x)∈R2 for all ε > 0.

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429

3.3. Approximate energy of the ν-component Let u(t, x) be a function in C 2 ([0, T ], H∞ (Rn )). We set uν (t, x) = ϕν (D)u(t, x). We obtain ∂t2 uν = ∂x (a(t, x)∂x uν ) + ∂x ([ϕν , a]∂x u) + (Lu)ν , where [ϕν , a] is the commutator between ϕν (Dx ) and a. We introduce the approximate energy of uν setting Z √ ∂t aε √ 1 eν,ε (t) = √ |∂t uν + √ uν |2 + aε |∂x uν |2 + |uν |2 dx. aε 2 aε R We have Z √ √ √ ∂t aε ∂t aε ∂t aε 2 d eν,ε (t) = √ (∂t ( √ ) − ( √ )2 )Re (uν ·(∂t uν + √ uν )) dx dt aε 2 aε 2 aε 2 aε Z √ ∂t aε 2 + √ Re (∂t2 uν ·(∂t uν + √ uν )) dx aε 2 aε Z √ √ + ∂t aε |∂x uν |2 + 2 aε Re (∂x uν · ∂x ∂t uν ) dx + −

Z

Z

2Re (uν ·(∂t uν + 2Re (uν ·

√ ∂t aε √ uν )) dx 2 aε

√ ∂t aε √ uν ) dx. 2 aε (13)

From (9) and (12) we deduce that Z

√ √ √ ∂t aε ∂t aε ∂t aε 2 √ (∂t ( √ ) − ( √ )2 )Re (uν ·(∂t uν + √ uν )) dx aε 2 aε 2 aε 2 aε 1 1 ≤ C log( + 1) 2−ν eν,ε (t), ε ε

where C > 0 does not depend on u or ν. Next we have Z √ ∂t aε 2 A1 := √ Re (∂t2 uν ·(∂t uν + √ uν )) dx aε 2 aε Z √ ∂t aε 2 = √ a Re (∂x2 uν ·(∂t uν + √ uν )) dx aε 2 aε Z √ ∂t aε 2 + √ ∂x a Re (∂x uν ·(∂t uν + √ uν )) dx aε 2 aε Z √ ∂t aε 2 + √ Re ((∂x ([ϕν , a]∂x u) + (Lu)ν )·(∂t uν + √ uν )) dx, aε 2 aε

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D. Del Santo

and

Z

√ √ ∂t aε |∂x uν |2 + 2 aε Re (∂x uν · ∂x ∂t uν ) dx Z √ ∂t aε 2 2 = − √ aε Re (∂x uν ·(∂t uν + √ uν )) dx aε 2 aε Z √ ∂t aε √ − 2 ∂x aε Re (∂x uν ·(∂t uν + √ uν )) dx 2 aε Z √ √ ∂x aε ∂t aε √ + ( − ∂x ∂t aε ) Re (∂x uν ·uν ) dx. √ aε

A2 :=

Since, from (9) and (10), there exists C > 0 not depending on u or ν such that Z √ ∂t aε 2 2 √ (a − aε ) Re (∂x uν ·(∂t uν + √ uν )) dx aε 2 aε √ ∂t aε 1 ≤ Cε log( + 1) 2ν k∂x uν kL2 k∂t uν + √ uν kL2 , ε 2 aε √ √ √ ∂x aε ∂t aε √ | + |∂x ∂t aε | ≤ C/ε, we obtain and remarking that sup(t,x)∈R2 | aε 2−ν 1 )eν,ε (t) |A1 + A2 | ≤ C(ε(1 + log( + 1)) 2ν + 1 + ε ε Z √ ∂t aε 2 + √ Re ((∂x ([ϕν , a]∂x u) + (Lu)ν )·(∂t uν + √ uν )) dx. aε 2 aε

Similarly, we have Z √ ∂t aε 2Re (uν ·(∂t uν + √ uν )) dx ≤ Ceν,ε (t), 2 aε

and

Z

2Re (uν ·

√ ∂t aε 2−ν eν,ε (t). √ uν ) dx ≤ C 2 aε ε

We choose ε = 2−ν and we set eν,2−ν (t) = eν (t), a2−ν = αν . As a conclusion we deduce that there exists C0 > 0 not depending on u or ν such that d eν (t) ≤ C0 (ν + 1)eν (t) dt Z √ ∂t αν 2 + √ Re (∂x ([ϕν , a]∂x u)·(∂t uν + √ uν )) dx αν 2 αν Z √ ∂t αν 2 + √ Re ((Lu)ν ·(∂t uν + √ uν )) dx. αν 2 αν

(14)

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431

3.4. Estimate of the total energy We define Em (t) =

∞ X

e−2β

∗

(ν+1)t 2νm

2

eν (t),

(15)

ν=0

with β ∗ to be fixed. It is possible to prove that there exists C1 , C2 > 0 such that Em (0) ≤ C1 (k∂t u(0, ·)k2Hm + ku(0, ·)k2Hm+1 )

(16)

Em (t) ≥ C2 (k∂t u(t, ·)k2Hm−βt + ku(t, ·)k2Hm+1−βt ),

(17)

and ∗

−1

where β = β (log 2)

∗

. Choosing β = C0 /2, from (14) we deduce

d Em (t) ≤ dt Z ∞ X ∗ 2 e−2β (ν+1)t 22νm √ Re (∂x ([ϕν , a]∂x u)·(∂t uν + αν ν=0 Z ∞ X ∗ 2 + e−2β (ν+1)t 22νm √ Re ((Lu)ν ·(∂t uν + αν ν=0

√ ∂t αν √ uν )) dx 2 αν √ ∂t αν √ uν )) dx. 2 αν (18) It is not difficult to show that there exists C3 > 0 such that Z √ ∞ X ∂t αν 2 −2β ∗ (ν+1)t 2νm e 2 √ Re ((Lu)ν ·(∂t uν + √ uν )) dx αν 2 αν (19) ν=0 ≤ C3 (Em (t))1/2 kLu(t, ·)kHm−βt .

3.5. Estimate of the commutator term The estimate of the first term in the right–hand side part of (18) is more delicate. We refer to [6, Lemmas 4.4 and 4.5] for the details, giving here only a sketch of the argument. First of all it is possible to prove that for all N ∈ N there exists CN > 0 such that  C0 2−ν if |µ − ν| ≤ 2, (20) k[ϕν , a]ϕµ kL(L2 ,L2 ) ≤ CN 2−N max{ν,µ} if |µ − ν| ≥ 3, where k · kL(L2 ,L2 ) denotes the norm as operator from L2 to L2 . Setting ϕ−1 ≡ 0 we have that [ϕν , a]∂x u =

+∞ X

µ=0

([ϕν , a](ϕµ−1 + ϕµ + ϕµ+1 ))∂x uµ ,

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D. Del Santo

so that Z √ ∂t αν 2 √ Re (∂x ([ϕν , a]∂x u)·(∂t uν + √ uν )) dx αν 2 αν Z ∞ X =− Re (([ϕν , a](ϕµ−1 + ϕµ + ϕµ+1 )∂x uµ ) µ=0

√ ∂t αν 2 ·∂x ( √ (∂t uν + √ uν )) dx, αν 2 αν

and, consequently, Z √ ∂t αν 2 √ Re (∂x ([ϕν , a]∂x u)·(∂t uν + √ uν )) dx αν 2 αν ≤3

√ ∂t αν 2 k([ϕν , a]ϕµ )∂x uµ kL2 k∂x ( √ (∂t uν + √ uν ))kL2 . αν 2 αν µ=0 ∞ X

It is possible to show that √ ∂t αν 2 k∂x ( √ (∂t uν + √ uν ))kL2 ≤ C 2ν e1/2 ν (t). αν 2 αν Finally the estimate (20) together with Schur’s lemma give that there exists C4 > 0 such that Z √ ∞ X ∗ ∂t αν 2 e−2β (ν+1)t 22νm √ Re (∂x ([ϕν , a]∂x u)·(∂t uν + √ uν )) dx αν 2 αν (21) ν=0 ≤ C4 Em (t).

3.6. End of the proof From (18), (19) and (21) we obtain d Em (t) ≤ C3 (Em (t))1/2 kLu(t, ·)kHm−βt + C4 Em (t), dt and (8) easy follows from (16) and (17). 4. Remarks and Problems Using the technique sketched in this communication it is possible to prove some local in time energy estimates in Sobolev spaces with “loss of derivatives” for operator having limited regularity in the x variable, e. g. if the coefficient a is C α with α > 1. Some problems on this topic remain open.

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• What to do with hyperbolic operators depending on more than one space variable? • What to do with systems? • What to do with a regularity condition like (6) (i. e. an integral condition) but with the coefficient a depending also on x? • What to do with an isotropic (i. e. involving both t and x) log–Zygmund regularity condition on a? All these questions will be considered in a forthcoming paper [5]. References 1. J.-M. Bony, Calcul symbolique et propagation des singularit´es pour les ´equa´ tions aux d´eriv´ees partielles non lin´eaires, Ann. Sci. Ecole Norm. Sup. (4) 14 (1981), no. 2, 209–246. 2. M. Cicognani, D. Del Santo and M. Reissig, A dyadic decomposition approach to a finitely degenerate hyperbolic problem, Ann. Univ. Ferrara, Sez. VII, Sci. Mat. 52 (2006), 281–289. 3. F. Colombini, E. De Giorgi and S. Spagnolo, Sur les ´equations hyperboliques avec des coefficients qui ne d´ependent que du temps, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 6 (1979), no. 3, 511–559. 4. F. Colombini and D. Del Santo, Strictly hyperbolic operators and approximate energies, in “Analysis and applications–ISAAC 2001, Proceedings of the 3rd international congress, Berlin, Germany, August 20–25, 2001”, pp. 253–277, H. G. W. Begehr ed., Kluwer Academic Publishers, Dordrecht 2003. 5. F. Colombini, D. Del Santo and G. M´etivier, in preparation. 6. F. Colombini and N. Lerner, Hyperbolic operators with non-Lipschitz coefficients, Duke Math. J. 77 (1995), no. 3, 657–698. 7. D. Del Santo, T. Kinoshita and M. Reissig, Energy estimates for strictly hyperbolic equations with low regularity in coefficients, Differential Integral Equations 20 (2007), no. 8, 879–900. 8. G. Glaeser, Racine carr´ee d’une fonction diff´erentiable, Ann. Inst. Fourier 13 (1963), 203–210. 9. T. Nishitani and S. Spagnolo, An extension of Glaeser inequality and its applications, Osaka J. Math. 41 (2004), no. 1, 145–157. 10. S. Tarama, Energy estimate for wave equations with coefficients in some Besov type class, preprint, 2007.

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AN OPTIMAL TRANSPORTATION METRIC FOR TWO NONLINEAR PDE’S M. FONTE Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences, Linz, A-4040, Austria E-mail: [email protected] An optimal transportation technique for nonlinear hyperbolic equations is discussed. Based on a Wasserstein-like metric in H 1 (R), a distance functional in defined, which yields continuous dependence with respect to the initial conditions of the energy conservative solutions of the Camassa-Holm equation. Keywords: Optimal transportation; Camassa-Holm equation.

1. Introduction In this paper there are summarized some global stability results and new techniques for nonlinear PDE equations, obtained by the author in collaboration with A. Bressan, partly contained in [1,2]. It is discussed the existence of a global solution after wave breaking for the Cauchy problem of the Camassa-Holm equation ut + 2κux − uxxt + 3uux = 2ux uxx + uuxxx, and the construction of a continuous semigroup of solutions in H 1 (R). This equation arises from a higher order level of approximation of the asymptotic expansion of the Euler’s equation for a shallow water wave model. Here we do not enter into details of the interpretation of such an equation, for the physical motivations we refer to [3–6]. The main part of the work is focused on the stability issue. It is described the construction of a metric on H 1 (R), obtained by an optimal transportation argument, for which continuous dependence w.r.t. the initial data is obtained. For a complete and detailed discussion on this problem we suggest [1,2,7]. Finally, an example on the optimal transportation technique for the Hunter-Saxton equation is discussed.

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435

In the following we focus our attention on the Camassa-Holm equation with parameter κ = 0. 1.1. Non-local formulation of the Camassa-Holm equation The Cauchy problem of the Camassa-Holm equation can be written as a scalar conservation law with an additional integro-differential term:   u2 1 . 1 (1) P (x) = e−|x| ∗ u2 + x ut + ( u2 )x + Px = 0 , 2 2 2 and with the initial data specified as u(0, x) = u0 (x),

u0 ∈ H 1 (R).

(2)

Earlier results on the existence and uniqueness of solutions can be found in [8,9]. One can regard (1) as an evolution equation on a space of absolutely continuous functions with derivatives ux ∈ L2 . Suppose that u is a smooth function, then differentiating (1) w.r.t. x and multiplying by u, ux one obtain the two conservation laws with source term   3  2   2  2 u uux u3 u ux + + = −ux P . +uP − = ux P, 2 t 3 2 t 2 3 x x As a consequence, for regular solutions the total energy Z  2  . u (t, x) + u2x (t, x) dx ≡ E0 E(t) = R

remains constant in time. As in the case of scalar conservation laws, by the strong nonlinearity of the equations, solutions with smooth initial data can lose regularity in finite time. For the Camassa-Holm equation (1), however, the uniform bound on kux kL2 guarantees that only the L∞ norm of the gradient can blow up, while the solution u itself remains H¨ older continuous at all times. In order to construct global in time solutions, several techniques have been developed. A first approach was introduced in [5,10]. Recently, two main methods have been developed. On one hand, one can add a small diffusion term in the right hand side of (1), and recover solutions of the original equations as a vanishing viscosity limit [11,12]. An alternative technique, developed in [13], relies on a new set of independent and dependent variables, specifically designed with the aim of “resolving” all singularities. In terms of these new variables, the solution to the Cauchy problem becomes regular for all times, and can be obtained as the unique fixed point of a contractive transformation.

436

M. Fonte

2. Multi-peakon solutions In the present paper, we consider yet another approach, related to the solitary solutions of the Camassa-Holm equation. As a starting point we consider all multi-peakon solutions, of the form u(t, x) =

N X

pi (t)e−|x−qi (t)| .

(3)

i=1

These are obtained by solving the system of O.D.E’s  X  pj e−|qi −qj | ,  q˙i =  j X  p ˙ = pi pj sign (qi − qj ) e−|qi −qj | . i  

(4)

j6=i

It is well known [14] that this system can be written in Hamiltonian form:  ∂   q˙i = H(p, q) , . 1X ∂pi H(p, q) = pi pj e−|qi −qj | . ∂  2  p˙i = − H(p, q) , i,j ∂qi

If all the coefficients pi are initially positive, then they remain positive and bounded for all times. The solution u = u(t, x) is thus uniformly Lipschitz continuous. We stress, however, that we are not making any assumption about the signs of the pi . In a typical situation, two peakons can cross at a finite time τ . As t → τ − their strengths pi , pj and center positions qi , qj will satisfy pi (t) → +∞ , qi (t) → q¯ ,

pj (t) → −∞ ,

pi (t) + pj (t) → p¯ ,

(5)

qj (t) → q¯ , qi (t) < qj (t) for t < τ , (6)

for some p¯, q¯ ∈ R. Moreover, ux (t) L∞ → ∞. In this case, we will show that there exists a unique way to extend the multi-peakon solution beyond the interaction time, so that the total energy is conserved. Having constructed a set of “multi-peakon solutions”, our main goal is to show that these solutions form a continuous semigroup, whose domain is dense in H 1 (R). Taking the unique continuous extension, we thus obtain a continuous semigroup of solutions of (1), defined on the entire space H 1 . One easily checks that the flow map Φt : u(0) 7→ u(t) cannot be continuous either as a map from H 1 into itself, or from L2 into itself. Distances defined in terms of convex norms perform well in connection with linear problems, but occasionally fail when nonlinear features become dominant (see [7, Example 2]).

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437

3. An optimal transportation metric In the present setting, we construct a new distance J(u, v) between functions u, v ∈ H 1 , defined by a problem of optimal transportation. Call F the family of all strictly increasing absolutely continuous maps ψ : R 7→ R which have an absolutely continuous inverse, and consider the measurable function     
 ′ . 2 2 ϕ(x) = sup θ ∈ [0, 1] s.t. θ · 1 + ux (x) ≤ 1 + vx ψ(x) ψ (x) ,

. the metric is defined as J(u, v) = minψ∈F J ψ , where Z  
. J ψ (u, v) = d x, u(x), 2 arctan ux (x) , ψ(x), v(ψ(x)), 2 arctan vx (ψ(x))
· ϕ(x) 1 + u2x (x) dx Z

(7) + 1 + u2x (x) − 1 + vx2 (ψ(x)) ψ ′ (x) dx ,

where d is the standard distance in the space R2 × T, T being the unit circle [0, 2π] with 0 and 2π identified. Roughly speaking, J(u, v) is the minimum cost in order to transport the mass distribution with density 1 + u2x located on the graph of u onto the mass distribution with density 1 + vx2 located on the graph of v, and the function ϕ represents the percentage of mass actually transported. With this definition of distance, the main result (Theorem 3.2) shows that d

J u(t), v(t) ≤ C J u(t), v(t) dt

for some constant C and any couple of multi-peakon solutions u, v. The distance functional J thus provides the ideal tool to measure continuous dependence on the initial data for solutions to the Camassa-Holm equation. For a multi-peakon solution, as long as all coefficients pi remain bounded, the solution to the system of ODE’s (4) is clearly unique. For each time t, call µt the measure having density u2 (t) + u2x (t) w.r.t. Lebesgue measure. Consider a time τ where a positive and a negative peakon collide, according to (5)-(6). As t → τ −, we have the weak convergence µt ⇀ µτ for some positive measure µτ which typically contains a Dirac mass at the point q¯. By energy conservation, we thus have Z Z   2
  2 2 u (τ, x)+u2x (τ, x) dx = E0 . q } = lim u (τ, x)+ux (τ, x) dx+µτ {¯ t→τ −

There are now two natural ways to prolong the multi-peakon solution beyond time τ : either a conservative solution, such that E(t) = E0 for t > τ ,

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or a dissipative solution, where all the energy concentrated at the point q¯ is lost. In this case, the dissipative solution is obtained by simply replacing the two peakons pi , pj with one single peakon of strength p¯, located at x = q¯. On the other hand, the conservative solution contains two peakons emerging from the point q¯. As t → τ +, their strengths and positions satisfy again (5), while (6) is replaced by qi (t) → q¯ ,

qj (t) → q¯ ,

qi (t) > qj (t) for t > τ .

(8)

The vanishing viscosity approach in [11,12] singles out the dissipative solutions. On the other hand, the coordinate transformation approach, based on optimal transport metrics, appear to be well suited for the study of both conservative and dissipative solutions. The following results are indeed proved (see [1,2]). Theorem 3.1. For each initial data u ¯ ∈ H 1 , there exists a solution u(·) of the Cauchy problem (1), (2). Namely, the map t 7→ u(t) is Lipschitz continuous from R into L2 , satisfies (2) at time t = 0, and the identity d u = −uux − Px dt

(9)

is satisfied as an equality between elements in L2 at a.e. time t ∈ R. This same map t 7→ u(t) is continuously differentiable from R into Lp and satisfies (9) at a.e. time t ∈ R, for all p ∈ [1, 2[ . The above solution is conservative in the sense that, for a.e. t ∈ R, Z  2  . u¯ (x) + u ¯2x (x) dx . (10) E(t) = E0 =

Theorem 3.2. Conservative solutions to (1) can be constructed so that they constitute a continuous flow Φ. Namely, there exists a distance functional J on H 1 such that 1 ku − vkL1 ≤ J(u, v) C ku − vkH 1 (11) C

for all u, v ∈ H 1 and some constant C uniformly valid on bounded sets of H 1 . Moreover, for any ¯, v(t) = Φt v¯ of (1), the
two solutions u(t) = Φt u map t 7→ J u(t), v(t) satisfies
J u(t), u ¯ ≤ C1 |t| , (12)
J u(t), v(t) ≤ J(¯ u, v¯) eC2 |t| (13) for a.e. t ∈ R and constants C1 , C2 , uniformly valid as u, v range on bounded sets of H 1 .

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Somewhat surprisingly, all the properties stated in Theorem 3.1 are still not strong enough to single out a unique solution. To achieve uniqueness, an additional condition is needed. Theorem 3.3. Conservative solutions t 7→ u(t) of (1) can be constructed with the following additional property: For each t ∈ R, call µ t the absolutely continuous measure having density u2 + u2x w.r.t. Lebesgue measure. Then, by possibly redefining µt on a set of times of measure zero, the map t 7→ µt is continuous w.r.t. the topology of weak convergence of measures. It provides a measure-valued solution to the conservation law wt + (uw)x = (u3 − 2uP u )x . The solution of the Cauchy problem (1), (2) satisfying the properties stated in Theorem 3.1 and this additional condition is unique. In the next section, we shall see how the optimal transportation technique can be useful for nonlinear equations like the Camassa-Holm equation. The key point is to consider transportation maps which yield Lipschitz continuity for the metric in a space of Radon measures. Since now, we consider the Hunter-Saxton [15] equation, and we give a brief heuristic idea for how this technique is involved in. 4. The Hunter-Saxton equation The Hunter-Saxton equation describes the propagation of waves in a massive vector field of a nematic liquid crystal. Since the physical interpretation and its derivation are beyond to the description of this work, we refer to [7,15]. It can be written in a non-local formulation as a conservation law with a source term: Z x  2 Z +∞  1 u . u2x (t, y) dy = Qu (t, x), (14) = − ut + 2 x 4 x −∞

where t ≥ 0 is the time variable, x ∈ R is the space variable in a reference frame, and u(t, x) ∈ R is related to the orientation of the liquid crystal molecules in the position x at time t. Results of existence of global dissipative solutions have been proved in [7] by using a method of characteristics, whereas in [16] conservative solutions are estabilished by using a geometric approach, obtaining then periodic solution in a manifold isometric to an open set of an infinite-dimensional L2 sphere. Here we want to discuss on the stability of these solutions w.r.t. the initial conditions. To our purposes, Hunter-Saxton equation has nothing but the non-local high order

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part of the Camassa-Holm’s source term. Adaptation is thus a matter of calculation. Suppose that there exists a smooth solution to (14). To the HunterSaxton equation we can associate the following two conservation laws u2x , 2 (u2x )t + (uu2x )x = 0,

(ux )t + (uux )x =

(15) (16)

which are obtained by computing the derivative of the equation (14) w.r.t. the x variable and then multiplying it by ux to achieve the second one. A further conservation law is satisfied by the source term Qu (t, x). Since for smooth solutions (16) yields the conservation of the energy Z . u2x (t, x) dx E0 ≡ E(t) = R

the function Qu can be expressed in the following way Z 1 x 2 E0 + u (t, y) dy. Q(t, x) = − 4 2 −∞ x

By deriving w.r.t. t we obtain

Qt + uQx = 0.

(17)

The function Qu is thus constant along the characteristic curves ξu (t, y) defined by ∂ ξu (t, y) = u(t, ξu (t, y)), ξu (0, y) = y. (18) ∂t Let us remark that the previous equation holds for all the time t in which u is a classical solution. It can be seen by the method of characteristics that if u0 6≡ 0 is a smooth initial data and for some x0 we have u0x (x0 ) < 0, along its outgoing characteristic the gradient blows up in finite time. Since the quantity E(t) remains bounded also at the time of blow up, we can think that a finite amount of energy will be concentrated at the point of blowup. In [7] the authors focus their attention on solutions that dissipate this quantity of energy. As far as the conservative solution is concerned, equation (16) will be satisfied in sense of measures, i.e. it means that thinking at a measure µt with absolutely continuous part which satisfies dµt = u2x (t, ·)dL (here with L we indicate the Lebesgue measure), it satisfies ∂t µt + ∂x (uµt ) = 0

in D′ .

To find a conservative solution to the Cauchy problem (14) with smooth initial condition and finite energy u0 means then to find a couple (u(t), µt )

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which satisfies the following system of conservation laws: let µ0 be the absolutely continuous measure w.r.t. Lebesgue measure defined by dµ0 = u0 2x dL, then   2 1 1 u  = − µt (R) + µt (] − ∞, x]) u(0, x) = u ¯(x), ∂t u + ∂x (19) 2 4 2  ∂t µt + ∂x (uµt ) = 0 µt |t=0 = µ0 .

Due to the nonlinearity of the problem, as shown in [7, Example 2] for the dissipative solution of the Hunter-Saxton equation, we can aspect that the usual “strong” distance stemming from convex norm is not useful in order to construct a continuous semigroup of solutions. Here we give a sketch of the construction of a metric which yields continuity of solution with respect to the initial data. 4.1. A transportation map

Let u0 and v0 be two initial data whose associated measures µ10 and µ20 have the same total mass µ10 (R) = µ20 (R). Suppose that such initial data are not constant in any interval of R, so that the functions Qu0 and Qv0 are absolutely continuous and increasing. We can thus define a continuous map Ψ0 for which at every x ∈ R it associates the unique point Ψ0 (x) such that Qu0 (x) = Qv0 (Ψ0 (x)).

(20)

Let us remark that the Ψ0 is an increasing function, in fact since Qu0 and Qv0 are increasing, then x < y implies Qv0 (Ψ0 (x)) = Qu0 (x) < Qu0 (y) = Qv0 (Ψ0 (y)), so Ψ0 (x) < Ψ0 (y) holds. Now we want to see how the map Ψ0 evolves in time. Since by (17) Qu , Qv are conserved along the characteristic curves (18), the equalities Qu (t, ξu (t, x)) = Qu0 (x),

Qv (t, ξv (t, x)) = Qv0 (x)

yield the definition of the transportation map . Ψt (ξu (t, x)) = ξv (t, Ψ0 (x)).

for all x ∈ R (21)

4.2. The stability of the nonlinear system of ODE As in [13, Section 3], we compute a change of variables in order to obtain a system of ODE with Lipschitz vector field. Let us suppose that the initial

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data u0 is in H 1 (R). Let us set . ω = 2 arctan(ux ), then ω belongs to the unit circle T. Computing the derivative of ω along the characteristics, having in mind the equation (15) for ux we have −u2x − tan2 (ω/2) (ux )t +u(ux)x d = = ω(t, ξ(t, y)) = 2 = − sin2 (ω/2). dt 1 + u2x 1+u2x 1+tan2 (ω/2) It is thus natural to consider the new unknowns   ξu (t, y) . Xu = Xu (t, y) =  u(t, ξu (t, y))  ωu (t, ξu (t, y))

which take values in the space R2 × T, and write the corresponding Cauchy problem     u(t, ξu (t, y))    d Xu (t, y) =   = f (Xu (t, y)),  Qu (t, ξu (t, y))    dt 2 − sin (ωu (t, ξu (t, y))/2)   (22)  y   u  .  X (0, y) =  u0 (y)    2 arctan(u0x (y))

We remark that by definition, Qu is far from to be Lipschitz continuous, then we cannot suppose in the previous system of ODE that the function f is a Lipschitz vector field. Hence, to overcome this lack of Lipschitz continuity, we shall make use of the function Ψt introduced in 4.1. Let Xu and Xv be two solution of the Cauchy problem (22), corresponding to the initial data u0 , v0 respectively. We allow ourselves to make an abuse of notation by defining the map Ψt in the following way: . Ψt (Xu (t, y)) = Xv (t, Ψ0 (y)). We gain a sort of Lipschitz continuity for the function f if we restrict it on the manifold located by Ψt . Let us compute the difference of the vector field f evaluated in the points Xu (t, y) and Ψt (Xu (t, y)). It holds   u(t, ξu (t, y)) − v(t, ξv (t, Ψ0 (y)))   |f (Xu (t, y))−f (Ψt (Xu (t, y)))| =  Qu (t, ξu (t, y))−Qv (t, ξv (t, Ψ0 (y)))  . u) sin2 ( ωv (t,ξ2v (Ψ0 )) ) − sin2 ( ωu (t,ξ ) 2 Since by definition of the map Ψ0 and by the equation (17) we have the identity Qu (t, ξu (t, y)) − Qv (t, ξv (t, Ψ0 (y))) ≡ 0,

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we deduce the estimate |f (Xu (t, y)) − f (Ψt (Xu (t, y)))| ≤ |Xu (t, y) − Ψt (Xu (t, y))| for the vector field f . From the previous inequality we can prove that the difference of the two solutions X u (t, y) and X v (t, Ψ0 (y)) can be estimated by the initial data. In fact, Gronwall Lemma applied to the inequality |Xu (t, y) − Xv (t, Ψ0 (y))| ≤ |Xu (0, y) − Xv (0, Ψ0 (y))| Z t |f (Xu (s, y)) − f (Xv (s, Ψ0 (y)))| ds + 0

u

v

yields the estimate |X (t, y) − X (t, Ψ0 (y))| ≤ et |Xu0 (y) − Xu0 (Ψ0 (y))|. The previous inequality suggests how to introduce the new distance J in order to obtain a stability result for solutions of the Hunter-Saxton equation. The function Ψ0 can be regarded as a transportation map which transports the measure µu into the measure µv . The distance will be thus a sort of Wasserstein [17] distance between measures. References 1. A. Bressan and M. Fonte, Methods Appl. Anal. 12, 191 (2005). 2. M. Fonte, Analysis of singular solutions for two nonlinear wave equations, Ph.D. thesis, International School of Advanced Studies, 2005. 3. R. Camassa and D. Holm, Phys. Rev. Lett. 71, 1661 (1993). 4. A. Constantin and H. McKean, Comm. Pure Appl. Math. 52, 949 (1999). 5. A. Constantin and L. Molinet, Comm. Math. Phys. 211, 45 (2000). 6. R. Johnson, J. Fluid Mech. 455, 63 (2002). 7. A. Bressan and A. Constantin, SIAM J. Math. Anal. 37, 996 (2005). 8. Z. Xin and P. Zhang, Comm. Pure Appl. Math. 53, 1411 (2000). 9. Z. Xin and P. Zhang, Comm. Partial Differential Equations 27, 1815 (2001). 10. A. Constantin and J. Escher, Indiana Univ. Math. J. 47, 1527 (1998). 11. G. Coclite, H. Holden and K. Karlsen, Discrete Contin. Dyn. Syst. 13, 659 (2005). 12. G. Coclite, H. Holden and K. Karlsen, SIAM J. Math. Anal. 37, 1044 (2005). 13. A. Bressan and A. Constantin, Arch. Rat. Mech. Anal. 183, 215 (2007). 14. R. Camassa, D. Holm and J. Hyman, Adv. Appl. Mech. 31, 1 (1994). 15. J. Hunter and R. Saxton, SIAM J. Appl. Math. 51, 1498 (1991). 16. J. Lenells, Discrete Contin. Dyn. Syst. 18, 643 (2007). 17. C. Villani, Topics in Optimal Transportation, American Mathematical Society, Providence, RI, 2003.

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GENERALIZED ENERGY CONSERVATION FOR WAVE EQUATIONS WITH TIME-DEPENDING COEFFICIENTS UNDER STABILIZATION PROPERTIES F. HIROSAWA Department of Mathematical Sciences, Yamaguchi University Yamaguchi 753-8512, Japan E-mail: [email protected] We present an overview of recent research about energy estimates for wave equations with time-depending coefficients taking account of the C m properties of the coefficients with stabilization properties, which are described by some integrals of the coefficients. By introducing these properties simultaneously, more precise analysis is possible. Keywords: Wave equation, generalized energy conservation, stabilization.

1. Introduction Let’s consider the asymptotic behavior of the energy in the Cauchy problem
∂t2 − a(t)2 ∆ u = 0, (t, x) ∈ (0, ∞) × Rn , (1) u(0, x) = u0 (x), ∂t u(0, x) = u1 (x), x ∈ Rn ,

where a(t) > 0 and a ∈ C 1 ([0, ∞)). Here the energy functional E(t) of (1) is given by  1 E(t) = (2) a(t)2 k∇u(t, ·)k2 + k∂t u(t, ·)k2 , 2 where k · k denotes the usual L2 norm in Rn . Evidently, the constant propagation speed a(t) ≡ const. concludes the energy conservation E(t) = E(0) for energy solutions for any t > 0, but it does not hold in general for variable coefficients. Here we introduce the following property as a generalization of the energy conservation law: E(t) ≃ E(0),

(3)

which is called GEC, generalized energy conservation, where we denote f (t) ≃ g(t) with two positive functions f and g if there exists a positive

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445

constant C such that C −1 g(t) ≤ f (t) ≤ Cg(t). Moreover we denote f (t) . g(t) if the estimate f (t) ≤ Cg(t) holds. We also consider the L2 well-posedness of the backward Cauchy problem
∂t2 − a(t)2 ∆ u = 0, (t, x) ∈ [0, T ) × Rn , (4) u(T, x) = u0 (x), ∂t u(T, x) = u1 (x), x ∈ Rn with a non-Lipschitz continuous coefficient a(t) at t = 0 satisfying a(t) > 0 and a ∈ C 1 ((0, T ]) ∩ L∞ ((0, T )), where T is a positive small constant. Here we identify the L2 well-posedness for (4) with the generalized energy conservation on [0, T ]: E(t) ≃ E(T ).

(5)

Evidently we have the following theorem for (1) and (4): Theorem 1.1. Let a ∈ C 1 ((0, ∞)) satisfy a(t) ≃ 1. Then GEC is valid if the L1 properties a′ ∈ L1 ((0, ∞)) for (1) and a′ ∈ L1 ((0, T )) for (4) hold. Noting E ′ (t) ≥ 0 for a′ (t) > 0 and E ′ (t) ≤ 0 for a′ (t) < 0, the energy can oscillate by effects of oscillating coefficients. However, such effects cannot be taken account by the L1 properties of a′ (t) in Theorem 1.1, thus the assumptions of a′ (t) should be relaxed. Indeed, GEC is valid for the following coefficients, though a′ (t) does not satisfy the L1 properties: ( γ 2 + cos ((log(1 + t)) ) for (1) a(t) = (6)

γ 2 + cos log t−1 for (4)

with γ ≤ 1; on the other hand, GEC does not hold in general for γ > 1 (see [4,6,12]). Actually, GEC is valid, though we generalize 2 + cos(·) in (6) to any positive and uniformly C 2 function p(·). The main subject in the hereafter is to set appropriate conditions to the coefficients for GEC. As might be expected, they should include a C 2 regularity condition of the coefficients, moreover, the critical number γ = 1 in the examples (6) must be derived from them. 2. C 2 property of the coefficients 2.1. Wave equations with variable propagation speeds

Let us introduce the following theorem, which is an improvement of Theorem 1.1 taking account of C 2 regularity of the coefficients; we shall call C m regularity of the coefficients and corresponding order of the derivatives by the C m property of the coefficients.

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Theorem 2.1 ([5] (cf. [12,16])). Let a ∈ C 2 ((0, ∞)) satisfy a(t) ≃ 1. If |a′ (t)| . t−1 and |a′′ (t)| . t−2 as t → ∞ for (1) and t → +0 for (4), then the same conclusions as in Theorem 1.1 are valid. Sketch of the proof for (1). We suppose that t is large without loss of generality. By partial Fourier transformation with respect to x, the equation of (1) is reduced to v ′′ + a(t)2 |ξ|2 v = 0, ′

′′

(7)

(k)

where v , v , · · · , v denote the derivatives of v with respect to t. Then the proof is concluded by a uniform estimate of the microlocal energy
1 E(t, ξ) = a(t)2 |ξ|2 |v(t, ξ)|2 + |v ′ (t, ξ)|2 . 2 The key of the proof is to estimate E(t, ξ) by different ways in the two zones ZΨ = {(t, ξ) ; t ≤ tξ } and ZH = {(t, ξ) ; t > tξ }, which are called the pseudo-differential zone and the hyperbolic zone, respectively, where tξ is determined by tξ |ξ| = N

(8)

with a large constant N . In ZΨ , the oscillation of a(t) has only a small effect, thus E(t, ξ) is estimated as a small perturbation of the constant coefficient case. On the other hand, the oscillations of a(t) are crucial in ZH , hence more precise analysis taking into account the C 2 property of a(t) may lead to a contribution for a better estimate. For this reason, we set a ˜(t, ξ) = χ (t/tξ ) a0 + (1 − χ (t/tξ )) a(t) as an approximation of a(t) in the respective zones, where a0 is a constant, χ ∈ C ∞ (R) is a monotone decreasing function satisfying χ(τ ) ≡ 1 for τ ≤ 1 and χ(τ ) ≡ 0 for τ ≥ 2. Here we introduce the symbol class in ZH by n S{m, p; k} = f (t, ξ) ∈ C k ([tξ , ∞); H −∞ (Rn )) ; o (9) |f (j) (t, ξ)| . t−(p+j) |ξ|m , 0 ≤ j ≤ k . ˜ it Then we verify that for any l > 0, f ∈ S{m, p; k} and g ∈ S{m, ˜ p˜; k} (j) ˜ follows that f ∈ S{m, p − j; k − j}, f g ∈ S{m + m, ˜ p + p˜; min{k, k}} and f ∈ S{m + l, p − l; k}. Moreover, we have a ˜, 1/˜ a ∈ S{0, 0; 2}. Let V = t (˜ a|ξ|v, −iv ′ ) be a solution of V ′ = AV . By the constant matrix M0 determined by M0−1 V = V1 := t (v ′ + i˜ a|ξ|v, v ′ − i˜ a|ξ|v), we have V1′ = (Φ1 + B1 ) V1 ,

(10)

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where Φ1 =



ϕ1+ 0 0 ϕ1−



,

B1 =



0 b1− b1+ 0



,

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ϕ1± = (˜ a′ ± i|ξ|(a2 + a ˜2 ))/(2˜ a) ∈ S{1, 0; 1} and b1± = (−˜ a′ ± i(a2 − 2 a ˜ )|ξ|)/(2˜ a) ∈ S{0, 1; 1}; thus the matrix M0 performs as a diagonalizer of A modulo S{0, 1; 1} by grace the C 1 property of a(t). Actually, Theorem 1.1 is trivially concluded from (10) since Re ϕ1± , b1± ∈ L1 ((0, ∞)), but Re ϕ1± , b1± ∈ S{0, 1; 1} are insufficient to conclude Theorem 2.1. Thus the second step of diagonalization procedure by the matrix   (˜ a′ )2 0 −1 M1 = I − 4 2 1 0 4˜ a |ξ| is required. Indeed, due to the C 2 property of a(t) (10) is reduced to V2′ = (Φ1 + R2 ) V2

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M1−1 V1 ,

for large N by the transformation V2 = where R2 satisfies R2 ∈ S{−1, 2; 0} and |R2 | . |a − a0 ||ξ| . |ξ| in ZΨ . Here we denote |R2 | = max1≤j,k≤2 {|(R2 )jk |}. By W2 = R t carrying out the transformation Rt Ψ−1 V with Ψ = diag(exp(− ϕ (s, ξ)ds), exp(− ϕ (s, ξ)ds)), we 2 1 1 0 1+ 0 1− 2 have W2′ = Ψ−1 R Ψ W . Consequently, noting |W (t, ξ)| ≃ E(t, ξ), 2 1 2 2 1 |Ψ−1 1 R2 Ψ1 | ≃ |R2 | and Z ∞ Z ∞ |R2 (s, ξ)|ds . tξ |ξ| + |ξ|−1 s−2 ds . 1, 0

tξ

we obtain E(t, ξ) ≃ E(0, ξ) uniformly with respect to (t, ξ). 2.2. Increasing or degenerating coefficients Let’s generalize Theorem 2.1 to equations with increasing or degenerating coefficients. We assume that a(t) is represented by the following product of increasing or degenerating function λ(t) and oscillating function ω(t): a(t) = λ(t)ω(t),

(13)

where ω(t) ≃ 1, λ′ (t) ≥ 0, limt→∞ λ(t) = ∞ for (1) and limt→+0 λ(t) = 0 for (4), respectively. Let us introduce the following conditions:  k λ(t) (k) (t) . λ(t) (14) λ Λ(t)

and

(k) ω (t) . Ξ(t)k

(15)

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Rt for k = 1, 2 as t → ∞ for (1) and t → +0 for (4), where Λ(t) = 0 λ(s)ds and Ξ(t) is a strictly decreasing function satisfying limt→∞ Ξ(t) = 0 and limt→+0 Ξ(t) = ∞. Then we have the following theorem: Theorem 2.2 ([6] (cf. [13,15,16])). Let λ, ω ∈ C 2 ((0, ∞)). Assume that λ′ (t) ≤ λ(t)2 /Λ(t), (14) for k = 2 and (15) for k = 1, 2 with Ξ(t) =

λ(t) . Λ(t)

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Then the following weighted generalized energy conservation is valid for (1): λ(t)E(0) ≃ E(t).

(17)

If tλ(t) ≤ Λ(t), then the following estimate is valid for (4): E(t) . E(T ).

(18)

The estimate of E(t, ξ) in ZH is an analogy of the proof of Theorem 2.1 after defining tξ by Λ(tξ )|ξ| = N . The estimate from above in ZΨ is derived by tλ(t) ≤ Λ(t); on the other hand, the degeneration of a(t) brings a crucial problem for the estimate from below. Remark 2.1. The estimate (18) does not ensure the usual meaning of the L2 well-posedness for (4); indeed, the boundedness of u(0, x) in H˙ 1 is not concluded by (18). Though the C ∞ well-posedness is valid without the assumption tλ(t) . Λ(t) (cf. [15,16]). 2.3. Dissipative wave equations By Liouville transformation, (1) can be reduced to the following Cauchy problem for dissipative wave equation:
∂t2 − ∆ + b(t)∂t u = 0, (t, x) ∈ (0, ∞) × Rn , (19) u(0, x) = u0 (x), ∂t u(0, x) = u1 (x), x ∈ Rn . In particular, if a(t) in the original equation is tending to infinity as t → ∞, ′ then the reduced coefficient b(t), R ∞which is represented by a(·) and a (·), has a positive sign. Moreover, if 0 b(s)ds = ∞, then the estimate (17) is reduced to a decay estimate of the energy for (19). Indeed, we have the following theorem:

Theorem 2.3 ([14]). Let b ∈ C 1 ([0, ∞)) satisfy b(t) ≥ 0, tb(t) < 1 and |b′ (t)| . t−2 as t → ∞. Then the following decay estimate holds for (19):  Z t 
E(t) . exp − b(s)ds E(0) + ku0 (·)k2 . (20) 0

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Remark 2.2. The C 1 property of b(t) corresponds to the C 2 property of a(t) for (1), hence this property is crucial for the estimate of the microenergy in the hyperbolic zone. 3. C m and stabilization properties of the coefficients It will be a natural expectation that some properties of C m regularity of the coefficients with m ≥ 3 contributes to some improvements of the previous results due to the C 2 property. Actually, such an expectation is not realized if we restrict ourselves to the order of γ in the example (6); indeed, Theorem 2.1 has already given the optimal order γ = 1 due to the C 2 property of the coefficients. However, if we do not persist in the examples (6), then we can derive some improvements from the C m properties of the coefficients by taking into account a new property of the coefficients, which will be introduced as the stabilization property. 3.1. Stabilization property for strictly hyperbolic equations Rt Let a(t) satisfy a(t) ≃ 1. We define Θ(t) = 0 |a(s) − a0 |ds with a positive constant a0 . Here we introduce the stabilization property as a non-trivial condition for Θ(t) by t−1 Θ(t) → 0

(21)

as t → ∞ for (1) and t → 0 for (4). Here we note that the constant a0 is uniquely determined since (21) is valid. The stabilization property (21) is introduced in [2,8] as an essential condition to the coefficients for the estimate in lower frequency part of the phase space to derive a benefit of the C m property of the coefficients. Indeed, Theorem 2.1 is improved by taking account of the C m property as follows: Theorem 3.1 ([2,8]). Let a ∈ C m ((0, ∞)) for m ≥ 2 satisfy a(t) ≃ 1. If Θ(t) ≃ tp with 0 ≤ p < 1 for (1) and p > 1 for (4), and (15) holds for k = 1, · · · , m with a(t) = ω(t) and Ξ(t) = t−p+(1−p)/m , then the same conclusions as in Theorem 2.1 are valid. The restriction to a′ (t) is weaker as m larger, and optimality of the order t−p for a′ (t) as m → ∞ is proved in [2,8]. The restriction to the order of a′ (t) in Theorem 3.1 is weaker than as in Theorem 2.1 since m 6= 1 and p 6= 1; thus we have an improvement of Theorem 2.1 under assuming both properties C m with m ≥ 2 and stabilization property simultaneously.

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Consequently, we have nothing new from this theorem for the examples (6), because the stabilization property does not hold for any γ > 0. Let us briefly introduce the crucial points in the proof of Theorem 3.1 in contrast to the proof of Theorem 2.1. We define tξ by Θ(tξ )|ξ| = N . Then the renewal hyperbolic zone is included in the previous one since the stabilization property (21) holds. Noting ϕ1+ = ϕ1− =: ϕ1 and b1+ = b1− := b1 , the diagonalizer M1 is represented by   1 0 −b1 . M1 = I + ϕ1 − ϕ1 b1 0 Let us define Φj and Bj inductively by Φj = diag((Aj )11 , (Aj )22 ) and Bj = Aj − Φj , where −1 −1 (Mj−1 )′ Aj = Mj−1 (Φj−1 + Bj−1 ) Mj−1 − Mj−1

for j = 2, · · · , m. Then we have the following lemma: Lemma 3.1 ([7,8]). For any j = 1, · · · , m we have the following: (i) Mj−1 are uniformly invertible and |Mj−1 | . 1. (ii) The entries of Φj and Bj are represented by (Φj )11 = (Φj )22 =: ϕj and (Bj )21 = (Bj )12 ∈ S{−j + 1, j, m − j}, thus Mj−1 is a diagonalizer modulo S{−j + 1, j, m − j}, where the symbol class is defined after replacing t −(p+j) by (t1/m Θ(t)1−1/m )R−(p+j) of (9). ∞ (iii) The estimate | 0 Re ϕj (s, ξ)ds| . 1 holds uniformly with respect to ξ.

Rt Rt Let us define Ψm = diag(exp(− 0 ϕm (s, ξ)ds), exp(− 0 ϕm (s, ξ)ds)). −1 −1 ′ Then Wm = Ψ−1 m Mm · · · M1 V1 is a solution of Wm = Qm Wm , where Qm ∈ S{−m + 1, m, 0} by Lemma 3.1. Therefore, we have Z ∞ Z ∞ m 1 1 ds . 1, |Qm (s, ξ)|ds . Θ(tξ )|ξ| + |ξ|−m+1 s− m Θ(s)−1+ m 0

tξ

it follows that E(t, ξ) ≃ |Wm (t, ξ)|2 ≃ |Wm (0, ξ)|2 ≃ E(0, ξ) uniformly with respect to (t, ξ). Thus Theorem 3.1 is proved. 3.2. Stabilization property for increasing or degenerating coefficients Let a(t) be represented by (13). Then some conclusions of Theorem 2.2 m can be improved by the C R t property and the stabilization property Θ(t)/Λ(t) → 0 with Θ(t) = 0 λ(s)|ω(s) − ω0 |ds.

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Theorem 3.2 ([11]). Let λ, ω ∈ C m ((0, ∞)) and λ(0) > 0. Assume that λ′ (t) ≤ λ(t)2 /Λ(t), (14) for k = 2, · · · , m and  1 λ(t) Θ(t) m (15) with Ξ(t) = for k = 1, · · · , m. (22) Θ(t) Λ(t) Then the estimate E(t) . λ(t)E(0) is valid for (1). The key of the proof is to separate the phase space by three zones, which are determined by Θ(tξ,1 )|ξ| = N and Λ(t2,ξ )|ξ| = N . Then the C m and the stabilization property perform in the zones of high and middle frequency parts, respectively (see [9,11]). Remark 3.1. C ∞ and the Gevrey well-posedness of (4) with λ(0) = 0 are considered by the C 2 property in [15,16] and by the L1 property of a′ (t) in [3], respectively. Both results are improved by taking account of the C m and the stabilization property in [9]. 3.3. Stabilization property for dissipative wave equations Let us suppose that the coefficient b(t) of (19) is represented by b(t) = b0 (1 + t)−1 + σ(t) with σ(t) ∈ C m ([0, ∞)) and a constant R t Rbτ0 ∈ (0, 1). Then the stabilization property is given by (21) with Θ(t) = 0 | 0 σ(s)ds−ω0 |dτ , where the constant ω0 is uniquely determined. Then we have the following theorem: m Theorem R t 3.3 ([10]). Let σ ∈ C ([0, ∞)) for m ≥ 1. Assume that (21), supt | 0 σ(s)ds| < ∞ and   1 !k−1 Θ(t) m+1 1 (k) for k = 0, · · · , m (23) σ (t) . Θ(t) t

and for any large t. Then (20) holds for (19).

4. Concluding remarks and future tasks The refined diagonalization procedure with the C m property of the coefficients makes it possible to derive more precise estimates of the energy under assuming suitable stabilization properties. Here the stabilization property describes an error made from the oscillating behavior of the coefficient. Precisely, it is represented by an integral of difference between a monotone model function and oscillating coefficient. Here we underline that the integral is not in the sense of L1 but a Riemann integral, therefore, we can

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derive some benefits of oscillation of the coefficient. Moreover, the error is not from a constant but a monotone function. This means that Theorem 3.2 and 3.3 can be understood as perturbations of results for equations with monotone coefficients. Thus, our argument is based on the properties for the wave equations with monotone coefficients, therefore, farther developments of the research to such equations are essential. Indeed, the conclusions of Theorem 3.2 and 3.3 may be not sharp because the estimates are only from the above. Concretely, we propose the following problems after clearing up the problems for non-oscillating cases ω(t) = σ(t) ≡ ω0 : (A) Estimates in both sides of E(t) for Theorem 3.2 (without the assumption λ′ (t) ≤ λ(t)2 /Λ(t) if it is possible). (B) Estimates in both sides of E(t) for Theorem 3.3 (without the assumption b0 ∈ (0, 1) if it is possible). (C) Generalized energy conservation for (4) under the assumptions (14) and (15) with (22). The problems (A) and (B) are very close. All the problems above may require introducing some modified energy functionals, which are defined appropriately to each equation. The equation of (19) can be reduced to the equation of (1) by Liouville transformation; thus Theorem 3.2 and 3.3 are essentially the same. Analogously, the Klein-Gordon equation with variable mass:
(24) ∂t2 − ∆ + m(t)2 u = 0

can be reduced to the equation of (1). Recently some generalized energy conservation for (24) with suitable monotone functions m(t) is considered in [1]. Thus the base to consider the following problem is already prepared:

(D) GEC for the Cauchy problem of the Klein-Gordon equation (24) with oscillating mass m(t) taking account of the C m and stabilization property of m(t). References 1. C. B¨ ohme, Generalized energy conservation for wave models with timedependent coefficients, Diploma Thesis, TU Bergakademie Freiberg, 2008. 2. M. Cicognani and F. Hirosawa, On the Gevrey well-posedness for strictly hyperbolic Cauchy problems under the influence of the regularity of the coefficients, to appear in Math. Scand.

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3. F. Colombini, D. Del Santo and T. Kinoshita, On weakly hyperbolic operators with non-regular coefficients and finite order degeneration, J. Math. Anal. Appl. 282, 410–420 (2003). 4. F. Hirosawa, On the Cauchy problem for second order strictly hyperbolic equations with non-regular coefficients, Math. Nachr. 256, 28–47 (2003). 5. F. Hirosawa, Loss of regularity for the solutions to hyperbolic equations with non-regular coefficients -an application to Kirchhoff equation-, Math. Methods Appl. Sci. 26, 783–799 (2003). 6. F. Hirosawa, Loss of regularity for second order hyperbolic equations with singular coefficients, Osaka J. Math. 42, 767–790 (2005). 7. F. Hirosawa, Global solvability for Kirchhoff equation in special classes of non-analytic functions, J. Differential Equations 230, 49–70 (2006). 8. F. Hirosawa, On the asymptotic behavior of the energy for the wave equations with time depending coefficients, Math. Ann. 339, 819–839 (2007). 9. F. Hirosawa, On second order weakly hyperbolic equations with oscillating coefficients and regularity loss of the solutions, preprint. 10. F. Hirosawa and J. Wirth, C m -theory of damped wave equations with stabilisation, to appear in J. Math. Anal. Appl. 11. F. Hirosawa and J. Wirth, Generalised energy conservation law for the wave equations with variable propagation speed, preprint. 12. M. Reissig and J. Smith, Lp -Lq estimate for wave equation with bounded time dependent coefficient, Hokkaido Math. J. 34, 541–586 (2005). 13. M. Reissig and K. Yagdjian, Lp -Lq decay estimates for the solutions of strictly hyperbolic equations of second order with increasing in time coefficients, Math. Nachr. 214, 71–104 (2000). 14. J. Wirth, Wave equations with time-dependent dissipation. I. Non-effective dissipation , J. Differential Equations 222, 487–514 (2006). 15. K. Yagdjian, The Cauchy problem for hyperbolic operators. Multiple characteristics, micro-local approach, Math. Topics, Akademie-Verlag, Berlin, 1997. 16. T. Yamazaki, Unique existence of evolution equations of hyperbolic type with countably many singular of degenerate points, J. Differential Equations 77, 38–72 (1989).

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A UNIFIED TREATMENT OF MODELS OF THEORMOELASTICITY, DECAY AND DIFFUSION PHENOMENA K. JACHMANN and M. REISSIG Fakult¨ at f¨ ur Mathematik und Informatik TU Bergakademie Freiberg, Germany We present a unified approach for studying qualitative properties of solutions to Cauchy problems for several thermoelasticity models, i.e., classical (or type-1), type-2, type-3 and second sound models, in one space dimension. The approach is based on a diagonalization procedure in phase space for an auxiliary problem. Keywords: Thermoelasticity; decay estimates; diffusion phenomena.

1. Introduction 1.1. Background Systems of thermoelasticity describe the elastic and thermal behavior of elastic and heat-conducting media. The linear classical (or type-1) model of thermoelasticity for homogeneous media is in the absence of external body forces and external heat supply given by  utt − αuxx + γ1 θx = 0, (1) θt − κθxx + γ2 utx = 0,

where u = u(t, x) ∈ R and θ = θ(t, x) ∈ R, (t, x) ∈ R>0 × R, denote the elastic displacement and the temperature difference to some equilibrium, respectively. For the constant coefficients we have the positiveness conditions α, κ, γ1 , γ2 > 0. The derivation of the classical thermoelasticity model is based on Fourier’s law of heat conduction, which implies that the heat equation for the coupled theory is a parabolic one and gives rise to the unphysical property of an infinite propagation speed of thermal disturbances. Replacing Fourier’s law by Cattaneo’s equation or even by a heat-flux equation of Jeffreys type leads to alternative models.

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455

Examples are thermoelasticity systems with second sound (cf. [1])   utt − αuxx + γ1 θx = 0, (2) θ + qx + γ2 utx = 0,  t τ qt + q + κθx = 0,

where τ > 0 denotes the relaxation time, type-2 models (cf. [3])  utt − αuxx + γ1 θx = 0, θtt − κθxx + γ2 uttx = 0

(3)

and the more dissipative type-3 models of thermoelasticity (cf. [2]), δ > 0,  utt − αuxx + γ1 θx = 0, (4) θtt − κθxx − δθtxx + γ2 uttx = 0. We are not only interested in studying the Cauchy problems for (1-4), but also in studying the corresponding ones with additional lower order terms in the first equation. One of these is a dissipation term, which can be motivated physically by taking the external force negatively proportional to the velocity. A lot of work on qualitative properties of solutions to the Cauchy problem of the classical system (1) has been done (cf. [4] and references within), and recently there have been some results on decay estimates for the alternative models (2-4) as well (cf. [5,7]). However, a rather new method of diagonalization in phase space (cf. [6]) can be applied to a more general problem including all above problems at once. Thus we are not only able to quickly reproduce all known results but to study further questions in an efficient way as well. Such questions could be ones related to well-posedness, Lp -Lq decay estimates, diffusion phenomena, propagation of singularities and the like. 1.2. The problem Subjects of our studies are Cauchy problems of the form  Ut + A0 U + A1 Ux − A2 Uxx = 0, U (0, x) = U0 (x)

(5)

for d-dimensional unknowns U = U (t, x) with (t, x) ∈ R≥0 ×R and Ai being complex and constant d-by-d matrices. Example 1.1. We consider the classical thermoelasticity model with an additional dissipation term (m > 0), that is  utt − αuxx + γ1 θx + mut = 0, (6) θt − κ θxx + γ2 utx = 0.

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With the use of U = (u+ , u− , θ)T , u± = ut ± form the above system into

√ α ux , we equivalently trans-

Ut + A0 U + A1 Ux − A2 Uxx = 0 with the matrices  √ m m   − α 0 γ1 2 2 0 √ m  , A1 =  0 A0 =  m α γ1  2 2 0 γ2 γ2 0 0 0 0 2 2

(7)

and A2 = diag(0, 0, κ).

Applying partial Fourier transformation we obtain for V = Fx→ξ (U ) the initial value problem  Vt + (A0 + iξA1 + ξ 2 A2 )V = 0, (8) V (0, ξ) = F (U0 )(ξ).

In Section 2 we will diagonalize the principal part of the system in (8) in different regions of the phase space correspondingly and derive solution representations for proving results on Lp -Lq decay estimates and some concerning diffusion phenomena in Section 3. We will complete our considerations by discussing some applications in Section 4. 2. Diagonalization in phase space 2.1. Diagonalization for small frequencies In the following we will describe a procedure, which allows us, under certain assumptions, to diagonalize the principal part of the system in (8) in Zint (σ) = {|ξ| ≤ σ ≪ 1}. The matrix A0 dominates the coefficient matrix A(ξ) = A0 +iξA1 +ξ 2 A2 , and we thus start the procedure with an Assumption: A0 is diagonalizable. We will gradually diagonalize the system in (8), starting off with Step 0: Diagonalization modulo O(ξ)-terms ˜ (0) V , with L ˜ (0) (R ˜ (0) ) being the matrix of left The vector function V˜ (0) = L (right) eigenvectors to the (in distinct groups of equal numbers ordered) eigenvalues λ0,1 = . . . = λ0,k1 ,

...

, λ0,kb1 −1 +1 = . . . = λ0,kb1 =d ,

(9)

˜ (0) R ˜ (0) = I, satisfies λ0,km 6= λ0,kn for all m 6= n, of A0 with L (0) (0) (0) V˜t + (Λ0 + iξ A˜1 + ξ 2 A˜2 )V˜ (0) = 0,

(10)

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457

(0) ˜ (0) . ˜ (0) Ai R where Λ0 = diag(λ0,1 , . . . , λ0,d ) and A˜i = L

Step 1: Diagonalization modulo O(ξ 2 )-terms The next step of the procedure consists of two substeps. Substep 1: We introduce V (1) = (I + iξK(1) )V˜ (0) , where K(1) = (kij )di,j=1 ∈ Cd×d with kii = 0 and non-diagonal entries to be determined later, which satisfies   (1) (1) (1) (1) (11) Vt + Λ0 + iξA1 + ξ 2 A2 + A3 V (1) = 0 with

(1)

(0)

A1 = [K(1) , Λ0 ] + A˜1 (1)

(12)

(1)

and matrices A2 and A3 = O(ξ 3 ) (i.e. its elements have the behavior O(ξ 3 ) for ξ → 0) for which we can write down explicit formulas (apart from (1) some remainder O(ξ l ) in case of A3 ). Here we have used that (I +iξK(1) ) is invertible for |ξ| ≤ σ ≪ 1 and that we can write V˜ (0) = (I+iξK(1) )−1 V (1) = (I − iξK(1) (I + iξK(1) )−1 )V (1) . For the commutator [K(1) , Λ0 ] = K(1) Λ0 − Λ0 K(1) we have 

k12 (λ0,2 −λ0,1 ) k21 (λ0,1 −λ0,2 ) 0  [K(1) , Λ0 ] =  .. ..  . . 0

 . . . k1d (λ0,d −λ0,1 ) . . . k2d (λ0,d −λ0,2 )   , (13) .. ..  . .

kd1 (λ0,1 −λ0,d ) kd2 (λ0,2 −λ0,d ) . . .

0

and we have ordered the eigenvalues of A0 in (9) in distinct groups of equal (1) numbers. Hence, we can choose K(1) such that A1 is block-diagonal, i.e. (1) (0) (1) (1) (1) A1 = [K(1) , Λ0 ] + A˜1 = diag(B1 , B2 , . . . , Bb1 ).

(14)

(1)

Note that if all eigenvalues of A0 are distinct, then A1 is diagonal if we choose K(1) appropriately. In general this is not the case, and we therefore go on by formulating another (1)

Assumption: A1

is diagonalizable.

Substep 2: ˜ (1) and R ˜ (1) of left and right eigenvectors of A(1) with There are matrices L 1 ˜ (1) R ˜ (1) = I and L ˜ (1) R ˜ (1) = Λ0 ˜ (1) = Λ0 L ˜ (1) Λ0 R L

(15) (16)

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so that the eigenvalues of Bk are arranged in distinct groups of equal numbers for each k. In (16) we have used that the entries in Λ0 in the (1) ˜ (1) V (1) blocks corresponding to Bk are always equal. The vector V˜ (1) = L satisfies   (1) (1) (1) (17) V˜t + Λ0 + iξΛ1 + ξ 2 A˜2 + A˜3 V˜ (1) = 0 (1) ˜ (1) . ˜ (1) A(1) R with Λ1 = diag(λ1,1 , . . . , λ1,d ) and A˜i = L i

For step n of the procedure we can, with an = i for odd and an = 1 for even n, we write down a scheme: Step n: Diagonalization modulo O(ξ n+1 )-terms in n + 1 substeps Substep 1:

1 V ((n−1) n ) = (I + an ξ n K((n−1) 1 ) )V˜ (n−1) generates n a coefficient matrix with block-diagonal part ((n−1) n1 ) Λ0 + iξΛ1 + . . . + an−1 ξ n−1 Λn−1 + an ξ n An .

.. . Substep n:

n−1 V (n) = (I + iξK(n) )V ((n−1) n ) generates a coefficient matrix with block-diagonal part (n) Λ0 + iξΛ1 + . . . + an−1 ξ n−1 Λn−1 + an ξ n An , (n) (n) (n) An = diag(B1 , . . . , Bbn ).

(n)

(0)

Assumption: An is diagonalizable (A0 := A0 ). ˜ (n) V (n) (V (0) := V ) generates a coefficient Substep n + 1: V˜ (n) = L matrix with diagonal part Λ0 + iξΛ1 + . . . + an ξ n Λn , B

(n)

B

(n)

B

(n)

B

(n)

Λn = diag(λ1 1 , . . . , λm11 , . . . , λ1 bn , . . . , λmbbnn ) B

(n)

ordered in distinct = diag(λn,1 , . . . , λn,d ), λj k groups of equal numbers for each k. Lemma 2.1. In substep k of step n, n ≥ 1, 1 ≤ k ≤ n + 1, we do not alter the diagonal part of the coefficient matrix when transforming the system k−1 k for V ((n−1) n ) (V˜ (n−1) for k = 1) into the one for V ((n−1) n ) (V˜ (n) for k = n + 1). Proof. The proof follows from the choice of the matrices involved in the transformations and the proposed ordering of the eigenvalues λi,j , 0 ≤ i ≤ n − 1, 1 ≤ j ≤ d. (0)

Reconsidering the procedure we should with A0 duce the assumptions

:= A0 at first intro-

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(An )

(k)

The matrices Ak

459

are diagonalizable for k = 0, 1, . . . , n.

Lemma 2.2. Suppose that the assumption (An ) holds. Then the characteristic roots µj = µj (ξ) of the coefficient matrix A(ξ) = A0 + iξA1 + ξ 2 A2 of (8) behave for |ξ| ≤ σ ≪ 1 with (in general) complex numbers λk,j as µj (ξ) = λ0,j + iλ1,j ξ + λ2,j ξ 2 + . . . + an λn,j ξ n + O(ξ n+1 ). Assume that (An ) holds. In generalization of the observation that everything works out especially fine, when all eigenvalues of A0 are distinct, we introduce the assumptions (Bn )

∀i, j ∈ {1, . . . , d} with i 6= j ∃k ∈ {0, 1, . . . , n} : λk,i 6= λk,j .

Note that the assumption (Bn ) is not satisfied for any n if and only if there are two identical eigenvalues of A(ξ) for small frequencies. We introduce a number c, assuming that (An ) and (Bn ) hold, by c = max cij i 0 ∀ξ ∈ Zmid (σ, N ) ∀µ(ξ) ∈ spec(A(ξ)) : Re µ(ξ) ≥ C > 0.

With (C) and the compactness of Zmid (σ, N ) we can prove Proposition 2.2. The solution V to (8) satisfies in Zmid (σ, N ) ˆ0 (ξ)|, |V (t, ξ)| . e−ct |U where c is a positive constant, provided (C) holds. Proof. The proof is a simple generalization of the one in [5], Prop. 3.3. Remark 2.1. The assumption (C) is sufficient to work with for the practical applications that we are discussing here. There are of course practical useful situations in which the eigenvalues may become imaginary. Those have to be discussed separately. 3. Results for solutions of (5) 3.1. Lp -Lq decay estimates We will restrict to the region of small frequencies, due to the fact that this is for all practical applications that we are discussing here the decaydetermining one, i.e., we will study decay estimates for solutions to the Cauchy problem (5) subjected to an operator ϕint (D) with a smooth symbol ϕint (ξ) supported in Zint (σ), ϕint (ξ) = 1 for |ξ| ≤ σ/2. We assume U0 ∈ S, that there exists a number n so that (An ) and (Bn ) hold and ∀ξ ∈ Zint (σ) ∀µ(ξ) ∈ spec(A(ξ)) : Re µ(ξ) ≥ 0.

(20)

For each eigenvalue µj = µj (ξ) we fix with the help of its asymptotic expansion from Lemma 2.2 a decay-determining number ns,j by ns,j = min(pj , qj ), where pj and qj shall be defined by the following:

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(i) We set pj = ∞ if µj is purely imaginary in Zint (σ). If the latter is not the case, then we choose pj to be the minimal one of all numbers m ∈ 2N0 with Re λm,j > 0. (ii) We set qj = ∞, unless there exists a natural number m ≥ 2 for which Im(am λm,j ) 6= 0. In that case we choose qj to be the minimal one of all such numbers m. Finally, we set ns := maxj=1,...,d ns,j . Theorem 3.1. Suppose that the above assumptions hold true. Then we have the following Lp -Lq decay estimates for solutions U of (5):  −ct  ns = 0,  e kU0 kLp , 1 1 1 − − ) ( n p q kϕint (D)U kLq . (1 + t) s kU0 kLp , 2 ≤ ns < ∞,   kU k p , ns = ∞. 0 L

Here 1 ≤ p ≤ 2,

1 p

+

1 q

= 1, and c is a positive constant.

Sketch of the proof. We use the representation for the solution of (8) from Proposition 2.1, derive L2 -L2 estimates, L1 -L∞ estimates and interpolate them via the Riesz-Thorin interpolation theorem. For the derivation of L 1 L∞ estimates we have to make different considerations for different cases for ns,j . Main tool for deriving results in the cases of finite ns,j and qj < pj , i.e. when the imaginary part of the eigenvalue µj gives an and determines the decay, is the Lemma of van der Corput. 3.2. Diffusion phenomena Included in our considerations are hyperbolic-parabolic coupled systems, like the classical thermoelasticity model (1). It is certainly of interest whether we can prove an underlying parabolic structure for large times t (from the viewpoint of decay estimates) in such a case. Observations of this type are referred to as diffusion phenomena. It is natural to prove diffusion phenomena for the decay-determining region of the phase space. We will therefore restrict our considerations here again to the region of small frequencies Zint (σ). Suppose that there exists a number n so that (An ) and (Bn ) hold, that all eigenvalues µj = µj (ξ) (we are looking for an underlying parabolic structure) have a positive real part in Zint (σ)\{0} and neglect the cases in which we obtain an exponential decay, i.e. in which the number ns from Section 3.1 vanishes.

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We define with the help of the finite numbers pj from Section 3.1 and ms := maxj=1,...,d pj , ns ≤ ms , a reference system:  ms X  W + ak Mk ∂xk W = 0, t (21) k=0   (c) (0) ˜ ˜ W (0, x) = L · . . . · L U0 (x). (k)

(k)

(k)

Here ak = (−1)⌊k/2⌋ , Mk = diag(m1 , . . . , md ) with mj = 0 if k > pj (k) ˜ (i) are from Section 2.1, and c is and mj = λk,j otherwise, the matrices L the number from (18). With a similar proof to the one for Theorem 3.1 we can show Theorem 3.2. Suppose that the above assumptions hold true. Then we ˜ (i) have for the solution U of (5), dual values q, 1 ≤ p ≤ 2 and the to L (i) ˜ inverse matrices R from Section 2.1 the estimate

  1 1 1 1

˜ (0) · . . . · R ˜ (c) W

ϕint (D) U − R

q . (1 + t)− ns ( p − q )− ms kU0 kLp . L

Taking the estimates from Theorem 3.1 for 2 ≤ ns < ∞ and kϕint (D)W kLq . (1 + t)− ns ( p − q ) kU0 kLp 1

1

1

into consideration, and thus that the difference, as stated in Theorem 3.2, is decaying faster, we can say that the asymptotic profile (at least from the point of view of decay estimates) of (5) is given by solutions to (21). 4. Applications Let us start by considering the Cauchy problem for classical thermoelasticity with an additional dissipation term, i.e. (6) with data u(0, x) = u0 , ut (0, x) = u1 and θ(0, x) = θ0 . The procedure works fine in all regions of the phase space. In particular, for small frequencies we obtain, (A2 ) and (B2 ) are satisfied, for the eigenvalues of A(ξ) = A0 + iξA1 + ξ 2 A2 with positive and distinct numbers a∓ the asymptotic behavior α + γ1 γ2 2 ξ + O(ξ 3 ). µ1,2 (ξ) = a∓ ξ 2 + O(ξ 3 ), µ3 (ξ) = m − m This together with the fact that we obtain exponential decay from the regions of bounded and large frequencies yields: Theorem 4.1. We assume u0 , u1 , θ0 ∈ S. Then we have

k(ut , ux , θ)kLq . (1 + t)− 2 ( p − q ) k(hDi u0 , u1 , θ0 )kLp,rp . 1

1

1

Unified Treatment of Thermoelasticity, Decay and Diffusion Phenomena

Here 1 < p ≤ 2, p1 + 1q = 1, rp = generalized Sobolev space.

1 p

−

1 q

−rp

and Lp,rp = hDi

463

Lp denotes the

When considering the Cauchy problem for classical thermoelasticity, i.e. (1) with data u(0, x) = u0 , ut (0, x) = u1 and θ(0, x) = θ0 , then we obtain the same result as in Theorem 4.1 (which is a known fact, cf. [4]). In particular, for the regions of large and bounded frequencies we obtain exponential decay, and for small frequencies (A1 ) and (B1 ) are satisfied, and we have for the eigenvalues of A(ξ) = iξA1 + ξ 2 A2 (Ai from Example 1.1) the asymptotic behavior √ κγ1 γ2 ξ 2 + O(ξ 3 ), µ1,3 (ξ) = ∓i α + γ1 γ2 ξ + 2(α + γ1 γ2 ) ακ ξ 2 + O(ξ 3 ). µ2 (ξ) = α + γ1 γ2 For a result on diffusion phenomena we define the reference system:  Wt + M1 Wx − M2 Wxx = 0, (22) ˜ (1) U0 (x), W (0, x) = L ˜ (1) ∈ R3×3 coming from the diagonalization procedure, with some matrix L √ κ diag (γ1 γ2 , 2α, γ1 γ2 ). M1 = α + γ1 γ2 diag (−1, 0, 1) and M2 = 2(α+γ 1 γ2 ) Theorem 4.2. Assume u0 , u1 , θ0 ∈ S. Then we obtain for the solution U ˜ (1) being the inverse of L ˜ (1) : to the initial problem of (7) with A0 ≡ 0 and R

−1 1−1 −1 ˜ (1) W

U − R

. (1 + t) 2 ( p q ) 2 k(hDi u0 , u1 , θ0 )kLp,rp Lq

for dual values q, 1 < p ≤ 2 and rp =

1 p

− q1 .

The asymptotic profile to the Cauchy problem for (1) (from the viewpoint of decay estimates) is thus a parabolic one. The procedure can not only be applied to all problems discussed in Section 1.1, but also - with some minor changes - to all Cauchy problems for the systems (1-4) with an additional mass term in the first equation. It can moreover be applied to all above mentioned problems in the 3D case for isotropic media by noting that after applying the Helmholtz decomposition and partial Fourier transformation we arrive at systems of the type Vt + (A0 + i|ξ|A1 + |ξ|2 A2 )V = 0,

(23)

where the two significant changes to (8) are that ξ ∈ R is replaced by |ξ| for ξ ∈ R3 and that the matrices Ai ∈ Cd×d depend smoothly on ξ/|ξ|, i.e. live on the unit sphere S2 .

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References 1. D. S. Chandrasekharaiah, Hyperbolic thermoelasticity: a review of recent literature, Appl. Mech. Rev. 51 (1998), 705–729. 2. A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid, J. Thermal Stresses 15 (1992), 253–264. 3. A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity 31 (1993), 189–208. 4. S. Jiang and R. Racke, Evolution Equations in Thermoelasticity, Chapman & Hall/CRC, Boca Raton, 2000. 5. M. Reissig and Y. G. Wang, Cauchy problems for linear thermoelastic systems of type III in one space variable, Math. Methods Appl. Sci. 11 (2005), 1359– 1381. 6. Y. G. Wang, A new approach to study hyperbolic-parabolic coupled systems, Banach Center Publications 60 (2003), 227–236. 7. L. Yang and Y. G. Wang, Lp -Lq Decay Estimates for The Cauchy Problem of Linear Thermoelastic Systems with Second Sound in One Space Variable, Quart. Appl. Math. 64 (2006), 1–15.

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SOME RESULTS ON SPECTRAL ANALYSIS OF NONSELFADJOINT PERTURBATIONS ¨ FOR SCHRODINGER AND WAVE EQUATIONS MITSUTERU KADOWAKI

∗

Department of Mechanical Engineering, Ehime University, Matsuyama, Ehime, 790-8577, Japan E-mail: [email protected] HIDEO NAKAZAWA

∗

Department of Mathematics, Chiba Institute of Technology, Narashino, Chiba, 275-0023, Japan E-mail: [email protected] KAZUO WATANABE Department of Mathematics, Gakushuin University, Toshima, Tokyo, 171-8588, Japan E-mail: [email protected] Some results presented in [5], [7], and [8] associated with non-selfadjoint perturbation of Schr¨ odinger and wave equations are summarized and a relation between the spectral structure and the asymptotic behavior of solutions are reviewed. Moreover, the issue of the degree of pole of resolvent is discussed. Keywords: Nonselfadjoint; spectral structure; Parseval equality; asymptotic behavior; spectral singularity.

1. Introduction—motivation and aim We shall consider the asymptotic behavior of the solutions of Schr¨ odinger and wave equations with some non-selfadjoint perturbations. Abstract form of our problem is given by
du(t) = (H0 + V )u(t) t ∈ I (⊆ R) , u(0) = u0 ∈ X (⊆ H). (1) i dt

∗ The first and second author are supported by Grant–in–Aid for Scientific Research (No. 19540189 and No. 17740082), Ministry of Education, Culture, Sports, Science and Technology, Japan.

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In the above, H0 denotes a self-adjoint unperturbed operator with domain D(H0 ) in a Hilbert space H. We consider the two cases (S) : H0 = −∆ with D(H0 ) = H 2 (RN ) in H = L2 (RN )   0 1 (W ) : H0 = i with D(H0 ) ⊆ H 2 (RN ) × H 1 (RN ) ∆0 in H = H˙ 1 (RN ) × L2 (RN ) Here H m represents the usual Sobolev space of order m, and H˙ 1 (RN ) denotes the completion of C0∞ (RN ) with respect to the norm ||f || = 1/2 R |∇f (x)|2 dx for f ∈ C0∞ (RN ). (S) and (W) denotes the RN Schr¨ odinger and wave case, respectively. The operator V represents a nonselfadjoint perturbation described below. The set X contains initial data and is given by X = D(H0 ) or D(H0 ) ⊂ X ⊂ H. Example 1.1. [Ikebe (1960): (S) with real potentials] Ikebe treated Schr¨ odinger equation with real short–range potentials [3]. We shall take in (1) as H = L2 (R3 ), H0 = −∆, X = D(H0 ) = H 2 (R3 ), I = R, and V is assumed to be a multiplication operator by a real–valued function V (x) satisfying “very short–range” conditions |V (x)| ≤ C(1 + |x|)−2 for any x ∈ R3 . Then the perturbed operator H = H0 + V becomes self-adjoint and its spectral structure is given by the following: the continuous spectrum σc (H) fills the non-negative real axis σc (H) = [0, ∞), and there exist at most finitely many negative eigenvalues σp (H) = {λj < 0 (j = 1, 2, · · · , m with m < ∞) }. The spectral decomposition for H is given by Z m X λdE(λ) λj Pj + H= j=1

σc (H)

and the solution u(t) = e−itH u(0) ∈ H of (1) is represented by Z m X e−itλ dE(λ)u0 . e−itλj Pj u0 + u(t) = j=1

σc (H)

Moreover the inverse wave(1) and the scattering operator exist in H: W± = s–lim eitH0 e−itH , t→±∞

S = (W+ )−1 W− .

(1) In the context of dissipative scattering which is the scope of our problem, the inverse wave operator is more useful than the usual one. See, for example, [14].

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Thus we find the solution which becomes asymptotically free: ||u(t) − e−itH0 W± u0 ||H → 0

(t → ±∞)(2) .

Indicating the usual spectral representation of H0 by F0 (λ) (e.g., [15]), we can also construct the spectral representation of H by F(λ) = F0 (λ)W+ . So, the generalized Parseval formula (cf., [18]) has the form Z m X (Pj f, g)H + (F(λ)f, F(λ)g)L2 (S 2 ) dλ(3) , (f, g)H = σc (H)

j=1

where Pj denotes the projection operator onto eigenspace associated with the eigenvalue λj . Note that the usual one is given by Z (f, g) = (F0 (λ)f, F0 (λ)g)L2 (S 2 ) dλ. σc (H0 )

Example 1.2 (Mochizuki (1967-68): (S) with complex potentials). Mochizuki [12,13] treated the similar problem under complex–valued function V (x) satisfying Im V (x) ≤ 0 (so H becomes maximal dissipative(4) ) and proved the existence of the local wave and the scattering operator. Moreover, the representation of the scattering operator and the uniqueness of the inverse scattering problem at high energy is obtained. However, as for the decaying mode, i.e., limt→+∞ ||u(t)||H = 0, the existence of such mode was not mentioned. Example 1.3 ((W) with dissipations). Consider the wave equation with dissipations. in (1) as H = H˙ 1 (RN ) × L2 (RN ), I =  We  shall take n 0 1 [0, ∞), H0 = i , D(H0 ) = (f = (f1 , f2 ) ∈ H | ∆f1 ∈ L2 (RN ), f2 ∈ ∆0   o 0 0 H 1 (RN ) , and V = i with b(x) > 0. The energy norm in 0 −b(x) r     w(t) 1 2 + ||w (t)||2 ||∇w(t)|| H is given by ||u(t)||H = = 2 t 2 L L , 2 wt (t) H

and w = w(t) = w(x, t) satisfies the equation wtt − ∆w + b(x)wt = 0 in RN × (0, ∞). Let us consider the following two cases: (2)

|| · ||H denotes the norm in H. (·, ·)X denotes the inner-product in X. (4) A closed linear operator A is maximal dissipative if and only if Im(Af, f ) H ≤ 0 for any f ∈ D(A) and R(H − zE) = H where R(H − zE) denotes the range of operator H −zE with some z belonging to resolvent set of H. Here E denotes the identity operator. (3)

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(a): 0 ≤ b(x) ≤ b0 (1 + |x|)−1−δ , (b): b1 (1 + |x|)−1 ≤ b(x) ≤ b2 , for some constants b0 , b1 , b2 and δ > 0. Then the following results are known: (i) Under assumption (a), there exist scattering states ([14] N 6= 2, [17] N ≥ 2): limt→+∞ ||u(t) − e−itH0 W+ u(0)||H = 0. (ii) Under assumption (b), there exists decaying states ([11], [15]) for suitable initial data: ||u(t)||H = O(t−1 ) (t → +∞). (iii) Under assumption (a) with small b0 the spectrum of H coincides with the one of H0 if N ≥ 3, and limiting absorption principle also holds for N 6= 2. ([16]). These examples suggest the possibility of characterization of asymptotic behavior of solutions by the spectral structure of the generator of H (for other results, see [6] and references therein). So we amass some evidence aiming to develop methodology to treat more general problems. Before closing this section we shall explain the content of the present paper. In section 2, we briefly introduce our results on Schr¨ odinger equations. One result on spectral singularity is outlined in section 3. Results for the wave equation are mentioned in section 4. About these matter so far, we omit detailed proofs. For them, please refer papers quoted in each section. In the final section 5, we shall argue on the existence of the wave operator for wave equations with dissipation in some layered region. A sketch of the proof is also given. Acknowledgments The authors thank to the referee who pointed out our mistakes. They also thank to the editor for his warm and kind response. 2. Schr¨ odinger equations with rank one perturbations In this section, we shall state some results from [5]. Before it, we should mention the book [1] by Albeverio, Hegh-Krohn and Holden. In it, they studied these types of perturbation systematically. Now let us consider equation (1) with H = L2 (R1 ), H0 = −d2 /dx2 , X = D(H0 ) = H 2 (R1 ), (5) I = R and V = Vα = α (·, δ)H δ, α = α1 + iα2 ∈ C, δ ∈ H−1 , where δ (5)

Here, (·, ·)H denotes the dual-coupling between H1 and H−1 . See [1].

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469

denotes the Dirac delta function. Then the operator H = Hα = H0 + Vα with the domain n D(Hα ) = U = u + aH0 (H02 + 1)−1 δ | u ∈ H2 , a ∈ C,

o (α 6≡ 0) (u, δ) = −a α−1 + δ, H0 (H02 + 1)−1 δ

becomes maximal dissipative for α2 < 0 (self-adjoint for α2 ≡ 0). The action of Hα is defined by Hα U = H0 u − a(H02 + 1)−1 ϕ for any U ∈ D(Hα ). Theorem 2.1 (Spectral structure). If α1 ≤ 0 and α2 ≤ 0, then the spectrum of Hα is given by σ(Hα ) = [0, ∞)∪ −α2 /4 if α1 < 0, = [0, ∞) if α1 = 0. More precisely, the spectral structure σ(Hα ) is given by σess (Hα ) = σc (Hα ) = [0, ∞), σr (Hα ) = ∅, σp (Hα ) = σd (Hα ) = −α2 /4 if α1 < 0, = ∅ if α1 = 0. The projection corresponding to the
eigenvalue −α2 /4 (α1 6= 0) is ¯ given by P−α2 /4 f (x) = −α/2 f, e(α|·|)/2 e(α|x|)/2 . Similarly to the selfH adjoint case, we can prove the existence of the wave operator W (α) = s- limt→+∞ eitH0 e−itHα under the assumptions α1 ≤ 0 and α2 < 0. Theorem 2.2 (Generalized Parseval formula, cf. Pavlov (1966)). (1) Assume α1 6= 0. For any f , g ∈ H ∩ L1 (R) and for any α ∈ {α = α1 + iα2 ; α1 < 0}, it holds ¯ (Fα f, Fα¯ g)H = (f, g)H + α/2(f, e(α|·|)/2 )H (e(α|·|)/2 , g)H ,

where Fα = F0 W (α) is the generalized Fourier transform (F0 is the usual one). (2) Assume α1 = 0. Then we have lim (Fiα2 f, χε F−iα2 g)H

ε→0

= (f, g)H + (iα2 )/4

Z

R

e

(iα2 |x|)/2

 Z  (iα2 |y|)/2 f (x)dx e f (y)dy , R

where χε is the characteristic function of {k ∈ R ; a ≤ ||k| + α2 /2|}. Theorem 2.3. The kernel of the wave operator is given by Ker W (α) = R(Pα ) if α1 < 0, α2 < 0, = {0} if α1 = 0, α2 < 0. Corollary 2.1. (1) (Decaying mode) u0 ∈ R(Pα ) if and only if limt→+∞ ||u(t)||H = 0. (2) (Scattering mode) (E − Pα )u0 6= 0 if and only if limt→+∞ ||u(t) − e−itH0 W (α)u0 ||H = 0.

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3. Spectral singularity for Schr¨ odinger equations with general rank one perturbations Why can we argue as above? Because the spectral singularity is the same as in the self-adjoint case. To explain it, we consider (1) with H = L2 (R1 ), H0 = −d2 /dx2 , X = D(H0 ) = H 2 (R1 ), and V = Vα = α (·, ϕ)H ϕ with α = α1 + iα2 ∈ C− where ϕ(x) = χ[0,1] (x) is the characteristic function of [0, 1] and C− denotes R √the complex lower half plain. Since √ (H0 − zE)−1 u(x)
= −(2i z)−1 R ei z|x−y| u(y)dy for Im z 6= 0, we have √ (H0 − zE)−1 ϕ, ϕ L2 = ξ −2 − ξ −3 (eξ − 1), where we put ξ = i z and √ for Im z > 0 we take Im z > 0. Note that the perturbed resolvent is ex
−1 pressed by (H − zE)−1 u = (H0 − zE)
−1 u − αΓ(z) (H0 − zE)−1 u, ϕ , where Γ(z) = 1 + α (H0 − zE)−1 ϕ, ϕ . Then we have the following two lemmas by explicit calculation [7]. √
Lemma 3.1. It holds that (H0 + iεE)−1 ϕ, ϕ = −iε
−1 −cε−3/2 (e−c ε −1) with c = e(iπ)/4 . Moreover, limε↓0 (H0 + iεE)−1 ϕ, ϕ = 0 holds. Lemma 3.2. It holds that √ √
(H0 − (λ − i0)E)−1 ϕ, ϕ = −λ−1 + λ−3/2 sin λ + iλ−3/4 (cos λ − 1).
Furthermore, limλ↓0 (H0 − (λ − i0)E)−1 ϕ, ϕ = 0.

From these lemmas we see that Γ(0) 6= 0. Hence z = 0 is not so singular for the resolvent of H. Next√we consider the degree of the zeros of Γ(λ− i0). For simplicity, we put µ = λ. Theorem 3.1. There is no µ > 0 fulfilling Γ(µ2 − i0) = Γ′ (µ2 − i0) = 0. 4. Wave equation with rank one dissipations In this section, we introduce some results [8] on wave equations with dissipative perturbations of rank one. Consider equation (1)with H = H˙ 1(R) ×   0 1 0 0 L2 (R), H = H0 + V with H0 = i 2 ,V =i and d /dx2 0 0 −(·, ϕ)L2 ϕ  D(H0 ) = D(H) = f = (f1R, f2 ) ∈ H | ∂x2 f1 ∈ L2 (R), f2 ∈ H 1 (R) , where ϕ ∈ L2,s (R) = u(x) | R (1 + |x|2 )s |u(x)|2 dx < ∞ (s > 1/2). In   w(t) this case, u(t) = ∈ H and w = w(t) = w(x, t) satisfies the equawt (t) tion wtt − ∆w + (wt , ϕ)ϕ = 0. Theorem 4.1 (cf. Mochizuki (1976)). (1) σp (H)∩R = ∅. (2) The wave operator W = s- lim eitH0 e−itH exists in H as a non-trivial operator. t→+∞

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This implies that the solution u(t) = e−itH u(0) does not have the bound mode. Here, we call the bound mode is neither the scattering mode nor the decaying mode. Assumption 4.1 (Assumption on ϕ). (A1) ϕ ∈ L2,s+1 (R), s > 1/2. 2 2 (A2) Φ(λ) ≤ Φ(µ) (0 ≤ µ ≤ λ), where Φ(λ) = |ϕ(λ)| ˆ + |ϕ(−λ)| ˆ . (To deal with Φ(−λ) for λ < 0, we extend Φ(λ) as a function on R.) Notation 4.1. Γ(z) = 1 − iz(r0 (z)ϕ, ϕ)0 , Γ(λ ± i0) = 1 − iλ(r0 (λ ± i0)ϕ, ϕ)0 , where r0 (z) = (−∂ 2 /∂x2 − z 2 E)−1 and z ∈ C \ [0, ∞), (·, ·)0 is inner product in L2 or dual coupling between L2,s and L2,−s for some s > 1/2, R0 (z) = (H0 − zE)−1 , R(z) = (H − zE)−1 , vϕ (z) = (ir0 (z)ϕ, zr0 (z)ϕ). We find that for z ∈ C \ R and f = (f1 , f2 ) ∈ H, R0 (z)f = (r0 (z)(zf1 + if2 ), i∂x r0 (z)∂x f1 +zr0 (z)f2 ). Similar to this, it holds that for any z ∈ C\R i(f,vϕ (¯ z )) vϕ (z). satisfying Γ(z) 6= 0, R(z)f = R0 (z)f + Γ(z) 4.2√(Spectral structure). It holds Theorem ∩ R that σ(H) √ C− = ∅ if R ϕ(x)dx ≤ 2, = {iκ0 } for some κ0 < 0 if ϕ(x)dx > 2. Moreover R R iκ0 is an eigenvalue and its multiplicity is one. Put ϕ0 = (0, ϕ). We define three operators F0 , F and G by    2−1/2 λfˆ1 (λ) + ifˆ2 (λ), λfˆ1 (−λ) + ifˆ2 (−λ) , if   (F0 f )(λ) = −1/2 2 −λfˆ1 (−λ) − ifˆ2 (−λ), −λfˆ1 (λ) − ifˆ2 (λ) , if

λ > 0, λ < 0,

(Ff )(λ) = (F0 f )(λ) + i(f, vϕ (λ − i0))(Γ(λ + i0))−1 (F0 ϕ0 )(λ),

(Gf )(λ) = (F0 u)(λ) − i(f, vϕ (λ − i0))(Γ(λ − i0))−1 (F0 ϕ0 )(λ).

Then we see that they become the spectral representation of H0 , H and H ∗ , respectively. Theorem 4.3 (Generalized Parseval formula). Assume (A1) and (A2). (1) If Γ(−i0) 6= 0, then it holds (R ((Ff )(λ), (Gg)(λ))C2 dλ, if Γ(−i0) > 0, (f, g)H = RR ((Ff )(λ), (Gg)(λ))C2 dλ + (P f, g)H , if Γ(−i0) < 0 R

for any f , g ∈ H, where P is the projection onto the eigenspace associated with the eigenvalue iκ0 of H.

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(2) If Γ(−i0) = 0, then it holds that for any f ∈ H, g ∈ E ≡ { g ∈ S(R) × S(R) : (vϕ (−i0), g) = 0 }, that Z ((Ff )(λ), (Gg)(λ))C2 dλ. (f, g)H = R

Remark 4.1. A related result can be seen in Gilliam–Schulenberger [2]. √ R Corollary 4.1. It holds that Ker W = {0} if R ϕ(x)dx ≤ 2, = R(P ) √ R if R ϕ(x)dx > 2. 5. Existence of wave operator for dissipative wave equations in layered region

In the final section, we shall consider the following problems: Fix N ∈ N and set Ω = {(x, y) : x ∈ RN , 0 < y < π}. Consider the following equations for w = w(x, y, t) on Ω(= {(x, y) : x ∈ RN , 0 ≤ y ≤ π}) × [0, ∞):   (x, y, t) ∈ Ω × (0, ∞),  wtt − ∆w + b(x, y)wt = 0, w(x, 0, t) = w(x, π, t) = 0,   w(x, y, 0) = w (x, y), w (x, y, 0) = w (x, y), 0 t 1

(x, t) ∈ RN × (0, ∞), (x, y) ∈ Ω,

PN where ∆ = j=1 ∂ 2 /∂x2j + ∂ 2 /∂y 2 . At this time, we can obtain only the existence of scattering states if the function b(x, y) is measurable and satisfies 0 ≤ b(x, y) ≤ b3 (1 + |x|2 )−θ/2 (b3 > 0, θ > 2 for N = 2, θ ≥ 2 for N ≥ 3). In order to prove this, we prepare with an abstract result from [14] and [4] that originated in [9]: Theorem 5.1. Let H be a Hilbert space, H0 a self-adjoint operator in H, and V a bounded, non-negative self-adjoint operator. Moreover assume the following three conditions: (a1): σ(H0 ) = σac (H0 ) = (−∞, −m] ∪ [m, ∞) or [m, ∞) for some m ≥ 0; (a2): V is compact; (a3): for any α, β satisfying m < α < β, there exist C = Cα,β and η = ηα,β > 0 such that √
√ (6) V ||L(H) ≤ C. (2) sup || V (H0 − z)−1 − (H0 − z¯)−1 α≤| Re z|≤β,0 α > 1, there exist C˜ = C˜α,β and η = ηα,β > 0 such that sup α≤| Re z|≤β,0 1/2. Then we have by well-known resolvent estimates (e.g., Kuroda [10]) for −∆x , ||Xs rn (λ ± i0)Xs ||L(L2 (Ω)) = O(1),

(|λ| → k + 0)

(7)

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for n = 1, 2, 3 · · · , k − 1. Using (4), (7) and ||Xθ/2 rk (λ ± i0)Xθ/2 ||L(L2 (Ω)) = O(1),

(|λ| → k + 0),

(8)

where θ ≥ 2 and N ≥ 3 (e.g., [4]), we have (6) with θ ≥ 2 and N ≥ 3. We shall prove (6) for θ > 2 and N = 2. For g ∈ C0∞ (R2 ), we find  Z   p
 i ′ 2 2 + E0 ( λ −k )(x , x) g(x′ )dx′ , rk (λ+i0) − rk (λ−i0) g (x) = R2 2 √ where E0 satisfies |E0 ( λ2 − k 2 )(x′ , x)| ≤ C(λ2 − k 2 )ε/2 |x′ − x|ε for any ε > 0 (Schlag [19]). Hence, we have for θ > 2, ||Xθ/2 (rk (λ + i0) − rk (λ − i0)) Xθ/2 ||L(L2 (Ω)) = O(1) as |λ| → k + 0. Thus (3), (7) and (8) imply (6) for θ > 2 and N = 2. Proof of Proposition 5.1. Note that for any f = (f1 , f2 ) ∈ H,  √ √  √  √ V (H0 −z)−1 −(H0 −z)−1 V f = 0, b {zρ(z) + Im zR0 (z)} bf2 .

By (3) there exist positive constants C1 and C2 independent of z such that √ √ || bρ(z) b||L(L2 (Ω)) ≤ C||Xθ/2 ρ(z)Xθ/2 ||L(L2 (Ω)) , √ √ || bR0 (z) b||L(L2 (Ω)) ≤ C||R0 (z)Xθ/2 ||L(L2 (Ω)) . Thus, we have the desired result if we note
Xθ/2 ρ(z)Xθ/2 f2 , f2 L2 (Ω) = 2 Re z Im z||R0 (z)Xθ/2 f2 ||L2 (Ω) .

References 1. S. F. G. Albeverio, R. Hegh-Krohn and H. Holden, Solvable models in quantum mechanics, Springer, Berlin, AMS, 1988/2005. 2. D. S. Gilliam and J. R. Schulenberger, Spectral analysis of a dissipative problem in electrodynamics: The Sommerfeld problem, Acta Appl. Math., 6, pp. 63–94 (1986). 3. T. Ikebe, Eigenfunction expansions associated with the Schr¨ odinger operators and their applications to scattering theory, Arch. Rational Mech. Anal., 5, pp. 1–34 (1960). 4. M. Kadowaki, Resolvent estimates and scattering states for dissipative systems, Publ. RIMS Kyoto Univ., 38, pp. 191–209 (2002). 5. M. Kadowaki, H. Nakazawa and K. Watanabe, On the asymptotics of solutions for some Schr¨ odinger equations with dissipative perturbations of rank one, Hiroshima. Math. J., 34, pp. 345–369 (2004).

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6. M. Kadowaki, H. Nakazawa and K.Watanabe, Non-selfadjoint perturbation of Schr¨ odinger and wave equations—Can one completely determine the behavior of solutions by only spectra?, Advanced studies in Pure Math., Asymptotic Analysis and Singularities—Hyperbolic and dispersive PDEs and fluid mechanics, Mathematical Society of Japan, Tokyo, 47-1, pp. 137–157 (2007). 7. M. Kadowaki, H. Nakazawa and K.Watanabe, On the rank one dissipative operator and the Parseval formula, submitted. 8. M. Kadowaki, H. Nakazawa and K.Watanabe, Parseval formula for wave equations with dissipative term of rank one, to appear. 9. T. Kato, Wave operators and similarity for some non–self adjoint operators, Math. Ann. 162, pp. 258–279 (1966). 10. S. T. Kuroda, An Introduction to Scattering Theory, Lecture Note Series N o 51, Matematisk Institut, Aarhus University, 1980. 11. A. Matsumura, Energy decay of solutions of dissipative wave equations, Proc. Japan Acad., 53, pp. 232–236 (1977). 12. K. Mochizuki, Eigenfunction expansions associated with the Schr¨ odinger operator with a complex potential and scattering inverse problem, Proc. Japan Acad., 43, pp. 638–643 (1967). 13. K. Mochizuki, Eigenfunction expansions associated with the Schr¨ odinger operator with a complex potential and the scattering theory, Publ. RIMS Kyoto Univ., 4, pp. 419–466 (1968). 14. K. Mochizuki, Scattering theory for wave equations with dissipative terms, Publ. RIMS Kyoto Univ., 12, pp. 383–390 (1976). 15. K. Mochizuki, Scattering theory for wave equations (Japanese), Kinokuniya, Tokyo, 1984. 16. H. Nakazawa, Principle of limiting absorption for the non-selfadjoint Schr¨ odinger operator with energy dependent potential, Tokyo. J. Math., 23, pp. 519–536 (2000). 17. H. Nakazawa, On wave equations with dissipations, Proccedings of 4th International conference “Analytical Methods of Analysis and Differential Equations” (AMADE-2006), vol. 3, Differential Equations, pp. 102–110 (2006). 18. B. S. Pavlov, On a non-selfadjoint Schr¨ odinger operator (Russian), Prob. Math. Phys. Spectral Theory and Wave Processes, Izdat. Leningrad Univ., Leningrad, 1, pp. 102–132 (1966). 19. W. Schlag, Dispersive estimates for Schr¨ odinger operators in dimension two, Comm. Math. Phys., 257, pp. 87–117 (2005).

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ON SCATTERING FOR EVOLUTION EQUATIONS WITH TIME-DEPENDENT SMALL PERTURBATIONS K. MOCHIZUKI Department of Mathematics. Chuo University Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan E-mail: [email protected] We first summarize an abstract scattering theory for time-dependent small perturbations. Our results give an extension of Kato’s results for time-independent smooth perturbations. In the latter half, as examples, we treat the Schr¨ odinger equation with time-dependent complex potential and the wave equation with time-dependent coefficients. Keywords: Scattering, time-dependent perturbation, Schr¨ odinger equation, wave equation.

1. An abstract result for time-dependent small perturbations Let H be a Hilbert space with inner product (·, ·) and norm k·k. We consider in H the evolution equation i∂t u − Λ0 u − V (t)u = 0,

t ∈ R,

(1)

where i denotes the imaginary unit, ∂t the partial derivative with respect to the time variable, Λ0 is a selfadjoint operator in H with dense domain D(Λ0 ), and V (t) is a Λ0 -compact operator which depends continuously on t ∈ R. We choose the initial condition at t = 0, u(x, 0) = f (x) ∈ H,

(2) −itΛ0

be the unitary and restrict ourselves to solutions in H. Let U0 (t) = e group in H which represents the solution of the free equation i∂t u0 −Λ0 u0 = 0. Then the perturbed problem (1), (2) reduces to the integral equation Z t u(t) = U0 (t)f − i U0 (t − τ )V (τ )u(τ )dτ. (3) 0

Scattering for Evolution Equations with Time-Dependent Small Perturbations

477

In the following we restrict ourselves to the class of perturbations with which the following two properties are satisfied. (I) For given f ∈ H, equation (3) has a unique solution u(t) ∈ C(R; H). We denote by U (t, s) ∈ B(H) the evolution operator which maps solutions at time s to those at time t: u(t) = U (t, s)u(s). The unique existence of solutions of (3) implies that for each fixed s and t, U (t, s) defines a bijection on H. (II) Moreover, the inequality |(V (t)u, v)| ≤ η(t)kukkvk + A(t, u)1/2 B(t, v)1/2

(4)

holds for u, v ∈ H, where η(t) is a nonnegative L1 -function of t ∈ R and A(t, u), B(t, v) are positive definite quadratic forms satisfying Z ±∞ A(t, U (t, s)f )dt ≤ C0 kf k2, (5) 0 Z ±∞ ≤ C1 kgk2 B(t, U (t)g)dt (6) 0 0

for some positive constants C0 , C1 independent of s ∈ R± and f , g ∈ H.

(II) may be called a time-dependent version of the Kato’s smooth perturbations (cf. [3,4]). Smallness of V (t) is implicitly required in (5). With these properties, we obtain the following theorem. Theorem 1.1. (i) { U (t, s) : t, s ∈ R } is a family of uniformly bounded operators. (ii) For every s ∈ R± , there exits the strong limit Z ± (s) = s − lim U0 (−t + s)U (t, s). t→±∞

(iii) The operator Z

±

±

= Z (0) satisfies

w − lim Z ± U (0, s)U0 (s) = I (weak limit). s→±∞

(iv) If C0 , C1 in (II) can be chosen to satisfy C0 C1 < 1, then Z ± : H −→ H is a bijection on H. Moreover, the scattering operator S = Z + (Z − )−1 is also a bijection. Proof. We put u(t, s) = U (t, s)f , u0 (t, s) = U0 (t − s)f0 . Then we have Z t (u(t, s), u0 (t, s)) = (f, f0 ) − i (V (τ )u(τ, s), u0 (τ, s))dτ s

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from (3). It then follows from (4) of (II) that for any σ, t ∈ R± , Z t |(u(t, s), u0 (t, s)) − (u(σ, s), u0 (σ, s))| ≤ η(τ )ku(τ, s)k ku0 (τ, s)kdτ σ

Z t 1/2 Z t 1/2 + A(τ, u(τ, s))dτ B(τ, u0 (τ, s))dτ . σ

(7)

σ

All the assertions of the theorem are verified from this inequality. (i) We put σ = s in (7). Then by (5), (6) of (II) Z t p |(u(t, s), u0 (t, s))−(f, f0 )| ≤ η(τ )ku(τ )k ku0 (τ )kdτ + C0 C1 kf k kf0k. s

Since U0 (t − s) is unitary, it follows that ku(t, s)k ≤ (1 + 1

Z t p C0 C1 )kf k + η(τ )ku(τ, s)kdτ. s

The requirement η(t) ∈ L (R) and the Gronwall inequality prove the assertion. (ii) Noting (i), we have from (7) |(u(t, s), u0 (t, s)) − (u(σ, s), u0 (σ, s))| Z t Z t 1/2   p C1 kf0 k. ≤ C2 kf k η(τ )dτ + A(τ, u(τ, s))dτ σ

σ

 Here, for any fixed s ∈ R± , · · · → 0 as σ, t → ±∞. Thus U0 (s − t)U (t, s) converges strongly in H as t → ±∞. (iii) Let σ = s and t → ±∞ in (7). Then noting (i), we have  Z ±∞ |(Z ± (s)f, f0 ) − (f, f0 )| ≤ kf k C2 η(τ )dτ kf0 k s

p Z + C0

s

±∞

1/2  . B(τ, u0 (τ, s))dτ

Choose here f = U0 (s)g and f0 = U0 (s)g0 . Then  Z ± |({U0 (−s)Z (s)U0 (s) − I}g, g0 )| ≤ kU0 (s)gk C2 +

p Z C0

s

(8)

η(τ )dτ kU0 (s)g0 k s 1/2  ∞ . B(τ, U0 (τ )g0 )dτ ±∞

Since g and g0 are arbitrary, this implies that as s → ±∞, Z ± U (0, s)U0 (s) = U0 (−s)Z ± (s)U0 (s) → I

Scattering for Evolution Equations with Time-Dependent Small Perturbations

weakly in H. (iv) Note that (8) implies  Z ± |({Z (s) − I}f, f0 )| ≤

s

±∞

479

 p η(τ )dτ C2 + C0 C1 kf kkf0k.

Since C0 C1 < 1, we can choose ±s > 0 sufficiently large to satisfy Z ±∞ p η(τ )dτ C2 + C0 C1 < 1. s

Thus, kZ± (s) − IkB(H) < 1 and Z ± (s) gives a bijection on H. The same property of Z ± then easily follows. 2. Schr¨ odinger equations with time dependent complex potentials We consider the Schr¨ odinger equation i∂t u + ∆u − V (x, t)u = 0

(9)

in (x, t) ∈ Rn × R, where ∆ is the n dimensional Laplacian and V (x, t) is a complex potential which is bounded and continuous in Rn × R. The Cauchy problem for (9) is reduced to the evolution equation (1) if we choose H = L2 (Rn ), Λ0 = −∆ with D(Λ0 ) = H 2 and V (t) = V (x, t). Properties (I) and (II) are verified based on space-time Lp -Lq estimates of solutions u(t). For this aim, we restrict ourselves to potentials like (A1) V (x, t) ∈ Lν (R; Lq ), where 0≤

1 2 ≤ q n

and

1 n =1− . ν 2q

V (x, t) also satisfies the smallness condition C3 kV kL∞ (R± ;Ln/2 ) < 1 when ν = ∞,

(10)

where C3 > 0 is given later in Lemma 2.2. A typical example of potentials is V (x, t) = c(1 + |x|)−α (1 + |t|)−β , where c ∈ C and α, β ≥ 0 with α + β > 1 and |c| is small if β = 0. 2 Note that potential (11) satisfies (A1) if we choose 1 1 2 = 0 when α = 0, = when β = 0 q q n

(11)

(12)

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K. Mochizuki

and 1 min{α, 2} max{0, 2(1 − β)} < < n q n when α, β > 0. The above potential is considered by Yafaev [13] when c is real and β > 0 (see also [1,2,6] and [14], where more general real potentials are studied). Lemma 2.1. Let 2 ≤ p ≤ ∞ and put

1 1 = 1 − . Then ′ p p

kU0 (t)f0 kLp ≤ (4π|t|)n/p−n/2 kf0 kLp′ . Lemma 2.2. Let n ≥ 3 and

  n−2 1 1 1 n 1 1 . ≤ ≤ and = − 2n p 2 r 2 2 p

Then there exists C3 > 0 such that

Z t



U (t − τ )h(τ )dτ 0

0

Lr (R± ;Lp )

≤ C3 khkLr′ (R± ;Lp′ ) .

Lemma 2.1 is classical. Lemma 2.2 is a direct consequence of Lemma n−2 1 n−2 1 . At the end point = , it is due to Keel-Tao [5]. As 2.1 if > p 2n p 2n a corollary of this lemma we have the following. Proposition 2.1. Let n, p and r be as in Lemma 2.2. Then (i) For any t ∈ R± ,

Z t

p

U0 (−τ )h(τ )dτ

≤ 2C3 khkLr′ (R± ;Lp′ ) . 0

(ii) For f0 ∈ L2 , we have U0 (t)f0 ∈ Lr (R± ; Lp ) and p kU0 (·)f0 kLr (R± ;Lp ) ≤ 2C3 kf0 k.

Similar estimates are obtained for perturbed solutions if we require (A1). For 1 ≤ γ, µ ≤ ∞ and ±s ≥ 0, we put γ,µ Y±,s = Lγ (R±,s ; Lµ ),

where R+,s = [s, ∞) for s ≥ 0 and R−,s = (−∞, s] for s ≤ 0. The γ,µ space Y±,0 = Lγ (R± ; Lµ ) is already used in this section. By (A1) we have ν,q V (x, t) ∈ Y±,s for any ±s ≥ 0. Moreover, as we see from (10), there exist ±s ≥ 0 such that ν,q < 1. C3 kV kY±,s

(13)

Scattering for Evolution Equations with Time-Dependent Small Perturbations

481

In the following we fix such an s, and choose the pair (p, r) related to (q, ν) as     1 1 1 1 1 1 1− and 1− . (14) = = p 2 q r 2 ν The condition for (q, ν) in (A1) is equivalent to that for (p, r) in Lemma 2.2. Proposition 2.2. ([10, Theorem 3]) Let n ≥ 3 and assume (A1). Then the following hold: (i) For each f ∈ L2 , the integral equation Z t u(t) = U0 (t − s)f + i U0 (t − τ )V (τ )u(τ )dτ s

r,p . Y±,s

has a unique solution u(t) ∈ (ii) This solution belongs to C(R±,s ; L2 ). Moreover, we have √ 2C3 r,p ≤ kf k kukY±.s ν,q 1 − C3 kV kY±,s

(15)

and

Z t

ν,q 2C3 kV kY±,s



≤ U (−τ )V (τ )u(τ )dτ 0

1 − C3 kV k ν,q kf k. Y±.s s

(16)

Now we are ready to verify properties (I), (II) of Section 1. (I) is obtained in Proposition 2.2. (II) is verified as follows. By (A1) and Proposition 2.1 (ii), Z t Z 1/2 p 1/2 2 ≤ kV k1/2 ku0 kY 2ν ′ ,2q′ ≤ 2C3 kV kY ν,q kf0 k, (17) |V ||u | dxdτ 0 Y ν,q ±,s

σ

±,s

±,s

where we have used the equalities   1 1 1 1 1− = , = ′ 2q 2 q p

1 1 = . ′ 2ν r

On the other hand, by (A1) and (15) of Proposition 2.2 we similarly have √ 1/2 Z t Z 1/2 2C5 kV kY ν,q ±,s 2 kf k. (18) ≤ |V ||u| dxdτ 1 − C5 kV k ν,q σ

Y±,s

So we can choose η(t) ≡ 0 and A(t)v = B(t)v = |V (x, t)|1/2 v in (5).

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K. Mochizuki

3. Wave equations with time dependent coefficients Let Ω be an exterior domain in Rn (n ≥ 3) with smooth boundary ∂Ω which is star-shaped with respect to the origin 0. We consider in Ω the wave equation ∂t2 w − ∆w + b0 (x, t)∂t w +

n X

bj (x, t)∂j w + c(x, t)w = 0

(19)

j=1

with Dirichlet boundary condition (x, t) ∈ ∂Ω × R,

w(x, t) = 0,

(20)

where bj (x, t) (j = 0, 1, · · · , n) and c(x, t) are real-valued bounded continuous functions. 1 Let HD denote the completion of C0∞ (Ω) with respect to the Dirichlet norm Z 1/2 kukD = |∇u|2 dx for u ∈ C0∞ (Ω), and let L2 be the usual L2 space in Ω. We define HE = 1 HD × L2 . Then HE forms a Hilbert space with the energy norm 1/2 1  , f = {f1 , f2 }. kf kE = √ kf1 k2D + kf2 k2 2

As an evolution equation in HE , the problem (19), (20) is rewritten in the form i∂t w ~ = Λ0 w ~ + V (t)w, w ~ = (w, wt )T ,

(21)

where Λ0 = i



0 1 ∆0



with b(x, t) · ∇ =

n X

and

V (t) = −i

0

0

b(·, t) · ∇ + c(·, t) b0 (·, t)

!

bj (x, t)∂j .

j=1

The operator Λ0 becomes selfadjoint in HE if the domain is defined by 2 1 2 1 D(Λ0 ) = HD × {HD ∩ L2 }, where HD is the set of functions f1 ∈ HD such 2 −itΛ0 ; t ∈ R} that ∆f1 ∈ L . Thus Λ0 generates a unitary group {U0 (t) = e on HE , and by use of this U0 (t), problem (21) is reduced to the integral equation Z t w(t) ~ = U0 (t)f + U0 (t − s)V (s)w(s)ds. ~ (22) 0

Scattering for Evolution Equations with Time-Dependent Small Perturbations

483

Let η = η(t) be a positive L1 -function of t ∈ R, and let ξ = ξ(r) be a smooth positive L1 -function of r > 0 which also satisfies ξ ′ (r) ≤ 0,

ξ ′ (r)2 ≤ 2ξ(r)ξ ′′ (r). −1−δ

(23) −1

Note that the functions ξ(r) = (1 + r) or ξ(r) = (1 + r) {log(e + r)}−1−δ satisfy (23). With these functions, we require the following conditions on the coefficients:   |bj (x, t)| ≤ ǫξ(r) + η(t) (j = 0, 1, · · · , n), (A2)  |c(x, t)| ≤ {ǫξ(r) + η(t)} n − 2 , 2r where ǫ is a small positive constant (see [7,8,11,12] for related conditions). A typical example of coefficients is ( |bj (x, t)| ≤ bj0 (1 + r)−αj (1 + |t|)−βj (j = 0, 1, · · · , n) (24) ˜ |c(x, t)| ≤ c0 r−1 (1 + r)−α˜ (1 + |t|)−β , where bj0 , c0 are positive constants, and αj , βj , α ˜ , β˜ ≥ 0 satisfy αj + βj > 1,

α ˜ + β˜ > 1.

˜ = 0. The constant bj0 (or c0 ) should be chosen sufficiently small if βj (or β) In order to verify properties (I), (II) to hold we shall make use of spacetime weighted energy estimates for solutions u(t). First we multiply by wt both sides of (19) and integrate by parts over Ω × [s, t]. Then noting the boundary condition w0t |∂Ω = 0, we obtain Z tZ 2 2 + kw(t)k ~ − k w(s)k ~ P1 dxdτ = 0, (25) E E s

where

P1 = {b0 wt + b · ∇w + cw}wt .   Z r n−1 w , ψ(r) = ξ(σ)dσ, both sides of Next, we multiply by ψ wr + 2r 0 (21), where wr = x ˜ ·∇w with x ˜ = x/|x|, and integrate over Ω ×[s, t] (s < t). Then noting the star-shapedness of ∂Ω, we have Z Z Z tZ X(t)dx − X(s)dx + {Z + P2 }dxdτ ≤ 0, (26) s

where

  n−1 X = ψ wt wr + wt w , 2r

484

K. Mochizuki



  1 (n − 1)(n − 3) 2 |∇w|2 − wr2 + w ψ − ψ′ r 4r2 n−1 2 1 w , + ψ ′ (wt2 + |∇w|2 ) − ψ ′′ 2 4r   n−1 P2 = ψ{b0 wt + b · ∇w + cw} wr + w . 2r Z=

By the use of (25) and (26), the following two propositions hold true. Proposition 3.1. ([9, Theorem 2]) Let ξ(r), r > 0, be a positive L 1 function satisfying (23). Then for any s < t we have  Z Z  1 t n−1 2 2 ~ 0 (0)k2E , ξ(|∇w0 |2 + w0t ) − ξ′ w0 dxdτ ≤ C4 kw 2 s 2r where C4 = C4 (ξ) =

4n − 6 kξkL1 . n−2

2 . Then 9C4 (i) There exists C5 = C5 (η) > 0 such that for any t ∈ R, we have

Proposition 3.2. ([9, Theorem 3]) Assume (A2) with ǫ
0 such that for any s < t we have  Z Z  1 t n−1 2 2 ~ w dxdτ ≤ C6 kw(0)k ξ(|∇w|2 + wt2 ) − ξ ′ E. 2 s Ω 2r These propositions prove property (II). In fact, since   Z n−2 |(V (t)u, v)E | ≤ {ǫξ(r) + η(t)} |∇u1 | + |u1 | + |u2 | |v2 |dx, 2r choosing  2   n−2 |u1 |2 + |u2 |2 dx, ǫξ(r) |∇u1 |2 + 2r Z B(t, v) = 3 ξ(r)|v2 |2 dx A(t, u) =

Z

in (4), we obtain inequalities (5) and (6) from Propositions 3.2 (ii) and 3.1, respectively. Property (I) is also verified by the use of the successive approximation based on these propositions.

Scattering for Evolution Equations with Time-Dependent Small Perturbations

485

References 1. J. Howland, Stationary scattering theory for time dependent Hamiltonians, Math. Ann. 207 (1974), 315-335. 2. A. Jensen, Results in Lp (Rd ) for the Schr¨ odinger equation with timedependent potential, Math. Ann. 299 (1994), 117-125. 3. T. Kato, Wave operators and similarity for some nonselfadjoint operators, Math. Ann. 162 (1966), 258-279. 4. T. Kato and K. Yajima, Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys. 1 (1989), 481-496. 5. M. Keel and R. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), 955-980. 6. H. Kitada and K. Yajima, A scattering theory for time-dependent long-range potentials, Duke Math. J. 49 (1982), 341-376 7. K. Mochizuki, Scattering theory for wave equations with dissipative terms, Publ. Res. Inst. Math. Sci. 12 (1976), 383-390. 8. K. Mochizuki, Scattering theory for wave equations, Kinokuniya Shoten, 1984 (in Japanese). 9. K. Mochizuki, On scattering for wave equations with time dependent coefficients, Tsukuba J. Math. 31 (2007), 327-342. 10. K. Mochizuki and T. Motai, On decay-nondecay and scattering for Schr¨ odinger equations with time dependent complex potentials, Publ. Res. Inst. Math. Sci. 43 (2007), 1183-1197. 11. K. Mochizuki and H. Nakazawa, Energy decay and asymptotic behavior of solutions to the wave equation with linear dissipation, Publ. Res. Inst. Math. Sci. 32 (1996), 401-414. 12. J. Wirth, Scattering and modified scattering for abstract wave equations with time-dependent dissipation, Adv. Differential Equations, 12 (2007), 11151133. 13. D. R. Yafaev, On the violation of the unitarity in time dependent potential scattering, Soviet Math. Dokl., 19 (1973), 1517-1521. 14. K. Yajima, Scattering theory for Schr¨ odinger equations with potentials periodic in time, J. Math. Soc. Japan 29 (1977), 729-743.

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COMPARISON OF ESTIMATES FOR DISPERSIVE EQUATIONS M. RUZHANSKY Department of Mathematics, Imperial College London London, United Kingdom E-mail: [email protected] M. SUGIMOTO Department of Mathematics, Osaka University Osaka, Japan E-mail: [email protected] This paper describes a new comparison principle that can be used for the comparison of space-time estimates for dispersive equations. In particular, results are applied to the global smoothing estimates for several classes of dispersive partial differential equations. Keywords: Comparison principle; dispersive equations; smoothing estimates; weighted estimates.

1. Introduction In this note we will present a new comparison principle that allows one to compare certain estimates for dispersive equations of different types based on expressions involving their symbols. In particular, a question is that if we have a certain estimate for one equation, whether we can derive a corresponding estimate for another equation. This question is of interest on its own, and it has several applications. Proofs of the statements of this paper can be found in the authors’ paper [14]. The main application of this technique that we have in mind is for the global smoothing estimates for dispersive equations. These smoothing estimates are essentially global space-time estimates in weighted Sobolev spaces over L2 , see for example [2,3,5,7,16,17]. There is a known method in the microlocal analysis on how to transform one equation into another, namely the canonical transforms realised in the form of Fourier integral

Comparison of Estimates for Dispersive Equations

487

operators [6]. In the global setting one needs to develop global weighted estimates in L2 for the corresponding classes of Fourier integral operators in order to apply them to the smoothing estimates. Such global estimates for Fourier integral operators have been established by the authors [12] and have been applied to derive new smoothing estimates for Schr¨ odinger equations [12,13]. These techniques allow one to reduce the analysis of dispersive equations to normal forms in one and two dimensions [11]. The comparison principles introduced below allow one to further relate estimates in normal forms, thus establishing comprehensive relations between smoothing estimates for dispersive equations with constant coefficient [14]. In this note, we denote x = (x1 , . . . , xn ), ξ = (ξ1 , . . . , ξn ), and Dx = ∂ , for all j = 1, 2, . . . , n, and (D1 , D2 . . . , Dn ), where Dj denotes Dxj = 1i ∂x j √ i = −1. 2. Comparison principles Let u(t, x) = eitf (Dx ) ϕ(x) and v(t, x) = eitg(Dx ) ϕ(x) be solutions to the following evolution equations, where t ∈ R and x ∈ Rn : ( (i∂t + f (Dx )) u(t, x) = 0, (1) u(0, x) = ϕ(x), and

(

(i∂t + g(Dx )) v(t, x) = 0, v(0, x) = ϕ(x).

(2)

First we state the following result relating several norms involving propagators for equations (1) and (2): Theorem 2.1. Let f ∈ C 1 (Rn ) be a real-valued function such that, for almost all ξ ′ = (ξ2 , . . . , ξn ) ∈ Rn−1 , f (ξ1 , ξ ′ ) is strictly monotone in ξ1 on the support of a measurable function σ on Rn . Then we have Z

2 |σ(ξ)|2

−n 2 dξ = (2π) | ϕ(ξ)| b

σ(Dx )eitf (Dx ) ϕ(x1 , x′ ) 2 |∂f /∂ξ1 (ξ)| L (Rt ×Rn−1 ) Rn x′

for all x1 ∈ R, where x′ = (x2 , . . . , xn ) ∈ Rn−1 .

The following comparison principle is a straightforward consequence of Theorem 2.1: Corollary 2.1. Let f, g ∈ C 1 (Rn ) be real-valued functions such that, for almost all ξ ′ = (ξ2 , . . . , ξn ) ∈ Rn−1 , f (ξ1 , ξ ′ ) and g(ξ1 , ξ ′ ) are strictly

488

M. Ruzhansky & M. Sugimoto

monotone in ξ1 on the support of a measurable function χ on Rn . Let σ, τ ∈ C 0 (Rn ) be such that, for some A > 0, we have |σ(ξ)|

|∂ξ1 f (ξ)|

1/2

≤A

|τ (ξ)|

|∂ξ1 g(ξ)|

(3)

1/2

for all ξ ∈ supp χ satisfying D1 f (ξ) 6= 0 and D1 g(ξ) 6= 0. Then we have



χ(Dx )σ(Dx )eitf (Dx ) ϕ(x1 , x′ ) n−1 L2 (Rt ×Rx′

≤ Akχ(Dx )τ (Dx )e

)

itg(Dx )

ϕ(e x1 , x′ )kL2 (Rt ×Rn−1 ) ′

(4)

x

for all x1 , x e1 ∈ R, where x′ = (x2 , . . . , xn ) ∈ Rn−1 . Consequently, for any measurable function w on R we have



w(x1 )χ(Dx )σ(Dx )eitf (Dx ) ϕ(x) L2 (Rt ×Rn x)

≤ Akw(x1 )χ(Dx )τ (Dx )eitg(Dx ) ϕ(x)kL2 (Rt ×Rnx ) . (5)

Moreover, if χ ∈ C 0 (Rn ) and w 6= 0 on a set of R with positive measure, the converse is true, namely, if we have estimate (4) for all ϕ, for some x1 , x e1 ∈ R, or if we have estimate (5) for all ϕ, and the norms are finite, then we also have inequality (3).

We remark that inequality (5) in Corollary 2.1 gives the comparison between different weighted estimates. The reason to introduce a cut-off function χ into the estimates is that the relation between symbols may be different for different regions of the frequencies ξ (for example this is the case for the relativistic Schr¨ odinger and for the Klein-Gordon equations, which are of order two for large frequencies and of order zero for small frequencies). In such case we can use this comparison principle to relate estimates for the corresponding ranges of frequencies, thus yielding more refined results, since then we have freedom to choose different σ for different types of behaviour of f ′ . The assumption σ, τ ∈ C 0 (Rn ) that was made in Corollary 2.1 is for the clarity of the exposition and can clearly be relaxed. In the case n = 1, we neglect x′ = (x2 , . . . , xn ) in a natural way and just write x = x1 , ξ = ξ1 , and Dx = D1 . Similarly in the case n = 2, we use the notation (x, y) = (x1 , x2 ), (ξ, η) = (ξ1 , ξ2 ), and (Dx , Dy ) = (D1 , D2 ). In both cases, we write x e=x e1 in the notation of Corollary 2.1. Then we have the following corollaries in lower dimensions:

Corollary 2.2. Suppose n = 1. Let f, g ∈ C 1 (R) be real-valued and strictly monotone on the support of a measurable function χ on R. Let σ, τ ∈ C 0 (R)

Comparison of Estimates for Dispersive Equations

489

be such that, for some A > 0, we have |σ(ξ)| |τ (ξ)| ≤A ′ ′ 1/2 |f (ξ)| |g (ξ)|1/2

(6)

for all ξ ∈ supp χ satisfying f ′ (ξ) 6= 0 and g ′ (ξ) 6= 0. Then we have kχ(Dx )σ(Dx )eitf (Dx ) ϕ(x)kL2 (Rt )

≤ Akχ(Dx )τ (Dx )eitg(Dx ) ϕ(e x)kL2 (Rt )

(7)

for all x, x e ∈ R. Consequently, for general n ≥ 1 and for any measurable function w on Rn , we have kw(x)χ(Dj )σ(Dj )eitf (Dj ) ϕ(x)kL2 (Rt ×Rnx )

≤ Akw(x)χ(Dj )τ (Dj )eitg(Dj ) ϕ(x)kL2 (Rt ×Rnx ) , (8)

where j = 1, 2, . . . , n. Moreover, if χ ∈ C 0 (R) and w 6= 0 on a set of Rn with positive measure, the converse is true, namely, if we have estimate (7) for all ϕ, for some x, x e ∈ R, or if we have estimate (7) for all ϕ, and the norms are finite, then we also have inequality (6). We have the following comparison principle in two dimensions:

Corollary 2.3. Suppose n = 2. Let f, g ∈ C 1 (R2 ) be real-valued functions such that, for almost all η ∈ R, f (ξ, η) and g(ξ, η) are strictly monotone in ξ on the support of a measurable function χ on R2 . Let σ, τ ∈ C 0 (R2 ) be such that, for some A > 0, we have |σ(ξ, η)|

|∂f /∂ξ(ξ, η)|

1/2

≤A

|τ (ξ, η)|

|∂g/∂ξ(ξ, η)|

1/2

(9)

for all (ξ, η) ∈ supp χ satisfying ∂f /∂ξ(ξ, η) 6= 0 and ∂g/∂ξ(ξ, η) 6= 0. Then we have



χ(Dx , Dy )σ(Dx , Dy )eitf (Dx ,Dy ) ϕ(x, y) 2 L (Rt ×Ry )

≤ Akχ(Dx , Dy )τ (Dx , Dy )e

itg(Dx ,Dy )

ϕ(e x, y)kL2 (Rt ×Ry )

(10)

for all x, x e ∈ R. Consequently, for general n ≥ 2 and for any measurable function w on Rn−1 we have kw(ˇ xk )χ(Dj , Dk )σ(Dj , Dk )eitf (Dj ,Dk ) ϕ(x)kL2 (Rt ×Rnx )

≤ Akw(ˇ xk )χ(Dj , Dk )τ (Dj , Dk )eitg(Dj ,Dk ) ϕ(x)kL2 (Rt ×Rnx ) , (11)

where j 6= k and x ˇk = (x1 , . . . , xk−1 , xk+1 , . . . , xn ). Moreover, if χ ∈ C 0 (R2 ) and w = 6 0 on a set of Rn−1 with positive measure, the converse is

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true, namely, if we have estimate (10) for all ϕ, for some x, x e ∈ R, or if we have estimate (10) for all ϕ, and the norms are finite, then we also have inequality (9). By the same argument as used in the proof of Theorem 2.1 and Corollary 2.1, we have a comparison result in the radially symmetric case. Below, we denote the set of the positive real numbers (0, ∞) by R+ . Theorem 2.2. Let f, g ∈ C 1 (R+ ) be real-valued and strictly monotone on the support of a measurable function χ on R+ . Let σ, τ ∈ C 0 (R+ ) be such that, for some A > 0, we have |σ(ρ)| |τ (ρ)| ≤A ′ ′ 1/2 |f (ρ)| |g (ρ)|1/2

(12)

for all ρ ∈ supp χ satisfying f ′ (ρ) 6= 0 and g ′ (ρ) 6= 0. Then we have kχ(|Dx |)σ(|Dx |)eitf (|Dx |) ϕ(x)kL2 (Rt )

≤ Akχ(|Dx |)τ (|Dx |)eitg(|Dx |) ϕ(x)kL2 (Rt )

(13)

for all x ∈ R . Consequently, for any measurable function w on R , we have n

n

kw(x)χ(|Dx |)σ(|Dx |)eitf (|Dx |) ϕ(x)kL2 (Rt ×Rnx )

≤ Akw(x)χ(|Dx |)τ (|Dx |)eitg(|Dx |) ϕ(x)kL2 (Rt ×Rnx ) . (14)

Moreover, if χ ∈ C 0 (R+ ) and w 6= 0 on a set of Rn with positive measure, the converse is true, namely, if we have estimate (13) for all ϕ, for some x ∈ Rn , or if we have estimate (14) for all ϕ, and the norms are finite, then we also have inequality (12). Theorem 2.2 provides an analytic alternative to computations for certain estimates in the radially symmetric case done with the help of special functions [18]. These comparison principles can be extended to provide the relation between Strichartz type norms, and the details and the meaning of the corresponding estimates can be found in authors’ paper [14]. Here we just give one corollary: Corollary 2.4. Let functions f, g, σ, τ be as in Theorem 2.2 and satisfy relation (12). Let 0 < p ≤ ∞. Then, for any measurable function w on Rn , we have the estimate kw(x)χ(|Dx |)σ(|Dx |)eitf (|Dx |) ϕ(x)kLp (Rnx ,L2 (Rt ))

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491

≤ Akw(x)χ(|Dx |)τ (|Dx |)eitg(|Dx |) ϕ(x)kLp (Rnx ,L2 (Rt )) . (15) From this, it √follows, for example, that for all 0 < p ≤ ∞, the quantities ||eit −∆ ϕ||Lp (Rnx ,L2 (Rt )) , || |Dx |1/2 e−it∆ ϕ||Lp (Rnx ,L2 (Rt )) , 3/2

and || |Dx |eit(−∆) ϕ||Lp (Rnx ,L2 (Rt )) for the propagators of the wave, Schr¨ odinger, and KdV-type equations are equivalent. 3. Some applications

Let us now give some examples of the use of these comparison principles. If both sides in expression (3) in Corollary 2.1 are equivalent, we can use the comparison in two directions, from which it follows that norms on both sides in (4) are equivalent. The same is true for Corollaries 2.2, 2.3 and Theorem 2.2. In particular, we can conclude that many smoothing estimates for the Schr¨ odinger type equations of different orders are equivalent to each other. Indeed, applying Corollary 2.2 in two directions, we immediately obtain that for n = 1 and l, m > 0, we have r

l l

(m−1)/2 it|Dx |m (16) e ϕ(x) 2 =

|Dx |(l−1)/2 eit|Dx | ϕ(x) 2

|Dx | m L (Rt ) L (Rt )

for every x ∈ R, assuming that supp ϕ b ⊂ [0, +∞) or (−∞, 0]. On the other hand, still in the case n = 1, we have easily

itD

e x ϕ(x) 2 = kϕkL2 (Rx ) for all x ∈ R, (17) L (Rt ) which is a straightforward consequence of the fact eitDx ϕ(x) = ϕ(x + t). These observations yield the following result: Theorem 3.1. Suppose n = 1 and m > 0. Then we have

m

≤ CkϕkL2 (Rx )

|Dx |(m−1)/2 eit|Dx | ϕ(x) L2 (Rt )

for all x ∈ R. Suppose n = 2 and m > 0. Then we have

m−1

≤ CkϕkL2 (R2 ) ϕ(x, y)

|Dy |(m−1)/2 eitDx |Dy | x,y L2 (Rt ×Ry )

(18)

(19)

for all x ∈ R. Each estimate above is equivalent to itself with m = 1 which is a direct consequence of equality (17).

Estimates (18) and (19) in Theorem 3.1 in the special case m = 2 were shown by Kenig, Ponce and Vega [9, p. 56] and by Linares and Ponce [10, p. 528], respectively. Theorem 3.1 shows that these results, together with their generalisation to other orders m, are in fact just corollaries of the

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elementary one dimensional fact eitDx ϕ(x) = ϕ(x + t) once we apply the comparison principle. By using the comparison principle in the radially symmetric and higher dimensional cases, we have also another type of equivalence of smoothing estimates, which can be found in authors’ paper [14]. Let us give one example: Theorem 3.2. For m > 0 (and any α, β) we have the following relations (in the first equality the left and the right hand sides are finite for the same values of α, β at the same time):

2

β−1

|Dx |β eit|Dx | ϕ

|x| L2 (Rt ×Rn ) r x

m m β−1 |Dx |m/2+β−1 eit|Dx | ϕ , =

|x| 2 L2 (Rt ×Rn x) and

m

α−m/2

|Dx |α eit|Dx | ϕ(x) 2

hxi L (Rt ×Rn x)

α−m/2

α it|Dx |m ≤ |x| |Dx | e ϕ(x) L2 (Rt ×Rn x)

α−m/2

α it|Dx |m |Dx | e ϕλ (x) ≤ sup hxi

L2 (Rt ×Rn x)

λ>0

,

where ϕλ (x) = λn/2 ϕ(λx), and we take α ≤ m/2 in the last esm timate. The operator norms of operators hxiα−m/2 |Dx |α eit|Dx | and m |x|α−m/2 |Dx |α eit|Dx | as mappings from L2 (Rn ) to L2 (Rt × Rnx ) are equal.

As a nice consequence, for n ≥ 3 and m > 0 we can conclude also the estimate s

m 2π

−1

kϕkL2 (Rnx ) , (20) ≤

|x| |Dx |m/2−1 eit|Dx | ϕ(x) 2 n m(n − 2) L (Rt ×Rx ) q 2π where the constant m(n−2) is sharp. This follows from the first equality

in Theorem 3.2 with β = 0 and the best constant in the case m = 2 (as shown by Simon [15] as a consequence of constants in Kato’s theory). As a consequence of Theorem 3.2, we have the following:

Corollary 3.1. Suppose m > 0 and (m − n)/2 < α < (m − 1)/2. Then we have

m

α−m/2

|Dx |α eit|Dx | ϕ(x) ≤ CkϕkL2 (Rn ) . (21)

|x| 2 n x L (Rt ×Rx )

Comparison of Estimates for Dispersive Equations

Suppose m > 0 and (m − n + 1)/2 < α < (m − 1)/2. Then we have

α−m/2 ′ α it(|D1 |m −|D′ |m )

|D | e ϕ(x) 2 ≤ CkϕkL2 (Rnx ) ,

|x| n

where D ′ = (D2 , . . . , Dn ).

L (Rt ×Rx )

493

(22)

Estimate (21) is known in the case m = 2 as the Kato–Yajima estimate [8]. The application of the comparison principles also yields some refinements for other equations, for example for the relativistic Schr¨ odinger equation [4], Klein-Gordon equations and wave equations [1]. We refer to [14] for further details. References 1. M. Ben-Artzi, Regularity and smoothing for some equations of evolution, Nonlinear partial differential equations and their applications. College de France Seminar, Vol. XI (Paris, 1989–1991), 1–12, Pitman Res. Notes Math. Ser., 299, Longman Sci. Tech., Harlow, 1994. 2. M. Ben-Artzi and A. Devinatz, Local smoothing and convergence properties of Schr¨ odinger type equations, J. Funct. Anal. 101 (1991), 231–254. 3. M. Ben-Artzi and S. Klainerman, Decay and regularity for the Schr¨ odinger equation, J. Analyse Math. 58 (1992), 25–37. 4. M. Ben-Artzi and J. Nemirovsky, Remarks on relativistic Schr¨ odinger operators and their extensions, Ann. Inst. H. Poincare Phys. Theor. 67 (1997), 29–39. 5. P. Constantin and J. C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc. 1 (1988), 413–439. 6. J. J. Duistermaat and L. H¨ ormander, Fourier integral operators. II, Acta Math. 128 (1972), 183–269. 7. T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in applied mathematics, 93–128, Adv. Math. Suppl. Stud., 8, Academic Press, New York, 1983. 8. T. Kato and K. Yajima, Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys. 1 (1989), 481–496. 9. C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), 33–69. 10. F. Linares and G. Ponce, On the Davey-Stewartson systems, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 10 (1993), 523–548. 11. M. Ruzhansky and M. Sugimoto, A new proof of global smoothing estimates for dispersive equations, Advances in pseudo-differential operators, 65–75, Oper. Theory Adv. Appl., 155, Birkh¨ auser, Basel, 2004. 12. M. Ruzhansky and M. Sugimoto, Global L2 -boundedness theorems for a class of Fourier integral operators, Comm. Partial Differential Equations 31 (2006), 547–569. 13. M. Ruzhansky and M. Sugimoto, A smoothing property of Schr¨ odinger equations in the critical case, Math. Ann. 335 (2006), 645-673.

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14. M. Ruzhansky and M. Sugimoto, Smoothing properties of evolution equations via canonical transforms and comparison principle, arXiv:math/0612274v4. 15. B. Simon, Best constants in some operator smoothness estimates, J. Funct. Anal. 107 (1992), 66–71. 16. P. Sj¨ olin, Regularity of solutions to the Schr¨ odinger equation, Duke Math. J. 55 (1987), 699–715. 17. L. Vega, Schr¨ odinger equations: Pointwise convergence to the initial data, Proc. Amer. Math. Soc. 102 (1988), 874–878. 18. B. G. Walther, Regularity, decay, and best constants for dispersive equations, J. Funct. Anal. 189 (2002), 325–335.

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DISPERSIVE ESTIMATES IN ANISOTROPIC THERMOELASTICITY J. WIRTH∗ Department of Mathematics, Imperial College London SW7 2AZ, UK We present a unified treatment of the type-1 model of anisotropic thermoelasticity in two space dimensions and derive dispersive estimates for its solutions. The approach is based on diagonalising the full symbol of a corresponding first order system and representing solutions in terms of Fourier integrals with complex phases. Keywords: Thermoelasticity, a-priori estimates, anisotropic media.

1. The thermo-elastic system 1.1. Introduction The system of thermo-elasticity couples the elastic behaviour of a medium with propagation of heat. There are different ways to model this coupling, see e.g. [1]; we are concerned with the so-called type-1 model based on Fourier’s law of heat conduction. It reads as Utt + A(D)U + γ∇θ = 0,

(1a)

θt − κ∆θ + γ∇ · Ut = 0,

(1b)

U (0, ·) = U1 ,

Ut (0, ·) = U2 ,

θ(0, ·) = θ0 ,

(1c)

where U (t, ·) : R2 → R2 denotes the elastic displacement and θ(t, ·) : R2 → R the temperature difference to some equilibrium. The thermal conductivity κ and the thermo-elastic coupling γ are assumed to be positive, the elastic operator A(D) is (in general) a self-adjoint and positive 2 × 2 system of (pseudo-) differential operators. While for the particular case of isotropic media, given by A(D) = −µ∆ − (λ + µ)∇ ⊗ ∇ ∗ Work

(2)

supported by the Royal Society, Grant CIJN 44364, and EPSRC, EP/E062873/1.

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with Lam´e constants λ and µ, decay results are well-known in arbitrary space dimension, the treatment for anisotropic media is much harder to tackle in general and only very few results are available for particular cases in two dimensions, [2,3]. We recall the main ideas to investigate the isotropic problem: If we use the Helmholtz decomposition of vector fields (which works in all space dimensions larger than one) and split the displacement into potential and solenoidal part the problem decouples. The solenoidal part satisfies √ a wave equation with propagation speed µ, while the potential part satisfies a simpler thermo-elastic system (which is in fact the system of onedimensional thermo-elasticity). The latter one can be studied in detail, the first one satisfies well known estimates. Decay rates are determined by the wave part (in dimensions larger than or equal to two). The result of this reasoning can be collected in the following statement. Theorem 1.1. Let U (t, x) and θ(t, x) be the solutions to (1) for an isotropic medium (2) with Schwarz data U1 , U2 , θ0 ∈ S(Rn ). Then the apriori estimate p (3) ||(Ut , A(D)U, θ)||L∞ ≤ Cr (1 + t)−δn ||(hDiU0 , U1 , θ0 )||H 1,r holds true for r > n and with δ1 = 1/2 and δn = (n − 1)/2, n ≥ 2.

If we analyse this reasoning we see that in the isotropic case solutions are decomposed into hyperbolic parts (solving free wave equations) and parabolic parts (solving a somewhat simpler parabolic type thermo-elastic system). Such a decomposition does not exist in the case of anisotropic media, however we can do it micro-locally in certain directions. Our main strategy is to find the hyperbolic components in the solutions, which might determine decay rates, and to show that all other parts behave more nicely. 1.2. Reformulation of the problem and basic assumptions We apply a partial Fourier transform to (1) and write it as system of ordinary differential equations parameterised by the frequency variable ξ ∈ R2 . This gives ˆtt + A(ξ)U ˆ + iγξ θˆ = 0, U (4a) ˆt = 0, θˆt + κ|ξ|2 θˆ + iγξ · U ˆ (0, ·) = U ˆ1 , ˆt (0, ·) = U ˆ2 , U U

(4b) ˆ ·) = θˆ0 , θ(0,

(4c)

where A(D) is replaced by its symbol A(ξ). For the following we assume that

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497

(A1) the symbol is 2-homogeneous, A(ξ) = |ξ|2 A(η), η = ξ/|ξ| and depends real-analytically on η ∈ S1 ; (A2) it is symmetric and positive definit, A(η) = A∗ (η) > 0; (A3) its eigenvalues are distinct, spec A(η) = {ω12 (η), ω22 (η)}. We denote the eigenvalues in ascending order, ω1 (η) < ω2 (η). These assumptions imply that they are real-analytic in η ∈ S1 and we can find real analytic normalised eigenvectors rj (η) : S1 → S1 corresponding to them. Furthermore, we denote by M (η) = (r1 (η)|r2 (η)) the unitary diagonaliser of A(η), D(ξ) = diag (ω1 (ξ), ω2 (ξ)) the square root of the corresponding diagonal matrix and aj (ξ) = ξ · rj (η) auxiliary coupling functions. We are now in a position to formulate (4) as equivalent system of first ˆ T . Then ˆ , (Dt −D(ξ))M ∗ (η)U ˆ , θ) order. Let for this V = ((Dt +D(ξ))M ∗ (η)U Dt V = B(ξ)V

(5)

 iγa1 (ξ)  ω2 (ξ) iγa2 (ξ)     B(ξ) =  −ω1 (ξ) iγa1 (ξ) ,    −ω2 (ξ) iγa2 (ξ) iγ iγ iγ 2 − iγ 2 a1 (ξ) − 2 a2 (ξ) − 2 a1 (ξ) − 2 a2 (ξ) −iκ|ξ|

(6)

with 

ω1 (ξ)

which splits into the sum of the 1-homogeneous part B1 (ξ) and the 2homogeneous part B2 (ξ). Note, that B2 (ξ) has only the lower right corner entry and the functions aj (ξ) = |ξ|aj (η) indeed describe the coupling between the B1 and B2 . Of particular importance are directions η ∈ S1 , where one of the coupling functions vanishes. Definition 1.1. We call a direction η¯ ∈ S1 hyperbolic with respect to the eigenvalue ωj2 (η) if the corresponding coupling function aj (η) vanishes in η = η¯. We denote all non-hyperbolic directions as parabolic. Assumption (A3) essentially guarantees that the upper 4 × 4 block has distinct eigenvalues. We need one more (slightly technical looking) assumption for our approach, which guarantees that B1 has distinct eigenvalues. We assume (A4) γ 2 6= ω22 (η) − ω12 (η) for all hyperbolic directions w.r.t. ω22 (η).

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1.3. Aims Assume for the moment that the matrix B(ξ) has only simple eigenvalues, i.e. #spec B(ξ) = 5. Then we can represent the solution V (t, ξ) to (5) as X eitν(ξ) Pν(ξ) V0 (7) V (t, ξ) = ν(ξ)∈spec B(ξ)

with eigenprojections Pν(ξ) to the eigenvalues ν. Applying the inverse Fourier transform yields expressions for U (t, x) and θ(t, x) in terms of Fourier integrals. We will see that the eigenvalues ν(ξ) are in general complex and at least the interesting ones are always separated from the others. Our main aims are twofold. On the one hand we want to prove properties of these eigenvalues ν(ξ), especially bounds on their imaginary parts together with asymptotic expansions near interesting ξ (i.e. near all points where real eigenvalues may occur). Results of this type are discussed in Section 2. On the other hand, we want to apply the above given representation of solutions to derive dispersive estimates. This is based on multiplier theorems and tools from harmonic analysis related to stationary phase method and results are given in Section 3. Finally in Section 4 we provide some examples. Full details of all proofs can be found in [4,5]. 2. Spectral properties of B(ξ) 2.1. General properties The special structure of matrix B(ξ) allows to consider its characteristic polynomial and to derive certain properties from it. We will use that under assumptions (A1) and (A2) the following statement holds true: Lemma 2.1. (1) The matrix B(ξ), ξ 6= 0, has real eigenvalues if and only if the direction η = ξ/|ξ| is hyperbolic. (2) Let η¯ be an isolated hyperbolic direction w.r.to ω j2 (η). Then for the corresponding ‘hyperbolic’ eigenvalue of B(ξ) and each fixed |ξ| the limits lim

η→¯ η

Im ν(ξ) >0 a2j (η)

(8)

exist and are positive. The first statement explains the notion of hyperbolic directions. Let’s start looking at the case of isotropic media again. The matrix A(η) = µI + (λ + µ)η ⊗ η has the eigenvalues µ and λ + µ with corresponding

Dispersive Estimates in Anisotropic Thermoelasticity

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eigenvectors η ⊥ and η. Thus the coupling functions are given by a1 (η) ≡ 0 and a2 (η) ≡ 1. All directions are hyperbolic and the system decouples, as already mentioned in the introduction. In general we have only a finite number of hyperbolic directions (due to (A1) the coupling functions are real-analytic on the circle) and there the matrix B(ξ) has two real eigenvalues. In the neighbourhood of these directions we can decompose our vector V into a hyperbolic component and a parabolic one. For the latter one we will see that the imaginary part of the corresponding eigenvalues satisfies certain bounds and essential decay properties come from the hyperbolic component. 2.2. Asymptotic expansion for small frequencies We decompose the matrix B(ξ) into homogeneous components, B(ξ) = B1 (ξ) + B2 (ξ). If ξ is small, B1 (ξ) determines the spectrum of B(ξ) up to terms of order O(|ξ|2 ), provided that the eigenvalues of B1 (ξ) are distinct. Assumption (A4) implies exactly that: Lemma 2.2. Assume (A1) and (A2). Then (A4) is equivalent to #spec B1 (ξ) = 5. Furthermore, spec B1 (ξ) = {0, ±˜ ν1 (ξ), ±˜ ν2 (ξ)} with ω1 (ξ) ≤ ν˜1 (ξ) ≤ ω2 (ξ) ≤ ν˜2 (ξ). The proof of this statement follows essentially from the characteristic equation of B1 (ξ), which can be written as ν˜(ξ) = 0

or

X a2j (ξ) 1 . = γ2 ν˜2 (ξ) − ωj2 (ξ) j=1,2 j

(9)

If we assume (A4) we can apply a standard diagonalisation scheme to obtain full asymptotic expansions of the eigenvalues as ξ → 0. Lemma 2.3. (Reissig-Wirth, [4]) Assume (A1), (A2) and (A4). Then the eigenvalues of the matrix B(ξ) have full asymptotic expansions as ξ → 0. The first terms of these expansions are given by ν0 (ξ) = iκ|ξ|2 b0 (η) + O(|ξ|3 ),

νj± (ξ)

2

(10a) 3

= ±|ξ|˜ νj (η) + iκ|ξ| bj (η) + O(|ξ| ),

where the functions bj ∈ C ∞ (S1 ) are given by  −1 γ 2 a21 (η) γ 2 a22 (η) b0 (η) = 1 + + 2 >0 κ1 (η) ω2 (η)

(10b)

(11)

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J. Wirth

and ν˜j2 + ω12 (η) ν˜j2 + ω22 (η) 2 2 bj (η) = 1+γ 2 a21 (η) 2 +γ a (η) 2 (˜ νj −ω12 (η))2 (˜ νj2 −ω22 (η))2

!−1

≥ 0.

(12)

Furthermore, bj (η) > 0 if η is parabolic and bj0 (η) = 0 if η is hyperbolic with respect to the eigenvalue ωj20 . One particular consequence of this statement is of interest. In combination with Lemma 2.1 (1) and the continuity of spec B(ξ) it follows that no eigenvalue is allowed to have a negative imaginary part. Corollary 2.1. All eigenvalues of B(ξ) have non-negative imaginary part. Therefore, we see that there exists a constant C such that for all t the a-priori estimate ||V (t, ·)||2 ≤ C||V0 ||2

(13)

is valid. The appearing constant C depends on the norms of the eigenprojections. 2.3. Asymptotic expansion for large frequencies For large |ξ| we employ a two-step diagonalisation procedure, generalising that of [6]. This is necessary since B2 (ξ) is the main part and this matrix has only two distinct eigenvalues. We can diagonalise in a first step to 4,1-block-diagonal structure. It is essential that (A3) guarantees that the 1-homogeneous component of the upper 4 × 4 block has distinct eigenvalues and can therefore be diagonalised in a second step. This in turn implies full asymptotic expansions. Lemma 2.4. (Reissig-Wirth, [4]) Assume (A1) – (A3). Then the eigenvalues of the matrix B(ξ) have full asymptotic expansions as ξ → ∞. The first terms of these expansions are given by iγ 2 + O(|ξ|−1 ), κ iγ 2 2 a (η) + O(|ξ|−1 ). νj± (ξ) = ±|ξ|ωj (η) + 2κ j ν0 (ξ) = iκ|ξ|2 −

(14a) (14b)

As consequence we obtain that for |ξ| ≥ c > 0 and away from hyperbolic directions the eigenvalues are uniformly separated from the real axis. Corollary 2.2. Let η = ξ/|ξ| be parabolic. Then the eigenvalues of B(ξ) satisfy Im ν(ξ) ≥ Cη > 0 for all |ξ| ≥ c > 0.

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501

3. Dispersive estimates of solutions We will give only the main ideas of the proof and cite the results. From Corollary 2.2 it follows that large parabolic frequencies (outside a conic neighbour of hyperbolic directions) lead to exponential decay. Since we expect only polynomial rates, we can concentrate on the remaining directions. For small parabolic directions the imaginary part of the eigenvalues behaves essentially like |ξ|2 and therefore estimates can be done via H¨ older inequality and boundedness of Fourier transform ||eitν(D) f ||L∞ ≤ ||e−tIm ν(ξ) hξi−r ||L1 ||f ||H 1,r ≤ C(1 + t)−1/2 ||f ||H 1,r , (15) like for the heat equation. Near hyperbolic directions we have two different influences. On the one hand the imaginary part vanishes like a2j (η) ∼ ϕ2ℓ such that integration over a conic neighbourhood yields from Z cZ ǫ 2 2ℓ 1 (16a) e−t|ξ| ϕ dϕ|ξ|d|ξ| ∼ t− 2ℓ 0 −ǫ Z ∞Z ǫ 2ℓ 1 e−tϕ dϕ|ξ|1−r d|ξ| ∼ t− 2ℓ , r>2 (16b) −ǫ

c

1/(2ℓ)

as decay rate t where ℓ denotes the vanishing order of the coupling function in the hyperbolic direction. On the other hand, the real part of the eigenvalue leads to oscillations in the representation of solutions as Fourier integrals. These oscillations may be studied by means of stationary phase method and may also lead to a decay rate. However, only one of these two strategies can be used. The above mentioned one works for all vanishing orders ℓ, the stationary phase estimate presented below is only possible if the vanishing order ℓ stands in some relation to the contact order γ¯ of the Fresnel curve [ S = { ξ ∈ R2 : 1 ∈ spec A(ξ) } = { ωj−1 (η)η : η ∈ S1 } (17) j

to its tangent in the corresponding direction η and for the corresponding sheet.

Lemma 3.1. (Reissig-Wirth, [4]) Let η¯ be hyperbolic with respect to the η ) the contact order between the curve eigenvalue ωj20 (η) and denote γ¯j0 (¯ ωj−1 η ) be the vanishing order of (η)η and its tangent in η ¯ . Let further ℓj0 (¯ 0 η ), then the estimate η ) < 2ℓj0 (¯ aj0 (η) in η¯. If γ¯j0 (¯ ±

1

||eitνj0 (D) f ||L∞ ≤ C(1 + t)− γ¯ ||f ||H 1,r is valid for any r > 2.

(18)

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The method of proof is an application of van der Corput Lemma, while the restriction on the contact order stems from the fact that we use the imaginary part of νj±0 (ξ) as part of the amplitude. The following statement collects our main results on dispersive estimates. Theorem 3.1. (Reissig-Wirth, [4]) Let η¯ ∈ S1 be a fixed direction, χ ∈ C0∞ (S1 ) a cutoff function for a small neighbourhood of η¯ and χ(ξ) = χ(ξ/|ξ|). Then the following micro-local dispersive estimates hold true: (1) If η¯ is hyperbolic with respect to ωj2 (¯ η ) and 2ℓj (¯ η ) ≤ γ¯j (¯ η ), then for any r > 2 we find a constant C such that p kχ(D)(Ut , A(D)U, θ)(t, ·)k∞ p
− 1¯ kχ(D)(Ut , A(D)U, θ)(0, ·)kH 1,r . (19) ≤ C(1 + t) 2ℓj (η)

(2) If η¯ is hyperbolic with respect to ωj2 (¯ η ) and 2ℓj (η) > γ¯j (¯ η ), then for any r > 2 we find a constant C such that p kχ(D)(Ut , A(D)U, θ)(t, ·)k∞ p
− 1¯ kχ(D)(Ut , A(D)U, θ)(0, ·)kH 1,r . (20) ≤ C(1 + t) γ¯j (η) (3) If η¯ is not hyperbolic, then for any r > 2 we find a constant C such that p kχ(D)(Ut , A(D)U, θ)(t, ·)k∞ p
≤ C(1 + t)−1 kχ(D)(Ut , A(D)U, θ)(0, ·)kH 1,r . (21) 4. Examples We will give some concrete applications of our general results. For this we consider the class of elastic operators defined by A(ξ) = D(ξ)SD∗ (ξ) with     τ1 λ σ1 ξ1 and S =  λ τ2 σ2  . (22) D(ξ) =  ξ2  σ1 σ2 µ ξ2 ξ1

For physical reasons one assumes that S is positive. The choice τ1 = τ2 = λ + 2µ and σ1 = σ2 = 0 leads to isotropic media. We discussed them in the introduction. We will concentrate on anisotropic examples here. They as well as the following pictures are taken from [5]. Depicted are the Fresnel curves for the media together with the coupling cunctions (as curves (2 + aj (η)η for η ∈ S1 ).

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Cubic media are defined by the conditions τ1 = τ2 and σ1 = σ2 = 0. For cubic media we obtain eight hyperbolic directions corresponding to the symmetry axes of a square. In all these directions the coupling functions vanish to first order and therefore Corollary 4.1. Cubic media have the dispersive decay rate t−1/2 .

Rhombic media satisfy σ1 = σ2 . They may have either four or eight hyperbolic directions. Except for the case where (λ+2µ−τ1 )(λ+2µ−τ2 ) = 0 the coupling functions vanish to first order and the dispersive decay rates are t−1/2 . Only in the latter case, where one coupling function vanishes to third order, we have to check the contact orders for the Fresnel curve in the corresponding direction. We assume τ1 = λ + 2µ. Following [5] we obtain the contact order 2 unless τ2 = λ + µ, where the contact order is 4. Corollary 4.2. Rhombic media have the dispersive decay rate t−1/2 except for the case τ1 = λ + 2µ, τ2 = λ + µ (or the values interchanged), where the dispersive decay rate is t−1/4 .

A special anisotropic medium. We chose τ1 = τ2 = λ + µ, σ1 = 0 and σ2 = µ. In this case the medium has six hyperbolic directions (forming a regular hexagon), for two of them the coupling function vanishes to second

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order. A close examination of the eigenvalues of A(ξ) in these directions shows that the Fresnel curve has an inflection point such that the order of contact is 3. Therefore Corollary 4.3. This medium has the dispersive decay rate t−1/3 .

5. Concluding remarks We will conclude this note with some short remarks on problems occurring for higher dimensions. Unlike for isotropic media, where all dimensions larger than one can be treated similarly, the complexity of the anisotropic problem increases with dimension. While in two dimensions assumption (A3) is not very restrictive and excludes only some particular choices of parameters, starting from three dimensions all interesting cases of anisotropic media have directions where (A3) is violated. Away from these multiplicities our strategy applies and yields asymptotic expansions of eigenvalues (of a similar form) and micro-localised dispersive estimates. The hard problem to tackle is the study of the neighbourhood of degenerate directions. References 1. A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermomechanics, Proc. R. Soc. London Ser. A 432, 171–194 (1991). 2. J. Borkenstein, Lp –Lq -Absch¨ atzungen der linearen Thermoelastizit¨ atsgleichungen f¨ ur kubische Medien im R2 , Diplomarbeit, Universit¨ at Bonn (1993). 3. M. S. Doll, Zur Dynamik (magneto-)thermoelastischer Systeme im R2 , Dissertation, Universit¨ at Konstanz (2004). 4. M. Reissig and J. Wirth, Anisotropic thermo-elasiticity in 2D – Part I: A unified treatment, Asymptotic Anal. (to appear). 5. J. Wirth, Anisotropic thermo-elasiticity in 2D – Part II: Applications, Asymptotic Anal. (to appear). 6. Y.-G. Wang, Microlocal analysis in nonlinear thermoelasticity, Nonlinear Anal. 54, 683–705 (2003).

Session 11

Oscillation of Functional Differential and Difference Equations

SESSION EDITORS L. Berezansky A. Zafer

Ben-Gurion University of the Negev, Be’er-Sheva, Israel Middle East Technical University, Ankara, Turkey

507

OSCILLATION CRITERIA FOR A CLASS OF THIRD-ORDER NONLINEAR DELAY DIFFERENTIAL EQUATIONS ˙ M. F. AKTAS ¸ † and A. TIRYAK I˙ ∗ Department of Mathematics, Gazi University, Faculty of Arts and Sciences, Teknikokullar, Ankara, 06500, Turkey † E-mail: [email protected] ∗ E-mail: [email protected] Using a Riccati-type transformation and the integral averaging technique, some new oscillation criteria for third-order delay differential equations of the form 
′ ′ + p (t) y ′ + q (t) f (y (g (t))) = 0 r2 (t) r1 (t) y ′ are established. The results obtained essentially improve earlier results [11]. Keywords: Oscillation; Third order; Delay differential equation; Asymptotic behavior; Riccati type transformation and integral averaging.

1. Introduction In this paper we consider nonlinear third-order functional differential equations of the form   ′ ′ r2 (t) (r1 (t) y ′ ) + p (t) y ′ + q (t) f (y (g (t))) = 0, t ≥ T0 , (1) where T0 > 0 is a fixed real number; and r1 , r2 , p, q ∈ C[T0 , ∞) such that r1 > 0, r2 > 0, p(t) ≥ 0, q (t) > 0, g ∈ C 1 [T0 , ∞) satisfies g ′ (t) ≥ 0, g (t) ≤ t, and g (t) → ∞ as t → ∞ and f ∈ C (R, R) satisfies f (u) /u ≥ K > 0 for u 6= 0. We consider only those solutions of Eq. (1) which are defined and nontrivial for all sufficiently large t. Such a solution is called oscillatory if it has arbitrarily large zeros, otherwise it is called nonoscillatory. Note that if y is a solution of Eq. (1), then −y is a solution of   ′ ′ r2 (t) (r1 (t) y ′ ) + p (t) y ′ + q (t) f ∗ (y (g (t))) = 0,

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where f ∗ (y) = −f (−y) and yf ∗ (y) > 0 for all y 6= 0. Since f ∗ and f are of the same class, we may restrict our attention only to a positive solution of Eq. (1) whenever a nonoscillatory solution of Eq. (1) is concerned. In the paper, we improve the results obtained in Tiryaki and Akta¸s [11]. For example, we see that their results are not applicable to Euler equation q0 p0 y ′′′ + 2 y ′ + 3 y = 0. t t By using a Riccati type transformation and the integral averaging technique, we establish some new sufficient conditions which insure that every solution of Eq. (1) is oscillatory or converges to zero. For some recent results on these topics we refer the reader to references [1–5,8,9,12,13] and the references cited therein. The paper is organized as follows. In next section, we shall present some lemmas which are useful in the proof of our main results. In the last section, we establish sufficient conditions and also conditions of Philos-type for oscillation of Eq. (1) and some examples are considered to illustrate the importance of the results. 2. Some Preliminary Lemmas For the sake of brevity, we define ′

L0 y (t) = y (t) ,

Li y (t) = ri (t) (Li−1 y (t)) ,

i = 1, 2,

′

L3 y (t) = (L2 y (t)) for t ∈ [T0 , ∞). So Eq. (1) can be written as L3 y (t) + p (t) y ′ + q (t) f (y (g (t))) = 0. Define the functions R1 (t, s) =

Z

s

t

du , r1 (u)

R2 (t, s) =

Z

s

for T0 ≤ s ≤ t < ∞. Throughout this paper, we assume that

t

du r2 (u)

R1 (t, T0 ) → ∞ as t → ∞

(2)

R2 (t, T0 ) → ∞ as t → ∞.

(3)

and

Lemma 2.1. Suppose that ′

(r2 (t)z ′ ) +

p (t) z=0 r1 (t)

(4)

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509

is nonoscillatory. If y is a nonoscillatory solution of Eq. (1), then there exists a T1 ≥ T0 such that either y (t) L1 y (t) > 0 or y (t) L1 y (t) < 0 for all t ≥ T1 . Lemma 2.2. Let the assumption (3) holds. If y be a nonoscillatory solution of Eq. (1) which satisfies y (t) L1 y (t) ≥ 0 for all large t, then there exists a T1 ≥ T0 such that L0 y (t) L3 y (t) ≤ 0

L0 y (t) Lk y (t) > 0, k = 0, 1, 2;

(5)

for all t ≥ T1 . A nonoscillatory solution y of Eq. (1) is said to have property V2 if it satisfies the inequalities (5). Lemma 2.3. Assume that there exists a continuous function g∗ (t) defined on [T0 , ∞) such that g∗ (t) ≤ g (t) for all t ≥ T0 and that lim g∗ (t) = ∞. t→∞

(6)

If y is a solution of Eq. (1) with property V2 , then there exists a T1 ≥ T0 such that L1 y (g (t)) ≥ R2 (g (t) , g∗ (t)) L2 y (t) for all t ≥ T1 . The proofs of above lemmas proceed along the lines of that of Tiryaki and Akta¸s [11, Lemmas 1-3] and hence are omitted. Lemma 2.4. Let ρ2 be a sufficiently smooth positive function defined on [T0, ∞) and set ′

ϕ (t) = r1 (t) (r2 (t) ρ′2 (t)) + ρ2 (t) p (t) .

Suppose that there exists a T1 ≥ T0 such that Z

∞

T1

ρ′2 (t) ≥ 0,

ϕ (t) ≥ 0,

(7)

(Kρ2 (s) q (s) − ϕ′ (s)) ds = ∞,

(8)

where Kρ2 (t) q (t) − ϕ′ (t) ≥ 0 for all t ∈ [T1 , ∞) and not identically zero in any subinterval of [T1 , ∞). If (2) holds and y is a nonoscillatory solution of Eq. (1) which satisfies y (t) L1 y (t) ≤ 0 for all t ≥ T1 , then lim y (t) = 0. t→∞

Proof. Let y be a nonoscillatory solution of Eq. (1). Without loss of generality, we may assume that y (t) > 0 and y (g (t)) > 0 for t ≥ t0 for some t0 sufficiently large. Then L1 y (t) becomes negative for all t ≥ t1 for some t1 ≥ t0 . Let lim y (t) = λ ≥ 0. Assume that λ 6= 0. t→∞

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Multiplying (1) through by ρ2 (t) and integrating from t1 to t, we obtain ρ2 (t) L2 y (t) = C1 + ρ′2 (t) r2 (t) L1 y (t) − ϕ (t) y (t)   Z t f (y (g (s))) ′ ρ2 (s) q (s) − ϕ (s) ds, y (s) − y (s) t1 where C1 is a constant. It follows in view of (7) that Z t (Kρ2 (s) q (s) − ϕ′ (s)) ds. ρ2 (t) L2 y (t) ≤ C1 − λ

(9)

t1

Employing (8) we see from (9) that L2 y (t) must take on negative values for t sufficiently large. By using (2) we see that y (t) must be eventually negative, a contradiction. Hence λ = 0. 3. Main Results Theorem 3.1. Assume that (2)-(3) and (6)-(8) hold, and that Eq. (4) is nonoscillatory. If there exists an eventually positive function ρ 1 (t) ∈ C 1 [T0 , ∞) such that Z t Kρ1 (s) q (s) lim sup t→∞

−

r1 (g (s)) r12 (s)

(ρ′1

T

 (s) r1 (s) − ρ1 (s) p (s) R2 (g (s) , g∗ (s)))2 ds = ∞ 4ρ1 (s) R2 (g (s) , g∗ (s)) g ′ (s)

(10)

for every T > T0 , then every solution y (t) of Eq. (1) is either oscillatory or satisfies y (t) → 0 as t → ∞. Proof. Let y be a nonoscillatory solution of Eq. (1). Without loss of generality, we may assume that y (t) > 0 and y (g (t)) > 0 for t ≥ t0 ≥ T0 . From Lemma 2.1 it follows that L1 y (t) > 0 or L1 y (t) < 0 for t ≥ t1 ≥ t0 . If L1 y (t) > 0 for t ≥ t1 , then y has property V2 for large t from Lemma 2.2. We define L2 y (t) (11) ω (t) = ρ1 (t) y (g (t)) for t ≥ t1 . In view of Eq. (1) by employing Lemma 2.3 (increasing the size of t1 if necessary) we have   ω ′ (t) ≤ −Kρ1 (t) q (t) − A (t) ω 2 (t) − ω (t) B (t) , (12) where

A (t) =

R2 (g (t) , g∗ (t)) g ′ (t) , r1 (g (t)) ρ1 (t)

B (t) =

ρ′1 (t) R2 (g (t) , g∗ (t)) − p (t) . ρ1 (t) r1 (t)

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511

Completing the square in (12) we obtain ω ′ (t) ≤ −Kρ1 (t) q (t) +

B 2 (t) . 4A (t)

(13)

Integrating (13) from t1 to t leads to  Z t B 2 (s) ds ≤ ω (t1 ) Kρ1 (s) q (s) − 4A (s) t1 a contradiction with (10). Let y (t) > 0, L1 y (t) < 0, t ≥ t1 . By Lemma 2.4 we have limt→∞ y (t) = 0. The proof is complete. Example 3.1. Consider the third order differential equation p0 q0 y ′′′ (t) + α y ′ (t) + β y (λt) = 0, t ≥ 1, 0 < λ ≤ 1, t t

(14)

where p0 , q0 , α, and β are some constants. If we choose ρ1 (t) = ρ2 (t) = t2 , λt and g∗ (t) = , from Theorem 3.1, every solution y (t) of Eq. (14) is either 2 1 oscillatory or satisfies y (t) → 0 as t → ∞ for 0 ≤ p0 ≤ , q0 > 0, α ≥ 2, 4 and β < 3. On the other hand, if α = 2 and β = 3 we have the same conclusion provided that 2

(4 − λp0 ) . 8λ2 1 25 1 In fact, if we take λ = 1, p0 = and q0 = , y1 (t) = , y2 (t) = 4 t   4  3 3 2 2 ln t , and y3 (t) = t sin ln t are solutions of Euler Eq. (14). t cos 2 2 q0 >

Note that Eq. (14) is a Euler equation, and the results of Tiryaki and Akta¸s [11] are not applicable to it. Example 3.2. Consider the third order delay differential equation 1 1 ′ y (t) + 3 y (t − 1) (4 − cos y (t − 1)) = 0, t ≥ 2. (15) t3 t t If we choose ρ1 (t) = ρ2 (t) = t2 , and g∗ (t) = − 1, from Theorem 3.1, 2 every solution y of (15) is either oscillatory or satisfied y (t) → 0 as t → ∞. R∞ Remark 3.1. When ϕ′ (t) ≤ 0, we can take ρ2 (s) q (s) ds = ∞ to replace (8). y ′′′ (t) +

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We will establish new oscillation criteria for Eq. (1) by using the integral averaging technique due to Philos [7]. We need the class of functions Re. Let D0 = {(t, s) : t > s ≥ T0 } and D = {(t, s) : t ≥ s ≥ T0 } .

A function H ∈ C (D, R) is said to belongs to the class Re if (i) H (t, t) = 0 for all t ≥ T0 ; H (t, s) > 0 for all (t, s) ∈ D0 ; (ii) H (t, s) has a continuous and nonpositive partial derivative on D0 with respect to the second variable, and for a positive continuous function h (t, s) p ∂H (t, s) = h (t, s) H (t, s) for all (t, s) ∈ D0 . − ∂s n For the choice H (t, s) = (t − s) , (n ≥ 1) , the Philos-type conditions reduce to the Kamenev-type ones. Other choices of H include n  √
n √ n t , t− s , et−s − es−t . ln s

Theorem 3.2. Assume that (2)-(3) and (6)-(8) hold, and that Eq. (4) is nonoscillatory. If there exist an eventually positive function ρ 1 (t) ∈ C 1 [T0 , ∞), and a function H ∈ Re such that  Z t Q2 (t, s) 1 ds = ∞ (16) Kρ1 (s) H (t, s) q (s) − lim sup 4A (s) t→∞ H (t, T ) T for every T > T0 , where

R2 (g (t) , g∗ (t)) g ′ (t) , and r1 (g (t)) ρ1 (t)   ′ p R2 (g (s) , g∗ (s)) ρ1 (s) , − p (s) Q (t, s) = h (t, s) − H (t, s) ρ1 (s) r1 (s) A (t) =

then every solution y (t) of Eq. (1) is either oscillatory or satisfies y (t) → 0 as t → ∞. Corollary 3.1. Assume that (2)-(3) and (6)-(8) hold, and that Eq. (4) is nonoscillatory. If there exist an eventually positive function ρ 1 (t) ∈ C 1 [T0 , ∞), and a function H ∈ Re such that for every T > T0 , Z t 1 lim sup ρ1 (s) H (t, s) q (s) ds = ∞, t→∞ H (t, T ) T Z t 2 Q (t, s) 1 ds < ∞, lim sup A (s) t→∞ H (t, T ) T

then every solution y of Eq. (1) is oscillatory or satisfies limt→∞ y (t) = 0.

Oscillation Criteria for Third-Order Nonlinear Delay Differential Equations

Example 3.3. Consider the third-order differential equation  ′′   1 1 1 1 ′ = 0, t ≥ 1. y + 3 y′ + 2 y 1 + t 4t t 1 + y2

513

(17)

t , from 2 Theorem 3.2, every solution y of (17) is either oscillatory or satisfied y (t) → 0 as t → ∞. 2

If we choose ρ1 (t) = ρ2 (t) = t, H (t, s) = (t − s) , and g∗ (t) =

Example 3.4. Consider the third-order delay differential equation   ′ ′ 1 1 ′ 1 √  y(√t) 1 ′ t e = 0, t > 1, (18) y (t) y (t) + y + t t4 t9 t2 ′  1 1 ′ u z + 5 z = 0 is nonoscillawhere f (u) = ue with K = 1. Equation t t tory (see Parhi [6] and Swanson [10, rp. 71]. If we choose ρ1 (t) = ρ2 (t) = t 2 t, H (t, s) = (t − s) , and g∗ (t) = , from Theorem 3.2, every solution 2 y of (18) is either oscillatory or satisfied y (t) → 0 as t → ∞. The following two results provide alternative oscillation criteria. Theorem 3.3. Let all the assumptions, except (16), of Theorem 3.2 hold. Further, for every T > T0 , let   H (t, s) 0 < inf lim inf ≤ ∞, t→∞ H (t, T ) s≥T and lim sup t→∞

1 H (t, T )

Z

t

T

Q2 (t, s) ds < ∞ . A (s)

(19)

Let also Ψ ∈ C ([T0 , ∞), R) be such that Z ∞ Ψ2+ (s) A (s) ds = ∞, where Ψ+ (s) = max {Ψ (t) , 0} and  Z t 1 Q2 (t, s) lim sup ds ≥ supΨ (t) . Kρ1 (s) H (t, s) q (s) − 4A(s) t→∞ H (t, T ) T t≥T Then every solution y (t) of Eq. (1) is either oscillatory or satisfies y (t) → 0 as t → ∞.

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Theorem 3.4. Let all the assumptions of Theorem 3.3 hold, except condition (19), which is replaced by Z t 1 H (t, s) ρ1 (s) q (s) ds < ∞ lim sup t→∞ H (t, T ) T for every T > T0 . Then, every solution y (t) of Eq. (1) is either oscillatory or satisfies y (t) → 0 as t → ∞. The proofs of above Theorems proceed along the lines of those of Tiryaki and Akta¸s [11] and hence are omitted. References 1. R. P. Agarwal, S. R. Grace and D. O’Regan, Oscillation Theory for Difference and Functional Differential Equations, Kluwer Acad. Publ., Dordrecht, 2000. 2. M. Cecchi and M. Marini, On the oscillatory behavior of a third order nonlinear differential equation, Nonlinear Anal. 15 (1990), 141–153. 3. T. Kusano and H. Onose, Oscillation theorems for delay equations of arbitrary order, Hiroshima Math. J. 2 (1972), 263–270. 4. N. Parhi and P. Das, Oscillatory and asymptotic behavior of a class of nonlinear functional differential equations of third order, Bull. Calcutta Math. Soc. 86 (1994), 253–266. 5. N. Parhi and P. Das, Oscillatory and asymptotic behaviour of a class of nonlinear differential equations of third order, Acta Math. Sci. 18 (1998), 95–106. 6. N. Parhi and S. K. Nayak, Nonoscillation of second-order nonhomogeneous differential equations, J. Math. Anal. Appl. 102 (1984), 62–74. 7. C. G. Philos, Oscillation theorems for linear differential equation of second order, Arch. Math. 53 (1989), 482–492. 8. S. H. Saker, Oscillation criteria of third-order nonlinear delay differential equations, Math. Slovaca 56 (2006), 433–450. ˇ 9. A. Skerlik, An integral condition of oscillation for equation y ′′′ + p(t)y ′ + q(t)y = 0 with nonnegative coefficients, Arch. Math. 31 (1995), 155–161. 10. C. A. Swanson, Comparison and Oscillation Theory of Linear Differential Equations, Academic Press, New York, 1968. 11. A. Tiryaki and M. F. Akta¸s, Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping, J. Math. Anal. Appl. 325 (2007), 54–68. 12. A. Tiryaki and S. Yaman, Asymptotic behaviour of a class of nonlinear functional differential equations of third order, Appl. Math. Lett. 14 (2001), 327– 332. 13. A. Tiryaki and S. Yaman, Oscillatory behaviour of a class of nonlinear differential equations of third order, Acta Math. Sci. B 21 (2001), 182–188.

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A CRITERION FOR EXISTENCE OF POSITIVE SOLUTIONS OF THE EQUATION y(t) ˙ = −p(t)y α (t − r) J. DIBL´IK Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technick´ a 8, Brno, 616 00, Czech Republic E-mail: [email protected] ˇ CKOV ˇ ´ M. R˚ UZI A Department of Applied Mathematics, Faculty of Science, ˇ Zilina University, Hurbanova 15, ˇ Zilina, 010 26, Slovak Republic E-mail: [email protected] Z. SVOBODA Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technick´ a 8, Brno, 616 00, Czech Republic E-mail: [email protected] We consider the first-order differential equation containing delay y(t) ˙ = −p(t)y α (t − r) with p : (t0 , ∞) → (0, ∞), α ∈ R, α ≥ 1 and r ∈ R, r > 0. A criterion for the existence of positive solutions (for t → ∞) is derived. Connections between the nonlinear case (α 6= 1) and the linear case (α = 1) are discussed. Keywords: Positive solution; delay; positivity criterion.

1. Preliminaries Let C([a, b], Rn ) where a, b ∈ R, a < b, R = (−∞, +∞) be the Banach space of continuous functions mapping the interval [a, b] into Rn . In the case a = −r < 0, b = 0, we shall denote this space as C, that is, C = C([−r, 0], Rn ). Let y(t) ˙ = f (t, yt )

(1)

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J. Dibl´ık, M. R˚ uˇ ziˇ ckov´ a & Z. Svoboda

be a system of retarded functional differential equations where f : Ω 7→ Rn is a quasibounded continuous mapping satisfying a local Lipschitz condition with respect to the second argument and Ω is an open subset in R × C. If σ ∈ R, A ≥ 0 and y ∈ C([σ−r, σ+A], Rn ), then, for each t ∈ [σ, σ+A], we define yt ∈ C by yt (θ) = y(t+θ), θ ∈ [−r, 0]. If necessary, we shall assume that the derivatives in the system (1) are right-sided. In accordance with [14], a function y(t) is said to be a solution of system (1) on [σ − r, σ + A) if there are σ ∈ R and A > 0 such that y ∈ C([σ − r, σ + A), Rn ), (t, yt ) ∈ Ω and y(t) satisfies the system (1) for t ∈ [σ, σ + A). For given σ ∈ R, ϕ ∈ C, we say that y(σ, ϕ) is a solution of the system (1) through (σ, ϕ) ∈ Ω if there is an A > 0 such that y(σ, ϕ) is a solution of the system (1) on [σ − r, σ + A) and yσ (σ, ϕ) = ϕ. In view of the above conditions, each element (σ, ϕ) ∈ Ω determines a unique solution y(σ, ϕ) of the system (1) through (σ, ϕ) ∈ Ω on its maximal interval of existence Iσ,ϕ = [σ, a), σ < a ≤ ∞ which depends continuously on the initial data [14]. A solution y(σ, ϕ) of the system (1) is said to be positive if yi (σ, ϕ) > 0

(2)

on [σ −r, σ]∪Iσ,ϕ for each i = 1, 2, . . . , n. A nontrivial solution y(σ, ϕ) of (1) is said to be oscillatory on Iσ,ϕ (subject to Iσ,ϕ = [σ, ∞)) if at least one inequality (2) does not hold on any subinterval [σ1 , ∞) ⊂ [σ, ∞), σ1 ≥ σ. 2. The Aim of the Paper Let us consider the following scalar equation with a single delay of the form y(t) ˙ = −p(t)y α (t − r)

(3)

where p : (t0 , ∞) → (0, ∞), t0 ∈ R is a continuous function, α ∈ R, α ≥ 1 and r ∈ R, r > 0. Let t∗ ∈ R, t∗ > t0 + r. We will derive a new necessary and sufficient criterion for the existence of positive solutions of retarded differential equation (3) for t → ∞. Namely, we prove the following Theorem 2.1. For the existence of a positive solution y = y(t) on [t∗ − r, ∞) of the equation (3), the existence is necessary and sufficient of a positive constant k and a locally integrable function λ : [t∗ − r, ∞) → R continuous on [t∗ − r, t∗ ) ∪ [t∗ , ∞) and satisfying the integral inequality  Z t  Z t λ(t) ≥ k α−1 exp α p(s)λ(s) ds + (1 − α) p(s)λ(s) ds (4) t−r

where t ∈ [t∗ , ∞).

t∗

A Criterion for Existence of Positive Solutions

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In recent literature, one can find numerous results concerning positive solutions (or their absence) of retarded differential equations. We mention e.g., papers by L. Berezansky and E. Braverman [2], L. Berezansky, Yu. Domshlak and E. Braverman [3], J. Dibl´ık [4,5], J. Dibl´ık and N. Koksch ´ [6], J. Dibl´ık and Z. Svoboda [7,8] Y. Domshlak and I.P. Stavroulakis [9], A. Elbert and I.P. Stavroulakis [10] and books by R.P.Agarwal, M.Bohner and Wan-Tong Li [1], L.H. Erbe, Q. Kong and B.G. Zhang [11], K. Gopalsamy [12], and I. Gy¨ ori and G. Ladas [13]. If α = 1, then (3) turns into a linear equation. The necessary and sufficient condition for the existence of positive solutions for (3) with α = 1 is well known. We recall it here. From Theorem 5 in [8] (compare with [1,4,11] as well) it follows: Theorem 2.2. For the existence of a positive solution y = y(t) on [t∗ − r, ∞) of the equation (3) with α = 1 the existence is necessary and sufficient of a locally integrable function λ : [t∗ −r, ∞) → R continuous on [t∗ −r, t∗ )∪ [t∗ , ∞) and satisfying the integral inequality Z t  λ(t) ≥ exp p(s)λ(s)ds (5) t−r

∗

for t ≥ t . We see that (4) reduces to (5) just for α = 1. An interesting fact in this connection is discussed in the paper. It is surprising that the inequality (5) with the function λ having almost the same properties is sufficient for the existence of a positive solution of (3). At the same time, an inequality similar to (5) is necessary for the existence of a positive solution of (3) if α ∈ (0, 1). 3. Auxiliary Results In [8], problems are considered on positive solutions of what is called ptype retarded functional differential equations. In the proof of Theorem 2.1, we will employ two results (Lemma 3.1 and Lemma 3.2 below) created by adapting of Lemma 1 and Lemma 2 from [8] for the case of retarded functional differential equations (i.e., in the sense described in the Section 1 (Preliminaries)). Let us introduce the vectors ρ(t) = (ρ1 (t), ρ2 (t), . . . , ρn (t))T : [t∗ − r, ∞) → Rn δ(t) = (δ1 (t), δ2 (t), . . . , δn (t))T : [t∗ − r, ∞) → Rn

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continuous on [t∗ − r, ∞) and continuously differentiable on [t∗ , ∞), such that ρi (t) < δi (t), i = 1, 2, . . . , n. In the sequel, such type of inequalities between coordinates of vectors will be abbreviated to ρ(t) ≪ δ(t) etc. Let us, moreover, define the set ω := {(t, y) : t ≥ t∗ − r, ρ(t) ≪ y ≪ δ(t)} and let Ω = (t0 , ∞) × C be put in the next. Lemma 3.1. Assume the following: (i ) For all j = 1, 2, . . . , p and all π ∈ C for which (t + θ, π(θ)) ∈ ω, θ ∈ [−r, 0), δj′ (t) < fj (t, π) when πj (0) = δj (t)

(6)

ρ′j (t) > fj (t, π) when πj (0) = ρj (t).

(7)

and

(ii ) For all j = p + 1, p + 2, . . . , n and all π ∈ C for which (t + θ, π(θ)) ∈ ω, θ ∈ [−r, 0), ρ′j (t) < fj (t, π) when πj (0) = ρj (t) and δj′ (t) > fj (t, π) when πj (0) = δj (t). Then there exists an uncountable set Y of solutions of the system (1) on [t∗ − r, ∞) such that, for each y(t) ∈ Y, ρ(t) ≪ y(t) ≪ δ(t),

t ∈ [t∗ − r, ∞)

(8)

holds. Remark 3.1. The number p in Theorem 3.1 and in the sequel can be equal, apart from other values, to zero or to n, too. In the first case, the condition (i) is omitted and, in the second one, the condition (ii) is omitted. By analogy, the equality p = 0 or p = n is understood in further investigations. Note, moreover, that, in the case ρ(t) ≥ 0, we deal, as it follows from (8), with positive solutions. Definition 3.1. We say that a continuous functional g : Ω → Rn is istrongly decreasing (or i-strongly increasing), i ∈ {1, 2, . . . , n} if, for each (t, ϕ) ∈ Ω and (t, ψ) ∈ Ω such that ϕ(θ) ≪ ψ(θ) where θ ∈ [−r, 0) and ϕi (0) = ψi (0),

A Criterion for Existence of Positive Solutions

519

the inequality gi (t, ϕ) > gi (t, ψ) (or gi (t, ϕ) < gi (t, ψ)) holds. Lemma 3.2. Let the functional f : Ω → Rn be continuous, locally Lipschitzian with respect to the second argument, quasibounded and, moreover, (i ) f be i-strongly decreasing if i = 1, . . . , p and i-strongly increasing if i = p + 1, . . . , n. (ii ) fi (t, 0) ≤ 0 for i = 1, . . . , p and fi (t, 0) ≥ 0 for i = p + 1, . . . , n if (t, 0) ∈ Ω. If the system (1) has a positive solution y = y(t) on [t∗ − r, ∞), then it also has a positive solution y = y ∗ (t) on [t∗ , ∞) that is continuously differentiable on [t∗ − r, t∗ ). Proof of Theorem 2.1. Necessity. Let y be a positive solution of (3) on [t∗ − r, ∞). We set f (t, π) := −p(t)π α (−r) where (t, π) ∈ Ω. Then f is a strongly decreasing functional and f (t, 0) = 0. It is easy to verify that all (remaining) assumptions of Lemma 3.2 (with i = p = 1, i.e., we omit the condition (ii)) hold. Following this conclusion, there exists a positive solution y = y ∗ (t) of (3) on [t∗ − r, ∞) that is continuously differentiable not only on [t∗ , ∞) but on on [t∗ − r, t∗ ), too. We define  y ∗ ′ (t)   if t ∈ [t∗ − r, t∗ ) ∪ (t∗ , ∞), −   y ∗ (t) L(t) :=  y ∗ ′ (t∗ + 0)   if t = t∗ . − y ∗ (t∗ )

It is easy to verify that, on [t∗ − r, ∞),  Z t  y ∗ (t) ≡ y ∗ (t∗ ) exp − L(s)ds .

(9)

t∗

We substitute the representation (9) into (3). On [t∗ , ∞), we get     Z t Z t−r L(s)ds L(s)ds ≡ −p(t)(y ∗ (t∗ ))α exp −α −y ∗ (t∗ )L(t) exp − t∗

t∗

or, equivalently, λ(t) ≡ k

α−1

 Z exp α

t t−r

p(t)λ(s)ds + (1 − α)

Z

t

p(t)λ(s)ds t∗



(10)

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where k := y ∗ (t∗ ) > 0 and L(t) := p(t)λ(t). It means that there are such positive k and a locally integrable function λ : [t∗ − r, ∞) → R continuous on [t∗ − r, t∗ ) ∪ [t∗ , ∞) that (4) holds on [t∗ , ∞). Sufficiency. This part of the proof uses Lemma 3.1. In its formulation we put j = 1, p = n = 1 (i.e., we omit condition (ii) in Lemma 3.1), π ≡ π1 (t), f (t, π) ≡ f1 (t, π) := −p(t)π(−r) and we define continuous on [t∗ − r, ∞) and continuously differentiable on [t∗ , ∞) functions ρ(t) ≡ ρ1 (t) := 0,   Z t p(s)λ(s) ds δ(t) ≡ δ1 (t) := k exp − t∗

where the constant k and the function λ, mentioned in Theorem 2.1, are used. In this case, the set ω turns into   Z t   ω := (t, y) : t ≥ t∗ − r, ρ(t) = 0 < y < k exp − p(s)λ(s) ds = δ(t) . t∗

We will verify inequality (6). Assuming πj (0) = δj (t) we get for t ≥ t∗ : f (t, π) − δ ′ (t) = − p(t)π α (−r) − δ ′ (t)

  Z t p(s)λ(s) ds . = − p(t)π α (−r) + kp(t)λ(t) exp −

(11)

t∗

Since, because of the assumptions of Lemma 3.1,

(t + θ, π(θ)) ∈ ω, θ ∈ [−r, 0), we have

 Z −π(−r) > −δ(t − r) = −k exp −

t−r

t∗

 p(s)λ(s) ds .

In this case we can continue to estimate the difference (11):  Z t  f (t, π) − δ ′ (t) > − p(t)δ α (t − r) + kp(t)λ(t) exp − p(s)λ(s) ds t∗   Z t−r p(s)λ(s)) ds = − k α p(t) exp −α t∗   Z t + kp(t)λ(t) exp − p(s)λ(s) ds t∗

and

  Z t p(s)λ(s) ds f (t, π) − δ (t) > kp(t) exp − t∗    Z t Z t α−1 p(s)λ(s)) ds + (1 − α) p(s)λ(s) ds . × λ(t) − k exp α ′

t−r

t∗

A Criterion for Existence of Positive Solutions

521

Taking into account assumption (4), we finalize: f (t, π) − δ ′ (t) > 0 for t ∈ [t∗ , ∞) and inequality (6) holds. Finally, we verify inequality (7). Due to the assumptions of Lemma 3.1, inclusion (t + θ, π(θ)) ∈ ω, θ ∈ [−r, 0) holds and, consequently, we have π(−r) > ρ(t − r) = 0. Therefore ρ′i (t) − f (t, π) = p(t)π(−r) > 0

for all t ≥ t∗ . All conditions of Lemma 3.1 are satisfied. This conclusion ends the proof of this part and Theorem 2.1 is proved. 4. Concluding Remarks As mentioned before, Theorem 2.1 generalizes Theorem 2.2 into the nonlinear equation (3). Now we show that the inequality for the linear case (5) with the function λ positive not only on interval [t∗ , ∞) (as it follows from (5)), but on interval [t∗ − r, t∗ ), too, gives a sufficient condition for the existence of a positive solution for nonlinear equation (3). This statement is formulated as: Theorem 4.1. For the existence of a positive solution y = y(t) on [t∗ − r, ∞) of the equation (3) the existence is sufficient of a locally integrable function λ : [t∗ − r, ∞) → (0, ∞) continuous on [t∗ − r, t∗ ) ∪ [t∗ , ∞) and satisfying the integral inequality Z t  λ(t) ≥ exp p(s)λ(s)ds (12) t−r

∗

for t ≥ t . Proof. We will trace the “sufficient” part of the proof of Theorem 2.1 omitting some technical details. We define continuous on [t∗ − r, ∞) and continuously differentiable on [t∗ , ∞) functions ρ(t) := 0 and  Z t  δ(t) := exp − p(s)λ(s) ds t∗ −r

where the function λ, mentioned in Theorem 4.1 is used. In this case,   Z t   ∗ ω := (t, y) : t ≥ t − r, ρ(t) = 0 < y < exp − p(s)λ(s) ds = δ(t) . t∗−r

We will verify inequality (6). Assuming πj (0) = δj (t) we get for t ≥ t∗ : f (t, π) − δ ′ (t) = − p(t)π α (−r) − δ ′ (t)

(13)

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 Z = − p(t)π (−r) + p(t)λ(t) exp − α



t

p(s)λ(s) ds .

t∗ −r

Since

 Z −π(−r) > −δ(t − r) = − exp −

t−r

t∗ −r

 p(s)λ(s) ds ,

we can continue to estimate the difference (13):  Z f (t, π) − δ ′ (t) > − p(t)δ α (t − r) + p(t)λ(t) exp − 

= − p(t) exp −α

Z

t−r

p(s)λ(s)) ds

t∗ −r Z t

 + p(t)λ(t) exp −

 Z =p(t) exp −

t

p(s)λ(s) ds

× λ(t) − exp +(1 − α)

p(s)λ(s) ds

t∗ −r

t∗ −r



Z

Z



t



t

p(s)λ(s) ds

t∗ −r





p(s)λ(s) ds

t−r t−r

p(s)λ(s) ds t∗ −r



.

Using (12), we get  Z f (t, π) − δ ′ (t) >p(t) exp − 

t−r

p(s)λ(s) ds

t∗ −r



× 1 − exp (1 − α) Since 1−α ≤0

and due to the positivity

of λ, for t ≥ t∗ , we have

 Z 1 − exp (1 − α)

Z

t−r

p(s)λ(s) ds t∗ −r

Z

t∗ −r



.

t−r

t∗ −r

t−r



p(s)λ(s) ds ≥ 0

 p(s)λ(s) ds ≥ 0.

Consequently, f (t, π) − δ ′ (t) > 0 for t ∈ [t∗ , ∞) and (6) holds. Inequality (7) can be verified in a way similar to the proof of Theorem 2.1. Now we show that an inequality, similar to inequality (5) is necessary for the existence of a positive solution of (3) with α ∈ (0, 1).

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523

Theorem 4.2. Let y = y(t) be a positive solution of (3) with α ∈ (0, 1) on [t∗ − r, ∞). Then there exists a positive constant k and a locally integrable function λ : [t∗ − r, ∞) → R continuous on [t∗ − r, t∗ ) ∪ [t∗ , ∞) satisfying the integral inequality  Z t  λ(t) ≥ k α−1 exp α p(t)λ(s)ds (14) t−r

for t ≥ t∗ .

Proof. We will trace the “necessary” part of the proof of Theorem 2.1 omitting some technical details again. Proceeding as indicated in this part of the proof, we get the identity (10). Since Z t p(t)λ(s)ds ≥ 0, t ≥ t∗ (1 − α) t∗

we get

 Z λ(t) ≥ k α−1 exp α

t

t−r t

 Z ≥ k α−1 exp α

p(t)λ(s)ds + (1 − α) p(t)λ(s)ds

t−r



Z

t

p(t)λ(s)ds t∗



and (14) holds. In connection with the above investigation, an interesting open problem arises. We formulate it as a conjecture. Conjecture 4.1. For the existence of a positive solution y = y(t) on [t∗ − r, ∞) of the equation (3) the existence is necessary and sufficient of a locally integrable function λ : [t∗ − r, ∞) → R continuous on [t∗ − r, t∗ ) ∪ [t∗ , ∞) and satisfying the integral inequality Z t  λ(t) ≥ exp p(s)λ(s)ds t−r

for t ≥ t∗ .

Acknowledgment This research was supported by the Grant 1/3238/06 of the Grant Agency of Slovak Republic (VEGA), by the Grant 201/08/0469 of Czech Grant Agency (Prague), and by the Council of Czech Government MSM 0021630503 and MSM 0021630529.

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References 1. R. P. Agarwal, M. Bohner, W.-T. Li, Nonoscillation and Oscillation: Theory for Functional Differential Equations, Marcel Dekker, Inc., 2004. 2. L. Berezansky, E. Braverman, Positive solutions for a scalar differential equation with several delays, Appl. Math. Lett., to appear. 3. L. Berezansky, Y. Domshlak, E. Braverman, On oscillation properties of delay differential equations with positive and negative coefficients, J. Math. Anal. Appl. 274 (2002), 81–101. 4. J. Dibl´ık, A criterion for existence of positive solutions of systems of retarded functional differential equations, Nonlinear Anal. 38 (1999), 327–339. 5. J. Dibl´ık, Positive and oscillating solutions of differential equations with delay in critical case, J. Comput. Appl. Math. 88 (1998), 185–202. 6. J. Dibl´ık, N. Koksch, Positive solutions of the equation x(t) ˙ = −c(t)x(t − τ ) in the critical case, J. Math. Anal. Appl. 250 (2000), 635-659. 7. J. Dibl´ık, Z. Svoboda, Positive solutions of retarded functional differential equations, Nonlinear Anal. 63 (2005), e813–e821. 8. J. Dibl´ık, Z. Svoboda, Positive solutions of p-type retarded functional differential equations, Nonlinear Anal. 64 (2006), 1831–1848. 9. Y. Domshlak, I. P. Stavroulakis, Oscillation of first-order delay differential equations in a critical case, Appl. Anal. 61 (1996), 359–371. ´ Elbert, I. P. Stavroulakis, Oscillation and non-oscillation criteria for delay 10. A. differential equations, Proc. Amer. Math. Soc. 123 (1995), 1503–1510. 11. L. H. Erbe, Q. Kong, B. G. Zhang, Oscillation Theory for Functional Differential Equations, Marcel Dekker, Inc., 1995. 12. K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic Publishers, 1992. 13. I. Gy¨ ori, G. Ladas, Oscillation Theory of Delay Differential Equations, Clarendon Press, 1991. 14. J. K. Hale, S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993.

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DISCRETENESS OF THE SPECTRUM OF A NONSELFADJOINT SECOND-ORDER DIFFERENCE OPERATOR ∗ ¨ EBRU ERGUN Department of Physics, Ankara University, 06100 Tando˘ gan, Ankara, Turkey E-mail: [email protected] GUSEIN S. GUSEINOV ˙ Department of Mathematics, Atılım University, 06836 Incek, Ankara, Turkey E-mail: [email protected] We investigate a non-Hermitian second-order linear difference operator and establish simple conditions for the discreteness of its spectrum.

1. Introduction In [1] the following problem was introduced and partly investigated in the Hilbert space L2 (−∞, ∞): −y ′′ (x) + q(x)y(x) = λρ(x)y(x), −

+

y(0 ) = y(0 ),

′

−

x ∈ (−∞, 0) ∪ (0, ∞),

y (0 ) = e

2iδ ′

+

y (0 ),

(1) (2)

where y(x) is a desired solution, q(x) is a real-valued function, λ is a spectral √ parameter, δ ∈ [0, π2 ), i is the imaginary unit i = −1, and  2iδ e if x < 0, ρ(x) = (3) e−2iδ if x > 0. The main distinguishing features of problem (1), (2) are that it involves a complex-valued coefficient function ρ(x) of the form (3) and that transition conditions of the form (2) are presented which also involve a complex coefficient. Such a problem is non-Hermitian with respect to the usual inner product of space L2 (−∞, ∞). Note that non-Hermitian Hamiltonians and ∗ This

work is supported by Grant 106 T549 from the Scientific and Technological Re¨ ITAK). ˙ search Council of Turkey (TUB

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E. Erg¨ un & G. S. Guseinov

complex extension of quantum mechanics have recently received a lot of attention [2]. Let Z denote the set of all integers. For any l, m ∈ Z with l ≤ m, [l, m] will denote the discrete interval being the set {l, l + 1, . . . , m}. Semiinfinite intervals of the form (−∞, l] and [l, ∞) will denote the discrete sets {. . . , l − 2, l − 1, l} and {l, l + 1, l + 2, . . .}, respectively. Throughout the paper all intervals will be discrete intervals. Let us set Z0 = Z \{0, 1} = {. . . , −2, −1} ∪ {2, 3, . . .} = (−∞, −1] ∪ [2, ∞).

(4)

In this paper we offer a discrete version of problem (1), (2) to be −∆2 yn−1 + qn yn = λρn yn , y−1 = y1 ,

∆y−1 = e

2iδ

n ∈ Z0 ,

∆y1 ,

(5) (6)

where y = (yn )n∈Z is a desired solution, ∆ is the forward difference operator defined by ∆yn = yn+1 − yn so that ∆2 yn−1 = yn+1 − 2yn + yn−1 , the coefficients qn are real numbers given for n ∈ Z0 , δ ∈ [0, π2 ), and ρn are given for n ∈ Z0 by  2iδ e if n ≤ −1, ρn = (7) −2iδ e if n ≥ 2. For a long time past, mathematicians and physicians are interested in discrete equations (difference equations) [3,4] . Discrete equations arise when differential equations are solved approximately by discretization. On the other hand they often arise independently as mathematical models of many practical events. Discrete equations can easily been algorithmized to solve them on computers. There are only a few works concerning discrete non-Hermitian quantum systems [5–7]. The main result of the present paper is that if qn ≥ c > 0 for

n ∈ Z0

and

lim qn = ∞,

|n|→∞

then the spectrum of problem (5), (6) is discrete. In order to introduce the concept of spectrum for problem (5), (6), define the Hilbert space l02 of complex sequences y = (yn )n∈Z0 such that P P |yn |2 < ∞ with the inner product hy, zi = n∈Z0 yn z n and norm n∈Z0p kyk = hy, yi , where the bar over a complex number indicates the complex conjugate. Now we try to rewrite problem (5), (6) in the form of an equivalent operator equation in l02 . Introduce the linear set  D = y = (yn )n∈Z0 ∈ l02 : (qn yn )n∈Z0 ∈ l02 .

Discreteness of the Spectrum of a Nonselfadjoint Difference Operator

527

Taking this set as the domain of L, L : D ⊂ l02 → l02 is defined by (Ly)n = −∆2 yn−1 + qn yn

for n ∈ Z0 ,

where y0 and y1 are defined from the equations y−1 = y1 ,

∆y−1 = e2iδ ∆y1 .

Note that y0 and y1 are needed when we evaluate (Ly)n for n = −1 and n = 2, respectively. Next, define the operator M : l02 → l02 by (M y)n = ρn yn for n ∈ Z0 , where ρn is given by (7). Since |ρn | = 1, M is a unitary operator. Therefore problem (5), (6) can be written as Ly = λM y, y ∈ D, or M −1 Ly = λy, y ∈ D. This motivates to introduce the following definition. Definition 1.1. By the spectrum of problem (5), (6) is meant the spectrum of the operator A = M −1 L with the domain D in the space l02 . Remember that (see [8]) if A is a linear operator with a domain dense in a Banach space, then a complex number λ is called a regular point of −1 the operator A if the inverse (A − λI) exists and represents a bounded operator defined on the whole space. The set of all nonregular points λ is called the spectrum of the operator A. Obviously the eigenvalues λ of the −1 operator A belong to its spectrum, since the operator (A − λI) does not exist for such points (the operator A − λI is not one-to-one). The set of all eigenvalues is called the point spectrum of the operator. The spectrum of the operator A is said to be discrete if it consists of a denumerable (i.e., at most countable) set of eigenvalues with no finite point of accumulation. A linear operator acting in a Banach space and defined on the whole space is called completely continuous if it maps bounded sets into relatively compact sets. Any completely continuous operator is bounded and hence its spectrum is a compact subset of the complex plane. As is well known, every nonzero point of the spectrum of a completely continuous operator is an eigenvalue of finite multiplicity (that is, to each eigenvalue there correspond only a finite number of linearly independent eigenvectors); the set of eigenvalues is at most countable and can have only one accumulation point λ = 0. It follows that if a linear operator A with a domain dense in a Banach space is invertible and its inverse A−1 is completely continuous, then the spectrum of A is discrete. In this paper we show that the operator L is invertible and its inverse L−1 is a completely continuous operator. This implies that the operator A = M −1 L is invertible and A−1 = L−1 M is a completely continuous operator. Hence the spectrum of the operator A is discrete.

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The paper is organized as follows. In Section 2 we present general properties of solutions of equations of type (5), (6). In Section 3 we show that the operator L is invertible and we describe the structure of the inverse operator L−1 . Finally, in Section 4 we show that the operator L−1 is completely continuous. 2. Second-Order Linear Difference Equations with Impulse Consider the second order linear homogeneous difference equation with impulse −∆2 yn−1 + pn yn = 0,

n ∈ Z0 = Z {0, 1} = (−∞, −1] ∪ [2, ∞),

y−1 = d1 y1 ,

∆y−1 = d2 ∆y1 ,

(8) (9)

where y = (yn ) with n ∈ Z is a desired solution, the coefficients pn are complex numbers given for n ∈ Z0 ; d1 , d2 presented in the “impulse conditions” in (9) are nonzero complex numbers. Using the definition of ∆-derivative we can rewrite problem (8), (9) in the form

where

−yn−1 + pen yn − yn+1 = 0,

n ∈ (−∞, −1] ∪ [2, ∞),

y−1 = d1 y1 ,

y0 − y−1 = d2 (y2 − y1 ),

pen = pn + 2,

n ∈ (−∞, −1] ∪ [2, ∞).

(10) (11)

Theorem 2.1. Let n0 be a fixed point in Z and c0 , c1 be given complex numbers. Then problem (8), (9) has a unique solution (yn ), n ∈ Z, such that yn0 = c0 ,

∆yn0 = c1 , that is,

yn0 = c0 ,

yn0 +1 = c0 + c1 = c′1 . (12)

Proof. First assume that n0 ∈ (−∞, −1]. Solving equation (10) on (−∞, −1] under the initial conditions (12), we find yn uniquely for n ∈ (−∞, 0]. Then we find y1 and y2 uniquely from (11) and then we solve equation (10) uniquely on [2, ∞). Let now n0 ∈ [1, ∞). Solving equation (10) on [2, ∞) subject to the initial conditions (12), we find yn uniquely for n ∈ [1, ∞). Then we find y−1 and y0 uniquely from (11) and then we solve equation (10) uniquely on (−∞, −1]. Finally, if n0 = 0, then we find y0 and y1 uniquely from the initial condiations (12) with n0 = 0. Then we find y−1 and y2 from the impulse

Discreteness of the Spectrum of a Nonselfadjoint Difference Operator

529

conditions (11). Next, solving equation (10) at first on (−∞, −1] we find yn uniquely for n ∈ (−∞, −2] and then solving (10) on [2, ∞) we find yn uniquely for n ∈ [3, ∞). Definition 2.1. For two sequences y = (yn ) and z = (zn ) with n ∈ Z, we define their Wronskian (or Casoratian) by Wn (y, z) = yn ∆zn − (∆yn )zn = yn zn+1 − yn+1 zn ,

n ∈ Z.

Theorem 2.2. The Wronskian of any two solutions y and z of problem (8), (9) is constant on each of the intervals (−∞, −1] and [1, ∞) :  − ω , n ∈ (−∞, −1], Wn (y, z) = (13) ω + , n ∈ [1, ∞). In addition, ω − = d1 d2 ω +

(14)

W0 (y, z) = −d2 ω + .

(15)

and

Proof. Suppose that y = (yn ) and z = (zn ), where n ∈ Z, are solutions of (8), (9). Let us compute the ∆ -derivative of Wn (y, z). Using the product rule for ∆-derivative ∆(fn gn ) = (∆fn )gn + fn+1 ∆gn = fn ∆gn + (∆fn )gn+1 , we find that ∆Wn (y, z) = yn+1 ∆2 zn − (∆2 yn )zn+1

= yn+1 pn+1 zn+1 − pn+1 yn+1 zn+1 = 0

for n ∈ (−∞, −2] ∪ [1, ∞). The latter implies that Wn (y, z) is constant on (−∞, −1] and on [1, ∞). Thus we have (13), where ω − and ω + are some constants (depending on the solutions y and z). Next using (13) and the impulse conditions in (9) for yn and zn , we get ω − = W−1 (y, z) = y−1 ∆z−1 − (∆y−1 )z−1

= d1 d2 [y1 ∆z1 − (∆y1 )z1 = d1 d2 W1 (y, z) = d1 d2 ω +

so that (14) is established.

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Finally, from the impulse conditions in (11) we find that y0 = (d1 − d2 )y1 + d2 y2 . Substituting this expression for y0 and z0 into W0 (y, z) = y0 z1 − y1 z0 , we get W0 (y, z) = −d2 W1 (y, z) = −d2 ω + . Therefore (15) is also proved. Corollary 2.1. If y and z are two solutions of (8), (9), then either Wn (y, z) = 0 for all n ∈ Z or Wn (y, z) 6= 0 for all n ∈ Z. By using Theorem 2.1, the following two theorems can be proved in exactly the same way when equation (8) does not include any impulse conditions [3]. Theorem 2.3. Any two solutions of (8), (9) are linearly independent if and only if their Wronskian is not zero. Theorem 2.4. Problem (8), (9) has two linearly independent solutions and every solution of (8), (9) is a linear combination of these solutions. We say that y = (yn ) and z = (zn ), where n ∈ Z, form a fundamental set of solutions for (8), (9) provided that they are solutions of (8), (9) and their Wronskian is not zero. Let us consider the nonhomogeneous equation −∆2 yn−1 + pn yn = hn ,

n ∈ (−∞, −1] ∪ [2, ∞),

(16)

with the impulse conditions y−1 = d1 y1 ,

∆y−1 = d2 ∆y1 ,

(17)

where hn is a complex sequence defined for n ∈ (−∞, −1] ∪ [2, ∞). We will extend hn to the values n = 0 and n = 1 by setting h0 = h1 = 0.

(18)

Theorem 2.5. Suppose that u = (un ) and v = (vn ) form a fundamental set of solutions for the homogeneous problem (8), (9). Then a general solution of the corresponding nonhomogeneous problem (16), (17) is given by yn = c1 un + c2 vn + xn ,

n ∈ Z,

Discreteness of the Spectrum of a Nonselfadjoint Difference Operator

where c1 , c2 are arbitrary constants and ( P0 s vn n ≤ 0, − s=n unWvss −u (u,v) hs , Pn un vs −u xn = s vn n ≥ 1. s=1 Ws (u,v) hs ,

531

(19)

Proof. Taking into account (18) it is not difficult to verify that the sequence xn defined by (19) is a particular solution of (16), (17), namely, xn satisfies equation (16) and the conditions x−1 = ∆x−1 = 0,

x1 = ∆x1 = 0.

This implies that the statement of the theorem is true. 3. The Inverse Operator L−1 Let L be the operator defined above in Introduction. In this section we describe the structure of the inverse operator L−1 . The following theorem is crucial for doing this. Theorem 3.1. Let qn ≥ c > 0

for

n ∈ Z0 = Z\ {0, 1} .

(20)

Then the equation −∆2 yn−1 + qn yn = 0,

n ∈ Z0 ,

(21)

subject to the conditions y−1 = y1 ,

∆y−1 = e2iδ ∆y1 ,

(22)

has two linearly independent solutions χ = (χn )n∈Z and ψ = (ψn )n∈Z such that n=0 X −∞

2

|χn | < ∞

and

∞ X

n=0

2

|ψn | < ∞.

(23)

Note that ker L = {y ∈ D : Ly = 0} consists only of the zero element. Indeed, if y ∈ D and Ly = 0, then y satisfies (21), (22). Since χ and ψ form a fundamental set of solutions of (21), (22), we can write yn = C1 χn + C2 ψn ,

n ∈ Z,

with some constants C1 and C2 . Hence Wn (y, ψ) = C1 Wn (χ, ψ) + C2 Wn (ψ, ψ),

n ∈ Z.

(24)

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E. Erg¨ un & G. S. Guseinov

Next, by y ∈ l02 and (23) we have yn → 0 as |n| → ∞ and ψn → 0 as n → ∞. Besides Wn (ψ, ψ) = 0 for all n and Wn (χ, ψ) is equal to a nonzero constant for n ≥ 1. Therefore taking the limit in (24) as n → ∞ we get that C1 = 0. It can similarly be shown that C2 = 0. Thus y = 0. It follows that the inverse operator L−1 exists. To present its explicit form we introduce the discrete Green function  1 χk ψn if k ≤ n, Gnk = Wk (ψ, χ) χn ψk if k ≥ n, where k and n are in Z.

Theorem 3.2. The inverse operator L−1 is defined on the whole space l02 and for every f ∈ l02 X Gnk fk , n ∈ Z0 . (25) (L−1 f )n = k∈Z0

Besides,

−1

L f ≤

1 kf k for all f ∈ l02 , (26) c cos δ where c is a constant from condition (20) and δ is from (22), k·k denotes the norm of space l02 . 4. Complete Continuity of the Operator L−1 In this section we will show that the operator L−1 is completely continuous, that is, it is continuous and maps bounded sets into relatively compact sets. Theorem 4.1. Let qn ≥ c > 0

for

n ∈ Z0

(27)

and lim qn = ∞.

|n|→∞

(28)

Then the operator L−1 is completely continuous. Proof. The operator L−1 is continuous by virtue of (26) that holds under the condition (27). In order to show that L−1 maps bounded sets into relatively compact sets consider any bounded set S in l02 , S =  2 f ∈ l0 : kf k ≤ d , and prove that L−1 (S) = Y is relatively compact in 2 l0 . To this end, we use the following known criterion for relative compactness in l02 : A set Y ⊂ l02 is relatively compact if and only if Y is bounded

Discreteness of the Spectrum of a Nonselfadjoint Difference Operator

533

and for every ε > 0 there exists a positive integer n0 (depending only on ε) such that X 2 |yn | ≤ ε for all y ∈ Y. |n|>n0

Take an arbitrary f ∈ S and set L−1 f = y. Then Ly = f or explicitly −∆2 yn−1 + qn yn = fn ,

n ∈ Z0 ,

(29)

where y0 and y1 are defined from the equations y−1 = y1 ,

∆y−1 = e2iδ ∆y1 .

(30)

Note that y0 and y1 are needed when we write out equation (29) for n = −1 and n = 2, respectively. It follows from equations (29), (30) that for any integers α < 0 and β > 1, ! β −1   X X
β 2 2 |∆yn | + qn |yn | + (cos δ) + Re −σn yn+1 ∆yn α−1 n=α n=2 ! β −1 X X (σn fn y n ) , = Re + (31) n=α

n=2

where σn = exp(−iδ) if n ≤ −1 and σn = exp(iδ) if n ≥ 0. Since f, y ∈ l02 and |σn | = 1, the sum on the right-hand side of (31) converges as α → −∞, β → ∞. Also from y ∈ l02 it follows that yn → 0 as |n| → ∞. Therefore
β → 0 as α → −∞, β → ∞. −σn y n+1 ∆yn α−1

Consequently, we arrive at the equality  X X 2 2 σn fn y n . |∆yn | + qn |yn | = Re (cos δ) n∈Z0

n∈Z0

Hence (cos δ)

X

n∈Z0

2

qn |yn | ≤ Re

X

σn fn y n

(32)

n∈Z0

and therefore using (27) and kf k ≤ d we get kyk ≤

d c cos δ

for all y ∈ Y.

This means that the set Y = L−1 (S) is bounded.

(33)

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From (32) we also have, using (33), X d2 2 qn |yn | ≤ c cos2 δ n∈Z0

for all y ∈ Y.

(34)

Take now an arbitrary ε > 0. By condition (28) we can choose a positive integer n0 such that qn ≥

d2 εc cos2 δ

for

|n| > n0 .

Then we get from (34) that X 2 |yn | ≤ ε for all y ∈ Y. |n|>n0

Thus the completely continuity of the operator L−1 is proved. Corollary 4.1. The operator A = M −1 L is invertible and its inverse A−1 = L−1 M is a completely continuous operator. Therefore the spectrum of the operator A is discrete. References 1. A. Mostafazadeh, Pseudo-Hermitian description of P T -symmetric systems defined on a complex contour, J. Phys. A 38 (2005), 3213–3234. 2. C. M. Bender, Making sense of non-Hermitian Hamiltonians, Rep. Progr. Phys. 70 (2007), 947–1018. 3. W. Kelley and A. C. Peterson, Difference Equations: An Introduction with Applications, Academic Press, New York, 1991. 4. G. Teshl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Math. Surveys Monogr., Vol. 72, Amer. Math. Soc., Providence, R.I., 2000. 5. C. M. Bender, P. N. Meisinger and N. Wang, Finite dimensional P T -symmetric Hamiltonians, J. Phys. A 36 (2003), 6791–6797. 6. S. Weigret, How to test for digonalizability: The discretized P T -invariant square-well potential, Czech. J. Phys. 55 (2005), 1183–1186. 7. M. Znojil, Matching method and exact solvability of discrete P T -symmetric square wells, J. Phys. A 39 (2006), 10247–10261. 8. M. A. Naimark, Linear Differential Operators, Part II, Ungar, London, 1968.

535

EVENTUALLY POSITIVE SOLUTIONS OF SECOND-ORDER SUPERLINEAR DYNAMIC EQUATIONS R. MERT† and A. ZAFER∗ Department of Mathematics, Middle East Technical University, Ankara, 06531, Turkey † E-mail: [email protected] ∗ E-mail: [email protected] A time scale T is an arbitrary nonempty closed subset of the real numbers R. We study the existence and nonexistence of positive solutions of a class of second-order superlinear equations defined on T. Keywords: Time Scale; Delay Dynamic Equation; Oscillation; Eventually Positive/Negative Solution.

1. Introduction Consider the forced superlinear second order delay dynamic equation x∆∆ (t) + q(t) |x(τ (t))|

α−1

x(τ (t)) = g(t),

t ∈ [t0 , ∞)T ,

(1)

where T is a time scale, t0 ∈ T and α > 1 are fixed, and q, g, τ ∈ Crd ([t0 , ∞)T ). It is assumed that τ is increasing, τ (t) ≤ t, τ (t) ∈ T for all t ∈ T , and limt→∞ τ (t) = ∞. By [t0 , ∞)T we mean a time scale interval; see the next section for some basics about a time scale. A. H. Nasr in [7] has studied (1) extensively when T = R generalizing the well-known theorems of Atkinson (α > 1) and Belohorec (0 < α < 1) from unforced equations to forced ones. If T = Z, then (1) becomes the superlinear forced delay difference equation ∆2 x(n) + q(n) |x(τ (n))|α−1 x(τ (n)) = g(n),

n ∈ {n0 , n0 + 1, . . .},

(2)

which may be considered as a discrete analogue of (1). We should note that (2) has not been sufficiently investigated when compared (1). Moreover, other special cases such as T = q Z , T = hZ, T = Pa,b (see [1] for definitions

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and other special types of time scales) lead to different equations with almost no work so far. Definition 1.1. A nontrivial solution x(t) of (1) is said to be oscillatory if and only if x(t) is neither eventually positive nor eventually negative; i.e., if and only if given any t1 ∈ [t0 , ∞)T there exists t2 ∈ [t1 , ∞)T such that x(t2 )x(σ(t2 )) ≤ 0. 2. Preliminaries A time scale T is an arbitrary nonempty closed subset of the real numbers R. The most well-known examples are T = R and T = Z. For details, see the monographs [1,2]. On a time scale T, the forward and backward jump operators are defined by σ(t) = inf{s ∈ T : s > t}

and ρ(t) = sup{s ∈ T : s < t},

where inf ∅ := sup T and sup ∅ := inf T. A point t ∈ T, t > inf T, is said to be left-dense if ρ(t) = t , right-dense if t < sup T and σ(t) = t, left-scattered if ρ(t) < t and right-scattered if σ(t) > t. The graininess function µ for a time scale T is defined by µ(t) := σ(t) − t and for any function f : T → R the notation f σ (t) denotes f (σ(t)). For a, b ∈ T with a ≤ b the closed interval [a, b]T is defined to be the set {t ∈ T : a ≤ t ≤ b}. Other types of intervals are defined similarly. The set Tκ := T\{m} if T has a left-scattered maximum m, otherwise Tκ := T. Definition 2.1. A function f : T → R is called ∆-differentiable at a point t ∈ Tκ if there exists a real number f ∆ (t) with the property that for any given ε > 0 there is a neighborhood U of t such that |f (σ(t)) − f (s) − f ∆ (t) (σ(t) − s)| ≤ ε|σ(t) − s|

for all s ∈ U.

The number f ∆ (t) is called the delta (or Hilger) derivative of f at t. It follows that f ∆ (t) = f ′ (t) when T = R, while f ∆ (t) = ∆f (t) := f (t + 1) − f (t) if T = Z. In fact, f ∆ (t) = lim

s→t

and f ∆ (t) =

f (s) − f (t) s−t

f (σ(t)) − f (t) σ(t) − t

if µ(t) = 0

if µ(t) > 0.

Eventually Positive Solutions of Second-Order Superlinear Dynamic Equations

537

It is easy to see that if f and g are differentiable then (f g)∆ = f ∆ g + f σ g ∆ = f g ∆ + f ∆ g σ and

 ∆ −g ∆ 1 = g gg σ

(where gg σ 6= 0).

A simple useful formula is f σ = f + µf ∆ .

(3)

Definition 2.2. A function f : T → R is called rd-continuous, if it is continuous at every right-dense point and if the left-sided limit exists (finite) at every left-dense point. The set of all such rd-continuous functions defined on a set D ⊂ T is denoted by Crd (D). As usual the space of functions that are differentiable with rd-continuous derivatives is denoted by C1rd (D). It is noteworthy to mention that every rd-continuous function f has an antiderivative F . As in the continuous case, a function F is called an antiderivative of f on T if F ∆ (t) = f (t) holds for all t ∈ Tκ . The usual chain rule from calculus is no longer valid on an arbitrary time scale (see [1, p. 31]). One form of the following extended chain rule, due to S. Keller [5] and generalized to measure chains by C. P¨ otzche [6], is as follows. Theorem 2.1 (1, Theorem 1.90). Let f : R → R be continuously differentiable and suppose g : T → R is delta differentiable. Then f ◦ g : T → R is delta differentiable and the formula Z 1  (f ◦ g)∆ (t) = f ′ (g(t) + hµ(t)g ∆ (t)) dh g ∆ (t) (4) 0

holds. The substitution rule and the Mean Value Theorem on an arbitrary time scale are given as follows. Theorem 2.2 (1, Theorem 1.98). Assume ν : T → R is strictly increase := ν(T) is a time scale. If f : T → R is an rd-continuous function ing and T and ν is delta differentiable with rd-continuous derivative , then for a, b ∈ T, Z ν(b) Z b ∆ e (f ◦ ν −1 )(s) ∆s. (5) f (t)ν (t) ∆t = a

ν(a)

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R. Mert & A. Zafer

Theorem 2.3 (4, Theorem A.2). Let f be a continuous function on [a, b]T which is delta differentiable on [a, b)T . Then there exit ξ, τ ∈ [a, b)T such that f ∆ (ξ) ≤

f (b) − f (a) ≤ f ∆ (τ ). b−a

(6)

3. Positive solutions Our first theorem is concerned with the existence of an eventually positive solution of (1). Next we give conditions under which there is neither an eventually positive solution nor an eventually negative solution. The combination of the two theorems gives a necessary and sufficient condition for oscillation of solutions of (1). Theorem 3.1. Assume that (i) q(t) is nonnegative; (ii) there exists a bounded function h ∈ C2rd ([t0 , ∞)T ) such that h∆∆ (t) = g(t). If Z

∞

σ(s)q(s) ∆s < ∞,

(7)

then (1) has an eventually positive solution. Proof. In view of (3.1), let T ∈ T be sufficiently large so that   Z ∞ 1/α 1 1 , , σ(s)q(s) ∆s ≤ min 2 (2M + 1)α (2M + 1)α−1 T

(8)

where M = sup{|h(t)| : t ∈ [t0 , ∞)T }. Consider the Banach space BT of all bounded rd-continuous functions x(t), t ∈ [T, ∞)T endowed with the norm kxk =

sup t∈[T,∞)T

|x(t)| .

Let   1 X := x ∈ BT : ≤ x(t) ≤ 2M + 1 . 2

Eventually Positive Solutions of Second-Order Superlinear Dynamic Equations

539

Clearly X is a closed and bounded subset of BT . Pick a point T ∗ ∈ T in such a way that T ∗ > T and τ −1 (T ∗ ) ∈ T. Define a map A : X → BT by  R∞ α   (M + 1) + h(t) − t (σ(s) − t)q(s)x (τ (s)) ∆s,   t ∈ [τ −1 (T ∗ ), ∞)T R∞ (Ax)(t) = (M + 1) + h(t) − τ −1 (T ∗ ) (σ(s) − t)q(s)xα (τ (s)) ∆s,     t ∈ [T, τ −1 (T ∗ ))T . A maps X into itself. Let x ∈ X. It is clear that (Ax)(t) is rd-continuous (in fact continuous) and (Ax)(t) ≤ 2M + 1. Moreover, in view of (8) we get Z ∞ (Ax)(t) ≥ 1 − σ(s)q(s)xα (τ (s)) ∆s τ −1 (T ∗ )

≥ 1 − (2M + 1)

α

Z

∞

T

σ(s)q(s) ∆s ≥

1 . 2

A is a contraction mapping. For this, let x, y ∈ X. Then Z ∞ |(Ax)(t) − (Ay)(t)| ≤ (σ(s) − t)q(s) |xα (τ (s)) − y α (τ (s))| ∆s ≤

τ −1 (T ∗ ) ∞

Z

τ −1 (T ∗ )

σ(s)q(s) |xα (τ (s)) − y α (τ (s))| ∆s.

Applying the mean value theorem (continuous version), we see that Z ∞ σ(s)q(s)z α−1 (τ (s)) ∆s, kAx − Ayk ≤ α kx − yk τ −1 (T ∗ )

where z(τ (s)) lies between x(τ (s)) and y(τ (s)). It follows that Z ∞ kAx − Ayk ≤ α kx − yk (2M + 1)α−1 σ(s)q(s) ∆s, T

and hence by (8), kAx − Ayk ≤

1 kx − yk . 2

Employing the Contraction Mapping Theorem, we conclude that A has a unique fixed point x ∈ X. It is easy to verify that this fixed point is a solution of (1). Remark 3.1. If q(t) is nonpositive and (3.1) holds with q replaced by −q, then (1) has an eventually negative solution.

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Theorem 3.2. Let g(t) ≡ 0, q(t) ≥ 0, q(t) 6≡ 0 eventually, and σ(t) = O(τ (t)) as t → ∞. If Z ∞ σ(s)q(s) ∆s = ∞, (9) then (1) has no eventually positive/negative solution.

Proof. Suppose to the contrary that there is an eventually positive solution x(t) of (1). If x(t) is eventually negative, we write y = −x and get back to the previous case. Choose t1 sufficiently large so that x(t) > 0 and x(τ (t)) > 0 for t ≥ t1 > t0 . From (1) we see that x∆∆ (t) ≤ 0 for t ≥ t1 . Consequently for some t2 ≥ t1 we have x(t) > 0,

x∆ (t) > 0,

x∆∆ (t) ≤ 0,

t ≥ t2 .

Define the function r(t) by the Riccati substitution r(t) := −t

x∆ (t) , xα (t)

t ≥ t2 .

Then r(t) satisfies  ∆ ∆ x∆ (t) x (t) r (t) = − α , − σ(t) α x (t) x (t) ∆

t ≥ t2 .

In view of (1), we have r∆ (t) = −

xα (τ (t)) x∆ (t)(xα )∆ (t) x∆ (t) + σ(t)q(t) + σ(t) , xα (t) xα (σ(t)) xα (t)xα (σ(t))

t ≥ t2 . (10)

On the other hand, by the chain rule (4) (with f (t) = tα , g(t) = x(t)) we have Z 1  α ∆ ∆ α−1 (x ) (t) = α(x(t) + hµ(t)x (t)) dh x∆ (t) 0

≥ αxα−1 (t)x∆ (t),

t ≥ t2 ,

(11)

where we have used x(t) > 0 and x∆ (t) > 0. Using (11) in (10) we obtain xα (τ (t)) ασ(t) x2α−1 (t) 2 x∆ (t) + σ(t)q(t) α + 2 r (t) α x (t) x (σ(t)) t xα (σ(t)) x∆ (t) xα (τ (t)) c xα (t) 2 ≥− α + σ(t)q(t) α + r (t), t ≥ t2 , (12) x (t) x (σ(t)) t xα (σ(t))

r∆ (t) ≥ −

where c = αxα−1 (t2 ).

Eventually Positive Solutions of Second-Order Superlinear Dynamic Equations

541

Let s ∈ T, s ≥ t2 be a fixed point. By the mean value theorem (6) applied to x(t) on [s, t]T , t > s, we have x∆ (t) 1 < , x(t) t−s

(13)

since x(t) > 0, x∆∆ ≤ 0. From (13) it follows that x(t) − (t − s)x∆ (t) > 0,

t ≥ s.

(14)

Dividing both sides of (14) by x(t)x(σ(t)), we see that  ∆ t−s > 0, t ≥ s. x(t)

(15)

Integrating (15) from τ (t) to σ(t), for sufficiently large t we get τ (t) − s x(τ (t)) ≥ x(σ(t)) σ(t) − s

and hence

x(τ (t)) ≥ K, x(σ(t))

(16)

t ≥ t3

(17)

for sufficiently large t3 ≥ t2 and some constant K > 0 since σ(t) = O(τ (t)) as t → ∞. Similar computations yield that for sufficiently large t3 x(t)) ≥ L, x(σ(t))

t ≥ t3

(18)

for some constant L > 0. Using (17) and (18) in (12) we see that r∆ (t) ≥ −

x∆ (t) r2 (t) + βσ(t)q(t) + γ , xα (t) t

t ≥ t3

for some constants β, γ > 0. Integrating from t3 to t we obtain Z t 2 Z t Z t ∆ r (s) x (s) σ(s)q(s) ∆s + γ ∆s + β ∆s. r(t) ≥ r(t3 ) − α (s) x s t3 t3 t3 To prove the convergence of the integral Z ∞ ∆ x (s) ∆s, xα (s)

we consider the following two cases seperately Case 1: limt→∞ x(t) < ∞. Case 2: limt→∞ x(t) = ∞.

(19)

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R. Mert & A. Zafer

First consider Case 1. In this case, it follows from 0
t3 such that Z t 2 r (s) ∆s, t ≥ t4 . (20) r(t) ≥ γ s t3 Setting z(t) := γ

Z

t

t3

we get z ∆ (t) γ ≥ , 2 z (t) t

where

r2 (s) ∆s s

z ∆(t) = γ

r2 (t) . t

Integrating from t4 to t, we obtain Z t ∆ Z t z (s) ∆s ∆s ≥ γ , 2 t4 z (s) t4 s which leads to a contradiction as t → ∞. Since we already know that Z ∞ ∆s =∞ s (see [3, Theorem 5.11]), to get a contradiction we will show that Z ∞ ∆ z (s) ∆s < ∞. z 2 (s)

Eventually Positive Solutions of Second-Order Superlinear Dynamic Equations

543

For this, we repeat the previous steps for z(t) instead of x(t). As in the previous case, it can easily be shown that the integral is convergent if limt→∞ z(t) < ∞. And when limt→∞ z(t) = ∞, we have z([t4 , ∞)T ) is a b := z([t4 , ∞)T ) and σ time scale unbounded from above. Let T b denotes the b To prove the convergence, we will show that forward jump operator on T. σ b(z(t)) = O(z ν (t)) as t → ∞ is satisfied for any ν ∈ [1, 2) (see [3, Corollary 5.14]). If z(t) is right-dense, then σ b(z(t)) = z(t). Also σ b (z(t)) = z(σ(t)) when z(t) is right-scattered. In this case, from (13) we have r2 (t) t  ∆ 2 x (t) = z(t) + γtµ(t) xα (t) 1 ≤ z(t) + γtµ(t) , (t − s)2 x2α−2 (t)

σ b(z(t)) = z(σ(t)) = z(t) + γµ(t)

where the last expression is O(z ν (t)) as t → ∞.

The following theorem is a direct consequence of the above results. Theorem 3.3. Let g(t) ≡ 0, q(t) ≥ 0, q(t) 6≡ 0 eventually, and σ(t) = O(τ (t)) as t → ∞. Then every solution of (1) is oscillatory if and only if Z ∞ σ(s)q(s) ∆s = ∞.

By considering special time scales T = R, T = N, and T = q N we easily obtain the following oscillation criteria.

Corollary 3.1. Let q(t) ≥ 0, q(t) 6≡ 0 eventually. Then every solution of the delay equation x′′ (t) + q(t) |x(t − r)|

α−1

is oscillatory if and only if Z

∞

x(t − r) = 0,

(r > 0),

t ∈ [t0 , ∞)R ,

σ(s)q(s) ds = ∞.

Corollary 3.2. Let q(n) ≥ 0, q(n) 6≡ 0 eventually. Then every solution of the delay difference equation ∆2 x(n) + q(n) |x(n − k)|

α−1

x(n − k) = 0,

(k > 0),

is oscillatory if and only if ∞ X

n=n0

(n + 1)q(n) = ∞.

n ∈ {n0 , n0 + 1, . . .},

544

R. Mert & A. Zafer

Corollary 3.3. Let Q(t) ≥ 0, Q(t) 6≡ 0 eventually. Then every solution of the delay q-differnce equation ∆q x(t) + Q(t) |x(t/q)|

α−1

x(t/q) = 0,

(q > 1),

t ∈ [t0 , ∞)qN ,

where ∆q x(t) = is oscillatory if and only if ∞ X

x(qt) − x(t) , (q − 1)t

q n Q(q n ) = ∞.

References 1. M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Boston, Birkhauser, 2001. 2. B. Kaymak¸calan, V. Lakshmikantham, and S. Sivasundaram, Dynamic Systems on Measure Chains, Dordrecht, Kluwer, 1996. 3. M. Bohner and G. Guseinov, Dynamic Systems and Applications 12, 45 (2003). 4. G. Guseinov, J. Math. Anal. Appl. 285, 107 (2003). 5. S. Keller, Asymptotisches Verhalten Invarianter Faserb¨ undel bei Diskretisierung und Mittelwertbildung im Rahmen der Analysis auf Zeitskalen, Ph.D. thesis, Universit¨ at Augsburg, Augsburg, Germany, 1999. 6. C. P¨ otzche, J. Comput. Appl. Math. 141, 249 (2002). 7. A. H. Nasr, J. Math. Anal. App. 212, 51 (1997).

545

OSCILLATION CRITERIA FOR SECOND-ORDER NONLINEAR IMPULSIVE DIFFERENTIAL EQUATIONS ¨ A. OZBEKLER Department of Mathematics, Atılım University ˙ Incek, Ankara, 06836, Turkey E-mail: [email protected] We establish sufficient conditions for oscillation of second-order nonlinear impulsive differential equations by using the integral averaging technique. In particular, we extend Philos-type oscillation criteria to impulsive differential equations. Keywords: Oscillation, Impulse, Nonlinear, Second-order, Integral averaging.

1. Introduction In this study, we are concerned with the problem of oscillation of secondorder nonlinear impulsive differential equation of the form (r(t)ψ(x)ϕα (x′ ))′ + p(t)ψ(x)ϕα (x′ ) + q(t)f (x) = 0, t 6= θi ; ∆(rψ(x)ϕα (x′ )) + qi f (x) = 0 t = θi ,

(1)

where ϕα (s) := |s|α−1 s and ∆z(t) := z(t+ ) − z(t− ), z(t± ) = limτ →t± z(τ ). We assume without further mention that (i) α > 0 is a constant; (ii) {θi } is a strictly increasing unbounded sequence of real numbers; {qi } is a real sequence;  (iii) r, p, q ∈ PLC [t0 , ∞) := u : [t0 , ∞) → R is continuous on each interval (θi , θi+1 ), u(θi± ) exist, u(θi ) = u(θi− ) for i ∈ N ; r(t) > 0; (iv) ψ ∈ C(R), f ∈ C 1 (R) with ψ(s) > 0, sf (s) > 0 for s 6= 0 and f ′ (s) ≥ Kα (ψ(s)|f (s)|α−1 )1/α is satisfied; Kα > 0 is a constant.

(2)

546

¨ A. Ozbekler

By a solution of (1), we mean a continuous function x defined on [t0 , ∞) such that x′ , (rψ(x)ϕα (x′ ))′ ∈ PLC [t0 , ∞) and x(t) satisfies (1) for t ≥ t0 . Existence of such solutions can be proved in a similar manner performed for equations without impulse effect [1]. As usual, a solution of (1) is called oscillatory if it has arbitrarily large zeros. The equation is called oscillatory if every solution is oscillatory. Equation (1) without impulse effect can be considered as a natural generalizations of the linear, selfadjoint, half-linear or nonlinear equations x′′ + q(t)x = 0,

(3)

′ ′

(r(t)x ) + q(t)x = 0,

(4)

′

(5)

′

(r(t)ϕα (x )) + q(t)ϕα (x) = 0, ′

′

(r(t)ψ(x)ϕα (x )) + q(t)f (x) = 0,

(6)

respectively, which have been the object of intensive studies in recent years. During the last two decades there has been a great deal work on oscillatory behavior of solutions for equations (3)-(6) [2–21]. In this study, by using the generalized Riccati technique and the averaging method, we obtain two oscillation criteria for equation (1). As far as averaging method is considered; the first general result in the literature was given by Philos [16] in 1989. He proved two oscillation criteria [16, Theorems 1 and 2] for equation (3) which were considered as extension of Kamenev’s criteria [4] and improvement of the result of Yan [20]. Later, the results of Philos have been extended by Li [22], Grace [23] and Manojlovic [14] by use of Riccati technique and the averaging method. In 2000, Manojlovic [15] developed oscillation theory for a general case of the equation (6) in which f (s) is not necessarily of the form ϕα (s) and ψ(s) 6= 1, without any restriction on the sign of q(t). In the last decade an intensive investigation of the impulsive differential equations is observed [24–27]. Although numerous aspects of qualitative theory were studied, the first investigation of oscillatory properties of these equations was worked by Gopalsamy and Zhang [27] in 1989. Afterwards, many authors have studied the oscillatory properties of various classes of impulsive differential equations; the results obtained were presented in [26] by Bainov and Simeonov. In 1996, Bainov, Domshlak and Simeonov [24] (see also [26, p. 30]) studied the second order linear impulsive differential equations of the form x′′ (t) + q(t)x(t) = 0, t 6= θi ; ∆x′ + qi x(t) = 0, t = θi ,

(7)

Oscillation Criteria for Nonlinear Impulsive Differential Equations

547

and they have given conditions for oscillation and nonoscillation for equation (7), see [24, Theorem 2]. In present work, we are interested in extending Philos type oscillation criteria [16] to equation (1), and we give some analogous results existing in the literature; namely [16, Theorems 1 and 2] and [24, Theorem 2]; also cf. [14,15,21]. 2. Main Results In order to prove our results we use the following well-known inequality which is due to Hardy, Littlewood and P´ olya [28]. Lemma 2.1. If A, B are nonnegative numbers, then Aγ − γAB γ−1 + (γ − 1)B γ ≥ 0,

γ > 1,

(8)

and the equality holds if and only if A = B. The following theorem is one of the main result of this study. Theorem 2.1. Let D0 = {(t, s) : t > s > t0 } and D = {(t, s) : t ≥ s ≥ t0 }. Assume H(t, s) ∈ C 1 (D : (0, ∞)), h(t, s) ∈ C(D0 , R) satisfy the conditions (i) H(t, t) = 0 for t ≥ t0 and H(t, s) > 0 on D0 ; (ii) H has a continuous and nonpositive partial derivative on D0 with respect to the second variable; p(s) ∂H (t, s) + H(t, s) = h(t, s)H α/(α+1) (t, s), (t, s) ∈ D0 . (iii) − ∂s r(s) If

lim sup t→∞

1 H(t, t0 )

Z t   H(t, s)q(s) − Γα r(s)|h(t, s)|α+1 ds t0  X + H(t, θi )qi = ∞,

(9)

t0 ≤θi γ||u||. Proof. The choice of γ = min{q(t) : t ∈ [ξ, w]} guaranties that γ > 0 and u(t) ≥ q(t)||u|| > γ||u||, ∀t ∈ [ξ, w]. To make use of the fixed point theorems we consider the cone P = {u ∈ B : u(t) > 0, t ∈ [ρ(0), σ(1)], and min u(t) ≥ γ||u||} (16) t∈[ξ,w]

on the Banach space B, and set Pr = {x ∈ P : ||x|| < r}. Theorem 2.1. [5] (Krasnoselskii Fixed Point Theorem) Let E be a Banach space, and let K ⊂ E be a cone. Assume Ω1 and Ω2 are open, bounded subsets of E with 0 ∈ Ω1 , Ω1 ⊂ Ω2 , and let A : K ∩ (Ω2 \ Ω1 ) → K be a completely continuous operator such that either (i) kAuk ≤ kuk for u ∈ K ∩ ∂Ω1 , kAuk ≥ kuk for u ∈ K ∩ ∂Ω2 ; (ii) kAuk ≥ kuk for u ∈ K ∩ ∂Ω1 , kAuk ≤ kuk for u ∈ K ∩ ∂Ω2 , hold. Then A has a fixed point in K ∩ (Ω2 \ Ω1 ).

or

560

S. G. Topal & A. Yantır

Theorem 2.2. [11] (Legget-Williams Fixed Point Theorem) Let P be a cone in a real Banach space E. Set P(ψ, a, b) = {x ∈ P : a ≤ ψ(x), kxk ≤ b}. Suppose A : Pr → Pr be a completely continuous operator and ψ be a nonnegative, continuous, concave functional on P with ψ(u) ≤ kuk for all u ∈ Pr . If there exist 0 < p < q < l ≤ r such that (i) {u ∈ P(ψ, q, l) : ψ(u) > q} = 6 ∅ and ψ(Au) > q for all u ∈ P(ψ, q, l), (ii) kAuk < p for all kuk ≤ p, (iii) ψ(Au) > q for u ∈ P(ψ, q, r) with kAuk > l hold, then A has at least three positive solutions u 1 , u2 and u3 in Pr satisfying ku1 k < p, ψ(u2 ) > q, p < ku3 k with ψ(u3 ) < q. 3. Main Results We are concerned with determining values of λ, for which there exist positive solutions of three point BVP (1)-(2). We use Krasnoselskii fixed point theorem and Legget-Williams fixed point theorem to prove the main results. From Lemma 2.3 it is clear that the solutions of (1)-(2) are the fixed points of the operator n Z σ(1) o T u(t) = λ G(t, s)h(s)f (s, u(s))∇s + Aϕ1 (t) . (17) ρ(0)

To state the main results we need to define the extended real numbers f (t, u) f (t, u) , f 0 = lim sup max , f0 = lim inf min + u u u→0 t∈[ρ(0),σ(1)] u→0+ t∈[ρ(0),σ(1)] f (t, u) f (t, u) f∞ = lim inf min , f ∞ = lim sup max . u→∞ t∈[ρ(0),σ(1)] u u u→∞ t∈[ρ(0),σ(1)] Let K and L be defined by Z w K = min G(t, s)h(s)∇s, t∈[ξ,w]

L=

max

ξ

t∈[ρ(o),σ(1)]

Z

σ(1)

ρ(0)

G(t, s)h(s)∇s =

(18) Z

σ(1)

G(s, s)h(s)∇s.

(19)

ρ(0)

Theorem 3.1. Assume that (C1)-(C4) are satisfied. Then there exists at least one positive solution of BVP (1)-(2) for each λ satisfying either one of the following: 1 1  1 − αϕ(η)  1  1 − αϕ(η)  1 , (b) 0 such that λL(f 0 + ǫ)(1 + α)/(1 − αϕ1 (η)) ≤ 1. The use of the definition of f 0 guarantees that there exists r1 > 0, sufficiently small such that f (t, u)/u < f 0 + ǫ, ∀u ∈ [0, r1 ]. It follows that f (t, u) < (f 0 + ǫ)u for 0 ≤ u ≤ r1 and t ∈ [ρ(0), σ(1)]. If u ∈ ∂Pr1 then n Z σ(1) o T u(t) = λ G(t, s)h(s)f (s, u(s))∇s + Aϕ1 (t) ρ(0)

Z 1 + α  σ(1) G(s, s)h(s)f (s, u(s))∇s 1 − αϕ1 (η) ρ(0) Z σ(1)  1+α  0 (f + ǫ)||u|| G(s, s)h(s)∇s ≤λ 1 − αϕ1 (η) ρ(0)  1+α  (f 0 + ǫ)||u||L ≤ ||u||. ≤λ 1 − αϕ1 (η)  ≤λ

Hence if Ω1 = {u ∈ P : ||u|| < r1 } then

||T u|| ≤ ||u||, ∀u ∈ ∂Pr1 = P ∩ ∂Ω1 .

(20)

Now we use the other part of the inequality in part (a). First we consider the case f∞ < ∞. In this case pick ǫ1 such that λγK(f∞ − ǫ1 ) ≥ 1. The use of the definition of f∞ guarantees that there exists r > r1 , sufficiently large so that f (t, u)/u > f∞ − ǫ1 , ∀u ≥ r. It follows that f (t, u) > (f∞ − ǫ1 )u

562

S. G. Topal & A. Yantır

for 0 ≤ u ≤ r1 and t ∈ [ρ(0), σ(1)]. Pick r2 such that r2 ≥ r/γ > r1 and let Ω2 = {u ∈ P : ||u|| < r2 }. If u ∈ ∂Pr2 then Lemma 2.5 ensures that n Z σ(1) o T u(t) = λ G(t, s)h(s)f (s, u(s))∇s + Aϕ1 (t) ≥λ

Z

ρ(0)

σ(1)

ρ(0)

G(t, s)h(s)f (s, u(s))∇s ≥ λ(f∞ − ǫ1 )γ||u||

≥ λ(f∞ − ǫ1 )γ||u||K ≥ ||u||.

Z

w

G(t, s)h(s)∇s

ξ

(21)

Finally we consider the case f∞ = ∞. In this case the inequality becomes λ > 0. If we choose M sufficiently large so that Z w λM γ G(t, s)h(h)∇s ≥ 1 (or λM γK ≥ 1), ξ

for any t ∈ [ρ(0), σ(1)] then there exists r > r1 so that f (t, u) > M u for u ≥ r. Let r2 be defined as above and let u ∈ ∂Pr2 . Then for all t ∈ [ρ(0), σ(1)] we have Z σ(1) Z w T u(t) ≥ λM G(t, s)h(s)u(s)∇s ≥ λM γ||u|| G(t, s)h(s)∇s ρ(0)

ξ

= λM γK||u|| ≥ ||u||.

(22)

From the inequalities (21) and (22) ||T u|| ≥ ||u||, ∀u ∈ ∂Pr2 = P ∩ ∂Ω2 .

(23)

Thus Krasnoselskii fixed point theorem implies that T u has a fixed point in P ∩ (Ω2 \ Ω1 ). Theorem 3.2. Let f (t, u) satisfy (C1). Assume that there exist two positive constants r2 > r1 > 0 satisfying the following conditions (C5) f (t, u) ≤ λ−1 M r2 for (t, u) ∈ [ρ(0), σ(1)] × [0, r2 ], (C6) f (t, u) ≥ λ−1 N r1 for (t, u) ∈ [ρ(0), σ(1)] × [0, r1 ], where M=

1 − αϕ1 (η)  1+α N= γ

Z

ξ

Z

σ(1)

G(s, s)h(s)∇s

ρ(0)

w

G(t0 , s)h(s)∇s

−1

,

−1

,

(24)

(25)

and t0 ∈ (ρ(0), σ(1)) such that ||u|| = u(t0 ). Then the problem (1)-(2) has at least one positive solution u satisfying r1 ≤ ||u|| ≤ r2 .

Positive Solutions of a Nonlinear Three-Point BVP on Time Scales

563

Now we state the sufficient conditions which guarantee the existence of at least three solutions for (1)-(2). Theorem 3.3. Let f (t, u) satisfy (C1) and there exist constants 0 < r1 < r2 < r3 such that following assumptions hold: (C7) f (t, u) < λ−1 M r1 for all (t, u) ∈ [ρ(0), σ(1)] × [0, r1 ], (C8) f (t, u) ≥ λ−1 N r2 for all (t, u) ∈ [ξ, w] × [r2 , r3 ], (C9) f (t, u) ≤ λ−1 M r3 for all (t, u) ∈ [ρ(0), σ(1)] × [ρ(0), r3 ]. Then (1)-(2) has at least three positive solutions u 1 , u2 and u3 satisfying ||u1 || < r1 , r2 < min |u2 (t)| ≤ r3 , r1 < ||u3 || ≤ r3 , t∈[ξ,w]

min |u3 (t)| < r2 .

t∈[ξ,w]

Proof. We first define a nonnegative, continuous, concave functional ψ : P → [0, ∞), ψ(u) := min |u(t)|. The cone P as in Eq. (16), M as in t∈[ξ,w]

Eq. (24) and N as in Eq. (25). Then ψ(u) ≤ kuk for all u ∈ P. If u ∈ Pr3 , then kuk ≤ r3 . So by the assumption (C9), we obtain Tu ≤ λ ≤λ





1+α 1 − αϕ1 (η) 1+α 1 − αϕ1 (η)

Z 

σ(1)

G(s, s)h(s)f (s, u(s))∇s

ρ(0) −1

λ

M r3

Z

σ(1)

G(s, s)h(s)∇s = r3 .

ρ(0)

Hence T : Pr3 → Pr3 . In the same way, if u ∈ Pr1 assumption (C7) yields ||T u|| < r1 . Therefore (ii) of Theorem 2.2 is satisfied. To check the condition (i) of Theorem 2.2 we choose u(t) = r3 , ∀t ∈ [ρ(0), σ(1)]. It is clear that u(t) = r3 ∈ P(ϕ, r2 , r3 ). Consequently, since ϕ(u) = ϕ(r3 ) = r3 > r2 then {u ∈ P(ϕ, r2 , r3 ) : ϕ(u) > r2 } = 6 ∅. Moreover by using assumption (C8) and Lemma 2.5 we obtain Z w G(t0 , s)h(s)f (s, u(s))∇s ϕ(T u) = min |T u(t)| ≥ λγ t∈[ξ,w]

−1

≥ λγλ

N r2

Z

ξ

w

G(t0 , s)h(s)∇s = r2 . ξ

Therefore (i) of Theorem 2.2 holds. Since all the conditions of Theorem 2.2 hold T u has at least three fixed points u1 , u2 and u3 , satisfying ||u1 || < r1 , r2 < min |u2 (t)| ≤ r3 , r1 < ||u3 || ≤ r3 and min |u3 (t)| < r2 . t∈[ξ,w]

t∈[ξ,w]

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References 1. D. R. Anderson, J. Difference Equations, 47, 1-12 (2004). 2. R. I. Avery and J. Henderson, Comm. Appl. Nonlinear Anal., 8, 27-36 (2001). 3. M. Bohner and A. Peterson, Dynamic Equations on time scales, An Introduction with Applications, Birkh¨ auser, Boston, 2001. 4. M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkh¨ auser, Boston, 2003. 5. D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, San Diego, 1988. 6. C. P. Gupta, J. Math. Anal. Appl., 168, 540-551 (1992). 7. C. P. Gupta, Appl. Math. Comp., 89, 133-146 (1998). 8. V. A. Il’in and E. I. Moiseev, Differential Equations, 23, 803-810 (1987). 9. W. Jiang and Y. Guo, J. Math. Anal. Appl., 327, 415-424 (2007). 10. M. A. Krasnoselskii, SIAM Review, 8, 122-123 (1964). 11. R. W. Legget and L. R. Williams, Indiana Univ. Math. J., 28, 673-688 (1979). 12. R. Ma and H. Wang, J. Math. Anal. Appl., 279, 216-227 (2003). 13. H.-R. Sun and W.-T. Li, J. Math. Anal. Appl., 299, 508-524 (2004). ˙ Yaslan, J. Comput. Appl. Math., 206, 888-897 (2007). 14. I.

Session 13

Reproducing Kernels and Related Topics

SESSION EDITORS D. Alpay A. Berlinet S. Saitoh D.-X. Zhou

Ben-Gurion University of the Negev, Be’er-Sheva, Israel Universit´e des Sciences et Techniques de Languedoc, Montpellier, France Gunma University, Kiryu, Japan City University of Hong Kong, Kowloon, China

567

REPRODUCING KERNELS FOR HARMONIC FUNCTIONS ON SOME BALLS K. FUJITA Faculty of Culture and Education, Saga University, Saga, 840-8502, Japan E-mail: [email protected] In our previous papers we have proved a theorem on harmonic extension of (complex) harmonic functions on “Np -balls” by using the “harmonic Bergman kernel”. Here we consider the “Cauchy integral representation” and give another proof of the theorem on harmonic extension by using the “Cauchy integral kernel”. Keywords: Reproducing kernel; Harmonic function; Integral representation.

Introduction We have studied holomorphic functions and (complex) harmonic functions on “Np -balls” (for the definition see Section 1.1). In [4], we gave the “Cauchy integral representation” for harmonic functions on the N∞ -ball (Lie ball) whose integral is taken over a boundary of the “complex light cone” (Section 2.1). Then we have considered the “harmonic Bergman kernel” for harmonic functions on the Np -ball. By using the harmonic Bergman kernel, in [2], we proved that harmonic functions on a Np -ball in C2 can be continued harmonically to a larger Lie ball, then in [3], we proved that result in general dimension (Section 1.2). In this paper, we will consider the Cauchy integral representation for harmonic functions on the Np -ball and will give another proof on harmonic continuation by using the “Cauchy kernel” (Section 2.2).

1. Harmonic functions on the Np -ball In this Section, we will introduce our definitions and some notation.

568

K. Fujita

1.1. Np -norm and Np -ball For z = (z1 , · · · , zn+1 ), w = (w1 , · · · , wn+1 ) ∈ Cn+1 we denote by z · w = z1 w1 + · · · + zn+1 wn+1 , z 2 = z · z, kzk2 = z · z. Consider the function 

2  kzk + Np (z) = 

p/2  p/2 1/p p p + kzk2 − kzk4 − |z 2 |2 kzk4 − |z 2 |2   . 2

Then, Np (z) is a norm on Cn+1 if p ≥ 1, and Np is monotone increasing in p; that is, we have N1 (z) ≤ Np (z) ≤ Nq (z) ≤ L(z), 1 ≤ p < q, where L(z) = lim Np (z) is the Lie norm: p→∞

L(z) = Note that N1 (z) =

q p kzk2 + kzk4 − |z 2 |2 .

p (kzk2 + |z 2 |)/2 = L∗ (z) = sup{|z · w|; L(w) ≤ 1}

is the dual Lie norm, N2 (z) = kzk is the complex Euclidean norm and 2−1/p L(z) ≤ Np (z) ≤ L(z).

(1)

For more details on the Np -norm, see [5]. ˜p (r) with radius r by For p ≥ 1, we define the Np -ball B   ˜p (r) = z ∈ Cn+1 ; Np (z) < r , B(r) ˜ ˜∞ (r) = z ∈ Cn+1 ; L(z) < r . B =B

˜p (r) is an open convex subset and balanced in Cn+1 . By the definition, B ˜ In particular, Bp (r) is a domain of holomorphy in Cn+1 . By (1), ˜ ˜p (r) ⊂ B(2 ˜ 1/p r). B(r) ⊂B We denote the closed Np -ball with radius r by   ˜p [r] = z ∈ Cn+1 ; Np (z) ≤ r , B[r] ˜ =B ˜∞ (r) = z ∈ Cn+1 ; L(z) ≤ r . B

Reproducing Kernels for Harmonic Functions on Some Balls

569

1.2. Harmonic functions and harmonic Bergman kernel The Lie ball is closely related to the harmonic functions because harmonic functions on a real ball can be continued analytically to the Lie ball with the same radius. We denote the complex Laplacian by   2 ∂2 ∂ + · · · + . ∆z ≡ 2 ∂z12 ∂zn+1 If f satisfies the differential equation ∆z f (z) = 0, then we call f the ˜p (r)) the space of holomor(complex) harmonic function. We denote by O(B ˜ phic functions on Bp (r) equipped with the topology of uniform convergence on compact sets. Put n o ˜p (r)) = f ∈ O(B ˜p (r)); ∆z f (z) = 0 . O∆ (B In [1], we considered the Bergman kernel on ) ( Z 2 ˜p (r)) = f ∈ O∆ (B ˜p (r)); |f (w)| dVB˜p (r) (w) < ∞ , HO∆ (B ˜ p (r) B

˜p (r). We dewhere dVB˜p (r) denotes the normalized Lebesgue measure on B ∆ ˜ note the Bergman kernel on HO∆ (Bp (r)) by Bp,r (z, w) and call it harmonic ˜p (r)) we have the following integral Bergman kernel; that is, for f ∈ HO∆ (B representation:

f (w) =

Z

˜p (r) B

∆ (z, w)dV ˜ f (z)Bp,r ˜ p (r) (z), w ∈ Bp (r). B

˜p (r) is a domain of holomorphy, there is a holomorphic function Since B ˜p (r). However harmonic which can not be continued analytically outside of B ˜p (r), 1 ≤ p < ∞, can be continued harmonically outside of functions on B ˜p (r). By using the harmonic Bergman kernel we proved the following B theorems in [2] and [3]. ˜p (r)). Define Theorem 1.1. Let 1 ≤ p < ∞ and f ∈ HO∆ (B Z ∆ (z, w)dV f (z)Bp,r F (w) = ˜p (r) (z). B ˜ p (r) B

˜ 1/p r)) and F (w) = f (w) on B ˜p (r). Then F ∈ O∆ (B(2 Theorem 1.2. Let 1 ≤ p < ∞. The restriction mapping r1 gives the following linear topological isomorphism: ∼ ˜ 1/p r)) −→ ˜p (r)). r1 : O∆ (B(2 O∆ (B

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K. Fujita

We remark that the inverse mapping of r1 is given by the mapping Z ∆ (z/ρ, w)dV f (ρz)Bp,r f 7→ F (w) = ˜p (r) (z), B ˜ p (r) B

where ρ < 1 is sufficiently close to 1. In the next section, we will give another proof of this theorem by using the Cauchy integral representation. 2. Cauchy integral representation for harmonic functions Harmonic functions are also closely related to the complex light cone because harmonic functions can be represented by an integral which is taken over a boundary of the complex light cone. 2.1. Complex light cone and Cauchy integral representation We denote the complex light cone by ˜ = {z ∈ Cn+1 ; z 2 = 0}. M Put ˜ (r) = {z ∈ M ˜ ; L(z) < r}, M

˜ [r] = {z ∈ M ˜ ; L(z) ≤ r}, M

˜ ; L(z) = r}. Mr = {z ∈ M

In [4], we proved the following theorems: ˜ (r)). For ρ < 1 sufficiently close to 1, we have Theorem 2.1. Let f ∈ O(M Z ˜ (ρr), f (ρz)Kr (z/ρ, w)dMr (z), w ∈ M f (w) = Mr

where Kr (z, w) =

r2(n−1) (r2 + 2z · w) (r2 − 2z · w)n

is the Cauchy kernel and dMr is the normalized invariant measure on Mr . Furthermore, if we define Z ˜ (2) f (ρz)Kr (z/ρ, w)dMr (z), w ∈ B(ρr), F (w) = Mr

˜ then F ∈ O∆ (B(ρr)).

Reproducing Kernels for Harmonic Functions on Some Balls

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Theorem 2.2. The restriction mapping r0 gives the following linear topological isomorphism: ∼

˜ ˜ (r)). r0 : O∆ (B(r)) −→ O(M

(3)

The inverse mapping of (3) is given by the mapping f 7→ F in (2). ˜ Corollary 2.1. Let f ∈ O∆ (B(r)). For ρ < 1 sufficiently close to 1, we have Z ˜ f (ρz)Kr (z/ρ, w)dMr (z), w ∈ B(ρr). f (w) = Mr

2.2. Another proof on harmonic continuation Using the above theorems, we can give another proof of Theorem 1.2. Put ˜ p (r) = {z ∈ M ˜ ; Np (z) < r}, M

˜ p [r] = {z ∈ M ˜ ; Np (z) ≤ r}, M

˜ ; Np (z) = r}. Mp,r = {z ∈ M

˜ , we have Np (z) = 21/2−1/p kzk and L(z) = 21/2 kzk. Thus For z ∈ M

˜ p (r) = M ˜ (21/p r), Mp,r = M21/p r , O(M ˜ (21/p r)) = O(M ˜ p (r)). M

˜ ˜p (r) ⊂ B(2 ˜ 1/p r) and Recall that we have B(r) ⊂B

˜ ˜p (r)) ⊃ O∆ (B(2 ˜ 1/p r)). O∆ (B(r)) ⊃ O∆ (B

˜p (r)). For ρ < 1 sufficiently close to 1, we Lemma 2.1. Let f ∈ O∆ (B have the following integral representation: Z ˜p (ρr), f (ρz)K21/pr (z/ρ, w)dMp,r (z), w ∈ B f (w) = (4) Mp,r

where dMp,r = dM21/p r . When p = ∞, Lemma 2.1 is nothing but Corollary 2.1. Proof. We denote by r2 the restriction mapping ˜p (r)) −→ O(M ˜ p (r)). r2 : O∆ (B

˜p (r)). Since B ˜p (r)| ˜ = Mp,r , we have Let f ∈ O∆ (B M Z ˜ p (ρr) f (ρz)K21/p r (z/ρ, w)dMp,r (z), w ∈ M f (w) = Mp,r

˜ 1/p ρr) define by Theorem 2.1. For w ∈ B(2 Z f (ρz)K21/p r (z/ρ, w)dMp,r (z), F (w) = Mp,r

(5)

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K. Fujita

˜ 1/p ρr)) by Theorem 2.1. Since B ˜p (r) ⊂ B(2 ˜ 1/p r), F = f then F ∈ O∆ (B(2 ˜p (r) by Corollary 2.1. Thus we have (4). on B Now for p with 1 ≤ p < ∞, we will prove that the restriction mappings r1 and r2 in the following diagram give the linear topological isomorphisms: ∼ ˜ 1/p r)) −→ ˜ (21/p r)) r0 : O∆ (B(2 O(M ↓ r1 ˜p (r)) −→ r2 : O∆ (B

|| ˜ p (r)) O(M

˜p (r)). We denote by α Another proof of Theorem 1.2. Let f ∈ O∆ (B the mapping f 7→ F in (5). Noting that ρ < 1 in (5) is arbitrarily close ˜p (r)) and to 1 and Mp,r = M21/p r , we have (r1 ◦ α)(f ) = f for f ∈ O∆ (B 1/p ˜ (α ◦ r1 )(F ) = F for F ∈ O∆ (B(2 r)) by Lemma 2.1 and Corollary 2.1. Thus we have ∼ ˜ 1/p r)) −→ ˜p (r)). r1 : O∆ (B(2 O∆ (B

Since r0 = r2 ◦ r1 , by Theorems 2.2 and 1.2 we have Corollary 2.2. The restriction mapping r2 gives the following linear topological isomorphism: ∼ ˜p (r)) −→ ˜ p (r)). r2 : O∆ (B O(M

When p = ∞, Corollary 2.2 is nothing but Theorem 2.2. 3. Cauchy kernel and Szeg¨ o kernel ˜ p [r]) the space of continuous functions on M ˜ p [r] and by We denote by C(M L2 (Mp,r , dMp,r ) the space of square integrable functions on Mp,r with re˜ p (r)) spect to the normalized invariant measure dMp,r on Mp,r . Let HO(∂ M 2 be the space of closure in L (Mp,r , dMp,r ) of the restriction to Mp,r of the ˜ p (r)) ∩ C(M ˜ p [r]): elements of O(M   Z 2 ˜ ˜ ˜ |f (w)| dMp,r (w) < ∞ . HO(∂ Mp (r)) = f ∈ O(Mp (r)) ∩ C(Mp [r]); Mp,r

As a corollary of Lemma 2.1, we have the following result. Corollary 3.1. The Cauchy kernel K21/p r (z, w) gives the Szeg¨ o kernel on ˜ p (r)). That is, for f ∈ HO(∂ M ˜ p (r)) we have HO(∂ M Z ˜ p (r). f (z)K21/p r (z, w)dMp,r (z), w ∈ M f (w) = Mp,r

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References 1. K. Fujita, Harmonic Bergman kernel for some balls, Univ. Iagel. Acta Math. 41 (2003), 225–234. 2. K. Fujita, Topics on the Bergman kernel for some balls, Proceedings of 5th International ISAAC Congress, to appear. 3. K. Fujita, Reproducing kernels for holomorphic functions on some balls related to the Lie ball, Ann. Polon. Math. 91 (2007), 219–234. 4. M. Morimoto and K. Fujita, Analytic functionals and entire functionals on the complex light cone, Hiroshima Math. J. 25 (1995), 493-512. 5. M. Morimoto and K. Fujita, Between Lie norm and dual Lie norm, Tokyo J. Math. 24 (2001), 499–507.

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NUMERICAL REAL INVERSION OF THE LAPLACE TRANSFORM BY USING A HIGH-ACCURACY NUMERICAL METHOD H. FUJIWARA Graduate School of Informatics, Kyoto University, Kyoto, 606-8501, Japan E-mail: [email protected] T. MATSUURA Department of Mechanical System Engineering, Graduate School of Engineering, Gunma University, Kiryu, 376-8515, Japan E-mail: [email protected] S. SAITOH Department of Mathematics, Graduate School of Engineering, Gunma University, Kiryu, 376-8515, Japan E-mail: [email protected] & [email protected] Y. SAWANO Department of Mathematics and Information Sciences Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji-shi, Tokyo, 192-0397, Japan E-mail: [email protected] We introduce our three types of numerical inversion formulas of the Laplace transform based on a Fredholm integral equation of the second kind, the Sinc functions (the Sinc method) and the numerical singular value decomposition of the Laplace transform in a certain reproducing kernel Hilbert space. However, we will need the power of the high-accuracy numerical method with multipleprecision arithmetic for difficult situations. We shall give computational experiments for some very difficult situations for the real inversion. Keywords: Laplace transform, real inversion, numerical inversion, reproducing kernel, Fredholm integral equation, sinc method, compact operator, singular value decomposition, computational experiment, high-accuracy numerical method, multiple-precision.

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1. Introduction We introduce very natural and numerical real inversion formulas of the Laplace transform Z ∞ e−pt F (t)dt, p > 0 (1) (LF )(p) = f (p) = 0

for functions F in some natural function space. The real inversion is very involved and see the recent related books [2] and [6]. We use the reproducing kernel Hilbert space HK comprised of absolutely continuous functions F on the positive real line R+ with finite norms Z

∞

0

1 |F ′ (t)|2 et dt t

1/2

and satisfying F (0) = 0. This Hilbert space admits the reproducing kernel K(t, t′ ) =

Z

min(t,t′ )

ξe−ξ dξ

(2)

0

(see [11, pp. 55-56]). Then we see that the linear operator (LF )(p)p on HK into L2 (R+ , dp) = L2 (R+ ) is bounded [12]. Therefore, we obtain the following result. Proposition 1.1. ([12]) For any g ∈ L2 (R+ ) and for any α > 0, the best ∗ approximation Fα,g in the sense  Z α inf

F ∈HK

0

∞

′

|F (t)|

21 t

t

e dt + k(LF )(p)p −

gk2L2 (R+ )



exists uniquely and we obtain the representation Z ∞ ∗ Fα,g (t) = g(ξ) (LKα (·, t)) (ξ)ξdξ.

(3)

0

Here, Kα (·, t) is determined by the functional equation Kα (t, t′ ) =

1 1 K(t, t′ ) − ((LKα,t′ )(p)p, (LKt )(p)p)L2 (R+ ) , α α

where Kα,t′ = Kα (·, t′ )

and

Kt = K(·, t).

(4)

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H. Fujiwara, T. Matsuura, S. Saitoh & Y. Sawano

2. By a Conventional Method for the Fredholm Integral Equation We take the Laplace transform of (4) in t and change the variables t and t′ as in (LKα (·, t))(ξ) 1 1 = (LK(·, t))(ξ) − ((LKα,t )(p)p, (L(LK· )(p)p))(ξ))L2 (R+ ) . α α

(5)

Therefore, by setting (LKα (·, t))(ξ)ξ = Hα (ξ, t), we obtain the Fredholm integral equation of the second type Z ∞ 1 αHα (ξ, t) + Hα (p, t) dp (p + ξ + 1)2 0   1 1 e−tξ e−t t+ + . (6) =− ξ+1 ξ+1 (ξ + 1)2 We gave a numerical experiment for the typical example  −te−t − e−t + 1 for 0 ≤ t ≤ 1 F0 (t) = 1 − 2e−1 for 1 ≤ t, whose Laplace transform is (LF0 )(p) =

i h 1 −(p+1) . 1 − (p + 2)e p(p + 1)2

(7)

For fixed t, we calculated the integral (6) over [0, 50] with span 0.01 by the Gauss-Kronrod method. For t, we took the values over [0, 5] with span 0.01. By (3), we calculated the inversion by the Gauss-Kronrod method and we obtained quite good results in [8]. 3. By the Sinc Method for the Fredholm Integral Equation In order to solve the integral equation (6), numerically, we employ the sinc method. See [7] for the basic relations of the sampling theory and the theory of reproducing kernels. We set   π z−u 1 1 sin (z − u) =: sinc π(z − u) h h h by the notations in [13]. We shall use the double exponential transform following the idea [14] ξ = ϕ(x) = exp(

π sinh x), 2

ϕ′ (x) =

π π cosh x exp( sinh x). 2 2

Numerical Real Inversion of the Laplace Transform by a Numerical Method

577

e α (x, t), and so Then, we have Hα (ξ, t) = Hα (ϕ(x), t) = H e α (x, t) = H

and e α (x, t) + αH

Z

∞

−∞

We approximate as

Then, by setting

e α (z, t) H

e α (x, t) ≃ H Alk ≡

we obtain the equation e α (lh, t) + αH

j=∞ X

e α (jh, t) sinc( x − j) H h j=−∞

1 ϕ′ (z)dz = f (ϕ(x), t). (ϕ(z) + ϕ(x) + 1)2 n=N X

n=−N

(8)

e α (jh, t) sinc( x − j). H h

1 ϕ′ (kh)h, (ϕ(kh) + ϕ(lh) + 1)2

X k

e α (kh, t)Alk = f (ϕ(lh), t) ≡ f˜(lh, t) H

(9)

and the representation Z ∞ Z ∞ g(ϕ(x))Hα (ϕ(x), t)ϕ′ (x)dx g(ξ)Hα (ξ, t)dξ = F ∗ (t) = −∞ 0 X e α (ih, t)ϕ′ (ih). g(ϕ(ih))H ≃h i

We used h = 0.05 and for t, we took the span with 0.01. For the simultaneous equations (9), we took from l = −200 to l = 799, that is, 1000 equations. We solved such equations for [0, 4.99] with the span 0.01 for t. For the numerical experiments, see [9]. 4. By Using a Singular System The singular value decomposition is applicable not only for reconstruction of solutions, but also for Hilbert scales and noise reduction of measurement data. See, for example, [6]. Though the singular value decomposition has various applications, its concrete treatments are hard both mathematically [10] and numerically [4]. We first recall that for the operator Lf (p) = p(Lf )(p), f ∈ HK → Lf ∈ L2 (R+ ) is an injective and compact linear operator [5]. Since the operator Lf (p) has the singular system, let {λn } be singular values of the operator L,

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H. Fujiwara, T. Matsuura, S. Saitoh & Y. Sawano

{vn } and {un } be complete orthonormal systems of N (L)⊥ (the orthogonal complement of the null space) and R(L) (the closure of the range space), respectively satisfying Lvn = λn un and L∗ un = λn vn . Then we obtain the following real inversion formula (see [6]). Proposition 4.1. We consider the Laplace transform Lf = F . If the original function f belongs to HK , then the real inversion of the Laplace transform L−1 is given by Z ∞  ∞ X 1 F (p)p un (p) dp vn (t), (L−1 F )(t) = λ 0 n=1 n

in the sense of norm convergence in HK .

Proposition 4.1 implies that for any F with F (p)p ∈ L2 (R+ ) and a natural number M , spectral cut-off regularization is given by Z ∞  M X 1 (L−1 F )(t) = F (p)p u (p) dp vn (t). n M λ 0 n=1 n

It is an approximation of the original function in HK by a function that belongs to HK , and for f ∈ HK we have the convergence lim L−1 M F = f M→∞

in HK .

5. Numerical Examples Now we present a discretization of the singular value decomposition of L. Employing a numerical integration scheme with a discretization parameter N , quadrature points xi and weights wi , 0 ≤ i ≤ N , i.e., Z ∞ N X f (xi )wi , f (x)dx ≈ 0

i=0

we discretize the eigenvalue problem and obtain a linear eigenvalue problem of a matrix whose (i, j)-entry is, for 0 ≤ j, k ≤ N N X  o xk e−xj xk n −xi (xk +1) aij = wj wk . x (x + 1) + 1 1 − e i k (xk + 1)2 k=0

Here we suppose that v˜n,i corresponds to vn (x

i ) and 2that+they

satisfy the

following normalization condition implied by un : L (R ) = 1: N X j=0

2

|˜ un,j | wj = 1,

Numerical Real Inversion of the Laplace Transform by a Numerical Method

579

where u ˜n,j corresponds to un (xj ) and is given by u˜n,j

N X 1 = x v˜n,k e−xj xk wk . ˜n j λ k=0

In analogy with the Nystr¨ om method, the discretized singular system ˜ {λn , v˜n,j , u ˜n,j } gives approximations of singular functions as ) u(N n (p)

N 1 X p v˜n,k e−pxk wk , = ˜n λ k=0

vn(N ) (t) =

N  o ˜n,k n 1 X u wk . 1 − e−t(xk +1) t(xk + 1) + 1 2 ˜n (xk + 1) λ k=0

The numerical real inversion formula with spectral cut-off regularization is given by ! M N X 1 X −1 (LM,N F )(t) = (10) F (xk )xk u ˜n,k wk vn(N ) (t). ˜n λ n=1

k=0

1 singular values 1e-05

1e-10

lambda n

1e-15

1e-20

1e-25

1e-30

1e-35

1e-40 0

200

400

600

800

1000

1200

1400

n

(a) Singular values λn 1

1.6 n=1 n=2 n=3 n=4 n=5

0.8

n=1 n=2 n=3 n=4 n=5

1.4 1.2 1 singular function u(p)

singular function v(t)

0.6

0.4

0.8 0.6 0.4 0.2

0.2 0 -0.2

0

-0.4 -0.2

-0.6 0

2

4

6

8

t

0

2

4

6

8

p

(b) Singular function Fig. 1.

10

(N) vn

∈ HK

(c) Singular function

(N) un

Numerical singular system for the operator L

∈ L2 (R+ )

10

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H. Fujiwara, T. Matsuura, S. Saitoh & Y. Sawano

Fig. 1 shows numerical singular systems obtained in 400 decimal-digit computation by multiple-precision arithmetic software ‘exflib’ designed by one of the authors ([3]). In the computation, we adopt the the double exponential rule with the discretization parameter N = 12000, and we solve the linear eigenvalue problem by the Householder transformation and the QL method. Example 1 We consider the function   if 0 ≤ t < 1;  t, 1 3 f (t) = 2 − 2 t, if 1 ≤ t < 3;   0, if t ≥ 3,

whose Laplace transform is

F (p) =

1 (2 − 3e−p + e−3p ). 2p2

Numerical reconstruction with M=50 Exact solution

1 0.8 0.6 0.4 0.2 0 0

1

2

3

4

5

t

Fig. 2.

Numerical real inversion for a continuous function (Example 1)

Fig. 2 shows the numerical real inversion fM for the spectral number M = 50 through the numerical singular value decomposition of L. In this example the original function f belongs to HK , and the proposed method gives good approximations. See [6], in particular, Fig. 2.2 and Fig. 2.3, where the truncated singular value decomposition for the Laplace transform operator L appears. Example 2 The next example is the case when f is a characteristic function of [ 12 , 1]∪[ 23 , 52 ]∪[3, 27 ]. Fig. 3 is the numerical spectral cut-off regularized solutions. The dotted curve is that for M = 300, and the solid curve is that

Numerical Real Inversion of the Laplace Transform by a Numerical Method

581

for M = 1500. We remark that the singular value λ300 ≈ 3 × 10−16 in our numerical computations, and this suggests the M = 300 is a limitation in the standard double precision numerical computation. In this case the high-accurate numerical computation is required to large spectral cut-off parameters M . 1.4 spectral cut-off for M=300 spectral cut-off for M=1500 1.2

1

f M(t)

0.8

0.6

0.4

0.2

0

-0.2 0

1

2

3

4

5

t

Fig. 3.

Numerical real inversion for a characteristic function (Example 2)

Example 3 Finally we consider F (p) = exp(−p), which is the Laplace transform of the Dirac delta δt−1 in the distribution sense. Our scheme produces Fig. 4 in which the dotted curve and the solid curve show spectral cut-off solutions f300 and f1500 , respectively. We can realize an effective numerical real inversion by the high-accuracy numerical computation technique and large spectral cut-off parameters. Acknowledgements H. Fujiwara is supported in part by the Grant-in-Aid for Young Scientists (B) (No. 17740057). T. Matsuura is supported in part by the Grant-in-Aid for Scientific Research (C)(No.18540110) from the Japan Society for the Promotion Science. S. Saitoh is supported in part by the Grant-in-Aid for Scientific Research (C)(2)(No. 19540164) from the Japan Society for the Promotion of Science. Y. Sawano is supported by Research Fellowships of

582

H. Fujiwara, T. Matsuura, S. Saitoh & Y. Sawano

the Japan Society for the Promotion of Science for Young Scientists. 20 spectral cut-off for M=300 spectral cut-off for M=1500

15

f M(t)

10

5

0

-5 0

1

2

3

4

5

t

Fig. 4.

Numerical real inversion for the Dirac delta (Example 3)

References 1. K. T. Atkinson, The numerical solution of the eigenvalue problem for compact integral operators, Trans. Amer. Math. Soc. 129 (1967), 458–465. 2. A. M. Cohen, Numerical methods for Laplace transform inversion, Springer, 2007. 3. H. Fujiwara, exflib, a multiple-precision arithmetic software, http://www-an.acs.i.kyoto-u.ac.jp/∼fujiwara/exflib 4. H. Fujiwara, High-accurate numerical method for integral equation of the first kind under multiple-precision arithmetic, Theoret. Appl. Mech. Japan 52 (2003), 192–203. 5. H. Fujiwara, T. Matsuura, S. Saitoh and Y. Sawano, Real inversion of the Laplace transform in numerical singular value decomposition, J. Anal. Appl. 6 (2008), 55–68. 6. J. P. Kaipio and E. Somersalo, Statistical and computational inverse problems, Springer, 2004. 7. T. Matsuura and S. Saitoh, Analytical and numerical inversion formulas in the Gaussian convolution by using the Paley-Wiener spaces, Appl. Anal. 85 (2006), 901–915. 8. T. Matsuura, A. Al-Shuaibi, H. Fujiwara and S. Saitoh, Numerical real inversion formulas of the Laplace transform by using a Fredholm integral equation of the second kind, J. Anal. Appl. 5 (2007), 123–136.

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9. T. Matsuura, A. Al-Shuaibi, H. Fujiwara, S. Saitoh and M. Sugihara, Numerical real inversion of the Laplace transform by using the sinc functions, Far East J. Math. Sci. 27 (2007), 1–14. 10. S. G. Mikhlin and K. L. Smolitskiy, Approximate methods for solution of differential and integral equations, American Elsevier, 1967. 11. S. Saitoh, Integral transforms, reproducing kernels and their applications, Pitman Res. Notes in Math. Series, 369, Addison Wesley Longman, 1997. 12. S. Saitoh, Approximate real inversion formulas of the Laplace transform, Far East J. Math. Sci. 11 (2003), 53–64. 13. F. Stenger, Numerical methods based on sinc and analytic functions, Springer Series in Computational Mathematics, 20, 1993. 14. H. Takahasi and M. Mori, Double exponential formulas for numerical integration, Publ. Res. Inst. Math. Sci. 9 (1974), 511–524.

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PRACTICAL INVERSION FORMULAS FOR LINEAR PHYSICAL SYSTEMS M. YAMADA and S. SAITOIH∗ Graduate School of Engineering, Gunma University, Kiryu 376-8515, Japan ∗ E-mail: [email protected] & [email protected] We shall give a new inversion formula for a linear system based on physical experimental data and by using reproducing kernels and Tikhonov regularization. In particular, we will not make any analytical assumption on the linear system, but will use physical experimental data for obtaining an approximate inversion formula for the linear system. Keywords: Reproducing kernel, linear system, inversion formula, physical system, Tikhonov regularization.

Introduction Inverse problems in mathematics which are expected to be applied to practical problems will, sometimes, have weak points in the viewpoint that the background theories are not faithful for practical and physical problems. For example, equations are the representations of some ideal models and are not those of faithful models in the real physical world. Sometimes, boundary conditions for the equations are involved in physical units and sometimes their physical realizations and observations are very difficult. Here, we shall give a new inversion formula for a linear system based on physical experimental data and by using reproducing kernels and the Tikhonov regularization. In particular, we will not make any analytical assumption on the linear system, but we use physical experimental data for obtaining an approximate inversion formula for the linear system L. We shall apply the following fundamental theory for this purpose. Let E be an arbitrary set, and let HK be a reproducing kernel Hilbert space (RKHS) admitting the reproducing kernel K(p, q) on E. For any Hilbert space H we consider a bounded linear operator L from HK into H.

Practical Inversion Formulas for Linear Physical Systems

585

We are generally interested in the best approximate problem inf kLf − dkH

(1)

f ∈HK

for a vector d in H. However, this extremal problem is involved in the both senses of the existence of the extremal functions in (1) and their representations. See [7] for the details. So, we shall apply its Tikhonov regularization. For any fixed positive α > 0, by introducing the inner product (f, g)HK (L;α) = α(f, g)HK + (Lf, Lg)H ,

(2)

we shall construct the Hilbert space HK (L; α) comprising functions of HK . This space, of course, admits a reproducing kernel. Furthermore, we obtain the following fundamental result. Proposition 0.1. ([8,5,6]) The extremal function fd,α (p) in the Tikhonov regularization inf {αkf k2HK + kd − Lf k2H}

f ∈HK

(3)

exists uniquely and it is represented in terms of the kernel K L (p, q; α) as fd,α (p) = (d, LKL (·, p; α))H ,

(4)

where KL (p, q; α) is the reproducing kernel for the Hilbert space HK (L; α) ˜ q; α) of the equation and is determined as the unique solution K(p, ˜ q , LKp )H = 1 K(p, q) ˜ q; α) + 1 (LK K(p, α α

(5)

with ˜ q = K(·, ˜ q; α) ∈ HK K

for

q ∈ E,

(6)

and Kp = K(·, p) ∈ HK

for

p ∈ E.

The reproducing kernel KL is, of course, represented in the operator form as KL (·, p; α) = (L∗ L + αI)−1 K(·, p), where L∗ is the adjoint operator of L. In (4), when d contains error or noise, we need its error estimate. For this, we can obtain the following general result.

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M. Yamada & S. Saitoh

Proposition 0.2. ([4],[5]). In (4), we have the estimate 1 p |fd,α (p)| ≤ √ K(p, p)kdkH . α

Following Proposition 0.1, we shall look for the approximate inversion fd,α (p) of the linear system Lf = d

for the bounded linear operator L from HK into H. Here, when α tends to zero, the approximate inversion fd,α (p) tends to the Moore-Penrose generalized inverse of the operator equation in a good way, when it exists, and some detailed behaviour of the convergence is examined. For more details of the convergence rate or the results for noisy data, see ([2],[3]). 1. Approach looking for the inversion Physically or by computers we can observe only discrete data, so, as a very general algorithm, we shall consider the discrete point data case. In (3), we shall consider the corresponding problem   N X (7) |(Lf )(Pj ) − dj |2 , inf αkf k2HK + f ∈HK

j=1

for fixed discrete points {Pj }j of the set E and for given values d = {dj }j ; that is, H is the usual Euclidean space RN . In order to use the representation (4), we need LKL (·, p; α)) and it is determined by (5). In (5), we operate L as functions in p and we have ˜ q; α) + Lp (LK ˜ q , LKp )H = Lp K(p, q). αLp K(p,

(8)

Here, when we can take α = 0 in the sense of numerical, we can take, of course, α = 0 in those arguments. As stated in Introduction, in many practical problems for the linear system, L is not given analytically and so, here, we shall assume that the system may be used many times, experimentally. However, in order to use the previous section method, we must realize some physical objects as the N data d = {dj }j , N ×N values Lp K(p, q) and N × N values Lp LKp of real values; that is, f and d = {dj }j are numerical representations of some physical objects in the system Lf = d. Since the reproducing kernel Hilbert space HK is the function space approximating the solution of the operator equation Lf = d, we can take many simple reproducing kernel Hilbert spaces as in ([7]), however, from

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587

the present situation, the reproducing kernel K(p, q) must be realized as the physical object for the present system. 2. Physical viewpoints We see in our inversion formula (4) that we use a concrete reproducing kernel K(p, q) through (5) but we do not use any Hilbert space structure of the reproducing kernel K(p, q). By the theory of reproducing kernels, for any positive matrix there exists a uniuely determined reproducing kernel Hilbert space ([1,7]). Furthermore, in our inversion formula (4), we are looking for approximations of the inversion in the function space HK , so, in general, the space HK is a sufficient large class of functions in the sense that we can approximate the inverse by the functions in HK . For example, for any characteristic function on any interval, we can approximate it by the Sobolev Hilbert space of 1 dimension uniformly. This will mean that for the input, we can consider a suitable positive matrix, here, by a suitable positive matrix, we mean that the positive matrix may be realized as the physical data and it will also depend on its physical system. In connection with these points of view, for example, for the 2 dimensional Sobolev space in ([7]), we shall use the more simple reproducing kernel 1 (9) K(x1 , x2 , y1 , y2 ) = exp(−|x1 − y1 |) exp(−|x2 − y2 |), 4 which is the usual product of the 1 dimensional Sobolev reproducing kernels and its reproducing kernel Hilbert space is the tensor product of the two Hilbert spaces of the one dimensional Sobolev Hilbert space (see [1,7] for this structure). We shall introduce several simple reproducing kernels on the whole real line space. Note here that for multidimensional spaces, we can consider the products as in (9). Furthermore, the restriction of a reproducing kernel to any subset is again a reproducing kernel. The sum and the usual product of two reproducing kernels on a same set are again reproducing kernels. For these elementary facts, see, ([1,7]). On the whole real space R, the following are reproducing kernels: (1) Any positive semidefinite matrix. (2) δ(x − y) (δ(0) = 1 and δ(x) = 0 for x 6= 0). (3) For any α > 0, exp(−α|x − y|). (4) exp(αxy) ( α > 0 ). (5) exp(−α(x − y)2 ) ( α > 0 ).

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(6) exp(−|x − y|)(1 + |x − y|). (7) min(x, y). sin(α(x − y)) . (8) For any α > 0, x−y On the half space {x > 0}, the following are reproducing kernels: 1 (1) (q ≥ 12 ). (x + y)2q 1 (q ≥ 12 ). (2) 2 (x + y 2 )2q (3) exp{min(x, y)}. On the interval {−1 < x < 1}, the following are reproducing kernels: 1 (q ≥ 12 ). (1) (1 − xy)2q 1 (2) log . 1 − xy Furthermore, note that any reproducing kernel K(p, q) on an arbitrary set E for a separable reproducing kernel Hilbert space is represented in the form, for some functions {ϕj (p)} on E X ϕj (p)ϕj (q), K(p, q) = j

that converges absolutely on E × E. Conversely, any function K(p, q) which is represented in this way for arbitrary complex-valued functions {ϕj (p)} on E is a reproducing kernel. 3. Exact algorithm We shall state the exact algorithm looking for the extremal function f d,α (p) in (4) clearly in the setting (7). 1) We set ˜ q; α))(P ), X(P, q) = (Lp K(p, k(P, q) = (Lp K(p, q))(P )

(10)

and κ(P, Q) = (Lq Lp K(p, q))(P, Q).

(11)

2) As the solution of the regular linear equations (8) αX(Pj , q) +

N X

j ′ =1

X(Pj ′ , q)κ(Pj , Pj ′ ) = k(Pj , q); j = 1, 2, ..., N,

(12)

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589

we determine X(Pj , q). Then we obtain the approximate inverse fd,α (p) =

N X

dj X(Pj , p).

(13)

j=1

Therefore, for some concrete problem for its inversion, we need the experimental data (10) and (11) of the two types in 1) and the procedure 2) is a mathematical problem. By Proposition 0.2, we note that in (13), the following estimate holds: 1/2 X N 1 p d2j . K(p, p) |fd,α (p)| ≤ √ α j=1

The simplest and the most typical case of the above algorithm is that the system L is any type matrix of type m and n (without loss of generality we assume that n ≥ m), and the positive matrix is the identity matrix of size n. Acknowledgements S. Saitoh is supported in part by the Grant-in-Aid for Scientific Research (C)(2)(No. 19540164) from the Japan Society for the Promotion of Science. References 1. N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404. 2. H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems Mathematics and Its Applications, 376, Kluwer Academic Publishers, 2000. 3. C. W. Groetsch, Inverse Problems in the Mathematical Sciences, Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden, 1993. 4. H. Itou and S. Saitoh, Analytical and numerical solutions of linear singular integral equations, Int. J. Appl. Stat. 12 (2007), 77–89. 5. T. Matsuura and S. Saitoh, Analytical and numerical inversion formulas in the Gaussian convolution by using the Paley-Wiener spaces, Appl. Anal. 85 (2006), 901–915. 6. T. Matsuura, S. Saitoh and D. D. Trong, Approximate and analytical inversion formulas in heat conduction on multidimensional spaces, J. Inverse Ill-posed Probl. 13 (2005), 479–493. 7. S. Saitoh, Integral Transforms, Reproducing Kernels and their Applications, Pitman Res. Notes in Math. Series, 369, Addison Wesley Longman, UK, 1997. 8. S. Saitoh, Approximate real inversion formulas of the Gaussian convolution, Appl. Anal. 83 (2004), 727–733.

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Session 14

Spaces of Differentiable Functions of Several Real Variables and Applications

SESSION EDITORS V. I. Burenkov S. G. Samko

Cardiff University, Cardiff, Wales, UK Universidade do Algarve, Faro, Portugal

593

¨ WEIGHTED HOLDER ESTIMATES OF SINGULAR INTEGRALS GENERATED BY A GENERALIZED SHIFT OPERATOR SADIG K. ABDULLAYEV∗ and A. A. AKPEROV Systematic investigation of multi-dimensional singular integrals generated by the generalized shift operator connected with the Laplace-Bessel differential equations started with the works of N. A. Kipriyanov and M. I. Klyuchantsev, where Privalov-type theorems were in particular proved. We study such γ integrals in the weighted H¨ older spaces Hαβ . We obtain sufficient conditions on the parameters α, β, and γ for the boundedness of such singular integral operators in these spaces. Note that we consider the case when the generalized shift is taken with respect to an arbitrary number of variables. Keywords: Singular integral operator, generalized shift operator, weighted H¨ older space.

1. Introduction Let Rn be the Euclidean space of dimension n ≥ 1,

+ Rm+k,k = {(x1 , ..., xm , xm+1 , ..., xm+k ) ∈ Rm+k : xm+1 > 0, ..., xm+k > 0},  + + Sm+k,k = x ∈ Rm+k,k : |x| = 1 ,

and let s

T u (x) = cν

Z

0

π

Z π  q u x′ − s′ , x2m+1 − 2xm+1 sm+1 cos α1 + s2m+1 , ..., ... 0

k q Y sin2νm+j −1 αj dα1 ...dαk x2m+k − 2xm+k sm+k cos αk + s2m+k

(1)

j=1

be the generalized shift operator (GSO) generated by the Laplace-Bessel operator [5–7]  m m+k X X  ∂2 ∂2 2νi ∂ ∆B = , + + ∂x2i ∂x2i xi ∂xi i=1 i=m+1 ∗ Partially

supported by INTAS (Grant 05-1000008-8157).

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Qk where νm+1 > 0, . . . , νm+k > 0, cν = π −k/2 j=1 Γ(νm+j + 1/2)Γ−1 (νm+j ) is a normalized constant. We use the notation x = (x′ , xm+1 , ..., xm+k ), s = (s′ , sm+1 , ..., sm+k ), x′ , s′ ∈ Rm . We consider the singular integral Z f (θ) 2νm+k 2νm+1 Au(x) = lim ...sm+k ds, (2) [T s u(x)] · sm+1 m+k+2ν ε→0+ |s| {s∈R+ m+k,k :|s|>ε} where θ = s/|s|, ν = νm+1 + ... + νm+k . 2. Preliminaries The starting point in solving the problem of the existence of singular integral (2) is the easy-to-check representation   Z s 2νm+1 2νm+k −m−k−2ν · |s| [T s u (x)] · sm+1 ...sm+k ds f |s| {s∈R+ m+k,k :a 0, ∀x ∈ Rm+k,k , and |h| ≤ rΓ8(x)

|u(x) − u(x + h)| ≤ c2 (u) ρ−1 (x) · (1 + |x|)−2γ · |h|γ . In this case kuk ≍ (min c1 (u) + min c2 (u)). Introduce the sets     rΓ (x) rΓ (x) + , ωx′ = y ∈ Rm+2k : |˜ x −y| < , ωx = s ∈ Rm+k,k : |s−x| < 2 2 For further reasonings we need the following.   γ + Corollary 2.1. If u ∈ Hαβ Rm+k,k , then

 q  kuk γ (A (y, y˜))γ−α q Hαβ 2 2 2 2 ≺ + y˜m+1 , ..., ym+k + y˜m+k a) u y ′ , ym+1 , l (1 + |y ′ | + B (y, y˜))

596

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 and B (y, y˜) where A (y, y˜) = min |ym+i | + |˜ ym+i |, i = 1, k k P (|ym+i | + |˜ ym+i |).

=

i=1

+ b) ∀x ∈ Rm+k,k , ∀y ∈ ωx′ ,  q  q γ dγ (˜ x, y) kukHαβ u y ′ , y 2 + y˜2 , . . . , y 2 + y˜2 −u(x) ≺ . m+1 m+1 m+k m+k ρ(x)

3. Main result

In this paper we consider a problem on the boundedness   of a singular inγ + tegral operator A : u 7→ Au in the spaces Hαβ Rm+k,k , where Au(x) is determined by formula (2). Theorem 3.1. Let f (θ) be bounded and suppose ∃C(f ) > 0, ∀θ1 , θ2 ∈ + Sm+k,k , |f (θ1 ) − f (θ2 )| ≤ Cf |θ1 − θ2 |δ , δ > 0, and Z 2νm+k 2νm+1 . . . θm+k ds(θ) = 0. f (θ)θm+1 + Sm+k,k

Then γ + + 1) for each u ∈ Hαβ (Rm+k,k ) singular integral exists for all x ∈ Rm+k,k ; 2) if conditions (5) and γ < δ are fulfilled, then a singular operator A γ + is bounded in Hαβ (Rm+k,k ). + Proof. Let x ∈ Rm+k,k . Then allowing for the reduction Z 2νm+k 2νm+1 . . . θm+k ds(θ) = 0, f (θ)θm+1 + Sm+k,k

we have |Au (x) | ≤ c ( |τ1 (x; ωx′ ) | + |τ2 (x; Rm+2k \ωx′ ) | ), where     q Z q ˜ f (θ) ′ 2 2 2 2 y + y ˜ , ..., y + y ˜ −u(x) u y , τ1 (x; B) = m+1 m+1 m+k m+k m+k+2ν B r 2νm+k −1 2νm+1 −1x˜,y dy ; ... y˜ × y˜m+1   q Z m+k q ˜ f (θ) ′ 2 2 2 2 y + y ˜ , ..., y + y ˜ τ2 (x; B)= u y , m+1 m+1 m+k m+k rxm+k+2ν ˜,y ′ )∩B (Rm+2k \ωx 2ν 2νm+1 −1 −1 × y˜m+1 ... y˜m+k m+k dy , and B ⊂ ωx′ . Taking into account inequalities b) of Corollary 2.1 we get −1 γ ρ(x) |τ1 (x; ωx′ )| ≺ kf kkukHαβ (1 + |x|)−2γ

Weighted H¨ older Estimates of Singular Integrals Generated by a Shift Operator

×

Z

B

−(m+k+2ν−γ)

rx˜,y

2νm+1 −1

|˜ ym+1 |

2νm+k −1

. . . |˜ ym+k |

597

dy

whence γ |τ1 (x; ωx′ )| ≺ kf k · kukHαβ · rΓγ−α (x) · (1 + |x|)−ℓ .

(7)

Now, let us estimate from above τ2 (x, Rm+2k \ωx′ ). Introduce the sets   |y| ∩ (Rm+2k \ωx′ ) Ax = y ∈ Rm+2k : |˜ x − y| ≤ 2   3|y| |y| ∩ (Rm+2k \ωx′ ) , ≤ |˜ x − y| ≤ Bx = y ∈ Rm+2k : 2 2 and Cx = {y ∈ Rm+2k : 3|y| ≤ |˜ x − y|} ∩ (Rm+2k \ωx′ ). Obviously γ (τ (x; A )+τ (x; B )+τ (x; C )) . |τ2 (x, Rm+2k \ωx′ )| ≺ kf k kukHαβ (8) 2 x 2 x 2 x

First of all we note that |˜ x − y| ≥ rΓ (x) for y ∈ Rm+2k \ωx′ . Now estimate from above the integrals at the right hand side of (7). Note that if y ∈ Ax , then (1 + |x|) ≍ (1 + |y|) and |˜ x − y|≍|x′ − y ′ | + rΓ (x) +

k X i=1

|xm+i − ym+i | + |˜ ym+1 | + . . . + |˜ ym+k |.

Then

Z ∞ Z ∞ τ2 (x; Ax ) ≺ (1 + |x|)−ℓ |˜ ym+k |2νm+k −1 d˜ ym+k . . . |˜ ym+1 |2νm+1 −1 d˜ ym+1 0 0 Z ∞ Z ∞ Z ∞ × dym+k . . . dym+2 (A(y, y˜))γ−α dym+1 ×

Z

Rm

0

0

0

−m−k−2ν  k X |xm+i −ym+i |+|˜ ym+1 |+. . .+|˜ ym+k | dz |z|+rΓ (x)+ i=1

≺ rΓγ−α (x) · (1 + |x|)−ℓ ,

(9)

where |z| = |x′ − y ′ |. Let us estimate from above τ2 (x; Bx ). Let |x| ≥ 1 and y ∈ Bx . Then |˜ x − y| ≍ |y| + |x|. Put µ = (γ + β) + (k + γ − α). By (6) µ > 0. Allowing for this fact we get Z ∞ Z ∞ τ2 (x; Bx ) ≺ |˜ ym+k |2νm+k −1 d˜ ym+k . . . |˜ ym+1 |2νm+1 −1 d˜ ym+1 0 0 Z ∞ Z ∞ Z ∞ γ−α × dym+k . . . dym+2 (A(y, y˜)) dym+1 0

0

0

598

S. K. Abdullayev & A. A. Akperov

Z

×

Rm

(|y ′ | + |x| + B(y, y˜))

−(m+k+µ)

dy ′ ≺ rΓγ−α (x) · (1 + |x|)−ℓ .

Let |x| < 1 and y ∈ Bx . Then for |y| ≥ 1 |y| ≍ |y| + |x| + 1, and also for |y| < 1 (1 + |y|) ≍ 1 and |˜ x − y| ≍ |y| + |x|. Therefore Z 2ν −1 2ν −1 γ−α |˜ ym+k | m+k ...|˜ ym+1 | m+1 (A (y, y˜)) dy τ2 (x; Bx ) ≺ m+k+2ν ′ (|y | + rΓ (x) + B (y, y˜)) {y∈Rm+2k :|y| 1). In this paper we are interested in Besov spaces with generalized smoothness. Roughly speaking, we obtain these spaces by replacing the usual regularity parameter s in (2) by a function with certain properties. Precisely, ϕ the space Bpq (Rn ), ϕ ∈ B, 0 < p, q ≤ ∞, is defined as the space of all tempered distributions quasi-normed by !1/q ∞ X ∨ n q j q ϕ n b (3) ϕ(2 ) k(ϕj f ) | Lp (R )k kf | B (R )k := pq

j=0

(usual modification if q = ∞), where B denotes the class of all continuous functions ϕ : (0, ∞) → (0, ∞), such that ϕ(1) = 1 and ϕ(t) := sup{ ϕ(tu) ϕ(u) : u > 0 } is finite for every t > 0 (see [10] for some useful properties of this class). Examples of such functions are ϕ(t) = ta (1 + | log t|)b , with ϕ s ϕ(t) = ta (1 + | log t|)|b| , a, b ∈ R. Clearly, Bpq (Rn ) = Bpq (Rn ) if ϕ(t) = ts . ϕ n The spaces Bpq (R ) appear naturally from real interpolation with function parameter between classical Bessel potential spaces [10]. In particular, s,Ψ they cover the spaces Bpq (Rn ) introduced by Edmunds and Triebel, where Ψ represents a certain perturbation of the smoothness s (more details may be found in [12]). The study of function spaces with generalized smoothness goes back to the 1970’s in connection with some problems of approximation theory investigated by Russian mathematicians [9]. A unified approach to generalized spaces can be found in the recent paper [6], including further references and applications. Below we shall make use of general weighted sequence spaces as follows. If {Xi }i∈I is a countable family of quasi-Banach spaces, ω ≡ {ωi }i∈I is a non-negative “sequence” and 0 < q ≤ ∞, then we will denote by ℓq (I, Xi , ωi ) the weighted sequence space of all families a ≡ {ai }i∈I , ai ∈ Xi , i ∈ I, with the quasi-norm !1/q X q ka | ℓq (I, Xi , ωi )k := , (ωi kai | Xi k) i∈I

604

A. Almeida

where ka | ℓ∞ (I, Xi , ωi )k must be interpreted as sup ωi kai | Xi k if q = ∞. i∈I

We shall omit the index set I if it is clear from the context. When Xi = X for every i ∈ I then we shall write ℓq (I, X, ω), where ω ≡ {ωi }i∈I . Moreover, in the particular case X = C we will also omit the “X” writing only ℓq (I) n n j if ωi = 1 for all i ∈ I. Putting ℓϕ q (Lp (R )) := ℓq (N0 , Lp (R ), ϕ(2 )) then

b ∨ }j∈N | ℓϕ (Lp (Rn )) . we have kf | B ϕ (Rn )k = {(ϕj f) pq

0

q

Lorentz sequence spaces over Z (the lattice of all points in Rn with integer components) will be also of interest in the sequel. Given ϕ ∈ B and 0 < q ≤ ∞, the Lorentz space λq (ϕ, Zn ) consists of all bounded complexvalued sequences a ≡ {am }m∈Zn having a finite quasi-norm !1/q ∞ X   q ka | λq (ϕ, Zn )k := , ϕ(k) a∗k−1 k −1 n

k=1

with the modification for sup [ϕ(k) a∗k−1 ] if q = ∞, where {a∗k }k∈N0 is the k∈N

decreasing rearrangement of {am }m∈Zn , given by a∗k = inf{δ ≥ 0 : #{m ∈ Zn : |am | > δ} ≤ k},

k ∈ N0 .

If ϕ(t) = t1/p then λq (ϕ, Zn ) is the Lorentz sequence space ℓpq (Zn ), which in turn is the classical space ℓq (Zn ) of all q-summable sequences when p = q. 2.2. Wavelet bases Let Lj = L = 2n − 1 if j ∈ N and L0 = 1. For any r ∈ N, there are real compactly supported functions ψ0 ∈ C r (Rn ), with

Z

Rn

such that

ψ l ∈ C r (Rn ),

xβ ψ l (x) dx = 0,

l = 1, . . . , L,

β ∈ Nn0 ,

|β| ≤ r,

 jn l 2 2 ψjm : j ∈ N0 , 1 ≤ l ≤ Lj , m ∈ Zn

is an orthonormal basis in L2 (Rn ), where  ψ0 (· − m) , j = 0, m ∈ Zn , l = 1, l ψjm (·) = l j−1 ψ (2 · −m) , j ∈ N, m ∈ Zn , 1 ≤ l ≤ L.

(4)

(5)

(6)

(7)

An example of such a system is the Daubechies wavelet basis [5]. Further details can be found in [17], [11, Sections 3.8, 3.9] and [18, Chapter 4]. In the

On Interpolation Properties of Generalized Besov Spaces

605

l sequel Ψr , r ∈ N, will stand for a Daubechies wavelet system {ψjm }(l,j,m)∈I with the properties (4)–(7) above, where I = {(l, j, m) : j ∈ N0 , 1 ≤ l ≤ Lj , m ∈ Zn }. Below we also consider the notation I ′ = {(l, j) : j ∈ N0 , 1 ≤ l ≤ Lj }. The properties of this system are sufficiently good to provide unconditional bases in many classical function spaces. It is known that Ψr provides an unconditional Schauder basis in the Bessel potential spaces Hps (Rn ) if 1 < p < ∞, r > |s| (in particular in Lp (Rn ) if 1 < p < ∞), and in the s Besov spaces Bpq (Rn ), with 1 ≤ p, q < ∞, r > |s| (see, for instance, [11, Chapter 6]). Based on atomic decompositions, local means and duality theory, Triebel [16] extended these assertions to the entire scales of Besov and Triebel-Lizorkin spaces (including the weighted case, see 8]). In particular, the Daubechies system gives unconditional bases in   2n n s + −s . Bpq (Rn ) if s ∈ R, 0 < p, q < ∞, r > r(s, p) := max s, p 2

Using suitable interpolation techniques, the author showed that Ψr also ϕ provides an unconditional basis in the generalized Besov spaces Bpq (Rn ), which is a corollary of the following statement proved in [1]. Theorem 2.1. Let ϕ ∈ B, 0 < p < ∞ and 0 < q ≤ ∞. Then there exists r(ϕ, p) such that, for any natural r > r(ϕ, p), there holds: given f ∈ S ′ (Rn ), ϕ then f ∈ Bpq (Rn ) if and only if it can be represented as X l (8) µljm ψjm with µ = {µljm }(l,j,m)∈I ∈ bϕ f= pq (l,j,m)∈I

(summability in S ′ (Rn )). Moreover, the wavelet coefficients µljm are uniquely determined by l µljm = µljm (f ) := 2jn hf, ψjm i,

Further,

(l, j, m) ∈ I.



f | B ϕ (Rn ) ∼ µ(f ) | bϕ , µ(f ) ≡ {µl (f )}(l,j,m)∈I . pq pq jm

Here bϕ pq denotes the sequence space quasi-normed by !q/p !1/q q X X  ϕ j −jn/p l p kµ | bpq k := ϕ(2 ) 2 |µjm | (l,j)∈I ′

(9)

(10)

(11)

m∈Zn

(usual modifications if p = ∞ and/or q = ∞) and r(ϕ, p) is given by   2n n (12) + − βϕ , r(ϕ, p) := max αϕ , p 2

606

A. Almeida

where αϕ and βϕ are, respectively, the upper and lower Boyd indices αϕ = lim

t→+∞

log ϕ(t) log t

and βϕ = lim

t→0

log ϕ(t) . log t

s For instance, if ϕ(t) = ts then αϕ = βϕ = s and bϕ pq = bpq is the sequence space considered by Triebel in [16].

Remark 2.1. The previous statement does not make use of duality theory ϕ l for the spaces Bpq (Rn ). In fact, the symbol hf, ψjm i should be interpreted l s as the sum of two quantities of the type hg, ψjm i, with g ∈ Bpq (Rn ) and l ψjm ∈ C r (Rn ) (see [1, Remarks 10, 15] for further details). 3. Interpolation with function parameter 3.1. Preliminaries Real interpolation with function parameter is convenient to study interϕ polation properties of spaces Bpq (Rn ), since we can take the interpolation parameter from the same class B. Recall that the interpolation space (A0 , A1 )γ,q , γ ∈ B, of compatible quasi-Banach spaces A0 , A1 is the space of  Z ∞  q dt 1/q all f ∈ A0 + A1 quasi-normed by kf kγ,q := γ(t)−1 K(t, f ) t 0 (modification if q = ∞), where K(t, f ) is the well-known Peetre Kfunctional (see [3,10] for details). As usual, if γ(t) = tθ then we simply write (A0 , A1 )θ,q to denote the corresponding interpolation space. For ϕ0 , ϕ1 , γ ∈ B, 0 < p, q0 , q1 , q ≤ ∞, we have
ϕ1 ϕ0 ϕ (Rn ) γ,q = Bpq (Rn ), Bpq Bpq (Rn ), (13) 1 0 where 0 < βγ ≤ αγ < 1, β ϕ0 > 0 (or α ϕ0 < 0), and ϕ(t) := ϕ1

ϕ1

γ

ϕ0 (t)  . ϕ0 (t) ϕ1 (t)

This

statement was proved by Cobos and Fernandez [4] in the Banach case by reducing the problem to the interpolation of appropriate sequence spaces. ϕ n They observed that Bpq (Rn ) is a retract [3,15] of ℓϕ q (Lp (R )), where R{fj }j∈N0 :=

∞ X j=0

F −1 (ϕ ej F fj ),

with ϕ ej =

1 X

r=−1

ϕj+r

(ϕ−1 ≡ 0), (14)

n ϕ n (convergence in S ′ (Rn )) is the retraction from ℓϕ q (Lp (R )) to Bpq (R ) and

J f := {F −1 (ϕj F f )}j∈N0

(15)

ϕ n gives the corresponding co-retraction from Bpq (Rn ) to ℓϕ q (Lp (R )). It was also observed in [4, Remark 5.4] that the result above remains true in the

On Interpolation Properties of Generalized Besov Spaces

607

quasi-Banach case. For this one should replace the space Lp (Rn ) by the Hardy space hp (Rn ), since the mapping R above is meaningless if 0 < p < 1. We refer to [1] for a more detailed description how this question in the general case can be dealt with. 3.2. Interpolation with change of p ϕ Real interpolation between spaces Bpq (Rn ) with p fixed produces a Besov space of the same type (cf. formula (13)). However, the change of the parameter p leads to different spaces, which are not included in the scale defined by (3) (cf. (1)). This question was studied in [4, Theorem 5.8] using the same mappings (14), (15). For 1 ≤ p0 6= p1 < ∞,
1 ≤ q < ∞, ϕ, γ ∈ B, ϕ (Rn ), with with 0 < βγ ≤ αγ < 1, one has Bpϕ0 q (Rn ), Bpϕ1 q (Rn ) γ,q = Bρ,q   −1 1 1 1 ϕ , where Bρ,q (Rn ) is a certain Besov type space ρ(t) = t p0 γ t p0 − p1 modelled on a generalized Lorentz space instead on the usual Lp -space. The quasi-Banach case was not considered in [4]. In fact, a direct extension of the Banach case would require some interpolation results for local Hardy spaces which are not available. So, instead of taking hp (Rn ) in place of Lp (Rn ), we propose a new retraction based on wavelet decompositions obtained in [1]. This will be developed in the sequel. l Let Ψr ≡ {ψjm }(l,j,m)∈I be a Daubechies wavelet system. If f ∈ ϕ n ϕ Bp0 q (R ) + Bp1 q (Rn ) and f = f0 + f1 , fi ∈ Bpϕi q (Rn ), i = 0, 1 is any l decomposition of f , then the quantities hf, ψjm i, given by l l l i := hf0 , ψjm i + hf1 , ψjm i, hf, ψjm

l hfi , ψjm i

(l, j, m) ∈ I,

(16)

with interpreted in the sense of Remark 2.1, are well-defined if we take the natural number r large enough, e.g. r > max (r(ϕ, p0 ), r(ϕ, p1 )) ,

(17)

l hf, ψjm i

where r(ϕ, pi ) is defined by (12). Since the quantity in (16) does not depend on the decomposition of f , we can introduce the following spaces. Definition 3.1. Let ϕ ∈ B, 0 < p0 , p1 , q, ν < ∞ and Υp0 ,p1 ≡ {ρj }j∈N0 be a family of functions (depending on p0 , p1 ) “almost” in B (in the sense that l the assumption ρj (1) = 1 may be dropped). Let also Ψr ≡ {ψjm }(l,j,m)∈I be a Daubechies wavelet system with r fixed according to (17). We introduce o n ϕ,(ν) ϕ,(ν),[Ψ ] BΥp ,p ,qr (Rn ) = f ∈ Bpϕ0 q (Rn ) + Bpϕ1 q (Rn ) : µ(f ) ∈ bΥp ,p ,q 0

1

0

1

where in

l µ(f ) = {µljm (f )}(l,j,m)∈I := {2jn hf, ψjm i}(l,j,m)∈I ,

(18)

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A. Almeida

l hf, ψjm i is given by (16). This space is naturally quasi-normed by





ϕ,(ν),[Ψ ] ϕ,(ν)

f | BΥp0 ,p1 ,qr (Rn ) := µ(f ) | bΥp0 ,p1 ,q ,

where we consider

ϕ,(ν)

µ(f ) | bΥp ,p 0



:= 1 ,q

!1/q X 

q

l j −j n n

ϕ(2 ) 2 ν {µjm (f )}m∈Zn | λq (ρj , Z ) .

(l,j)∈I ′

Now we are able to formulate the main result.

Theorem 3.1. Let ϕ ∈ B, 0 < p0 6= p1 < ∞, 0 < q < ∞. Let also γ ∈ B with 0 < βγ ≤ αγ < 1. Then ϕ,(p0 ),[Ψr ] (Rn ) 0 ,p1 ,q

BΥp

ϕ,(p0 ),[Ψr ] (Rn ), 0 ,p1 ,q

֒→ (Bpϕ0 q (Rn ), Bpϕ1 q (Rn ))γ,q ֒→ BΓp

where Υp0 ,p1 ≡ {Aγ,p0 ,p1 (j) ρj }j∈N0 , Γp0 ,p1 1

and

(19)

 ≡ Aγ,p0 ,p1 (j)−1 ρj j∈N0 , with

t p0 , ρj (t) =  1 1 1 1 γ 2−jn( p0 − p1 ) t p0 − p1

j ∈ N0 ,

    −jn( p1 − p1 ) jn( 1 − 1 ) 0 1 γ 2 p0 p1 , Aγ,p0 ,p1 (j) : = γ 2

(20)

j ∈ N0 .

(21)

Remark 3.1. We cannot give an explicit formula since the quantities Aγ,p0 ,p1 (j) in (21) cannot be uniformly estimated with respect to j. In general, we only know that Aγ,p0 ,p1 (j) ≥ 1, which can be easily checked from the properties of the class B. However, at least in the (parameter) case γ(t) = tθ (0 < θ < 1), Theorem 3.1 gives an exact interpolation formula (see (22) below), since we have Aγ,p0 ,p1 (j) ≡ 1 in this case and, consequently, the corresponding endpoint spaces in (19) coincide. Corollary 3.1. Let ϕ ∈ B, 0 < p0 6= p1 < ∞, 0 < q < ∞. Then ϕ,(p0 ),[Ψr ] (Rn ), 0 ,p1 ,q

(Bpϕ0 q (Rn ), Bpϕ1 q (Rn ))θ,q = BΓp where Γp0 ,p1 ≡



2

  1 jn p1 − p 0

1

tp



j∈N0

, with

1 p

=

1−θ p0

+

θ p1 ,

(22)

0 < θ < 1.

For the classical smoothnes ϕ(t) = ts , formula (22) provides a chars acterization for the spaces Bpq(q) (Rn ) considered by Triebel [15] (cf. (1)), s n since we may write Bpq(q) (R ) = (Bps0 q (Rn ), Bps1 q (Rn ))θ,q .

On Interpolation Properties of Generalized Besov Spaces

609

s Corollary 3.2. Let s ∈ R, 0 < p, q < ∞. Then f ∈ Bpq(q) (Rn ) if and only s if µ(f ) ∈ bpq(q) , where



µ(f ) | bspq(q) :=

X h

iq n 2j (s− p ) {µljm (f )}m∈Zn | ℓpq (Zn )

(l,j)∈I ′

!1/q

.

In particular, bspq(p) = bspq are the sequence spaces used in [16].

Proof of Theorem 3.1. From Theorem 2.1 we may conclude that Bpϕi q (Rn ) is a retract of bϕ pi q , i = 0, 1. In fact, X l (summability in S ′ (Rn )) µljm ψjm R{µljm } = (l,j,m)∈I

ϕ n defines a retraction from bϕ pi q into Bpi q (R ), being the associate col l retraction given by Jf = {µjm (f )} with µljm (f ) = 2jn hf, ψjm i. Therefore, we have



ϕ

f | (Bpϕ q (Rn ), Bpϕ q (Rn ))γ,q ∼ Jf | (bϕ (23) p0 q , bp1 q )γ,q . 0 1

So it remains to interpolate the sequence spaces in the right-hand side of (23). To this end, we observe that bϕ pi q may be interpreted as the weighted
−j n sequence space ℓq ℓpi (Zn ), ω i , where ω i = {ϕ(2j ) 2 pi }(l,j)∈I ′ (cf. the notation from Section 2.1). From Proposition 3.2 in [13] we obtain



ℓq ℓp0 (Zn ), ω 0 , ℓq ℓp1 (Zn ), ω 1 γ,q = ℓq (ℓp0 (Zn ), ℓp1 (Zn ))γj ,q , ωj0 ,   1 1 γj ej = γj (1) where γj (t) = γ 2−jn( p0 − p1 ) t , j ∈ N0 . Putting γ

then γ ej ∈ B and (ℓp0 (Zn ), ℓp1 (Zn ))γj ,q = γj1(1) (ℓp0 (Zn ), ℓp1 (Zn ))γe j ,q , where γj1(1) (ℓp0 (Zn ), ℓp1 (Zn ))γe j ,q denotes the space (ℓp0 (Zn ), ℓp1 (Zn ))γe j ,q equipped with the quasi-norm γj1(1) k · kγe j ,q . But (ℓp0 (Zn ), ℓp1 (Zn ))γe j ,q = λq (ρj , Zn ),

(24)

where λq (ρj , Zn ) are Lorentz sequence spaces and ρj := γj (1) ρj , with 1

ρj (t) =

p  t1 0 1  , − p γj t 0 p 1

j ∈ N0 (see [10, Theorem 3]). Formula (24) can be

written in the “non-normalized form” as (ℓp0 (Zn ), ℓp1 (Zn ))γj ,q = λq (ρj , Zn ),

j ∈ N0 .

The properties of B (see [10]) allow us to arrive at the estimates

1 (Aγ,p0 ,p1 (j))−1 k · | λq (ρj , Zn )k ≤ k · kγj ,q ≤ c Aγ,p0 ,p1 (j) k · | λq (ρj , Zn )k, c

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the constants Aγ,p0 ,p1 (j) given by (21) and c > 1 independent of j. Thus λq (Aγ,p0 ,p1 (j) ρj , Zn ) ֒→ (ℓp0 (Zn ), ℓp1 (Zn ))γj ,q ֒→ λq (Aγ,p0 ,p1 (j)−1 ρj , Zn ), where now the “embedding constants” are independent of j. This yields ϕ,(p0 )

b Υp

0 ,p1 ,q

ϕ,(p0 )

ϕ ֒→ (bϕ p0 q , bp1 q )γ,q ֒→ bΓp

0 ,p1 ,q



,

with Υp0 ,p1 = {Aγ,p0 ,p1 (j) ρj }j∈N0 and Γp0 ,p1 = Aγ,p0 ,p1 (j)−1 ρj from which we obtain (19) taking into account the equivalence (23).



j∈N0

,

References 1. A. Almeida, Wavelet bases in generalized Besov spaces, J. Math. Anal. Appl. 304 (2005), 198–211. 2. A. Almeida, Function Spaces with Generalized Smoothness and Variable Integrability, PhD Thesis, University of Aveiro, 2005. 3. J. Bergh and J. L¨ ofstr¨ om, Interpolation Spaces. An Introduction, SpringerVerlag, Berlin, 1976. 4. F. Cobos and D. L. Fernandez, Hardy-Sobolev spaces and Besov spaces with a function parameter, Lecture Notes in Math., vol. 1302 (1988), 158–170. 5. I. Daubechies, Orthonormal bases of compactly supported wavelets, Commun. Pure Appl. Math. 41 (1988), no. 7, 909–996. 6. W. Farkas and H.-G. Leopold, Characterizations of function spaces of generalized smoothness, Annali Math. Pura Appl. 185 (2006), no. 1, 1–62. 7. D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 (1979), 27–42. 8. D. D. Haroske and H. Triebel, Wavelet bases and entropy numbers in weighted function spaces, Math. Nachr. 278 (2005), no. 1-2, 108–132. 9. G. A. Kalyabin and P. I. Lizorkin, Spaces of functions of generalized smoothness, Math. Nachr. 133 (1987), 7–32. 10. C. Merucci, Applications of interpolation with a function parameter to Lorentz, Sobolev and Besov spaces, Lecture Notes in Math., vol. 1070 (1984), 183–201. 11. Y. Meyer, Wavelets and Operators, Cambridge Univ. Press, Cambridge, 1992. 12. S. Moura, Function spaces of generalized smoothness, Dissert. Math. 398 (2001), 1–87. 13. L. E. Persson, Real Interpolation Between Cross-Sectional Lp -Spaces in Quasi-Banach Bundles, Technical Report 1, University of Lule˚ a, 1986. 14. H. Triebel, Theory of Function Spaces, Birkh¨ auser, Basel, 1983. 15. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Johann Ambrosius Barth, Heidelberg, 1995. 16. H. Triebel, A note on wavelet bases in function spaces, in: Orlicz Centenary Volume, Banach Center Publications, 64 (2004), 193–206. 17. H. Triebel, Theory of Function Spaces III, Birkh¨ auser, Basel, 2006. 18. P. Wojtaszczyk, A Mathematical Introduction to Wavelets, Cambridge Univ. Press, Cambridge, 1997.

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PARTIAL HYPOELLIPTICITY OF DIFFERENTIAL OPERATORS TSEGAYE G. AYELE Addis Ababa University, Faculty of Science, Addis Ababa, Ethiopia E-mail: [email protected] WORKU T. BITEW Temple University, Philadelphia, USA E-mail:[email protected] We describe the partial hypoellipticity of differential operators with constant coefficients in terms of fundamental solutions. Keywords: Convolution, Differential Operator, Hypoellipticity, Fundamental Solution.

1. Introduction Consider a nonzero linear differential operator P (D) with constant coefficients X P (D) = aα D α , (1) α∈K

where K is a finite set in Nn0 , the space of multi-indices, i.e., Nn0 = N0 ×· · ·× N0 (n copies) with N0 = N∪{0}. For α ∈ Nn0 , |α| = α1 +· · ·+αn , max |α| = l; αj α aα ∈ K are constants; D α = D1α1 D2α2 · · · Dnαn , and Dj j = 1i ∂ αj . ∂xj

Definition 1.1. Let Ω ⊂ Rn be an open set and u ∈ D ′ (Ω). A differential operator (1) is called hypoelliptic if P (D)u ∈ C ∞ (Ω) implies u ∈ C ∞ (Ω). For the first time some remarkable results about infinitely differentiability and analyticity of solutions of elliptic differential operators were obtained by Bernstein at the beginning of the 20th century. Later on Petrovskii proved that all classical solutions of equation P (D)u = f with analytic right hand side are analytic if and only if operator P (D) is elliptic.

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The conditions for ellipticity and hypoellipticity of operator P (D) in terms of fundamental solutions were described by H¨ ormander [2]. But the condition for partial hypoellipticity of operator P (D) in terms of fundamental solutions was not established. We look at the case when the solution u of the equation P (D)u = f is conditionally smooth whenever f is a smooth function. That is, we want to study the smoothness of u with respect to some variables. We will use the expression generalized functions for distributions. Let Ω ⊂ Rn be an open set m, n ∈ Z such that 0 ≤ m < n , for x ∈ Ω put x = (x′ , x′′ ) where x′ ∈ Rm and x′′ ∈ Rn−m . If we denote ψm (x) = ψ(x′ ) ⊗ δ(x′′ ), where δ is the Dirac delta function, then supp(ψm ) = supp(ψ) × supp(δ) = supp(ψ) × {0} and later we shall show that the set \ (Ω + y) Ωψm = {x : {x} − (supp(ψm )) ⊂ Ω} = y∈supp(ψm )

′

is open and if u ∈ D (Ω), then the convolution u ∗ ψm is defined on the set Ωψm and is a generalized function in D ′ (Ωψm ). Definition 1.2. Let Ω ⊂ Rn be an open set and u ∈ D ′ (Ω). A differential operator (1) is called partially hypoelliptic with respect to the plane x′′ = 0, if P (D)u ∈ C ∞ (Ω) implies that for any function ψ ∈ D(Rm ) the convolution u ∗ ψm ∈ C ∞ (Ωψm ). Remark 1.1. If m = 0, then ψ0 = δ and thus we obtain Definition 1.1. Definition 1.3. A generalized function E ∈ D ′ (Rn ) is called a fundamental solution for a differential operator (1) if P (D)E = δ.

(2)

The existence of a fundamental solution for an operator (1) in D ′ (Rn ) was first proved independently by Malgrange (1953) and Ehrenpresis (1954) whereas its existence in the space S ′ (Rn ) of generalized functions of slow growth (= tempered distributions) was proved by H¨ ormander (1958). ′ n Equation (2) in the space S (R ) is equivalent to the algebraic equation, n P (iξ)Eˆ = (2π)− 2 ,

(3)

ˆ Thus the problem of finding with respect to the Fourier transform FE = E. a fundamental solution of slow growth turns out to be a special case of the more general problem of dividing a generalized function of slow growth by

Partial Hypoellipticity of Differential Operators

613

a polynomial, that is, of the problem of finding a solution u ∈ S ′ (Rn ) of the equation P (iξ)u = f,

(4)

where P is a nonzero polynomial and f is a specified generalized function in S ′ (Rn ) . The solvability of the problem of division was proved in 1958 independently by H¨ ormander and Lojasiewicz. As a consequence it was proved that every nonzero linear differential operator with constant coefficients has a fundamental solution of slow growth. In this paper we prove the following theorem in which a description of the partial hypoellipticity of a differential operator (1) in terms of its fundamental solutions is given. Theorem 1.1. For a differential operator (1) to be partially hypoelliptic with respect to the plane x′′ = 0, it is necessary and sufficient that there exists a fundamental solution E of (1) such that for any function ψm ∈ D(Rn ) the convolution E ∗ ψm ∈ C ∞ (Rn \supp(ψm )). Remark 1.2. From the existence of such fundamental solution it follows that every fundamental solution E satisfies E ∗ ψm ∈ C ∞ (Rn \supp(ψm )). Remark 1.3. If m = 0, then ψ0 = δ and we obtain H¨ ormander’s theorem about description of the hypoellipticity of an operator (1) in terms of fundamental solutions. We will state and prove it later for the sake of completeness. 2. Preliminaries In this section we will describe the basic tools we need in the proof of our theorem and we prove them for the sake of completeness. For sets A and B we will use the following notations, A − B = {x − y : x ∈ A and y ∈ B} and the distance between A and B, ρ(A, B) = inf x∈A,y∈B |x − y|. Theorem 2.1. Let Ω ⊂ Rn be open , u ∈ D ′ (Ω), ψ ∈ D(Rn ). Then \ (Ω + y), Ωψ = {x : {x} − supp(ψ) ⊂ Ω} = y∈supp(ψ)

the convolution u∗ψ exists on Ωψ and (u∗ψ, ϕ)(x) = (u(x), (ψ(y), ϕ(x+y)) for all ϕ ∈ D(Ωψ ).

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Note that for the convolution u ∗ ψm to be defined for all ϕ ∈ D(Ωψm ) T it is important to show that the set Ωψm = y∈supp(ψm ) (Ω + y) is open. Proposition 2.1. If Ω ⊂ Rn and x, y ∈ Rn , then ρ(x−y, Ω) = ρ(x, Ω +y).

Proposition 2.2. If Ω ⊂ Rn and x, y, z ∈ Rn , then |ρ(x, Ω + y) − ρ(x, Ω + z)| ≤ |y − z|. Proof. |ρ(x, Ω + y) − ρ(x, Ω + z)| = |ρ(x − y, Ω) − ρ(x − z, Ω)| and |ρ(x − y, Ω) − ρ(x − z, Ω)| ≤ |x − y − (x − z)| ≤ |y − z| by Proposition 2.1. Corollary 2.1. Let Ω ⊂ Rn and x, y ∈ Rn . Then the function ρ(x, Ω + y) is continuous in y on Rn . Proposition 2.3. Let Ωα ⊂ Rn , where α ∈ A ⊂ R. Then for any x ∈ Rn the condition [ ρ(x, Ωα ) > 0 (5) α∈A

is equivalent to inf ρ(x, Ωα ) > 0 .

(6)

α∈A

Proof. Condition (6) is equivalent to the existence of c > 0, such that ρ(x, Ωα ) > c, for all α ∈ A. Then B(x, c) ⊂ Ωαc , for all α ∈ A, and this means \ [ [ B(x, c) ⊂ Ωαc = ( Ωα )c ⇒ ρ(x, Ωα ) > 0. α∈A

α∈A

α∈A

Conversely, let condition (5) be satisfied and suppose that condition (6) be not satisfied. If A is a finite set, then there exists α0 ∈ A such that ρ(x, Ωα0 ) = 0. Hence [ ρ(x, Ωα ) ≤ ρ(x, Ωα0 ) = 0. S

α∈A

That is, ρ(x, α∈A Ωα ) = 0. If A is an infinite set, then for each k ∈ N there exists αk ∈ A such that ρ(x, Ωαk ) ≤ 1/k. Consequently, [ 1 ρ(x, Ωα ) ≤ ρ(x, Ωαk ) ≤ k α∈A S and taking the limit we get ρ(x, α∈A Ωα ) = 0. In both cases we arrived at a contradiction to condition (5) and this completes the proof.

Partial Hypoellipticity of Differential Operators

615

Lemma 2.1. Let Ω ⊂ Rn be an open set and F compact, then the set T G = y∈F (Ω + y) is open.

Proof. That G is open means that ρ(x, Gc ) > 0 for all x ∈ G. That is, for all x ∈ G, \ [ [ ρ(x, [ (Ω + y)]c ) = ρ(x, (Ω + y)c ) = ρ(x, (Ω c + y)) > 0. y∈F

y∈F

y∈F

By Proposition 2.3 this is equivalent to inf y∈F ρ(x, (Ω c + y)) > 0 for all x ∈ G. Assume to the contrary that there exists x ∈ G such that inf y∈F ρ(x, (Ω c + y)) = 0. This implies that for every k ∈ N there exists yk ∈ F such that ρ(x, (Ω c + yk )) ≤ 1/k. We can select a convergent subsequence {yks } such that lims→∞ yks = y0 for y0 ∈ F . Then by Corollary 2.1, ρ(x, (Ω c + y0 )) = lims→∞ ρ(x, (Ω c + yks )) = 0. Since (Ω c + y0 ) is closed by the assumption, [ [ \ x ∈ (Ω c + yo ) ⊂ (Ω c + y) = (Ω + y)c = [ (Ω + y)]c = Gc . y∈F

y∈F

y∈F

This contradicts the fact that x ∈ G. Consequently, G = open.

T

y∈F (Ω

+ y) is

From Lemma 2.1 we can conclude that for ψ ∈ D(Rn ) and open set T Ω ⊂ Rn the set Ωψ = y∈supp(ψ) (Ω + y) is open. Next we want to show that (ψ(y), ϕ(x + y)) ∈ D(Ω), for all ψ ∈ D(Rn ), ϕ ∈ D(Ωψ ). To do this we need the following two propositions. Proposition 2.4. For ψ ∈ D(Rn ), ϕ ∈ D(Ωψ ), let Z ψ(y)ϕ(x + y)dy. Λ(x) = Rn

Then supp(Λ) ⊂ supp(ϕ) − supp(ψ).

Proof. Let H = {x : for all y ∈ supp(ψ), (x + y) ∈ / supp(ϕ)}. Observe that Z ψ(y)ϕ(x + y)dy = 0}. H ⊂ {x : supp(ψ)

This implies that supp(Λ) ⊂ H c . We know that x ∈ H c ⇔ there exists y ∈ supp(ψ) such that x ∈ (supp(ϕ) − y) ⇔ x ∈ supp(ϕ) − supp(ψ). Since H c = H c , we have supp(Λ) ⊂ supp(ϕ) − supp(ψ). Proposition 2.5. If supp(ϕ) ⊂ Ωψ , then supp(ϕ) − supp(ψ) ⊂ Ω.

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Proof. Since supp(ϕ) ⊂

\

y∈supp(ψ)

(Ω + y) ⇔ for all y ∈ supp(ψ),

supp(ϕ) ⊂ Ω + y. It follows that there exists y ∈ supp(ψ) such that supp(ϕ) − y ⊂ Ω ⇔ supp(ϕ) − supp(ψ) ⊂ Ω. Propositions 2.4 and 2.5 imply that (ψ(y), ϕ(x + y)) ∈ D(Ω) for all ψ ∈ D(Rn ) and ϕ ∈ D(Ωψ ). Remark 2.1. i) For any u ∈ D ′ (Ω) and ψ ∈ D(Rn ) the convolution u ∗ ψ is well defined on Ωψ . ii) Suppose Ω ∈ Rn is an open set, m, n ∈ Z are such that 0 ≤ m < n, and for any ψ ∈ D(Rm ) define ψm (x) = ψ(x′ ) ⊗ δ(x′′ ), where x = (x′ , x′′ ), where x′ ∈ Rm , x′′ ∈ Rn−m . Then supp(ψm ) = supp(ψ) × {0}, and the set T Ωψm = y∈supp(ψm ) (Ω + y) is open. Definition 2.1. Let Ω ∈ Rn be open, u ∈ D ′ (Ω) and ψ ∈ D(Rm ). Then for ϕ ∈ D(Ωψm ), we define Z (u ∗ ψm , ϕ) = (u(x), ψ(y ′ )ϕ(x + y ′ )dy ′ ). supp(ψ)

Proposition 2.6. Let Λ(x) = supp(ϕ) − (supp(ψ) × {0}).

R

Rn

ψ(y ′ )ϕ(x + y ′ )dy ′ . Then supp(Λ) ⊂

′ ′ Proof. Define H / supp(ϕ)}. R = {x′ : for all ′y ∈′ supp(ψ) × {0}, (x + y ) ∈ Then H ⊂ {x : Rn ψ(y )ϕ(x + y )dy = 0}. Hence supp(Λ) ⊂ H c . Now let x ∈ H c , that is, there exists y ′ ∈ supp(ψ)×{0} such that (x+y ′ ) ∈ supp(ϕ). This is equivalent to the existence of y ′ ∈ supp(ψ)×{0} : x ∈ (supp(ϕ)−y ′ ). Therefore x ∈ supp(ϕ) − supp(ψ) × {0}. Observe that supp(ϕ) ⊂ Ωψm and

supp(ϕ) − supp(ψ) × {0} ⊂ Ω.

(7)

This completes the proof. The proof of the following theorem is similar to that of Theorem 2.1 Theorem 2.2. Suppose m, n ∈ Z are such that 0 ≤ m < n, Ω ⊂ Rn is open and ψ ∈ D(Rm ). Then (ψ(y ′ ), ϕ(x + y ′ )) ∈ D(Ω) for all ϕ ∈ D(Ωψm ). Proof. We conclude from (7) and Proposition 2.6 that (ψ(y ′ ), ϕ(x + y ′ )) ∈ D(Ω) for all ψ ∈ D(Rm ) and ϕ ∈ D(Ωψm ).

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617

Remark 2.2. For any u ∈ D ′ (Ω) and ψ ∈ D(Rm ) the convolution u ∗ ψm is well defined on Ωψm . Lemma 2.2. Let Ω ⊂ Rn be an open set, m, n ∈ Z be such that 0 ≤ m < n and ψ ∈ D(Rm ). Let ϕ, ϕk ∈ D(Ωψm ) for k ∈ N and ϕk → ϕ in D(Ωψm ) as k → ∞. Then (ψ(y ′ ), ϕk (x + y ′ )) → (ψ(y), ϕ(x + y ′ )) in D(Ωψ ). Proof. ϕk → ϕ in D(Ωψm ) means ϕk − ϕ → 0 in D(Ωψm ). Hence (ψ(y ′ ), ϕk (x + y ′ )) − (ψ(y ′ ), ϕ(x + y ′ )) = (ψ(y ′ ), (ϕk (x + y ′ ) − ϕ(x + y))) → (ψ(y ′ ), 0) = 0

That is, (ψ(y ′ ), ϕk (x + y ′ )) → (ψ(y), ϕ(x + y ′ )) in D(Ωψ ). Lemma 2.3. Let Ω ⊂ Rn be an open set, m, n ∈ Z be such that 0 ≤ m < n and ψ ∈ D(Rm ). Then u ∗ ψm ∈ D′ (Ωψm ) for any u ∈ D ′ (Ω). Proof. From Remark 2.2 it follows that u∗ψm is defined on Ωψm , and from Lemma 2.2 it follows that the functional is continuous. Since it is linear on D(Ωψm ) we conclude that u ∗ ψm ∈ D′ (Ωψm ). Remark 2.3. If u ∈ C ∞ (Ω), then the convolution u ∗ ψm ∈ C ∞ (Ωψm ). 3. Description of the partial hypoellipticity in terms of fundamental solutions It was proved by Petrovskii that all classical solutions of equations P (D)u = f with analytic right hand side are analytic if and only if operator (1) is elliptic. In other words, if Ω ⊂ Rn is an open set, then for any u ∈ C l (Ω) such that P (D)u is an analytic function in Ω, u is an analytic function in Ω if and only if operator (1) is elliptic. In terms of the polynomial P this is equivalent to P(ξ) 6= 0 for all ξ 6= 0, where X aα D α P(D) = α∈K:|α|=l

is the principal part of operator (1). The corresponding description for hypoelliptic operators was obtained by H¨ ormander. That is, a differential operator (1) is hypoelliptic if and only if for all multi-indices α 6= 0, P (α) (ξ) →0 P (ξ)

as ξ → ∞.

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T. G. Ayele & W. T. Bitew

The following is H¨ ormander’s theorem about description of the hypoellipticity in terms of fundamental solutions. Theorem 3.1. For an operator (1) to be hypoelliptic it is necessary and sufficient that there exists a fundamental solution E of (1) such that E ∈ C ∞ (Rn \{0}). Remark 3.1. From the existence of such fundamental solution it follows that any fundamental solution E ∈ C ∞ (Rn \{0}). Proof. Necessity: Let Ω ∈ Rn be open and u ∈ D ′ (Ω) be a solution of P (D)u = f where f ∈ C ∞ (Ω). Since E is a fundamental solution of operator (2.1), one has P (D)E = δ where δ ∈ C ∞ (Rn \{0}). Consequently by Definition 2.1, E ∈ C ∞ (Rn \{0}). Sufficiency: Let E ∈ C ∞ (Rn \{0}) be a fundamental solution of operator (1) and u ∈ D ′ (Ω) be a solution to the equation P (D)u = f with f ∈ C ∞ (Ω). Let Ω ′ be an arbitrary open set such that Ω ′ ⊂⊂ Ω, and let η ∈ D(Ω) with η(x) = 1 for all x ∈ Ω ′ . The generalized function ηu is compactly supported in Ω and satisfies the equation P (D)(ηu) = ηf + f1 where ηf ∈ D(Ω) and f1 ∈ D′ (Rn ) such that supp(f1 ) ⊂ supp(η)\Ω ′ . Therefore ηu = δ ∗ ηu = P (D)E ∗ ηu = E ∗ P (D)ηu = E ∗ (ηf + f1 ) = E ∗ (ηf ) + E ∗ f1 . Since ηf ∈ D(Rn ), E ∗ ηf ∈ C ∞ (Rn ) and E ∗ f1 ∈ C ∞ (Rn \supp(f1 )), we have E ∗ f1 ∈ C ∞ (Rn \(Ω\Ω ′ )), that is, E ∗ f1 ∈ C ∞ (Ω ′ ). And since ηu = u in Ω ′ and Ω ′ ⊂⊂ Ω is arbitrary, u ∈ C ∞ (Ω). It is known that operator (1) is partially hypoelliptic with respect to the plane x′′ = 0 if and only if for all α 6= 0 P (α) (ξ ′ , ξ ′′ ) →0 P (ξ ′ , ξ ′′ )

as ξ ′′ → ∞ and ξ ′ remains bounded.

Lemma 3.1. Let G ⊂ Rn be a closed set, f ∈ D ′ (Rn ) ∩ C ∞ (Rn \G) and g ∈ E ′ (Rn ). Then f ∗ g ∈ C ∞ (Rn \(supp(g) + G)). Moreover, for all γ > 0, (f ∗ g)(x) = (g(y), η(y)f (x− y)) on Rn \((supp(g)γ )+ G), where η ∈ D(Rn ), η = 1 in some neighborhood of supp(g) and supp(η) ⊂ (supp(g))γ . Proof. Since f, g ∈ D ′ (Rn ) and supp(g) is compact, we have f ∗g ∈ D ′ (Rn ) and (f ∗ g, ϕ) = (f (x)g(y), η(y)ϕ(x + y)) = (g(y), (f (x), η(y)ϕ(x + y))) for all ϕ ∈ D(Rn ), where η is any function from D(Rn ) such that η = 1 in some neighborhood of supp(g) and vanishes outside (supp(g))γ . One can

Partial Hypoellipticity of Differential Operators

619

S observe that supp(g) + G = y∈G (supp(g) + y) and [ \ \ (supp(g) + G)c = [ (supp(g) + y)]c = (supp(g) + y)c = (supp(g)c + y). y∈G

y∈G

y∈G

T

Similarly, (((supp(g))γ )+G)c = y∈G ((supp(g))γ )c +y). Now let supp(ϕ) ⊂ T ((supp(g))γ ) + G)c . Then supp(ϕ) ⊂ y∈G ((supp(g))γ )c + y) if and only if supp(ϕ) − y ⊂ ((supp(g))γ )c for all y ∈ G if and only if supp(ϕ) − G ⊂ ((supp(g))γ )c . That is, (supp(ϕ)−G)∩((supp(g))γ )c = ∅. Then there exists δ0 = δ0 (γ) such that, for all δ satisfying 0 < δ ≤ δ0 (γ), we have (supp(ϕ) − Gδ ) ∩ ((supp(g))γ )c = ∅. Equivalently, we can say that (supp(ϕ) − Gδ ) ⊂ Rn \(supp(g))γ . Since supp(ϕ(x + .)) = (supp(ϕ) − x) ⊂ supp(ϕ) − Gδ , we have supp(η(.))∩supp(ϕ(x+.)) = ∅. Then for all x ∈ Gδ and for all y ∈ Rn , we have η(y)ϕ(x + y) = 0, That is, supp(η(y)ϕ(. + y)) ⊂ Rn \Gδ . Therefore \ Gδ = Rn \G. supp(η(y)ϕ(. + y)) ⊂ Rn \ 0 0 such that B(x0 , δ) ⊂ Ωψm , Ωx′ 0 = B(x0 , δ) − supp(ψm ) and Ωx′ 0 ⊂⊂ Ω. Let ηx0 ∈ D(Ω) be such that ηx0 (x) = 1 for all x ∈ Ωx′ 0 . Observe that ηx0 u ∗ ψm = u ∗ ψm on B(x0 , δ). That is, (ηx0 u ∗ ψm , ϕ) = (u ∗ ψm , ϕ) for all ϕ ∈ D(B(x0 , δ))). Indeed, (ηx0 u ∗ ψm , ϕ) = (ηxo (x)u(x), (ψm (y), ϕ(x + y))

= (ηxo (x)u(x), ((ψ(y ′ ) ⊗ δ(y ′′ ), ϕ(x′ + y ′ , x′′ + y ′′ )))) = (ηxo (x)u(x), (ψ(y ′ ), ϕ(x′ + y ′ , x′′ )))

= (u(x), (ηxo (x)ψ(y ′ ), ϕ(x + y ′ ))) Z ηxo (x)ψ(y ′ )ϕ(x + y ′ )dy ′ ) = (u(x), Rm

′

′′

′

since x = (x , x ) = (x + y ′ , x′′ ) − (y ′ , 0) ∈ B(x0 , δ) − supp(ψm ) for all y ′ ∈ supp(ψ). Then for all x with x + y ′ ∈ supp(ϕ) ⊂ B(x0 , δ), we have ηx0 (x) = 1. Consequently Z ϕ(x + y ′ )ψ(y ′ )dy ′ ) = (u ∗ ψm , ϕ) (ηx0 u ∗ ψm , ϕ) = (u(x), Rm

for all ϕ ∈ D(B(x0 , δ)). Hence u ∗ ψm ∈ C ∞ (Ωψm ) and we are done.

Acknowledgment The authors thank Prof. V. I. Burenkov, Prof. M. L. Goldman and Prof. S. E. Mikhailov for their valuable advice and suggestions. References 1. V. S. Vladimirov (1979), Generalized Functions in Mathematical Physics, Mir Publishers, Moscow. 2. L. H¨ ormander (1969), Linear Partial Differential Operators, New York. 3. T. G. Ayele (1999), Description of condition of partial hypoellipticity of operators in terms of their fundamental solutions, All-Union Institute of Scientific and Technical Information, Russian Academy of Sciences, No. 1027-B99 Moscow, 1999. 4. T. G. Ayele (1999), About smoothness of solutions and conditions of localization for spectral decomposition of operators with constant coefficients, Ph.D. thesis, Moscow, Peoples’ Friendship University of Russia.

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SOME EMBEDDINGS AND EQUIVALENT NORMS OF THE Lλ,s p,q SPACES DOUADI DRIHEM Department of Mathematics L.M.P.A., M’Sila University, P.O. Box 166, M’Sila 28000 Algeria E-mail: [email protected]. Our aim is to give some properties of the Lλ,s p,q spaces, especially concerning embeddings and equivalent norms based on maximal functions and local means. Keywords: Besov spaces, Campanato spaces, Triebel-Lizorkin spaces, L λ,s p,q spaces, local means, maximal functions.

1. Introduction G. Bourdaud [1] showed that an function f ∈ L2loc belongs to BM O (Rn ) (bounded mean oscillation) if and only if

1 X

F −1 (ϕj Ff ) | L2 (BJ ) 2 < ∞, (1) sup BJ |BJ | j≥J

where the supremum is taken over all J ∈ Z and all balls BJ of Rn with radius 2−J and {Ψ, ϕj }j∈N is a smooth dyadic resolution of unity in Rn (see Definition 2.1). This gives that the important space BM O (Rn ) can be described by the Littlewood-Paley decomposition. The idea of G. Bourdaud was used in [7] to get the Littlewood-Paley characterization for Campanato spaces L˙ 2,λ (which contain as a special case the space BM O (Rn )) and their local versions Lα 2 , where [7] showed that if 0 ≤ λ < n + 2, the Cam2,λ ˙ panato space L coincides algebraically and topologically with the space n (R ), which can be obtained by replacing in (1), |BJ | by |BJ |λ/n . L˙ λ,0 2,2 In this work, the focus is to give some properties for the spaces Lλ,s p,q (s ∈ R, λ ≥ 0, 0 < p, q < ∞), this new class of function spaces is defined as the set of all tempered distributions f , such that !1/q X

−1

q

1 jsq p λ,s

f | Lp,q = sup 2 F (ϕj Ff ) | L (BJ ) < ∞, λ/n BJ |BJ | + j≥J

Some Embeddings and Equivalent Norms of the Lλ,s p,q Spaces

623

where the supremum is taken over all J ∈ Z and all balls BJ of Rn with radius 2−J and {ϕj }j∈N0 is the smooth dyadic resolution of unity in Rn . In this work we will establish, under appropriate assumptions on the parameters, some embedding theorems between a pair of these spaces, between these spaces and the Besov spaces or the Triebel-Lizorkin spaces. Also we will give a useful characterization of Lλ,s p,q spaces based on the socalled local means and maximal functions. Furthermore we will give some equivalent norms of the Local approximation spaces Lα 2 for −n/2 ≤ α < 1. The proof has as a starting point the technique used by H.-Q. Bui, M. Paluszy´ nski and M. Taibleson (see [2,3]) and the simplified version of their papers given by V. S. Rychkov in [9]. 2. Definitions and basic properties As usual, Rn the n-dimensional real Euclidean space, N the collection of all natural numbers and N0 = N ∪ {0}. The letter Z stands for the set of all integer numbers. For v ∈ Z, we set v + = max {v, 0}. By S (Rn ) we denote the Schwartz space of all complex-valued, infinitely differentiable and rapidly decreasing functions on Rn and by S ′ (Rn ) the dual space of all tempered distributions on Rn . We define the Fourier transform of a function f ∈ S (Rn ) by Z ∧ −n/2 F (f ) (ξ) = f (ξ) = (2π) f (x) e−ix·ξ dx. Rn

∨

Its inverse is denoted by F −1 f or f. Both F and F −1 are extended to the dual Schwartz space S ′ (Rn ) in the usual way. If s ∈ R, 0 < q ≤ ∞ and J ∈ Z, then ℓsq,J is the set of all sequences {fk }k≥J + of complex numbers such that !1/q

X

q s ksq 2 |fk | < ∞,

{fk }k≥J + | ℓq,J + = k≥J +

with the obvious modification if q = ∞. Given two quasi-Banach spaces X and Y , we write X ֒→ Y if X ⊂ Y and the natural embedding of X in Y is continuous. We write X # Y if there is an f such that f ∈ X but f∈ / Y . We shall use c to denote positive constant which may differ at each appearance. In this section we recall some definitions and some necessary tools.

Definition 2.1. Let Ψ be a function in S (Rn ) satisfying 0 ≤ Ψ (x) ≤ 1, Ψ (x) = 1 for |x| ≤ 1 and Ψ (x) = 0 for |x| ≥ 23 . We put ϕ0 (x) = Ψ (x),

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D. Drihem


ϕ1 (x) = Ψ(x/2) − Ψ (x) and ϕj (x) = ϕ1 2−j+1 x for j ≥ 2, then we have supp ϕj ⊂ x ∈ Rn : 2j−1 ≤ |x| ≤ 3 · 2j−1 , ϕj (x) = 1 for 3 · 2j−2 ≤ |x| ≤ 2j and X Ψ (x) + ϕj (x) = 1 for all x ∈ Rn . j≥1

The system of functions {ϕj } is called a smooth dyadic resolution of unity. ′ For any f ∈ S (Rn ), we define the convolution operators ∆j by ∆j f = ∨

∨

ϕj ∗ f for j ∈ N and ∆0 f = Ψ ∗ f . Thus we obtain the Littlewood-Paley decomposition X f= ∆j f j≥0

′

′

of all f ∈ S (Rn ) (convergence in S (Rn )). Definition 2.2. Let s ∈ R, J ∈ Z, 0 < p, q ≤ ∞ and Ω ⊆ Rn . Then ℓsq,J + (Lp (Ω)) is the set of all sequences {fk }k≥J + of complex-valued Lebesgue measurable functions such that !1/q

X

q s p ksq p 2 kfk | L (Ω)k < ∞.

{fk }k≥J + | ℓq,J + (L (Ω)) = k≥J +

with the obvious modification if p = ∞ and/or q = ∞.

Note that when J = 0 and Ω = Rn we have ℓsq,0 (Lp (Rn )) = ℓsq (Lp ). We now define the spaces Lλ,s p,q which will be our main object of study. Definition 2.3. Let s ∈ R, λ ≥ 0 and 0 < p, q < ∞. The space Lλ,s p,q is the ′ n collection of all f ∈ S (R ) such that !

q 1/q

1

s p λ,s

f | Lp,q = sup < ∞, (2)

{∆j f }j≥J + | ℓq,J + (L (BJ )t) λ/n BJ |BJ |

where the supremum is taken over all J ∈ Z and all balls BJ of Rn with radius 2−J . Note that the space Lλ,s p,q equipped with the norm (2) is a quasi-Banach space. Now we recall the definition of local approximation spaces. Definition 2.4. Let 1 ≤ p < +∞, α ≥ − np and N = max(−1, [α]), where p n [α] is the integer part of α. We say f ∈ Lα p if and only if f ∈ Lloc (R )

Some Embeddings and Equivalent Norms of the Lλ,s p,q Spaces

625

and for some constant M = M (f ), for every cube Q of side length δ, there exists a polynomial PB of degree ≤ N (PB = 0 if N = −1) such that 1

Z

p

!1/p

|f − PB | dx < M, if 0 < δ < 1, |B|1+αp/n B  1/p Z 1 p sup < M, if δ = 1. |f (x)| dx B |B| B

We denote by f | Lα p the infimum of the constants M as above. sup B

The following theorem gives some important properties of the spaces Lλ,0 2,2 that are proved in [7].

Theorem 2.1. Let s ∈ R and −n/2 ≤ α < 1. Then 2α+n,0 L2,2 = Lα 2

and

2α+n,s L2,2 = I s (Lα 2 ).

α s Here I s (Lα 2 ) denotes the image of L2 under I (Riesz potential operator).

Remark 2.1. Some import properties of the homogeneous counterpart of the space Lλ,s p,q are given in [7]. s s Now we give a definition of the spaces Bp,q and Fp,q . s Definition 2.5. (i) Let s ∈ R and 0 < p, q ≤ ∞. The Besov space Bp,q is ′ n the collection of all f ∈ S (R ) such that



s

f | Bp,q = {∆j f }j≥0 | ℓsq (Lp ) < ∞. s (ii) Let s ∈ R, 0 < p < ∞ and 0 < q ≤ ∞. The Triebel-Lizorkin space Fp,q ′ is the collection of all f ∈ S (Rn ) such that




s

f | Fp,q = {∆j f }j≥0 | Lp ℓsq < ∞.


Here if s ∈ R, 0 < p < ∞ and 0 < q ≤ ∞, then Lp ℓsq is the set of all sequences {fk }k≥0 of complex-valued Lebesgue measurable functions such that












{fk }k≥0 | Lp ℓsq =

2ks fk k≥0 | ℓq

< ∞, p

with the obvious modification if q = ∞. The following lemma is from Young’s inequality in ℓ0q,J + .

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D. Drihem

Lemma 2.1. Let 0 < a < 1, J ∈ Z and 0 < q ≤ ∞. Let

{εk } be a

sequence of positive real numbers such that {εk }k≥J + | ℓ0q,J + = I < ∞. n o P∞ The sequence ηk : ηk = j=k aj−k εj is in ℓ0q,J + with



{ηk }k≥J + | ℓ0q,J + ≤ c I. c depends only on a and q.

3. Embeddings The following results are generalizations of the results of [6]. Theorem 3.1. Let 0 < t < p < ∞, s ∈ R, r − n/t = s − n/p and λ,s 0 < q, q1 < ∞. Then Lλ,r t,q1 ֒→ Lp,q if and only if 0 < q1 ≤ q < ∞. Theorem 3.2. Let s, r ∈ R, λ ≥ 0, 0 < p, q, q1 < ∞ and 0 < t ≤ ∞. s+ε (i) Let 0 < t ≤ p < ∞. Then Bt,q ֒→ Lλ,s p,q if and only if ε ≥ λ/q+n/t−n/p. 0,s s (ii) We have Bp,q1 ֒→ Lp,q if and only if 0 < q1 ≤ q < ∞. s+ε (iii) Let λ > 0. Then Bp,∞ ֒→ Lλ,s p,q if and only if ε ≥ λ/q. s+ε (iv) If λ ≥ nq/p and ε > λ/q+n/t−n/p, we have Lλ,s p,q # Bt,q , 1 < t ≤ ∞. r (v) If λ ≥ nq/p and r ∈ R, we have Lλ,s p,q # Bt,q , 0 < t ≤ 1. 4. Equivalent quasi-norms In order to formulate the main result of this paper, let us consider k0 , k ∈ S (Rn ) and an integer S ≥ −1 such that for some ε > 0 ∧ ∧ ε < |ξ| < 2ε, (3) k0 (ξ) > 0 for |ξ| < 2ε, k (ξ) > 0 for 2

and

Z

xα k (x) dx = 0

Rn

for any

|α| ≤ S.

(4)

Here (3) are Tauberian conditions, while (4)
are moment conditions on k. We recall the notation kj (x) = 2jn k 2j x , for j ≥ 1. For any a > 0, ′ f ∈ S (Rn ), j ∈ N0 and x ∈ Rn we denote (Peetre’s maximal functions) kj∗,a f (x) = sup

y∈Rn

|kj ∗ f (y)| a, (1+2j |x−y|)

∗,a kj,J f (x) = sup

y∈BJ

|kj ∗ f (y)| a . (5) (1+2j |x−y|)

Usually kj ∗ f is called a ‘local mean’. We are able now to state the main result of this paper.

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627

Theorem 4.1. Let λ ≥ 0, 0 < p, q < ∞, s < S + 1 and a > max(n/p, λ/q). Then !

q 1/q



1

∗,a p s

f | Lλ,s ′ = sup (6)

kj f j≥J + | ℓq,J + (L (BJ )) p,q λ/n BJ |BJ | and !

q 1/q

′′ 1

s p

f | Lλ,s

= sup (7)

{kj ∗ f }j≥J + | ℓq,J + (L (BJ )) p,q λ/n BJ |BJ |

are equivalent quasi-norms in Lλ,s p,q .

Theorem 4.2. Let λ ≥ 0, 0 < p, q < ∞, s < S + 1 − λ/q and a > n/p. Then

n

q !1/q o



1 ′′′ ∗,a p s

f | Lλ,s

= sup

k f

(8) | ℓ + (L (BJ )) p,q q,J j,J

λ/n j≥J + BJ |BJ | is an equivalent quasi-norm in Lλ,s p,q .

Remark 4.1. In view of the inequalities ∗,a kj,J f (x) ≤ kj∗,a f (x)

∗,a and kj ∗ f (x) ≤ kj,J f (x) ,

′

for any f ∈ S (Rn ) and any x ∈ BJ , it follows from Theorem 4.1 that in the case of a > max (n/p, λ/q), (8) is a simple consequence of (6) and (7).

′

2α+n,0 Corollary 4.1. Let −n/2 ≤ α < 1 and a > 0. Then f | L2,2

(in

′′′

′′



2α+n,0 2α+n,0 case of a > max (n/2, n/2 + α)), f | L2,2

(in

and f | L2,2 α case of a > n/2 and S > −1 + n/2 + α) are equivalent norms in L2 .

2α+n,0 The proof of Corollary 4.1 is immediate because L2,2 = Lα 2 for any −n/2 ≤ α < 1. Let µ0 , µ ∈ S (Rn ) be two positive functions on Rn such that

µ0 (ξ) = 1 if |ξ| ≤ 2 and supp µ0 ⊂ {ξ ∈ Rn : |ξ| ≤ 4}

(9)

and µ (ξ) = 1 if 1/2 ≤ |ξ| ≤ 2 and supp µ ⊂ {ξ ∈ Rn : 1/4 ≤ |ξ| ≤ 4} . (10)
For any j ≥ 1 and ξ ∈ Rn we define µj (ξ) = µ 2−j ξ . The results below will play a key role in the proof of Theorems 4.1 and 4.2.

628

D. Drihem

Theorem 4.3. Let s ∈ R, λ ≥ 0, 0 < p, q < ∞ and a > max (n/p, λ/q). Then !



q 1/q

1

∗,a

p s

f | Lλ,s

sup (11)

∆j f j≥J + | ℓq,J + (L (BJ )) p,q = λ/n BJ |BJ | is an equivalent quasi-norm in Lλ,s p,q .

Theorem 4.4. Let s ∈ R, λ ≥ 0, 0 < p, q < ∞ and a > n/p. Then

n

q !1/q o



1 s p

f | Lλ,s = sup

∆∗,a f | ℓq,J + (L (BJ )) p,q j,J

λ/n + j≥J BJ |BJ |

is an equivalent quasi-norm in Lλ,s p,q .

Theorem 4.5. Let s ∈ R, λ ≥ 0, 0 < p, q < ∞. Then under the above assumptions on the functions µ0 and µ,

n

q !1/q o

∨

1 p s λ,s

µj ∗ f

f | Lp,q = sup

| ℓ + (L (BJ )) q,J

λ/n µ j≥J + BJ |BJ |

is an equivalent quasi-norm in Lλ,s p,q .

Proof. We will give only the proof of Theorem 4.1. The idea is from V. S. Rychkov [9]. Step 1. Take any pair of functions ϕ0 and ϕ ∈ S (Rn ) so that for an ε´ > 0, ε´ ∧ ∧ < |ξ| < 2´ ε, ε, ϕ (ξ) > 0 for ϕ0 (ξ) > 0 for |ξ| < 2´ 2

and define for any a > 0 the functions ϕ∗,a j f as in (5). Then there is a ′ n constant c > 0 such that for any f ∈ S (R ) !



q 1/q

′

1

∗,a s p λ,s

f | Lp,q ≤ c sup . (12)

ϕj f j≥J + | ℓq,J + (L (BJ )) λ/n BJ |BJ |

To prove (12) we consider the cases J ≥ 1 (or J ≤ 0 and j ≥ 1) and (J ≤ 0 and j = 0) separately. This estimate can be obtained by the same arguments as in [9]. Step 2. We will prove in this step that there is a constant c > 0 such that for any f ∈ S ′ (Rn )

′

′′

f | Lλ,s

≤ c f | Lλ,s

. (13) p,q p,q

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629

Analogously to (3) find two functions Λ, ω ∈ S (Rn ), so that for an ε > 0, ∧

∧

supp Λ ⊂ {ξ ∈ Rn : |ξ| < 2ε}, supp ω ⊂ {ξ ∈ Rn : ε/2 < |ξ| < 2ε} and for all f ∈ S ′ (Rn ) and j ∈ N0 X kj ∗ f = Λj ∗ (k0 )j ∗ kj ∗ f + kj ∗ ωm ∗ km ∗ f. m≥j+1

By an argument similar to the one in [9], we deduce that for all f ∈ S ′ (Rn ), x ∈ BJ and j ∈ N that Z r X ∗,a |km ∗ f (z)| (j−m)N r+mn k f (x) r ≤ c 2 ar dz, j m Rn (1 + 2 |x − z|) m≥j

where 0 < r < ∞, N ∈ N can be still be taken arbitrarily large and c > 0 is independent of j. Together with the corresponding estimate for k0∗,a f . Let J ∈ Z and BJ a ball of Rn centered at x0 with radius 2−J . Then Z Z r XZ |km ∗ f (z)| (...) dz + dz = (...) dz. (14) ar m BJ−1 Rn (1 + 2 |x−z|) i≥0 BJ−i−2 BJ−i−1 It possible to choose r so that n max



1 1 a , a+n/p−λ/q



< r < p; we make such

1 a choice and fix r for the rest of the proof. Now the function z 7→ (1+|z|) ar   r 1 1 is in L and we may use the majorant property |g| ∗ (1+|·|)ar (x) ≤

cM (|g|r ) (x) for the Hardy-Littlewood maximal operator M (see Stein and Weiss [11]). It follows that the right-hand side of (14) is bounded by X
2−mn M |km ∗ f |r χBJ−1 (x) + 2Jn−(iar+mn) kkm ∗ f | Lr (BJ−i−2 )kr , i≥0

where we have used |x−z| ≥ |z −x0 |−|x−x0 | ≥ 2i−J and 2(m−J)(n−ar) ≤ 1. Since the Hardy-Littlewood maximal operator M maps Lρ (Rn ) into Lρ (Rn ) for 1 < ρ ≤ ∞ and since kkm ∗ f | Lr (BJ−i−2 )k ≤ c 2(i−J)(n/r−n/p) kkm ∗ f | Lp (BJ−i−2 )k, we get by Lemma 2.1 and the embedding ℓsq,(J−i−2)+ (Lp (BJ−i−2 )) ֒→ ℓsq,J + (Lp (BJ−i−2 )), for any i ∈ N0 ∪ {−1}, that



∗,a s,q

kj f j≥J + | ℓJ + (Lp (BJ )) !

r 1/r X +

s,q i (n(1−r/p)−ar) p ≤c 2

{km ∗ f }j≥(J−i−2)+ | ℓJ + (L (BJ−i−2 )) i≥−1



n o

′′

. ≤ c 2λJ/q 2−i(a−n(1/r−1/p)−λ/q) | ℓr f | Lλ,s p,q

630

D. Drihem

Whence, multiplying through by 2−λJ/q , using |BJ |−λ/n = 2λJ , and taking the sup over all J ∈ Z and all balls BJ of Rn with radius 2−J , we get the −i(a−n(1/r−1/p)−λ/q) estimate (12) by the fact that 2 ∈ ℓr . Step 3. We will prove in this step that for all f ∈ S ′ (Rn ) the following estimates are true:

′



λ,s ′′

f | Lλ,s

≤ c f | Lλ,s

and f | Lλ,s

. (15) p,q p,q p,q ≤ c f | Lp,q Let µ0 and µ ∈ S (Rn ) be two positive functions on Rn satisfying (9) and ∧ ∧ (10). Let ϕ0 = µ0 and ϕ = µ. The first inequality in (15) is proved by the chain of the estimates !

q 1/q



′ 1

∗,a s p

f | Lλ,s

≤ c sup

ϕj f j≥J + | ℓq,J + (L (BJ )) p,q λ/n BJ |BJ | !

q 1/q

1

p s ≤ c sup

{ϕj ∗ f }j≥J + | ℓq,J + (L (BJ )) λ/n BJ |BJ |

≤ c f | Lλ,s p,q , where the first inequality is (12) (see Step 1), the second inequality is (13) (with ϕ and ϕ0 instead of k and k0 (see Step 2), and finally the last inequality follows by Theorem 4.5. The proof of the second inequality in (15) is by the chain !



q 1/q

1

∗,a

s p

f | Lλ,s

ϕj f j≥J + | ℓq,J + (L (BJ )) p,q ≤ c sup λ/n BJ |BJ |

′

′′

≤ c f | Lλ,s

, ≤ c f | Lλ,s p,q p,q

where the first inequality is an obvious consequence of (11), the second inequality is (12) (see Step 1) with the roles of k0 and k, ϕ0 and ϕ, respectively, interchanged, and finally the last inequality is (13) (see Step 2). Hence the theorem is proved.

References 1. G. Bourdaud, Analyse fonctionnelle dans l’espace Euclidien, Publ. Math. Univ. Paris VII, 23 (1987). 2. H. Q. Bui, M. Paluszy´ nski and M. Taibleson, A maximal characterization of weighted Besov-Lipschitz and Triebel-Lizorkin spaces, Studia Math. 119 (1996), pp. 219-246. 3. H. Q. Bui, M. Paluszy´ nski and M. Taibleson, Characterization of the BesovLipschitz and Triebel-Lizorkin spaces. The case q < 1 , J. Fourier Anal. Appl. 3, special issue (1997), pp. 837-846.

Some Embeddings and Equivalent Norms of the Lλ,s p,q Spaces

631

4. S. Campanato, Propriet` a di H¨ olderianit` a di alcune classi di funzioni, Ann. Scuola Norm. Sup. Pisa 17 (1963), pp. 175-188. 5. S. Campanato, Propriet` a di una famiglia di spazi funzionali, Ann. Scuola. Norm. Sup. Pisa 18 (1964), pp. 137-160. 6. A. El Baraka, An embedding theorem for Campanato spaces, Electron. J. Diff. Eqns. 2002 (2002), N. 66, pp. 1-17. 7. A. El Baraka, Littlewood-Paley characterization for Campanato spaces, J. Funct. Spaces Appl. 4 (2) (2006), pp. 193-220. 8. M. Frazier, B. Jawerth and G. Weiss, Littlewood-Paley theory and the study of function spaces, CBMS, 79, 1991. 9. V. S. Rychkov, On a theorem of Bui, Paluszy´ n ski, and Taibleson, Proc. Steklov Inst. Math. 227 (1999), pp. 280-292. 10. W. Sickel and H. Triebel, H¨ older inequalities and sharp embeddings in funcs s tion spaces of Bp,q and Fp,q type, Z. Anal. Anwendungen 14 (1), (1995), pp. 105-140. 11. E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, NJ, 1971. 12. J.-O. Str¨ omberg and A. Torchinsky, Weighted Hardy spaces, Lecture Notes in Math. 1381, Springer, Berlin, 1989. 13. H. Triebel, Theory of function spaces, Birkh¨ auser, Basel, 1983. 14. H. Triebel, Theory of function spaces II, Birkh¨ auser, Basel, 1992.

632

D. Drihem

Session 15

Numerical Functional Analysis

SESSION EDITORS A. Ashyralyev P. E. Sobolevskii

˙ Fatih University, Istanbul, Turkey Universidade Federal do Cear´ a, Fortaleza, Brazil

635

GLOBAL EXPONENTIAL PERIODICITY FOR DISCRETE-TIME HOPFIELD NEURAL NETWORKS WITH FINITE DISTRIBUTED DELAYS AND IMPULSES H. AKC ¸A Mathematical Sciences Department, Faculty of Sciences, United Arab Emirates University, Al Ain, UAE E-mail: [email protected] V. COVACHEV Department of Mathematics & Statistics, College of Science, Sultan Qaboos University, Muscat, Sultanate of Oman & Institute of Mathematics, Bulgarian Academy of Sciences, Sofia, Bulgaria E-mail: [email protected] & [email protected] Z. COVACHEVA Higher College of Technology, Muscat, Sultanate of Oman & Higher College of Telecommunications and Post, Sofia, Bulgaria E-mail: [email protected] S. MOHAMAD Department of Mathematics, Faculty of Science, Universiti Brunei Darussalam, Brunei Darussalam E-mail: [email protected] The discrete counterpart of a class of Hopfield neural networks with periodic integral impulsive conditions and finite distributed delays is introduced. The continuation theorem of coincidence degree theory is used to obtain a sufficient condition for the existence of a periodic solution of the discrete system considered. By introducing an appropriate Lyapunov functional, a sufficient condition is obtained for the uniqueness and global exponential stability of the periodic solution. Keywords: Discrete Hopfield neural networks; Finite distributed delays; Periodic impulses; Global exponential periodicity.

636

H. Ak¸ca, V. Covachev, Z. Covacheva & S. Mohamad

1. Introduction In the present paper we introduce the discrete counterpart of a class of Hopfield neural networks with periodic integral impulsive conditions and finite distributed delays. We apply the continuation theorem of coincidence degree theory (Gaines and Mawhin [7]) to obtain a sufficient condition for the existence of a periodic solution of the discrete system considered. By introducing an appropriate Lyapunov functional we derive a sufficient condition for the uniqueness and global exponential stability of the periodic solution. For works proving the existence of a periodic solution of differential and difference equations by the coincidence degree theory the reader can see Fan and Agarwal [3], Fan and Wang [4,5], Fan et al. [6], Li [9], Li and Kuang [10], Li and Lu [11]. In particular, in Li and Lu [11] the existence of a periodic solution of Hopfield-type neural network with impulses is proved. In Zhou et al. [14] one proves the existence of a periodic solution of a discrete-time analogue of a bidirectional associative memory (BAM) neural network with periodic coefficients and finite distributed delays without impulses. 2. Statement of the problem. Main results We consider a class of Hopfield neural networks with periodic integral impulsive conditions and finite distributed delays, which are formulated in the form of a system of impulsive delay differential equations Z ω  m X dxi bij (t)fj gij (s)xj (t − s) ds + Ii (t), (1) = −ai (t)xi (t) + dt 0 j=1 t 6= tk ,

∆xi (tk ) ≡ xi (tk + 0) − xi (tk ) Z ω  m X Bijk Φj cij (s)xj (tk −s) ds + αik , = −γik xi (tk ) + j=1

i = 1, m,

(2)

0

k ∈ Z,

where m is the number of neurons in the network, xi (t) is the state of the i-th neuron at time t, ai (t) > 0 is the rate at which the i-th neuron resets its state when isolated from the system, bij (t) is the synaptic connection weight from the j-th neuron to the i-th one, fj (·) are signal transmission functions of the j-th neuron, ω is the maximum transmission delay from one neuron to another, gij (·) and cij (·) are nonnegative delay kernels, Ii (t) is the external input to the i-th neuron, tk (k ∈ Z) are the instants of impulse

Discrete-Time Neural Networks with Finite Distributed Delays and Impulses

637

effect which form a strictly increasing sequence, γik (i = 1, m, k ∈ Z) are positive constants. We assume that the above system (1), (2) satisfies the following periodicity conditions: ai (t), bij (t), Ii (t) are ω-periodic in t; tk+p = tk + ω, γi,k+p = γik , Bij,k+p = Bijk , αi,k+p = αik . Without loss of generality we can assume that 0 < t1 < t2 < · · · < tp < ω. The Hopfield neural network (1) is similar to the bidirectional associative memory neural network considered in Zhou et al. [14]. Combining some ideas of Mohamad and Gopalsamy [13], Ak¸ca et al. [1], Zhou et al. [14] we shall formulate the discrete counterpart of system (1), (2). For a positive integer N we choose the discretization step h = ω/N . For the moment we assume N so large that h < min (tk+1 − tk ). k=1,p

Then each interval [nh, (n + 1)h] contains at most one instant of impulse effect tk . For convenience we denote n = [t/h], the greatest integer in t/h, and nk = [tk /h]. Clearly, we will have nk+p = nk + N for all k ∈ Z. Let n ∈ Z, n 6= nk . This means that the interval [nh, (n + 1)h] contains no instant of impulse effect tk . We approximate the integral term in (1) by a sum: Z ω N X gij (s)xj (t − s) ds ≈ gij (ℓh)xj ((n − ℓ)h) ϕ(h), 0

ℓ=1

2

where ϕ(h) = h + O(h ). Next we approximate the differential equation (1) on the interval [nh, (n + 1)h] by ! N m X X dxi gij (ℓh)xj ((n−ℓ)h) ϕ(h) . bij (nh)fj +ai (nh)xi (t) = Ii (nh)+ dt j=1 ℓ=1

We multiply both sides of this equation by exp (ai (nh)t) and integrate over the interval [nh, (n + 1)h]. Thus we obtain   (3) xi ((n + 1)h) − xi (nh) = − 1 − e−ai (nh)h xi (nh) ! ) ( N m X X 1 − e−ai (nh)h bij (nh)fj gij (ℓh)xj ((n − ℓ)h) ϕ(h) . + Ii (nh)+ ai (nh) j=1 ℓ=1

638

H. Ak¸ca, V. Covachev, Z. Covacheva & S. Mohamad

Henceforth by abuse of notation we write xi (n) = xi (nh) and define ∆xi (n) = xi (n + 1) − xi (n) (i = 1, m, n ∈ Z). For convenience we adopt the notation Ai (n) = 1 − e−ai (nh)h Ii (n) = bij (n) =

−ai (nh)h

1−e ai (nh)

(i = 1, m, n ∈ Z \ {nk }k∈Z ), (i = 1, m, n ∈ Z \ {nk }k∈Z ),

Ii (nh)

1 − e−ai (nh)h bij (nh) ai (nh)

(i, j = 1, m, n ∈ Z \ {nk }k∈Z ),

(i, j = 1, m, ℓ = 1, N ).

gij (ℓ) = gij (ℓh)ϕ(h)

Clearly, we have 0 < Ai (n) < 1. In particular, if ai (t) < ω1 , then Ai (n) < N1 . With the above notation equation (3) takes the form ! N m X X bij (n)fj gij (ℓ)xj (n−ℓ) , (4) ∆xi (n) = −Ai (n)xi (n) + Ii (n) + j=1

ℓ=1

n 6= nk .

i = 1, m,

Next, for n = nk the interval [nh, (n + 1)h] contains the instant of impulse effect tk . On this interval we approximate the impulse condition (2) by ! N m X X Bijk Φj cij (ℓ)xj (nk −ℓ) , (5) ∆xi (nk ) = −γik xi (nk ) + αik + j=1

i = 1, m,

ℓ=1

k ∈ Z,

where cij (ℓ) = cij (ℓh)ϕ(h)

(i, j = 1, m,

ℓ = 1, N ).

For uniformity of notation we define Ai (nk ) = γik ,

Ii (nk ) = αik

(i = 1, m, k ∈ Z).

Now the difference system (4), (5) can be written in operator form as ∆x = Hx,

(6)

where (Hx)i (n) = −Ai (n)xi (n) + Ii (n) N  m P P   bij (n)fj gij (ℓ)xj (n − ℓ) , n 6= nk ,  ℓ=1   + j=1 N m P P    cij (ℓ)xj (nk − ℓ) , n = nk . Bijk Φj j=1

ℓ=1

(7)

Discrete-Time Neural Networks with Finite Distributed Delays and Impulses

639

In order to formulate our assumptions, we need some more notation. Let IN = {0, 1, . . . , N − 1}, Ai = min Ai (n),

Ai =

n∈IN

N −1 X

Ai (n),

i = 1, m.

n=0

Now we introduce the following conditions: H1. Ai (n + N ) = Ai (n), Ii (n + N ) = Ii (n) for i = 1, m, n ∈ Z; nk ∈ Z for all k ∈ Z and nk+p = nk + N ; bij (n + N ) = bij (n) (n 6= nk ), Bij,k+p = Bijk (k ∈ Z) for i, j = 1, m. H2. Ai > 0, Ai < 1 for i = 1, m. H3. The functions fj (·), Φj (·) (j = 1, m) are bounded on R and there exist positive constants Mj and Lj such that |fj (x) − fj (y)| ≤ Mj |x − y|,

|Φj (x) − Φj (y)| ≤ Lj |x − y|

for all x, y ∈ R. H4. gij (ℓ) ≥ 0, cij (ℓ) ≥ 0 for i, j = 1, m, ℓ = 1, N . We again introduce some notation: I i = max |Ii (n)|,

i = 1, m,

n∈IN

bij = sup |bij (n)|, n6=nk

B ij = max |Bijk |,

For an N -periodic sequence v(n) we denote v˜ = ρi = I i + Next we denote

1 N

NP −1

v(n); for i = 1, m

n=0

m  1 X (N − p)bij |fj (0)| + pB ij |Φj (0)| . N j=1

Mj = max{Lj , Mj }, Gij =

i, j = 1, m.

k=1,p

N X

gij (ℓ),

Cij =

ℓ=1

Bij = max{bij , B ij },

j = 1, m, N X

cij (ℓ), i, j = 1, m,

ℓ=1

Gij = max{Gij , Cij }, i, j = 1, m.

We introduce the m × m matrices !
Ai 1 − Ai , i = 1, m , A = diag 1 + N Ai Then we introduce the conditions

m

B = (Bij Mj Gij )i,j=1 .

640

H. Ak¸ca, V. Covachev, Z. Covacheva & S. Mohamad

H5. min i=1,m

A˜i − Mi

H6. Ai > Mi

m X j=1

m X j=1

Bji Gji

!

> 0.

Bji Gji for i = 1, m.

H7. The matrix A − B is an M -matrix (Fiedler [2], Horn and Johnson [8]). Theorem 2.1. Suppose that conditions H1–H5, H7 hold. Then the equation (6) has at least one N -periodic solution. Sketch of the proof. We shall prove Theorem 2.1 using Mawhin’s continuation theorem (Gaines and Mawhin [7, p. 40]). To state this theorem we need some preliminaries: Let X, Y be real Banach spaces, L : Dom L ⊂ X −→ Y be a linear mapping, and H : X −→ Y be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dim Ker L = codim Im L < +∞ and Im L is closed in Y. If L is a Fredholm mapping of index zero and there exist continuous projectors P : X −→ X and Q : Y −→ Y such that Im P = Ker L, Ker Q = Im L = Im (I − Q), then the mapping L|Dom L∩Ker P : (I − P )X −→ Im L is invertible. We denote the inverse of this mapping by KP . If Ω is an open bounded subset of X, the mapping H will be called L-compact on Ω if QH(Ω) is bounded and KP (I − Q)H : Ω −→ X is compact. Since Im Q is isomorphic to Ker L, there exists an isomorphism J : Im Q −→ Ker L. Now Mawhin’s continuation theorem can be stated as follows. Lemma 2.1. Let L be a Fredholm mapping of index zero, let Ω ⊂ X be an open bounded set and let H : X −→ Y be a continuous operator which is L-compact on Ω. Assume that the following conditions hold: (a) for each λ ∈ (0, 1), x ∈ ∂Ω ∩ Dom L, Lx 6= λHx; (b) for each x ∈ ∂Ω ∩ Ker L, QHx 6= 0; (c) deg (JQH, Ω ∩ Ker L, 0) 6= 0, where deg(·) is the Brouwer degree [12]. Then the equation Lx = Hx has at least one solution in Ω ∩ Dom L. Let us choose X = Y = {x(n) = (x1 (n), x2 (n), . . . , xm (n))T : x(n + m P |xi |, then X N ) = x(n), n ∈ Z}. If we define |xi | = max |xi (n)|, kxk = n∈IN

i=1

is a Banach space with the norm k · k. For x ∈ X, let Hx be defined by (7), Lx = ∆x and P x = Qx = x ˜ = (˜ x1 , x ˜2 , . . . , x ˜m )T .

Discrete-Time Neural Networks with Finite Distributed Delays and Impulses

641

Then Ker L = {x ∈ X : x = h ∈ Rm } (vectors with components indeNP −1 pendent of n), Im L = {x ∈ X : xi (n) = 0, i = 1, m} is a closed set n=0

in X, and codim L = m. Thus L is a Fredholm mapping of index zero. It is easy to see that P and Q are continuous projectors and Im P = Ker L, ¯ for any bounded Im L = Ker Q = Im (I − Q), and H is L-compact on Ω set Ω ⊂ X. Moreover, in condition (c) of Lemma 2.1 the isomorphism J can be taken as the identity operator I. First for the solutions x of the operator equation Lx = λHx for λ ∈ (0, 1), that is, n ∈ IN , i = 1, m,

∆xi (n) = λ(Hx)i (n),

after lengthy calculations we derive the estimate m X Ai (1 − Ai ) Bij Mj Gij |xj | ≤ ρi . |xi | − 1 + N Ai j=1

(8)

If we introduce the vectors |x| = (|x1 |, |x2 |, . . . , |xm |)T and ρ = (ρ1 , ρ2 , . . . , ρm )T , then the system of inequalities (8) for i = 1, m can be written in a matrix form (A − B)|x| ≤ ρ,

(9)

where the matrices A and B were introduced in §2. By virtue of condition H7 the inequality (9) implies |x| ≤ (A − B)−1 ρ. ∗ T If (A − B)−1 ρ = (C1∗ , C2∗ , . . . , Cm ) , this means that the components of m P Ci∗ , each solution of ∆x = λHx satisfy |xi | ≤ Ci∗ . If we denote C ∗ = i=1

then each solution of ∆x = λHx satisfies kxk ≤ C ∗ . Now we take Ω = {x ∈ X : kxk < C}, where C > C ∗ will be chosen later. Obviously Ω satisfies condition (a) of Lemma 2.1. Now let x ∈ ∂Ω ∩ Ker L = ∂Ω ∩ Rm , i.e., x is a constant vector in Rm with kxk = C. For such x we obtain ! m m X X ρi . Bji Gji C − kQHxk ≥ min A˜i − Mi i=1,m

By condition H5 min i=1,m

A˜i − Mi

i=1

j=1

m X j=1

Bji Gji

!

> 0.

642

H. Ak¸ca, V. Covachev, Z. Covacheva & S. Mohamad

Then we can choose C > C ∗ so large that min i=1,m

A˜i − Mi

m X j=1

!

Bji Gji C >

m X

ρi .

i=1

Hence for x ∈ ∂Ω ∩ Ker L we have kQHxk > 0 and QHx 6= 0, that is, condition (b) of Lemma 2.1 is satisfied. To prove (c), we define the mapping (QH)µ : Dom L × [0, 1] −→ X by ˜ = (A˜1 x1 , A˜2 x2 , . . . , A˜m xm )T . (QH)µ = −µA˜ + (1 − µ)QH, where Ax For x ∈ ∂Ω ∩ Ker L as above we obtain ! m m X X ˜ ρi > 0. k(QH)µ xk ≥ min Ai − Mi Bji Gji C − i=1,m

i=1

j=1

This means that (QH)µ x 6= 0 for x ∈ ∂Ω ∩ Ker L and µ ∈ [0, 1]. From the homotopy invariance of the Brouwer degree [12] it follows that ˜ Ω ∩ Ker L, 0) = (−1)m 6= 0. deg (QH, Ω ∩ Ker L, 0) = deg (−A, According to Lemma 2.1 the equation (6) has at least one N -periodic solution. This completes the proof of Theorem 2.1. Theorem 2.2. Suppose that conditions H1–H4, H6, H7 hold. Then the N -periodic solution of (6) is unique and globally exponentially stable. Sketch of the proof. Let g ij (ℓ) = max{gij (ℓ), cij (ℓ)}, i, j = 1, m, ℓ = 1, N . Clearly, N X ℓ=1

g ij (ℓ) = Gij ,

i, j = 1, m.

Now we shall use the following assertion. Lemma 2.2. Assume that condition H6 holds. Then there exists λ > 1 such that for any i = 1, m, n ∈ IN and λ ∈ (1, λ] we have λ(1 − Ai (n)) + Mi

m X j=1

Bji

N X ℓ=1

λℓ+1 g ji (ℓ) − 1 ≤ 0.

Now let us suppose that x∗ (n) = (x∗1 (n), x∗2 (n), . . . , x∗m (n))T is an N periodic solution of equation (6), and x(n) = (x1 (n), x2 (n), . . . , xm (n))T is any solution of (6) for n ≥ 0, defined at least for n ≥ −N . From (6) and (7) for n ∈ Z+ 0 = {n ∈ Z : n ≥ 0}, n 6= nk we derive |xi (n + 1) − x∗i (n + 1)| ≤ (1 − Ai (n))|xi (n) − x∗i (n)|

Discrete-Time Neural Networks with Finite Distributed Delays and Impulses

+

m X

bij Mj

j=1

N X ℓ=1

643

gij (ℓ)|xj (n − ℓ) − x∗j (n − ℓ)|,

while for n = nk we have |xi (nk + 1) − x∗i (nk + 1)| ≤ (1 − Ai (nk ))|xi (nk ) − x∗i (nk )| +

m X

B ij Lj

j=1

N X ℓ=1

cij (ℓ)|xj (nk − ℓ) − x∗j (nk − ℓ)|.

Now we introduce the quantities yi (n) = λn |xi (n) − x∗i (n)|,

λ ∈ (1, λ],

i = 1, m,

n ≥ −N.

Then we have yi (n + 1) ≤ λ(1−Ai (n))yi (n) + λ ∈ (1, λ],

m X j=1

Bij Mj

i = 1, m,

N X

λℓ+1 g ij (ℓ)yj (n−ℓ),

(10)

ℓ=1

n ∈ Z+ 0.

Now we consider a Lyapunov functional V (n) = V (y1 , y2 , . . . , ym )(n) defined by ) ( N n−1 m m X X X X ℓ+1 Bij Mj λ g ij (ℓ) yj (s) , n ∈ Z+ yi (n) + V (n) = 0. j=1

i=1

ℓ=1

s=n−ℓ

Taking into account (10), we estimate the difference ∆V (n) = V (n + 1) − V (n) for n ∈ Z+ 0: ) ( N m m X X X ℓ+1 Bji λ g ji (ℓ) − 1 yi (n). λ(1 − Ai (n)) + Mi ∆V (n) ≤ j=1

i=1

ℓ=1

By virtue of Lemma 2.2 we have ∆V (n) ≤ 0 for all n ∈ Z+ 0 , which implies that n ∈ Z+ 0.

V (n) ≤ V (0), On the other hand, we have V (n) ≥ and V (0) ≤

m X i=1

(

1 + Mi

m X

yi (n) =

m X

N X

i=1

i=1

j=1

m X

Bji

ℓ=1

λn |xi (n) − x∗i (n)|

ℓ+1

ℓλ

)

g ji (ℓ)

max |xi (s) − x∗i (s)|,

s∈I−N

(11)

644

H. Ak¸ca, V. Covachev, Z. Covacheva & S. Mohamad

where I−N = {−N, −N + 1, . . . , −1, 0}. Here we used that 1 < λ ≤ λ. Thus from inequality (11) we obtain m X i=1

|xi (n) −

x∗i (n)|

−n

≤ Mλ

where

m X i=1

M = max 1 + Mi i=1,m

max |xi (s) − x∗i (s)|,

s∈I−N

m X j=1

Bji

N X ℓ=1

ℓ+1

ℓλ

n ∈ Z+ 0,

!

g ji (ℓ) .

This completes the proof of Theorem 2.2. References 1. H. Ak¸ca, R. Alassar, V. Covachev and Z. Covacheva, Dyn. Syst. Appl. 13, 77 (2004). 2. M. Fiedler, Special Matrices and Their Applications in Numerical Mathematics, Martinus Nijhoff, Dordrecht, 1986. 3. M. Fan and S. Agarwal, Appl. Anal. 82, 801 (2002). 4. M. Fan and K. Wang, Discrete Contin. Dyn. Syst. Ser. B 4, 563 (2004). 5. M. Fan and K. Wang, Math. Comput. Modelling 35, 951 (2002). 6. M. Fan, K. Wang and D. Jiang, Math. Biosci. 160, 47 (1999). 7. R. E. Gaines and J. L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977. 8. R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991. 9. Y. K. Li, Proc. Amer. Math. Soc. 127, 1331 (1999). 10. Y. K. Li and Y. Kuang, J. Math. Anal. Appl. 255, 260 (2001). 11. Y. K. Li and L. Lu, Phys. Lett. A 333, 62 (2004). 12. J. W. Milnor, Topology from the Differentiable Viewpoint, The University Press of Virginia, Charlottesville, 1969. 13. S. Mohamad and K. Gopalsamy, Math. Comput. Simul. 53, 1 (2000). 14. T. Zhou, Y. Liu, Y. Liu and A. Chen, Internat. J. Math. Manuscripts 1, (2007).

645

´ DIFFERENCE A NOTE ON THE MODIFIED PADE SCHEMES A. ASHYRALYEV Department of Mathematics, Fatih University, ˙ Istanbul, 34500, Turkey E-mail: [email protected] The high order of accuracy of the modified Pad´ e difference schemes for approximate solution of the nonlocal boundary value problem for the differential equation v′ (t) + Av(t) = f (t)(0 ≤ t ≤ 1), v(0) = v(λ) + µ, 0 < λ ≤ 1 in an arbitrary Banach space E with a strongly positive operator A are considered. The almost coercive stability and the coercive stability estimates for the solutions of these difference schemes are established. Keywords: Parabolic equation; Pad´ e difference schemes; Well-posedness.

1. Introduction. Pad´ e difference schemes It is known that (see, e.g. Dehghan [1], Cannon, Perez-Esteva and Hoek [2], Gordeziani, Natani and Ricci [3], Dautray and Lions [4], Ashyralyev ¨ and Ozdemir [5] and the references given therein) many applied problems in fluid mechanics, other areas of physics and mathematical biology were formulated into nonlocal mathematical models. However, such problems were not well-investigated in general. The nonlocal boundary value problem v ′ (t) + Av(t) = f (t) (0 ≤ t ≤ 1), v(0) = v(λ) + µ, 0 < λ ≤ 1

(1)

for an abstract parabolic equation in arbitrary Banach space E with the strongly positive operator A is ill-posed in the Banach space C(E) = C([0, 1], E) with norm kϕkC(E) = max ||ϕ(t)||E . 0≤t≤1

In the papers Ashyralyev, Hanalyev and Sobolevskii [6], Ashyralyev [8], the well-posedness of the nonlocal boundary value problem (1) was established in various Banach spaces under strong positivity of the operator A.

646

A. Ashyralyev

Let [0, 1]τ = {tk = kτ, 0 ≤ k ≤ N, N τ = 1} and 2τ ≤ λ. We consider the Pad´e difference schemes of (p + q) − th order of accuracy q p X uk − uk−1 X βj (−A)j τ j−1 uk−1 = ϕk , αj (−A)j τ j−1 uk − + τ j=1 j=1

ϕk = − +

p X

αj

j=1

βj

(−A)j−i+1 f (i) (tk )τ j−1

i=0

j=1

q X

j−1 X

(2)

j−1 X i=0

(−A)j−i+1 f (i) (tk−1 )τ j−1 , 1 ≤ k ≤ N,

λ ∈Z τ   p+q−1 X (−A)m λ λ τ )m u[ λ ] + ϕ0 + µ, (λ − ∈ / Z, u0 = τ m! τ τ m=0 ( 0, λτ ∈ Z  λ  m Pm−1 ϕ0 = Pp+q−1 1 m−i+1 (i) f (t[ λ ] ), m=1 i=0 (−A) m! (λ − τ τ ) u0 = u λ + ϕ0 + µ, τ

τ

λ τ

∈ / Z,

where

(

αj = βj =

(p+q−j)!p!(−1)j (p+q)!j!(p−j)! (p+q−j)!q! (p+q)!j!(q−j)!

for any j, 1 ≤ j ≤ p,

for any j, 1 ≤ j ≤ q.

(3)

Let Fτ (E) be the linear space of mesh functions ϕτ = {ϕk }N 1 with values in the Banach space E. Next on Fτ (E), we introduce the Banach spaces Cτ (E) = C([0, 1]τ , E), Cτβ,γ (E) = C β,γ ([0, 1]τ , E)(0 ≤ γ ≤ β < 1), Lp,τ (E) = Lp ([0, 1]τ , E)(1 ≤ p < ∞) with the norms kϕτ kCτ (E) = max kϕk kE , 1≤k≤N

τ

kϕ kCτβ,γ (E) = kϕτ kCτ (E) + kϕτ kLp,τ (E) =

N X

K=1

sup 1≤k λ and u0 =

λ τ

651

∈ Z. Then from (10) and (13), it follows that

λ τ −r

r (τ A)u0 Rp,p (τ A)Rp−1,p

+

r X

λ

−r

r−j+1 τ Rp,p (τ A)Rp−1,p (τ A)ϕj τ

j=1

+

λ/τ X

λ

−j

τ (τ A)Rp−1,p (τ A)ϕj τ + ϕ0 + µ. Rp,p

j=r+1

By Lemma 2.2, we have that X r λ r−j+1 τ −r (τ A)ϕj τ (τ A)Rp−1,p Rp,p u0 = Tτ

(14)

j=1 λ

+

τ X

j=r+1

u0 =

2p−1 X m=0

λ τ

∈ / Z. Then from formula (14) and the condition   2p−1 X (−A)m λ τ )m u[ λ ] + ϕ0 + µ (λ − (15) u0 = τ m! τ m=0

Now, let rτ < λ and

it follows that

 λ τ −j (τ A)Rp−1,p (τ A)ϕj τ + ϕ0 + µ . Rp,p

   (−A)m λ [λ m τ ]−1 (τ A)R0,2p (τ A)u0 τ) (λ − Rp−1,p m! τ  +ϕ0 +µ.

λ

[λ τ ]−1 +Rp−1,p (τ A)Q−1 0,2p (τ A)ϕ1 τ +

[τ ] X

[λ τ ]−j−1 (τ A)Q−1 Rp−1,p p−1,p (τ A)ϕj τ

j=2

By Lemma 2.2, we have that (2p−1    X (−A)m λ [λ m τ ]−1 u0 = Tτ Rp−1,p τ) (λ − (τ A)Q−1 0,2p (τ A)ϕ1 τ m! τ m=0 ) [λ/τ ] X [ λ ]−j−1 −1 τ . Rp−1,p (τ A)Qp−1,p (τ A)ϕj τ + ϕ0 + µ + j=2

λ τ

∈ / Z. Then from (12) and (15) it follows that    2p−1 X (−A)m λ [λ r−1 τ ]−r u0 = (τ A)Rp−1,p (τ A)R0,2p (τ A)u0 τ )m × Rp,p (λ − m! τ m=0

Let rτ > λ and

r X [λ [λ r−j+1 r−1 τ ]−r τ ]−r Rp,p +Rp,p (τ A)Rp−1,p (τ A)Q−1 (τ A)Rp−1,p (τ A)ϕj τ 0,2p (τ A)ϕ1 τ + j=2

652

A. Ashyralyev λ

+

[τ ] X

[λ τ ]−j Rp,p (τ A)Rp−1,p (τ A)ϕj τ

j=r+1



+ ϕ0 + µ.

By Lemma 2.3, we have that (2p−1    λ X (−A)m λ [ τ ]−r r−1 (τ A)Rp−1,p (τ A)Q−1 τ )m Rp,p (λ− u0 = Tτ 0,2p (τ A)ϕ1 τ m! τ m=0 +

r X

[ λ ]−r

τ Rp,p

r−j+1 (τ A)Rp−1,p (τ A)ϕj τ

j=2

[λ/τ ]

+

X

[λ τ ]−j (τ A)Rp−1,p (τ A)ϕj τ Rp,p

+ ϕ0 + µ

j=r+1

) 

.

This completes the proof of Lemma 2.4. In fact, the following inequality

−1

τ

{τ (u −uk−1 )}N

1 Lp,τ (E) ≤ M [kϕ kLp,τ (E) + kAµkE ] k

does not, generally speaking, hold in an arbitrary Banach space E and for the general strongly positive operator A. Nevertheless, we can establish the almost coercivity and well- posedness of the difference schemes (5) in Banach spaces Lp,τ (Eα,p ), 1 ≤ p ≤ ∞. Theorem 2.1. Let 1 ≤ p ≤ ∞. The solutions of the difference schemes (5) in Lp,τ (E) obey the almost coercivity inequality k{τ −1 (uk − uk−1 )}N (16) 1 kLp,τ (E) i o h n 1 ≤ M (δ, λ) kA (µ + ϕ0 ) kE + min ln , 1 + |ln kAkE→E | kϕτ kLp,τ (E) . τ Here and in the future L∞,τ (E) = Cτ (E). Theorem 2.2. Let 1 ≤ p ≤ ∞ and 0 < α < 1. The difference schemes (5) is well posed in Lp,τ (Eα,p ) and the following coercivity inequality holds: k{τ −1 (uk −uk−1 )}N 1 kLp,τ (Eα,p ) h ≤ M (δ, λ) kA (µ + ϕ0 ) kEα,p +

(17) i 1 kϕτ kLp,τ (Eα,p ) . α(1 − α)

The proof of estimates (16) and (17) follows the scheme of proof of Ashyralyev [27] and is based on estimates (4), (6), (7), (8) and formula (9). In practice this abstract result permits to obtain the almost coercivity inequality and the coercive stability estimates for the solutions of difference

A Note on the Modified Pad´ e Difference Schemes

653

schemes of the high order of accuracy in the case of the nonlocal boundary value problem for the 2m-th order multidimensional parabolic equation. References 1. M. Dehghan, Math. Prob. Eng. 2003, 81 (2003). 2. J. R. Cannon, S. Perez-Esteva and J. van der Hoek, SIAM J. Num. Anal. 24, 499 (1987). 3. N. Gordeziani, P. Natani and P. E. Ricci, Comp. Math. Appl. 50, 1333 (2005). 4. R. Dautray and J. L. Lions, Analyse Mathematique et Calcul Numerique Pour les Sciences et les Technique, Masson, Paris, 1988. ¨ 5. A. Ashyralyev and Y. Ozdemir, Comp. Math. Appl. 50, 1443 (2005). 6. A. Ashyralyev, A. Hanalyev and P. E. Sobolevskii, AAA 6, 53 (2001). ˙ Karatay and P. E. Sobolevskii, Discrete Dyn. Nat. Soc. 2, 7. A. Ashyralyev, I. 273 (2004). 8. A. Ashyralyev, J. Evol. Equat. 6, 1 (2006). 9. A. Ashyralyev and P.E. Sobolevskii, Well-Posedness of Parabolic Difference Equations, Birkh¨ auser Verlag, Basel, Boston, Berlin, 1994. 10. A. Ashyralyev, Math. Probl. Eng. 2007, (2007). 11. A. Ashiraliev and P. E. Sobolevskii, Dopovidi Akademii Nauk Ukrainskoi RSR Seriya A, Fiziko-Matematichni ta Technichni Nauki 3, (1988) (Russian). 12. A. Ashyralyev and P. E. Sobolevskii, Nonlinear Anal. Theory Methods Appl. 24, 257 (2005). 13. A. Ashyralyev, P. E. Sobolevskii, New Difference Schemes for Partial Differential Equations, Birkh¨ auser Verlag, Basel, Boston, Berlin, 2004. 14. D. Guidetti, B. Karas¨ ozen, and S. Piskarev, J. Math. Sci. 122, 3013 (2004). 15. A. Ashyralyev, S. Piskarev, S. Wei, Numer. Func. Anal. Optim. 23, 669 (2002). 16. I. P. Gavrilyuk and V. L. Makarov, Comput. Meth. Appl. Math. 1, 333 (2001). 17. I. P. Gavrilyuk and V. L. Makarov, Math. Comput. 74, 555 (2005). 18. I. P. Gavrilyuk and V. L. Makarov, SIAM J. Numer. Anal. 43, 2144 (2005). 19. D. Gordeziani and G. Avalishvili, Differ. Equ. 41, 703 (2005). 20. V. B. Shakhmurov, J. Math. Anal. Appl. 292, 605 (2004). 21. J. I. Ramos, Linearly implicit, Appl. Math. Comput. 174, 1609 (2006). 22. A. V. Gulin and V. A. Morozova, Differ. Equ. 39, 962 (2003) (Russian). 23. A. V. Gulin, N. I. Ionkin and V. A. Morozova, Differ. Equ. 37, 970 (2001) (Russian). 24. M. Sapagovas, J. Computer Appl. Math. 88, 89 (2003). 25. M. Sapagovas, J. Comp. Appl.Math. 92, 77 (2005). 26. Y.G. Wang, M. Oberguggenberger, J. Math. Anal. Appl. 233, 644 (1999). 27. A. Ashyralyev, Discrete Cont. Dyn. Syst. Series B, 7, 29 (2007). 28. A. Ashyralyev, Abstracts, International Conference on Modern Analysis and Applications dedicated to the centenary of Mark Krein, Odessa, Ukraine (2007), 17.

654

NUMERICAL SOLUTION OF A ONE-DIMENSIONAL PARABOLIC INVERSE PROBLEM ∗ and E. DEMIRC ˘ ˙ A. ASHYRALYEV, A. S. ERDOGAN I˙

Department of Mathematics, Fatih University, ˙ Istanbul, 34500, Turkey ∗ E-mail: [email protected] The stable difference schemes for the approximate solutions of the parabolic equation with control parameter  ∂v(t,x) ∂ 2 v(t,x)   ∂t = ∂x2 + p (t) v (t, x) + f (t, x) ,     x ∈ (0, L) , t ∈ (0, T ] , v(0, x) = ϕ(x), x ∈ [0, L],    v(t, 0) = α(t), v(t, L) = β(t), t ∈ [0, T ],    v(t, x∗ ) = γ(t), t ∈ [0, T ], x∗ ∈ (0, L)

are considered. Here α(t), β(t), γ(t), f (t, x) and ϕ(x) are given sufficiently smooth functions, and v(t, x) and p(t) are unknown functions. Stable difference schemes are constructed. A procedure of modified Gauss elimination method is used for solving these difference schemes. The method is illustrated by numerical examples. Keywords: Parabolic equation; Difference schemes; Convergence; Inverse problems; Control parameter.

1. Introduction The parabolic equation with the control parameter is a class of parabolic inverse problems and the equation is non-linear. While determining the solution of these problems, we shall determinate some unknown control parameter. These problems play a very important role in many branches of science and engineering. Some examples were given in temperature overspecification by Dehghan [1], robotics, chemistry (chromatography) by Kimura and Suzuki [2], physics (optical tomography) by Gryazin, Klibanov and Lucas [3]. As a result, considerable efforts have been expanded in formulating numerical solution methods that are both accurate and efficient. Methods of solutions parabolic inverse problems have been studied by researchers

Numerical Solution of a One-Dimensional Parabolic Inverse Problem

655

Cannon, Lin and Xu [4], Chao-rong Ye and Zhi-zhong Sun [5] and Dehghan [6]. Our goal in this work is to investigate the first and second order of accuracy difference schemes for approximately solving this problem. A procedure of modified Gauss elimination method is used for solving these difference schemes. We consider the parabolic equation with control parameter  ∂ 2 u(t,x) ∂u(t,x)    ∂t = ∂x2 + p (t) u (t, x) + f (t, x) ,    x ∈ (0, L) , t ∈ (0, T ] , (1) u(0, x) = ϕ(x), x ∈ [0, L],    u(t, 0) = α(t), u(t, L) = β(t), t ∈ [0, T ],    u(t, x∗ ) = γ(t), t ∈ [0, T ], x∗ ∈ (0, L).

1.1. Evaluating the Approximate Value of the Control Parameter P (t) First, using the parabolic equation, we get utxx (t, x) = uxxxx (t, x) + p (t) uxx (t, x) + fxx (t, x) ,

utxxxx (t, x) = uxxxxxx (t, x) + p (t) uxxxx (t, x) + fxxxx (t, x) .

(2) (3)

From equations (2) and (3) , it follows ut (t, x) uxx (t, x) f (t, x) − − , u (t, x) u (t, x) u (t, x) utt (t, x) u (t, x) − u2t (t, x) p′ (t) = u2 (t, x) u (t, x) (uxxxx (t, x) + p (t) uxx (t, x) + fxx (t, x)) − u2 (t, x) uxx (t, x) ut (t, x) − ft (t, x) u (t, x) + f (t, x) ut (t, x) , + u2 (t, x) uttt (t, x) u (t, x) − utt (t, x) ut (t, x) p′′ (t) = u2 (t, x) ut (t, x) (uxxxx (t, x) + p (t) uxx (t, x) + fxx (t, x)) − u2 (t, x) uxxxxxx (t, x) + p (t) uxxxx (t, x) + fxxxx (t, x) − pt (t) uxx (t, x) − u (t, x) p (t) (uxxxx (t, x) + p (t) uxx (t, x) + fxx (t, x)) + ftxx (t, x) + u (t, x) (uxxxx (t, x) + p (t) uxx (t, x) + fxx (t, x)) ut (t, x) + u2 (t, x) p (t) =

656

A. Ashyralyev, A. S. Erdo˘ gan & E. Demirci

uxx (t, x) utt (t, x) − ftt (t, x) u (t, x) + f (t, x) utt (t, x) u2 (t, x)
2 utt (t, x) u (t, x) − u2t (t, x) − u3 (t, x) 2u (t, x) (uxxxx (t, x) + p (t) uxx (t, x) + fxx (t, x)) − u3 (t, x) 2 (−uxx (t, x) ut (t, x) + ft (t, x) u (t, x) − f (t, x) ut (t, x)) − . u3 (t, x) +

Using these formulas and putting x = x∗ , t = 0, we get ϕxx (x∗ ) f (0, x∗ ) γt (0) − − , ϕ(x∗ ) ϕ(x∗ ) ϕ(x∗ ) γtt (0) ϕ(x∗ ) − γt2 (0) p′ (0) = ϕ2 (x∗ ) ∗ ϕ(x ) (ϕxxxx (x∗ ) + p (0) ϕxx (x∗ ) + fxx (0, x∗ )) − ϕ2 (x∗ ) ∗ ϕxx (x )γt (0) − ft (0, x∗ ) ϕ(x∗ ) + f (0, x∗ ) γt (0) , + ϕ2 (x∗ ) γttt (0) ϕ(x∗ ) − γtt (0) γt (0) p′′ (0) = ϕ2 (x∗ ) γt (0) (ϕxxxx (x∗ ) + p (0) ϕxx (x∗ ) + fxx (0, x∗ )) − ϕ2 (x∗ ) ϕxxxxxx (x∗ ) + p (0) ϕxxxx (x∗ ) + fxxxx (0, x∗ ) − pt (0) ϕxx (x∗ ) − ϕ(x∗ ) ∗ p (0) (ϕxxxx (x ) + p (0) ϕxx (x∗ ) + fxx (0, x∗ )) + ftxx (0, x∗ ) + ϕ(x∗ ) ∗ − (ϕxxxx (x ) + p (0) ϕxx (x∗ ) + fxx (0, x∗ )) γt (0) − ϕ2 (x∗ ) ∗ −ϕxx (x )γtt (0) + ftt (0, x∗ ) ϕ(x∗ ) − f (0, x∗ ) γtt (0) − ϕ2 (x∗ )
∗ 2 2 γtt (0) ϕ(x ) − γt (0) − ϕ3 (x∗ ) ∗ 2ϕ(x ) (ϕxxxx (x∗ ) + p (0) ϕxx (x∗ ) + fxx (0, x∗ )) − ϕ3 (x∗ ) ∗ 2 (−ϕxx (x )γt (0) + ft (0, x∗ ) ϕ(x∗ ) − f (0, x∗ ) γt (0)) . − ϕ3 (x∗ ) p (0) =

Numerical Solution of a One-Dimensional Parabolic Inverse Problem

657

Second, the approximate formulas for p (tk ), tk = kτ , k = 1, 2, · · · will be constructed. For p (τ ), we have p (τ ) = p (0) + p′ (0) τ + p′′ (0) p (2τ ) can be calculated by using the formula


τ2 + o τ3 . 2


p (2τ ) = −3p (0) + 4p (τ ) − 2τ p′ (0) + o τ 3 .

(4)

(5)

Then, p (tk ) , k ≥ 3, can be calculated by using the iteration algorithm

(6) p (tk ) = (p (tk−3 ) − 3p (tk−2 ) + 3p (tk−1 )) 1 + τ 4 + o τ 3 , tk = kτ, k = 3, N, N τ = T.

The proofs of formulas (5) and (6) are based on the method of undetermined coefficients. Neglecting the last small terms, we get following formulas for the approximate value of p (t).   p0 = p (0) ,   2  p1 = p (0) + p′ (0) τ + p′′ (0) τ2 , (7)  p2 = −3p (0) + 4p1 − 2τ p′ (0) 
  p = (p 4 , 3 ≤ k ≤ N. k k−3 − 3pk−2 + 3pk−1 ) 1 + τ 1.2. Difference Schemes

Now, we will consider the stable difference schemes of the first order of accuracy  k k−1 k k uk un −un n+1 −2un +un−1  − − pk ukn = ϕkn ,  2 τ h   k   ϕn = f (tk , xn ), tk = kτ, 1 ≤ k ≤ N, N τ = T, (8) xn = nh, 1 ≤ n ≤ M − 1 , M h = L,   0  = ϕ (x ) , x = nh, 0 ≤ n ≤ M, u  n n n   k u0 = α (tk ) , ukM = β (tk ) , tk = kτ, 0 ≤ k ≤ N and second order of accuracy  k k−1 k−1 k k uk−1 +uk−1 uk n+1 −2un +un−1 n+1 −2un n−1  n  un −u − − 2 2  τ 2h 2h  k k−1  p u +p u  k n k−1 n  = ϕkn ,  2  −k τ ϕn = f (tk − 2 , xn ), tk = kτ, 1 ≤ k ≤ N, N τ = T,   x n = nh, 1 ≤ n ≤ M − 1 , M h = L,    0  u  n = ϕ (xn ) , xn = nh, 0 ≤ n ≤ M,   k u0 = α (tk ) , ukM = β (tk ) , tk = kτ, 0 ≤ k ≤ N for the numerical solution of problem (1) .

(9)

658

A. Ashyralyev, A. S. Erdo˘ gan & E. Demirci

2. Numerical Results For numerical analysis, we consider the problem  2 u(t,x) ∂u(t,x)  = ∂ ∂x + p (t) u (t, x) 2  ∂t  2  +(π 2 − (t + 1)2 )e−t sin πx, x ∈ (0, 1) , t ∈ (0, 1] ,  u(0, x) = sin πx, x ∈ [0, 1],  √  2  u(t, 0) = 0, u(t, 1) = 0, u(t, 14 ) = 22 e−t , t ∈ [0, 1].

(10)

2

The exact solution of the given problem is u (t, x) = e−t sin πx and for the control parameter p (t) is 1 + t2 . First, we will calculate the approximate value of the control parameter p (t). In our numerical process, by putting t = 0 and x = 41 , we can obtain p (0) = 1, p′ (0) = 0, p′′ (0) = 2. Then, by using equation (7), we can calculate all pk , k = 0, N . Second, applying the first order of accuracy difference scheme (8), we obtain (N + 1) × (M + 1) system of linear equations and we write them in the matrix form AUn−1 + BUn + CUn+1 = Rϕn , 1 ≤ n ≤ M − 1, U0 = UM = e 0,

where



000 x 0 0   0 x 0 C=A= . . .  0 0 0 000 

1 v   0 B= .  0 0

0 y1 v . 0 0

0 0 y2 . 0 0

··· 0 0 0 ··· 0 0 0 ··· 0 0 0 . . . . ··· 0 x 0 ··· 0 0 x

··· 0 0 ··· 0 0 ··· 0 0 . . . · · · v yN −1 ··· 0 v

 0 0   0 ,  .  0 0 (N +1)×(N +1)

 0 0    0  .  .   0  yN (N +1)×(N +1)

Here, x = − h12 , v = − τ1 , yk = τ1 + h22 − pk , 1 ≤ k ≤ N ,  0    us ϕ(t0 )  Us =  ...  f or s = n + 1, n, n − 1, ϕ =  ... uN s

(N +1)x1

(11)

ϕ(tN )

(N +1)×1

,

Numerical Solution of a One-Dimensional Parabolic Inverse Problem





1 0 ··· 0 0 1 ··· 0  R= . . . . 0 0 · · · 1 (N +1)×(N +1)

 sin πxn   ϕ1n    ... , ϕn =     ϕN −1  

659

.

n

ϕN n

(N +1)×1

So, we have the second order difference equation with respect to n with matrix coefficients. The solution of the system of linear equation can be obtained by using the modified Gauss elimination method. We will seek a solution of matrix equation of the form Un = αn+1 Un+1 + βn+1 , n = M − 1, · · · , 2, 1.

(12)

Here, 

0 0 α1 =  . 0

   0 ··· 0 0 0 0 ··· 0   and β1 =  . . . . 0 · · · 0 (N +1)×(N +1) 0 (N +1)×1

and αk and βk are derived from αn+1 = −(B + Cαn )−1 A,

βn+1 = (B + Cαn )−1 (Rϕn − Cβn ) , n = 1, 2, · · · , M − 1.

So, after calculating αj ’s and βj ’s, Un can be obtained by using the boundary condition at x = 1 as follows UM = e 0,

Un = αn+1 Un+1 + βn+1 , n = M − 1, · · · , 2, 1.

Finally, applying the second order of accuracy difference scheme (9), we have again (N + 1) × (M + 1) system of linear equations and we write them in the matrix form (11), where   0 0 0 ··· 0 0 0 q q 0 ··· 0 0 0     0 q q ··· 0 0 0 C=A= ,  . . . . . . .   0 0 0 ··· q q 0 0 0 0 · · · 0 q q (N +1)×(N +1)

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A. Ashyralyev, A. S. Erdo˘ gan & E. Demirci



0 z  1  0 B=  .  0 0

0 g1 z2 . 0 0

0 0 g2 . 0 0

··· 0 0 ··· 0 0 ··· 0 0 . . . · · · zN −1 gN −1 ··· 0 zN

 0 0    0  .  .   0  gN (N +1)×(N +1)

Here, q = − 2h1 2 , zk = − τ1 + h12 − pk−1 , 1 ≤ k ≤ N , gk = τ1 + h12 − pk , 1 ≤ k ≤ N . In order to get the solution of difference scheme (9), again we use modified Gauss elimination method and MATLAB programs. Now, we will give the results of the numerical analysis. The numerical solutions are recorded for different values of N and M and ukn represents the numerical solutions of these difference schemes at (tk , xn ). Table 1 gives the error analysis between the exact solution of p(t) and the solutions derived by the numerical process. Table 1. Error analysis for p (t) .

Max. Error

N=20

N=40

N=80

0.0083

0.0044

0.0022

Table 2 gives the error analysis between the exact solution and the solutions derived by difference schemes. Table 2 is constructed for N = M = 20, 40 and 80 respectively. For comparison, the errors are computed by E = max u(tk , xn ) − ukn . 1≤k≤N 1≤n≤M

Table 2. Error analysis for the exact solution u(t, x). Method

N=M=20

N=M=40

N=M=80

order of accuracy

0.0048

0.0030

0.0016

2nd order of accuracy

0.0026

0.0006

0.0002

1

st

To find the control parameter p (t) , we use u (t, x∗ ) = γ (t) for t = 0, values of its first and second derivatives and smoothness of γ (t) in t. Therefore, we will consider the error between γ (tk ) and ukx∗ . Table 3 gives [h]

Numerical Solution of a One-Dimensional Parabolic Inverse Problem

the maximum error for h =

1 M

661

and N = M = 20, 40 and 80 respectively.

Table 3. Error analysis between γ (tk ) and ukx∗ [h] Method

N=M=20

N=M=40

N=M=80

1st order of accuracy

0.0034

0.0021

0.0011

2nd order of accuracy

0.0018

4.5662×10−4

1.1529×10−4

Thus, the second order of accuracy difference scheme is more accurate comparing with the first order of accuracy difference scheme. Remark 2.1. In (6), we can put 1+τ p for p = 4, 5, 6. Nevertheless without 1 + τ p , we can use identity
(13) p (tk ) = p (tk−3 ) − 3p (tk−2 ) + 3p (tk−1 ) + o τ 3 , tk = kτ, k = 3, N , N τ = T.

Neglecting last terms, we can obtain the algorithm for pk . The results are given below. Table 4. Error analysis for p (t) . N=20 Max. Error

1.2212×10

N=40 −14

4.7695×10

N=80 −13

8.3393×10−13

Table 5. Error analysis for the exact solution u (t, x) . Method 1st order of accuracy 2

nd

order of accuracy

N=M=20

N=M=40

N=M=80

0.0049

0.0030

0.0016

0.0026

0.0006

0.0002

Table 6. Error analysis between γ (tk ) and ukx∗ [h] Method

N=M=20

N=M=40

N=M=80

order of accuracy

0.0034

0.0021

0.0011

2nd order of accuracy

0.0018

4.5333×10−4

1.1327×10−4

1

st

As is seen from Table 4, even though the errors are very small, the result is unstable in the approximate solution of the control parameter p (t). Since errors are very small, Tables 5-6 are nearly the same as Tables 2-3. Note

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A. Ashyralyev, A. S. Erdo˘ gan & E. Demirci

that we obtain the multiplication factor 1 + τ 4 experimentally and use it in (6) for the stable approximate solution of the control parameter p(t). 3. Conclusions In this study, we developed a new way to obtain the control parameter and also we use first and second order of accuracy difference schemes. For N=M=20, we found the maximum error as 0.0049 while in Chao-rong Ye and Zhi-zhong Sun’s work [5], it is 0.0077. In M. Dehghan’s article [1] the errors are given for h=1/50 and s=1/4. We reached more accurate results for small M and N numbers. Difference schemes of high order of accuracy can be investigated by using the operator approach in [7]. References 1. 2. 3. 4. 5. 6. 7.

M. Dehghan, Appl. Numer. Math 124, 17 (2001). T. Kimura, T. Suzuki, SIAM J. Appl. Math. 53, 1747 (1993). Y. A. Gryazin, M. V. Klibanov, T. R. Lucas, Inverse Probl. 15, 373 (1999). J. R. Cannon, Y. L. Lin, S. Xu, Inverse Probl. 10, 227 (1994). C.-R. Ye and Z.-Z. Sun, Appl. Math. Comput. 188, 214 (2007). M. Dehghan, Appl. Math. Comput. 135, 491 (2003). A. Ashyralyev, P. E. Sobolevskii, New Difference Schemes for Partial Differential Equations, Birkh¨ auser Verlag, Basel, Boston, Berlin, 2004.

663

NUMERICAL SOLUTION OF NONLOCAL BOUNDARY VALUE PROBLEMS FOR ELLIPTIC-PARABOLIC EQUATIONS A. ASHYRALYEV Department of Mathematics, Fatih University, ˙ Istanbul, 34500, Turkey E-mail: [email protected] O. GERC ¸ EK ˙Istanbul Vocational School, Fatih University, ˙ Istanbul, 34500, Turkey E-mail: [email protected] A numerical method is proposed for solving elliptic-parabolic partial differential equations with nonlocal boundary conditions. Stable difference schemes of first and second orders of accuracy are presented. Stability and almost coercive stability of these difference schemes are established. The method is illustrated by numerical examples. Keywords: Elliptic-parabolic equation; Difference schemes.

1. Introduction Methods of solving nonlocal boundary value problems for elliptic-parabolic differential equations have been studied by Bazarov and Soltanov [1], Glazatov [2], Ashyralyev and Soltanov [3] and Karatopraklieva [4]. Let Ω be the open unit cube in the n-dimensional Euclidean space Rn (0 < xk < 1, 1 ≤ k ≤ n) with boundary S, Ω = Ω ∪ S. In [−1, 1] × Ω, mixed boundary value problem P  −utt − nr=1 (ar (x)uxr )xr = g(t, x), 0 < t < 1, x ∈ Ω,   P n    ut + r=1 (ar (x)uxr )xr = f (t, x), −1 < t < 0, x ∈ Ω, (1) u(t, x) = 0, −1 ≤ t ≤ 1, x ∈ S,    u(−1, x) = u(1, x) + µ(x), x ∈ Ω,   u(0+, x) = u(0−, x), ut (0+, x) = ut (0−, x), x ∈ Ω for multidimensional mixed elliptic-parabolic equation is considered.The

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A. Ashyralyev & O. Ger¸cek

functions ar (x) > a > 0 (x ∈ Ω), g(t, x) (t ∈ (0, 1), x ∈ Ω) and f (t, x) (t ∈ (−1, 0), x ∈ Ω) are smooth. Here a is a constant. In present paper, we are interested in studying stable difference schemes for the numerical solution of the nonlocal boundary value problem for elliptic-parabolic differential equation (1). The first and second order of accuracy difference schemes are presented. The stability and almost coercive stability of these difference schemes are established. A procedure of modified Gauss elimination method is used for solving these difference schemes for the one-dimensional elliptic-parabolic differential equation. The method is illustrated by a numerical example. 2. Difference Schemes The discretization of problem (1) is carried out in two steps. In the first step, the grid sets eh = {x = xm = (h1 m1 , · · · , hn mn ), m = (m1 , · · ·, mn ) , Ω 0 ≤ mr ≤ Nr , hr Nr = 1, r = 1, · · · , n} , eh ∩ Ω, Sh = Ω eh ∩ S Ωh = Ω

are defined. To the differential space operator A by the formula Au = −

n X

(ar (x)uxr )xr ,

r=1

acting in the space of functions u(x), defined in Ω and satisfying the Dirichlet boundary condition: u(x) = 0 in S, we assign the difference operator Axh by the formula Axh uh

=−

n  X r=1

ar (x)uh−

xr



(2) xr ,jr

acting in the space of grid functions uh (x), satisfying the conditions uh (x) = 0 for all x ∈ Sh . With the help of Axh , we arrive at the nonlocal boundary value problem  2 h  + Axh uh (t, x) = g h (t, x), 0 ≤ t ≤ 1, x ∈ Ωh , − d udt(t,x)  2    duh (t,x) − Axh uh (t, x) = f h (t, x), −1 ≤ t ≤ 0, x ∈ Ωh , dt (3) eh ,  uh (−1, x) = uh (1, x) + µh (x), x ∈ Ω    h h h  eh u (0+, x) = uh (0−, x), du (0+,x) = du (0−,x) ,x ∈ Ω dt dt for an infinite system of ordinary differential equations.

Nonlocal Boundary Value Problems for Elliptic-Parabolic Equations

665

In the second step, we replace problem (3) by the first order of accuracy difference scheme  h h uh k+1 (x)−2uk (x)+uk−1 (x)  + Axh uhk (x) = gkh (x),  τ2 −  h h    gkh(x) =h g (tk , xn ), tk = kτ, 1 ≤ k ≤ N − 1, N τ = 1, x ∈ Ωh ,   uk (x)−uk−1 (x) − Axh uhk−1 (x) = fkh (x), τ (4) h h  fk (x) = f (tk−1 , xn ), tk−1 = (k−1) τ, −N + 1 ≤ k ≤ 0, x ∈ Ωh ,     eh ,  uh (x) = uhN (x) + µh (x), x ∈ Ω    h−N h h h eh u1 (x) − u0 (x) = u0 (x) − u−1 (x), x ∈ Ω

and the second order of accuracy difference scheme  h uh (x)−2uh k (x)+uk−1 (x)  + Axh uhk (x) = gkh (x), − k+1  2 τ   h h  gk (x) = g (tk , xn ), tk = kτ, 1 ≤ k ≤ N − 1, N τ = 1, x ∈ Ωh ,   
 uhk (x)−uhk−1 (x) Axh h − 2 uk (x) + uhk−1 (x) = fkh (x), τ (5) 1 h h  1 = (k− )τ, −N +1 ≤ k ≤ 0, x ∈ Ωh , 1 , xn ), t f (x) = f (t  k− k− k 2  2 2   eh ,  uh−N (x) = uhN (x) + µh (x), x ∈ Ω    eh . −uh2 (x) + 4uh1 (x) − 3uh0 (x) = 3uh0 (x) − 4uh−1 (x) + uh−2 (x), x ∈ Ω

eh ) To formulate the results, one needs to introduce the space L2h = L2 (Ω h eh , of all the grid functions ϕ (x) = {ϕ(h1 m1 , · · ·, hn mn )} defined on Ω equipped with the norm !1/2 X

h

ϕ e = ϕh (x) 2 h1 · · · hn . L2 (Ωh )

x∈Ωh

p Theorem 2.1. Let τ and |h| = h21 + · · · + h2n be sufficiently small numbers. Then the solutions of difference scheme (4) and (5) satisfy the stability and almost coercivity estimates  

h

h

h

h





max fk L + max gk L + µ L , max uk L ≤ C1 2h 2h 2h 2h −N+1≤k≤0 1≤k≤N−1 −N ≤k≤N−1

−2 h

uk+1 − 2uhk + uhk−1 L + max τ −1 uhk − uhk−1 L max τ 2h 2h −N +1≤k≤0 1≤k≤N −1 " n n n

X X X




µh xr x , jr

uh xr x , jr ≤ C2 +τ +

fh

r

L2h

r=1

1 + ln τ + |h|

L2h

r



r=1

r=1

max

−N ≤k≤−1

h

f k

L2h

+

max

1≤k≤N −1

h

g

0 xr xr , jr L2h

k L2h



,

where C1 and C2 do not depend on τ, h, µh (x) and gkh (x), 1 ≤ k ≤ N − 1, fkh , −N + 1 ≤ k ≤ 0.

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A. Ashyralyev & O. Ger¸cek

Proof. The proof of Theorem 2.1 is based on the symmetry properties of the difference operator Axh defined by the formula (2) in L2h and the following theorem on the coercivity inequality for the solution of the elliptic difference problem in L2h Sobolevskii [5]. Theorem 2.2. For the solution of the elliptic difference problem Axh uh (x) = ω h (x), x ∈ Ωh , uh (x) = 0, x ∈ Sh ,

(6)

the following coercivity inequality holds: n X

h

u x ¯r xr ,jr L r=1

2h

≤ C3 ω h L

2h

.

3. Numerical Results We have not been able to obtain a sharp estimate for the constants figuring in the stability inequality. Therefore, we will give the following results of numerical experiments of the nonlocal boundary value problem  ∂2 u ∂u   ∂t + ∂x2 = (1 − t) sin x, − 1 < t < 0, 0 < x < π,  2 2    ∂∂tu2 + ∂∂xu2 = −t sin x, 0 < t < 1, 0 < x < π, (7) u (1, x) = u (−1, x) + 2 sin x, 0 ≤ x ≤ π,    u(0+, x) = u(0−, x), ut (0+, x) = ut (0−, x), 0 ≤ x ≤ π,    u (t, 0) = u (t, π) = 0, −1 ≤ t ≤ 1

for elliptic-parabolic equation. In the first step, we apply the first order of accuracy difference scheme ( 4) to get system of equations in the matrix form  AUn+1 + BUn + CUn−1 = Dϕn , 1 ≤ n ≤ M − 1, (8) U0 = e 0, UM = e 0, where A, B, C are (2N + 1) × (2N + 1) matrices defined by    0 0 ·· 0 0 0 0 ·· 0 0 −1 0 0 0 ·· 0 0  a 0 ·· 0 0 0 0 ·· 0 0   b c 0 0 ·· 0 0     0 a ·· 0 0 0 0 ·· 0 0   0 b c 0 ·· 0 0     ·· ·· ·· ·· ·· ·· ·· ·· ·· ··   ·· ·· ·· ·· ·· ·· ··        0 0 ·· a 0 0 0 ·· 0 0   0 0 ·· b c 0 0 A=C= , B =   0 0 ·· 0 0 a 0 ·· 0 0   0 0 ·· 0 d e d     0 0 ·· 0 0 0 a ·· 0 0   0 0 ·· 0 0 d e     ·· ·· ·· ·· ·· ·· ·· ·· ·· ··   ·· ·· ·· ·· ·· ·· ··     0 0 ·· 0 0 0 0 ·· a 0   0 0 ·· ·· ·· ·· ·· 0 0 ·· 0 a 0 0 ·· 0 0 0 0 ·· 0 f g c

00 00 00 ·· ·· 0 ·· 0 ·· d ·· ·· ·· de 0 ··

 1 0  0  ··    0 , 0  0  ··   d 0

Nonlocal Boundary Value Problems for Elliptic-Parabolic Equations

667

and D is (2N + 1) × (2N + 1) identity matrix, ϕn , Us are (2N + 1) × 1 column vectors as 

 ϕ−N n  ···    0  ϕn =   ϕn  ,  ···  ϕN n



 Us−N · · ·    0  Us =   Us  · · · 

for s = n ± 1, n.

UsN

with  2 sin xn , k = −N,    (1 − tk−1 ) sin xn , −N + 1 ≤ k ≤ 0, ϕkn =  −t sin xn , 0 ≤ k ≤ N − 1,   k sin xn , k = N. So, we have the second order difference equation (8) with respect to n with matrix coefficients. To solve this difference equation, we have applied a procedure of modified Gauss elimination method. Hence, we obtain a solution of the matrix equation in the form 

Uj = αj+1 Uj+1 + βj+1 , UM = 0,

j = M − 1, · · ·, 2, 1,

where αj (j = 1, · · ·, M ) are (2N + 1) × (2N + 1) square matrices and βj (j = 1, · · ·, M ) are (2N + 1) × 1 column matrices defined by 

αj+1 = −(B + Cαj )−1 A, βj+1 = (B + Cαj )−1 (Dϕj − Cβj ),

where j = 1, · · ·, M − 1, α1 is the (2N + 1) × (2N + 1) zero matrix and β1 is the (2N + 1) × 1 zero matrix. In the second step, we apply the second order difference scheme (5) to get a system of linear equations in the matrix form 

AUn+1 + BUn + CUn−1 = Dϕn , U0 = e 0, UM = e 0,

1 ≤ n ≤ M − 1,

(9)

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A. Ashyralyev & O. Ger¸cek

where A, B, C are (2N + 1) × (2N + 1) matrices defined by 

00 x x  0 x   ·· ··   0 0 A=C = 0 0  0 0   ·· ··  0 0 00

0 0 ·· 0 0 ·· x 0 ·· ·· ·· ·· ·· x x ·· 0 0 ·· 0 0 ·· ·· ·· ·· 0 0 0 0 ··

·· 0 0 ·· 0 0 ·· 0 0 ·· ·· ·· 0 ·· ·· a ·· ·· 0 a ·· ·· ·· ·· 0 0 ·· ·· 0 ··

0 0 0 ·· 0 0 0 ·· a 0

 0 0  0  ··    0 , 0  0  ··   0



−1 0 0 0 ·· 0 0  y z 0 0 ·· 0 0   0 y z 0 ·· 0 0   ·· ·· ·· ·· ·· ·· ··    0 0 ·· y z 0 0 B=  0 0 ·· 0 d e d   0 0 ·· 0 0 d e   ·· ·· ·· ·· ·· ·· ··   0 0 ·· ·· ·· ·· ·· 0 0 ·· 1 −4 6 4

0

00 00 00 ·· ·· 0 ·· 0 ·· d ·· ·· ·· de 1 ··

 1 0  0  ··    0 , 0  0  ··   d 0

and D is the (2N + 1) × (2N + 1) identity matrix, ϕn , Us are (2N + 1) × 1 column vectors as  ϕ−N n  ···   0   ϕn =   ϕn  ,  ···  

ϕN n

 Us−N · · ·   0   Us =   Us  · · ·  

for s = n ± 1, n,

UsN

with  2 sin xn , k = −N,    (1 − (tk − τ2 )) sin xn , −N + 1 ≤ k ≤ 0, ϕkn =  −t sin xn , 1 ≤ k ≤ N − 1,   k 0, k = N.

So, we again have the second order difference equation (9) with respect to n with matrix coefficients. To solve this difference equation, we have applied the same procedure of modified Gauss elimination method. Finally, to compare the results, we give the errors of the numerical solution for first order of accuracy difference and second order accuracy difference schemes computed by N EM =

max

−N ≤k≤N

1≤n≤M−1

u(tk , xn ) − uk , n

where u (tk , xn ) and ukn represent the exact solution and the numerical solution at (tk , xn ) respectively. The results are shown in the following

Nonlocal Boundary Value Problems for Elliptic-Parabolic Equations

669

table. Table 1. Comparison of the errors Method Difference scheme (4) Difference scheme (5)

N=M=20

N=M=30

N=M=60

0.0468 0.00080

0.0319 0.00036

0.0163 0.00009

It is observed that, the second order of accuracy difference scheme is more accurate compared to the first order of accuracy difference scheme. References 1. D. Bazarov and H. Soltanov, Some Local and Nonlocal Boundary Value Problems for Equations of Mixed and Mixed-Composite Types, Ylym, Ashgabat, 1995 (Russian). 2. S. N. Glazatov, Nonlocal Boundary Value Problems for Linear and Nonlinear Equations of Variable Type, Sobolev Institute of Mathematics SB RAS, Novosibirsk, 1998 (Russian). 3. A. Ashyralyev and H. Soltanov, On elliptic-parabolic equations in a Hilbert space, Proc. of the IMM and CS of Turkmenistan, Turkmenistan, 1995 (Russian). 4. M. G. Karatopraklieva, Differensial’nye Uravneniya, 27, 68 (1991) (Russian). 5. P. E. Sobolevskii, On Difference Methods for the Approximate Solution of Differential Equations, Voronezh State University Press, Voronezh, 1975 (Russian).

670

ON ONE DIFFERENCE SCHEME OF SECOND ORDER OF ACCURACY FOR HYPERBOLIC EQUATIONS A. ASHYRALYEV Department of Mathematics, Fatih University, ˙ Istanbul, 34500, Turkey E-mail: [email protected] ¨ M. E. KOKSAL Department of Mathematics, Fatih University, ˙ Istanbul, 34500, Turkey & Department of Mathematics, Gebze Institute of Technology, Kocaeli, 41400, Turkey E-mail: [email protected] A numerical solution of a mixed problem for hyperbolic equations with dependent coefficients is considered. The difference scheme of the second order of accuracy is presented. The stability estimates for the solution of this difference scheme are constructed. The numerical analysis is given for the wave equation. Keywords: Hyperbolic equation; Difference schemes; Numerical solution.

1. Introduction. The Difference Scheme It is known (see, for example, Krein [1], Fattorini [2] and Piskarev [3]) that various initial-boundary value problems for the hyperbolic equations can be reduced to the initial value problem d2 u(t) + A(t)u(t) = f (t), 0 < t < T, u(0) = ϕ, u′ (0) = ψ dt2

(1)

for differential equations in a Hilbert space H. Here A(t) are the self-adjoint positive definite operators in H with a t-independent domain D = D(A(t)). Of great interest is the study of absolute stable difference schemes of a high order of accuracy for hyperbolic partial differential equations, in which stability was established without any assumptions to respect of the grid steps τ and h. Such type stability inequalities for the solutions of the

A Difference Scheme of Second Order of Accuracy for Hyperbolic Equations

671

first order of accuracy difference scheme  −2 τ (uk+1 − 2uk + uk−1 ) + Ak uk+1 = fk , Ak = A(tk ), fk = f (tk ), (2) tk = kτ, 1 ≤ k ≤ N − 1, N τ = T, τ −1 (u1 − u0 ) = ψ, u0 = ϕ for the approximate solution of problem (1) were established for the first time in Sobolevskii and Chebotaryeva [4]. Let Ω be the open unit cube in the n-dimensional Euclidean space n R (0 < xk < 1, 1 ≤ k ≤ n) with boundary S, Ω = Ω ∪S. We consider the initial-boundary value problem for the multidimensional hyperbolic equation  2 n P ∂ u(t,x)   (ar (t, x)uxr )xr = f (t, x),  ∂t2 − r=1 (3) x = (x1 , . . . , xn ) ∈ Ω, 0 < t < 1,    ∂u(0,x) = ψ(x), x ∈ Ω; u(t, x) = 0, x ∈ S, u(0, x) = ϕ(x), ∂t where ar (t, x) (x ∈ Ω), ϕ(x), ψ(x) (x ∈ Ω) and f (t, x) (t ∈ (0, 1), x ∈ Ω) are given smooth functions and ar (t, x) ≥ a > 0. The Hilbert space L2 (Ω) of all square integrable functions defined on Ω, equipped with the norm Z  21 Z 2 kf kL2 (Ω) = ··· |f (x)| dx1 · · · dxn x∈Ω

is introduced. Problem (3) has a unique smooth solution u(t, x) for smooth ar (t, x) and f (t, x) functions. This allows us to reduce the mixed problem (3) to the initial value problem (1) in a Hilbert space H = L2 (Ω) with a self-adjoint positive definite operators A(t) defined by (3). The discretization of problem (3) is carried out in two steps. In the first step, the grid sets −

Ω h = {x = xj = (h1 j1 , · · ·, hn jn ), j = (j1 , · · ·, jn ), 0 ≤ jr ≤ Mr , −

−

hr Mr = 1, r = 1, · · ·, n}, Ωh = Ω ∩ Ω, Sh = Ω ∩ S are defined.
The Banach space L2h = L2 Ω h of the grid functions ϕh (x) = {ϕ(h1 m1 , · · ·, hn mn )}

defined on Ω h , equipped with the norm h

kϕ kL2h =

X

x∈Ω h

h

2

|ϕ (x)| h1 · · · hn

! 21

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A. Ashyralyev & M. E. K¨ oksal

is introduced. To the differential operators Ax (t) generated by the problem (3), we assign the difference operators Axh (t) by the formula Axh (t)uh (t, x) = −

n X r=1

(ar (t, x)uh− )xr ,jr xr

(4)

acting in the space of grid functions uh (t, x), satisfying the conditions uh (t, x) = 0 for all x ∈ Sh . It is known that Axh (t) are self-adjoint positive definite operators in L2h . With the help of Axh (t) we arrive at the initial value problem ( 2 h d u (t,x) + Axh (t)uh (t, x) = f h (t, x), 0 < t < 1, x ∈ Ω h , dt2 (5) h (0,x) h u (0, x) = ϕh (x), du dt = ψ h (x), x ∈ Ω h for an infinite system of ordinary differential equations. In the second step, applying (3), one can construct the first order of accuracy in t and the second order of accuracy in the space variables  1 h h h x h h   τ 2h(uk+1h− 2uk + uk−1 ) + Ah (tk+1 )uk+1 = fk ,   f = f (tk+1 , x), tk = kτ, x ∈ Ωh , 1 ≤ k ≤ N − 1, N τ = 1, k (6) 1 h h h h h (u  τ 1 − u0 ) = ψ , ψ = ψ (x),   −  h u0 = ϕh , ϕh = ϕh (x), x ∈ Ω h ; uhk = 0, x ∈ Sh

for approximate solution (3). Abstract theorems given in Sobolevskii and Chebotaryeva [4] permit us to establish the stability of the difference scheme (6). We are interested in studying the high order of accuracy two-step difference schemes for the approximate solutions of the problem (3). In the present paper, the second order of accuracy difference scheme  1 x 1 x 1 h h h h h   τ 2 (uk+1 − 2uk + uk−1 ) + 2 Ah (tk )uk + 4 Ah (tk )uk+1   1 x h h h h  + 4 Ah (tk )uk−1 = fk , fk = f (tk , x),     tk = kτ, x ∈ Ωh , 1 ≤ k ≤ N − 1, N τ = 1, (7) 1 τ h h x h h h x h h  τ (u1 − u0 ) + τ Ah (0)(u1 − u0 ) = 2 (f0 − Ah (0)u0 ) + ψ ,   −    f0h = f h (0, x), ψ h = ψ h (x), uh0 = ϕh , ϕh = ϕh (x), x ∈ Ω h ;    uh = 0, x ∈ S k

h

is presented.

Theorem 1.1. Let τ and |h| be sufficiently small numbers. Then, for the solution of the difference schemes (6) and (7), the following inequality is valid:

A Difference Scheme of Second Order of Accuracy for Hyperbolic Equations

673



!

n

uh −2uh +uh h h X

uk+1 +uk

k+1 k k−1

max + max

1≤k≤N −1

1≤k≤N −1 τ2 2

r=1 xr xr ,jr L L2h 2h 

h

−1 h
h fk − fk−1 L + f0 L max τ ≤ C1 2h 2h 1≤k≤N −1

n n



i X X

, + +

ϕh−

ψ h− xr xr ,jr xr ,jr L2h r=1

r=1

L2h

where C1 is independent of τ , h, fkh , ψ h and ϕh .

The proof of Theorem 1.1 is based on the symmetry properties of the operators Axh (t) defined by (4) and the following theorem on the coercivity inequality for the solution of the elliptic difference problem in L2h . Theorem 1.2. For the solutions of the elliptic difference problem Axh uh (x) = wh (x), x ∈ Ωh , uh (x) = 0, x ∈ Sh

(8)

the following coercivity inequality holds (Sobolevskii [5]): n X r=1

||uhxr xr ,jr kL2h ≤ C2 wh L

2h

.

In the following section, applying the difference schemes (6) and (7), numerical methods are proposed for solving a one-dimensional hyperbolic partial differential equation with dependent coefficients. 2. Numerical Results For numerical analysis the following initial-boundary value problem  2 2 ∂ u(t,x) u(t,x)  = (1 + t + x) exp(−t) sin x, − (t + x) ∂ ∂x  2 2   ∂t 0 < t < 1, 0 < x < π,   u(0, x) = sin x, ut (0, x) = − sin x, 0 ≤ x ≤ π,   u(t, 0) = u(t, π) = 0, 0 ≤ t ≤ 1

(9)

is considered. First, applying the first order of accuracy difference scheme (6), the first

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order of accuracy difference scheme  k+1 k+1 k−1 uk+1 +uk+1 un −2uk n+1 −2un n−1  n +un  − g(t , x ) = f (tk , xn ), 2 k+1 n 2  τ h    t = kτ, x = nh, 1 ≤ k ≤ N − 1, 1 ≤ n ≤ M − 1, k n      u0n = ϕ(xn ), xn = nh, 0 ≤ n ≤ M, u1 −u0

n n = ψ(xn ), xn = nh, 0 ≤ n ≤ M,  τ   k k  u = u  0 M = 0, 0 ≤ k ≤ N,    g(t, x) = t + x, f (t, x) = 2 exp(−t) sin x,    ϕ(x) = sin x, ψ(x) = − sin x

(10)

for the approximate solutions of the problem (9) is presented. We have (M + 1) × (M + 1) system of linear equations (10) and we write them in the matrix form ( Ak U k+1 + BU k + C U k−1 = Dϑk , 1 ≤ k ≤ N − 1, (11) 1 0 ^ ^ U 0 = ϕ(x n ), U = U + τ ψ(xn ), 0 ≤ n ≤ M, where Ak , B, C, and D are (M + 1) × (M + 1) matrices given by   0 0 0 ... 0 0 1  ak+1 bk+1 ak+1 ... 0 0 0   1  1 1  0 ak+1 bk+1 ... 0 0 0    2 2   Ak =  ... ... ... ... ... ... ...  ,   k+1  0 0 0 ... bk+1 0  M−2 aM−2  k+1 k+1   0  0 0 ... ak+1 M−1 bM−1 aM−1 1 0 0 ... 0 0 0 

0 0  0   B =  ...  0  0 0 

1 0 D=  ... 0

Here

0 ... 1 ... ... ... 0 ...

0 c 0 ... 0 0 0

0 0 c ... 0 0 0

... ... ... ... ... ... ...

 0 0 , ...  1

ak+1 =− n

0 0 0 ... c 0 0

0 0 0 ... 0 c 0

 0 0  0   ...  ,  0  0 0



0 0  0   C =  ...  0  0 0

0 d 0 ... 0 0 0

0 ... 0 ... d ... ... ... 0 ... 0 ... 0 ...

0 0 0 ... d 0 0

0 0 0 ... 0 d 0

 0 0  0   ...  ,  0  0 0

 U0s  U1s   , where s = k ± 1, k. Us =   ...  s UM (M+1)×(1) 

tk+1 + xn , h2

bk+1 = n

1 tk+1 + xn + , τ2 h2

A Difference Scheme of Second Order of Accuracy for Hyperbolic Equations

0 ≤ k ≤ N − 1, 0 ≤ n ≤ M , c = − τ22 , d =

675

1 τ2 ,

e = τ1 ,  k   ϑ0  0, n = 0,  ϑk1   , ϑnk = f (tk , xn ), 1 ≤ n ≤ M − 1, , ϑk =   ...   0, n = M, ϑkM (M+1)×1 

   ψ0 ϕ0  ψ1   ϕ1    , ψn =  ϕn =  .  ...   ...  ψM (M+1)×1 ϕM (M+1)×1

So, we have the second order difference equation (11) with respect to k with matrix coefficients. To solve this difference equation we apply the following procedure: 
−1
−1
−1 k+1  = Ak Dϑk − Ak BU k − Ak CU k−1 , U k = 1, 2, · · ·, N − 1,   0 1 ^ ^ U = sin(x 0 ≤ n ≤ M. n ), U = (1 − τ )sin(xn ),

Second, applying the second order of accuracy difference scheme (7), the following second order of accuracy difference scheme for the approximate solutions of the problem (9)   k k k−1 uk+1 −2uk+1 +uk+1 uk  uk+1 −2uk n−1 n+1 −2un +un−1 n n +un  − g(tk , xn ) + n+1 4hn2  2 2  τ 2h     k−1  uk−1 +uk−1  n+1 −2un n−1  = f (tk , xn ), xn = nh, tk = kτ,  + 4h2      1 ≤ k ≤ N − 1, 1 ≤ n ≤ M − 1, (12) u0n = ϕ(xn ),xn = nh, 0 ≤ n ≤ M,   1 1 1  1 0 u −2u +u  un −un n+1 n n−1  = τ2 + f (0, xn ) + ψ(xn ),  τ h2    k  xn = nh, 1 ≤ n ≤ M − 1, u0 = ukM = 0, 0 ≤ k ≤ N,      g(t, x) = t + x, f (t, x) = 2 exp(−t) sin x,    ϕ(x) = sin x, ψ(x) = − sin x

is presented. We have again (M + 1) × (M + 1) system of linear equations (12) and we write them in the matrix form ( Ak U k+1 + B k U k + C k U k−1 = Dϑk , 1 ≤ k ≤ N − 1, (13) 1 0 ^ ^ U 0 = ϕ(x n ), EU = vU + γ(xn ), 0 ≤ n ≤ M, where

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A. Ashyralyev & M. E. K¨ oksal



0  ak  1  0   k A =  ...   0   0 1

0 bk1 ak2 ... 0 0 0

0 ak1 bk2 ... 0 0 0

0  ck  1 0   k B =  ...  0  0 0

0 dk1 ck2 ... 0 0 0

0 ck1 dk2 ... 0 0 0

0  ak  1  0   k C =  ...   0   0 0

0 bk1 ak2 ... 0 0 0

0 ak1 bk2 ... 0 0 0





 ... 0 0 1 ... 0 0 0   ... 0 0 0    , ... ... ... ...   k k  ... bM−2 aM−2 0  ... akM−1 bkM−1 akM−1  ... 0 0 0 (M+1)×(M+1)  ... 0 0 0 ... 0 0 0   ... 0 0 0    , ... ... ... ...   k k ... dM−2 cM−2 0   ... ckM−1 dkM−1 ckM−1  ... 0 0 0 (M+1)×(M+1)

 ... 0 0 0 ... 0 0 0   ... 0 0 0    , ... ... ... ...   k k ... bM−2 aM−2 0   ... akM−1 bkM−1 akM−1  ... 0 0 0 (M+1)×(M+1)

D is the (M + 1) × (M + 1) identity matrix, and   0 0 0 ... 0 0 1  j p j ... 0 0 0   1 1 1   0 j p ... 0 0 0   2 2    E =  ... ... ... ... ... . ... ...     0 0 0 ... jM−2 pM−2 0     0 0 0 ... jM−1 pM−1 jM−1  1 0 0 ... 0 0 0 (M+1)×(M+1)

Here

tk + xn k 1 tk + xn k tk + xn , bn = 2 + , cn = − , 4h2 τ 2h2 2h2 2 tk + xn τ xn 1 τ xn dkn = − 2 + , jn = − 2 , pn = 2 + 2 , τ h2 2h τ h

akn = −

A Difference Scheme of Second Order of Accuracy for Hyperbolic Equations

677

γn = (−1 + τ2 (1 + xn )) sin(xn ), v = τ1 , 0 ≤ k ≤ N , 0 ≤ n ≤ M ,  k   ϑ0  0, n = 0,  ϑk1   ϑnk = f (tk , xn ), 1 ≤ n ≤ M − 1, ϑk =  ,  ...   0, n = M, ϑkM (M+1)×1    ϕ0 γ0  γ1   ϕ1    , γn =  ϕn =  .  ...   ...  ϕM (M+1)×1 γM (M+1)×1 

So, we have the second order difference equation (13) with respect to k with matrix coefficients. To solve this difference equation we apply the procedure 
−1
−1 k k
−1 k k−1 k+1  = Ak Dϑk − Ak B U − Ak C U , U k = 1, 2, · · ·, N − 1,   0 ^ ^ U = ϕ(xn ), U 1 = E −1 vU 0 + E −1 γ(x 0 ≤ n ≤ M. n ), Now, the results of the numerical analysis are given. For their comparison, the errors computed by !1/2 M−1 X 2 k u(tk , xn ) − un h E0 = max , 1≤k≤N −1

n=1

of the numerical solutions are recorded for different values of N =M, where u(tk , xn ) represents the exact solution and ukn represents the numerical solution at (tk , xn ). The error for the approximate solutions obtained by using the first and second order of accuracy difference schemes are shown in Table 1 for N = M = 20, 30, 40 and 50 respectively. Table 1. Comparison of the errors E0 for the approximate solutions Method Scheme (6) Scheme (7)

N=M=20

N=M=30

N=M=40

N=M=50

0.0108 0.0027

0.0074 0.0012

0.0056 0.0006

0.0046 0.0004

Thus, the second order of accuracy difference schemes are more accurate when compared with the first order of accuracy difference scheme.

678

A. Ashyralyev & M. E. K¨ oksal

Remark 2.1. In papers by Ashyralyev and K¨ oksal [6,7], two types of second order of accuracy difference schemes generated by A1/2 (t) for the numerical solution of the abstract problem (1) are presented. The stability estimates for the solution of these difference schemes and for the first and second order difference derivatives are established. However, for the practical realization of these difference schemes it is necessary to construct an operator A1/2 (t). This action is very difficult for a computer. Therefore, in spite of the theoretical results, the role of their applications to the numerical solution of the boundary value problem is not great. References 1. S. G. Krein, Linear Differential Equations in a Banach Space, Nauka, Moscow, 1966 (Russian). 2. H. O. Fattorini, Second Order Linear Differential Equations in Banach Space, Elsevier Science Publishers B.V., North-Holland, 1985. 3. S. Piskarev, Differential Equations in a Banach Space and Their Approximation, Moscow State University Publishing House, Moscow, 2005 (Russian). 4. P. E. Sobolevskii and L. M. Chebotaryeva, Izv. Vyssh. Uchebn. Zav. Matematika 5, 103 (1977) (Russian). 5. P. E. Sobolevskii, Difference Methods for the Approximate Solution of Differential Equations, Voronezh State University Press, Voronezh, 1975 (Russian). 6. A. Ashyralyev and M. E. K¨ oksal, Numer. Func. Anal. Opt. 26, 739 (2005). 7. A. Ashyralyev and M. E. K¨ oksal, Discrete Dyn. Nat. Soc. 2007, 1 (2007).

679

ON WELL-POSEDNESS OF ABSTRACT HYPERBOLIC PROBLEMS IN FUNCTION SPACES A. ASHYRALYEV Department of Mathematics, Fatih University, ˙ Istanbul, 34500, Turkey E-mail: [email protected] M. MARTINEZ† and J. PASTOR‡ Department of Applied Mathematics, University of Valencia, Valencia, Spain † E-mail: [email protected] ‡ E-mail: [email protected] S. PISKAREV∗ Scientific Research Computer Center, Moscow State University, Vorobjevy Gory, Moscow 119899, Russia E-mail: [email protected] This paper is devoted to the analysis of abstract hyperbolic differential equations in C([0, T ]; E θ ), C α ([0, T ]; E) and Lp ([0, T ]; E θ ) spaces. The presentation is based on C0 -cosine operators theory and a functional analysis approach. For the solutions of second order differential equations, the weak maxima regularity estimates are established. Keywords: Hyperbolic problem; Banach spaces; Maximal regularity; Weak maximal regularity.

1. Introduction The importance of coercive (maximal regularity, well-posedness) inequalities is well-known. Ashyralyev, Piskarev and Weis [2], Ashyralyev and Sobolevskii [4,5], Guidetti [12], Keyantuo and Lizama [17], Piskarev [23]. There are a lot of papers which investigated the maximal regularity property for second order differential equations in time: Ashyralyev and ∗ Research partially supported by Russian Foundation for Basic Research 07-01-00269, 07-01-92104 and by University of Valencia.

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A. Ashyralyev, M. Martinez, J. pastor & S. Piskarev

Sobolevskii [3], Clement and Guerre-Delabrire [8], Hoppe [14], Keyantuo and Lizama [16]. Recently, Ashyralyev, Cuevas and Piskarev [1], considered the well-posedness of difference schemes for abstract elliptic equations in Lp ([0, T ]; E) spaces. The present paper is devoted to the analysis of abstract hyperbolic differential equations in C([0, T ]; E θ ), C α ([0, T ]; E) and Lp ([0, T ]; E θ ) spaces. Let B(E) denote the Banach algebra of all linear bounded operators on a complex Banach space E. The set of all linear closed densely defined operators in E will be denoted by C(E). In a Banach space E, let us consider the following inhomogeneous Cauchy problem: u′ (t) = Au(t) + f (t), t ∈ [0, T ],u(0) = u0 ,

(1)

where the operator A ∈ C(E) generates a C0 -semigroup and f (·) is some function from [0, T ] into E. Problem (1) can be considered in various functional spaces. The most popular situations are the following settings: the well-posedness in C([0, T ]; E), C α,0 ([0, T ]; E), and Lp ([0, T ]; E) spaces (see Ashyralyev and Sobolevskii [4]). We say that problem (1) is well posed, say in C([0, T ]; E), if, for any f (·) ∈ C([0, T ]; E) and any u0 ∈ D(A), (i) the problem (1) is uniquely solvable, i.e., u(·) satisfies the equation and boundary condition (1), u(·) is continuously differentiable on [0, T ], u(t) ∈ D(A) for any t ∈ [0, T ] and Au(·) is continuous on [0, T ]; (ii) the operator (f (·), u0 ) → u(·) as an operator from C([0, T ]; E) × D(A) to C([0, T ]; E) is continuous. The coercive well-posedness in the space Υ([0, T ]; E) for problem (1) means that it is well-posed in the space Υ([0, T ]; E) and ku′ (·)kΥ([0,T ];E) + kAu(·)kΥ([0,T ];E) ≤ C (kf (·)kΥ([0,T ];E) + ku0 kE˜ ),

(2)

˜ is some subspace of E. For results of the coercive well-posedness where E see Ashyralyev and Sobolevskii [4], Guidetti, Karas¨ ozen and Piskarev [11]. We have to note that if (1) is coercive well-posed in the space C([0, T ]; E), then (see Eberhardt and Greiner [18]) operator A should be bounded or the space E contains a subspace isomorphic to c0 . It means that in general problem (1) is not coercive well-posed in C([0, T ]; E) space. However, problem (1) is classically well-posed in C([0, T ]; E α ), (see Da Prato and Grisvard [9,10], Sobolevskii [24]), where E α = (E, D(A))α is a suitable interpolation space. It was proved ( see Ashyralyev and Sobolevskii [3] that coercive well-posedness in C α,0 ([0, T ]; E) space is equivalent to the condition that A generates analytic C0 -semigroup.

On Well-Posedness of Abstract Hyperbolic Problems in Function Spaces

681

In the mean time the situation in Lp ([0, T ]; E) space is not complete. One has only an extrapolation theorem and one could get coercive inequality just for the interpolation space instead of E. The necessary and sufficient conditions for coercive well-posedness of problem (1) in Lp ([0, T ]; E) with E to be a UMD space was obtained in Kalton and Lancien [15], Kunstmann and Weis [21] and Weis [27]. Theorem 1.1. (Weis [27]) Let A generate a bounded analytic semigroup exp(·A) on a UMD–space E. Then problem (1) is coercive well-posed in the space Lp (IR+ ; E) if and only if one of the sets i), ii) or iii) is R–bounded: i) {λ(λI − A)−1 : λ ∈ iIR, λ 6= 0}; ii) {exp(tA), tA exp(tA) : t > 0}; iii) {exp(zA) : | arg z| ≤ δ}. The case of a second order equation is very different from the first order equation case. In a Banach space E, let us consider the Cauchy problem u′′ (t) = Au(t) + f (t),

t ∈ [0, T ],

u(0) = u0 , u′ (0) = u1

(3)

with the operator A ∈ C(E) generating a C0 -cosine operator function C(·, A). We will write A ∈ C(M, ω) if kC(t, A)k ≤ M eω|t| , t ∈ IR. Definition 1.1. A function u(·) is called a classical solution of problem (3) if u(·) is twice continuously differentiable, u(t) ∈ D(A) for all t ∈ [0, T ], and u(·) satisfies relations (3). If f (·) ∈ C([0, T ]; E) and u(·) is a classical solution of (3), then Z t u(t) = C(t, A)u0 + S(t, A)u1 + S(t − s, A)f (s) ds, t ∈ [0, T ].

(4)

0

As in the case of C0 -semigroups of operators, the function u(·) given by (4) is not a classical solution in general. Proposition 1.1. (Fattorini [13]) Let A ∈ C(M, ω), and let either (i) f (·), Af (·) ∈ C([0, T ); E) and f (t) ∈ D(A) for t ∈ [0, T ] or (ii) f (·) ∈ C 1 ([0, T ]; E). Then the function u(·) from (4) with u0 ∈ D(A) and u1 ∈ E 1 is a classical solution of problem (3) on [0, T ]. Here E 1 is Kisynskii space, i.e. E 1 is the space with the norm kxkE 1 := kxk + sup kC ′ (t, A)xk. 0 0 one gets by (iv) from Proposition 3.1 that kC(t + h, A) − C(t − h, A)k ≤ k2AS(t, A)kkS(h, A)k → 0 as h → 0 and therefore A is bounded.

On Well-Posedness of Abstract Hyperbolic Problems in Function Spaces

687

From Theorem 1.3 the following results follow. Theorem 3.3. (F ) is satisfied and f (·) ∈
Assume that condition

C 1 [0, T ]; E θ ∩ C [0, T ]; D B 1+2θ .
Then the problem (3) is weakly



coercively solvable in the pair ( D A1+2θ , D B 1+2θ , C [0, T ]; E θ and
ku′′ (·)kC([0,T ];E θ ) + kAu(·)kC([0,T ];E θ ) ≤ M kAu0 kE θ + kBu1 kE θ
+ min kf (·)kC 1 ([0,T ];E θ ) , kf (·)kC([0,T ];D(B1+2θ )) Theorem 3.4. Assume that condition (F ) is satisfied and f (·) ∈ C 1+α ([0, T ]; E) ∩ C α ([0, T ]; D (B)) Then the
problem

(3) is weakly
coercively solvable in the pair α D A1+ 2 , D B 1+α , C α ([0, T ]; E) and
α ku′′ (·)kC α ([0,T ];E) + kAu(·)kC α ([0,T ];E) ≤ M kA1+ 2 u0 kE + kB 1+α u1 kE
+ min kf (·)kC 1+α ([0,T ];E) , kf (·)kC α ([0,T ];D(B))

Remark 3.1. As was mentioned in Theorem 1.5 the coercive wellposedness of (3) in Lp ([0, T ]; E) spaces holds in general if and only if A is bounded. Even in the case of periodic functions spaces α Lp2π (IR; E), C2π (IR; E) the situation is not changed: in Keyantuo and Lizama [16] it was shown that maximal regularity holds if and only if {−k 2 : k ∈ ZZ} ⊂ ρ(A), supk∈ZZ kk(−k 2 I − A)−1 k < ∞, which is not true even for general bounded C0 -cosine operator function in the Hilbert space E. In the particular case of UMD spaces E maximal regularity of Cauchy problem (3) for second order in time differential equations is defined by the location of the spectrum of operator A, but not by the smoothness of the space E, that is the function λ2 (λ2 I − A)−1 must be a Fourier multiplier which is not true in general. So we do not a coercive well-posedness of (3) in Lp ([0, T ]; E θ ) space in general. Theorem 3.5.
Assume that condition (F ) is satisfied and f (·) ∈

W 1,p [0, T ]; E θ ∩Lp [0, T ]; D B 1+2θ
. Then the problem (3) is weakly



coercively solvable in the pair D A1+2θ , D B 1+2θ , Lp [0, T ]; E θ and
ku′′ (·)kLp ([0,T ];E θ ) k + kAu(·)kLp ([0,T ];E θ ) ≤ M kAu0 kE θ + kBu1kE θ
+ min kf (·)kW 1,p ([0,T ];E θ ) , kf (·)kLp ([0,T ];D(B1+2θ )) References 1. A. Ashyralyev, C. Cuevas and S. Piskarev, Numer. Func. Anal. Opt. 29, 1 (2008).

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A. Ashyralyev, M. Martinez, J. pastor & S. Piskarev

2. A. Ashyralyev, S. Piskarev and L. Weis, Numer. Func. Anal. Opt. 23, 669 (2002). 3. A. Ashyralyev and P. E. Sobolevskii, Discrete Dyn. Nat. Soc. 183 (2005). 4. A. Ashyralyev and P. E. Sobolevskii, Well-Posedness of Parabolic Difference Equations, Birkh¨ auser Verlag, Basel, Boston, Berlin, 1994. 5. A. Ashyralyev and P. E. Sobolevskii, New Difference Schemes for Partial Differential Equations, Birkh¨ auser Verlag, Basel, Boston, Berlin, 2004. 6. D. K. Chyan, S. Y. Shaw and S. Piskarev, J. London Math. Soc. 59, 1023 (1999). 7. I. Cioranescu and V. Keyantuo, Semigroup Forum, 63, 429 (2001). 8. P. Clement and S. Guerre-Delabrire, Mat. Appl. 9, 245 (1999). 9. G. Da Prato and P. Grisvard, J. Math. Pures Appl. 54, 305 (1975). 10. G. Da Prato and P. Grisvard, C. R. Acad. Sci. Paris 283, 709 (1976). 11. D. Guidetti, B. Karas¨ ozen and S. Piskarev, J. Math. Sci. 122, 3013 (2004). 12. D. Guidetti, Numer. Func. Anal. Opt. 28, 307 (2007). 13. H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, Elsevier Science Publishers B.V., North-Holland, 1985. 14. H. W. R. Hoppe, SIAM J. Numer. Anal. 19, 1110 (1982). 15. N. J. Kalton and G. Lancien, Math. Z. 235, 559 (2000). 16. V. Keyantuo and C. Lizama, Math. Z. 235, 489 (2006). 17. V. Keyantuo and C. Lizama, Studia Math. 168, 25 (2005). 18. B. Eberhardt and G. Greiner, Acta Appl. Math. 27, 47 (1992). 19. J. Kisynski, Studia Math. 44, 93 (1972). 20. S. G. Krein, Linear Differential Equations in Banach Space, American Mathematical Society, Providence, R.I., 1971. 21. P. C. Kunstmann and L. Weis, Lecture Notes in Math., Springer, Berlin, 2004. 22. C. Palencia and S. Piskarev, Semigroup Forum 63, 127 (2001). 23. S. Piskarev, Differential equations in Banach space and it’s approximation, Moscow State University Publish House, Moscow 2005 (Russian). 24. P. E. Sobolevskii, Trudy Nauchn.-Issled. Inst. Mat. Voronezh. Gos. Univ. 68 (1975) (Russian). 25. M. Sova, Rozpr. Mat. 49, 1 (1966). 26. C. C. Travis and G. F. Webb, Acta Math. Acad. Sci. Hung. 32, 75 (1978). 27. L. Weis, Math. Ann. 319, 735 (2001).

689

A NOTE ON DIFFERENCE SCHEMES OF SECOND ORDER OF ACCURACY FOR HYPERBOLIC-PARABOLIC EQUATIONS ¨ ˙ ∗ A. ASHYRALYEV and Y. OZDEM IR Department of Mathematics, Fatih University, ˙ Istanbul, 34500, Turkey ∗ E-mail: [email protected] The second order of accuracy difference schemes for approximately solving the multipoint nonlocal boundary value problem for the differential equations  2 du(t) d u(t)   dt2 + Au(t) = f (t), (0 ≤ t ≤ 1), dt + Au(t) = g(t) (−1 ≤ t ≤ 0), N L N L P P P P  αj u(µj )+ βj u′ (λj )+ϕ, |αj |, |βj | ≤ 1, 0 < µj , λj ≤ 1  u(−1) = j=1

j=1

j=1

j=1

in a Hilbert space H with the selfadjoint positive operator A are presented. Stability estimates for the solution of these difference schemes are established. Stability estimates for the solutions of these difference schemes of nonlocal boundary value problems for hyperbolic-parabolic equations are obtained. Keywords: Hyperbolic-parabolic equation; Nonlocal boundary value problems; Stability.

1. Introduction Methods of solutions of nonlocal boundary value problems for hyperbolicparabolic differential equations have been studied extensively by Dzhuraev [1], Karatoprakliev [2], Karatopraklieva [3], Korzyuk and Lemeshevsky [4], Berdyshev and Karimov [5], Salakhitdinov and Urinov [6], Ashyralyev and ¨ Ozdemir [7]. The nonlocal boundary value problem for differential equation  d2 u(t)   dt2 + Au(t) = f (t), (0 ≤ t ≤ 1),   du(t) = g(t), (−1 ≤ t ≤ 0) dt + Au(t) PN PL  u(−1) = j=1 αj u(µj ) + j=1 βj u′ (λj ) + ϕ,   P P  N L j=1 |βj | ≤ 1, 0 < µj , λj ≤ 1 j=1 |αj |,

(1)

690

¨ A. Ashyralyev & Y. Ozdemir

in a Hilbert space H with self-adjoint positive definite operator A is considered. ¨ In the paper Ashyralyev and Ozdemir [7], the stability estimates for the solution of the problem (1) are established. In applications, the stability estimates for the solutions of the mixed type boundary value problems for hyperbolic-parabolic equations are obtained. In the present paper, the second order of accuracy difference schemes for the approximate solution of the boundary value problem (1) are presented. The stability estimates for the solution of these difference schemes and their first and second order difference derivatives are established. The stability estimates for the solutions of these difference schemes of nonlocal boundary value problem for hyperbolic-parabolic equation are obtained. 2. The Difference Schemes. Two Auxiliary Lemmas Throughout this paper, for simplicity, min µj , min λj > 2τ will be considered. Let us associate the boundary value problem (1) with the corresponding second order of accuracy difference scheme  −2 2 τ (uk+1 − 2uk + uk−1 ) + Auk + τ4 A2 uk+1 = fk , fk = f (tk ),     t = kτ, 1 ≤ k < N, τ −1 (I + τ 2 A)(u1 − u0 ) = Z1 ,    k   Z1 = τ2 (f (0) − Au0 ) + (g(0) − Au0 ),   −1  τ τ   τ (uk − uk−1 ) + A(I + 2 A)uk = (I + 2 A)gk , τ (2) gk = g(tk − 2 ), tk = kτ, −(N − 1) ≤ k ≤ 0,   N P u −u  [µ /τ ] [µ /τ ]−1 j j   )+ϕ αj (u[µj /τ ] + (µj − [µj /τ ]τ ) u−N =  τ   j=1   L  P u[λ /τ ] −u[λj /τ ]−1   + βj ( j + (λj −[λj /τ ]τ + τ2 )(f[λj /τ ] −Au[λj /τ ] )). τ j=1

First of all, let us give some lemmas that will be needed below. Lemma 2.1. The following estimates

2

kR(±τ A1/2 )kH→H ≤ 1, kτ A1/2 R(±τ A1/2 )kH→H ≤ 1,

kτ AR(±τ A

1/2

β

β

k

)kH→H ≤ 1, (kτ ) kA R kH→H ≤ 1,

hold, where k ≥ 1, 0 ≤ β ≤ 1, R(±τ A1/2 ) = (I ± iτ A1/2 −

−1 τ2 2 A)

and R = (I + τ A +

Lemma 2.2. We denote PN µ B = j=1 αj (1 + ( τj − [µj /τ ])){ 21 [R[µj /τ ]−1 (τ A1/2 )(I −

(3) (4)

τ 2 2 −1 . 2 A )

iτ A1/2 ) 2

Difference Schemes of Second Order of Accuracy iτ A1/2 )]RN 2

+ R[µj /τ ]−1 (−τ A1/2 )(I +

−

1 −1/2 (I 2i A
−1 2

+

691

τ 4 A2 4 )

τ × [R[µj /τ ] (−τ A1/2 ) − R[µj /τ ] (τ A1/2 )] I + τ A A( + 1)RN }) 2 P µj 1 iτ A1/2 [µj /τ ]−2 1/2 − N α ( − [µ /τ ]){ [R (τ A )(I − ) j j=1 j τ 2 2 iτ A1/2 )]RN 2

+ R[µj /τ ]−2 (−τ A1/2 )(I + −

[µj /τ ]−1 τ 4 A2 (−τ A1/2 ) − 4 )[R
P −1 L τ 2A A( τ2 + 1)RN + j=1 βj τ1 (I

1 −1/2 (I 2i A

× I+

+

× { 21 [R[λj /τ ]−1 (τ A1/2 )(I − + R[λj /τ ]−1 (−τ A1/2 )(I + −

− (λj − [λj /τ ]τ + τ2 )A)

iτ A1/2 )]RN 2

[λj /τ ] τ 4 A2 ((−τ A1/2 ) − R[λj /τ ] ((τ A1/2 )](I + 4 )[R P 1/2 1 1 [λj /τ ]−2 1)RN } − L (τ A1/2 )(I − iτ A2 ) j=1 βj τ { 2 [R

1 −1/2 (I 2i A

× A( τ2 +

iτ A1/2 ) 2

R[µj /τ ]−1 (τ A1/2 )]

+

+ R[λj /τ ]−2 (−τ A1/2 )(I +

iτ A1/2 1 −1/2 )]RN − 2i A (I 2 [λj /τ ]−1 1/2 2

× [R[λj /τ ]−1 (−τ A1/2 ) − R

(τ A

)](I + τ

The operator I − B has an inverse Tτ = (I − B)

−1

τ 4 A2 4 ) −1 A) A( τ2 +

τ 2 A)−1

+

1)RN .

and the estimate

kTτ kH→H ≤ M

(5)

holds, where M does not depend on τ. 3. The Main Theorem Theorem 3.1. Suppose that ϕ ∈ D(A), g0 ∈ D(A1/2 ) and f0 ∈ D(A1/2 ). Then, for the solutions of the difference scheme (2), the stability estimates max

−N ≤k≤N

kuk kH ≤ M1 [kϕkH + kA−1/2 f0 kH + kA−1/2 g0 kH

+ max kA−1/2 (fk − fk−1 )τ −1 k + 1≤kmα −p =E , dµgα >mα for any γ ∈ Λmα −p (Mα , E), where dµgα is the invariant measure on Mα , induced by the dual imbedding mapping i∗α : Λ(Mα ) → Λ(Mα ) from the corresponding invariant measure dµg on the Riemannian manifold M, endowed with the positive definite Riemannian metrics g : T (M )×T (M ) → R; the scalar product p

(1)

(1)

(1)

(1)

(1)

(1)

< β1 ∧ β2 ∧ ... ∧ βk , γ1 ∧ γ2 ∧ ... ∧ γk >k (1)

(1)

:= det{< βi , γj

>1 : i, j = 1, k},

(1) (1) >1 :=< gˆα−1 βi , gˆα−1 γj >gα for gˆα : T (Mα ) → T ∗ (Mα ) is the

(1) (1) βi , γj

(19) (1) (1) βi , γj

∈ Λ1 (Mα ), any where < i, j = 1, k, and canonical isomorphism, generated by the corresponding metrics < ·, · >gα on T (Mα ); ii) (mα − p)-dimensional volume |*β| of the form *β ∈ Λmα −p (Mα , E) equals p -dimensional volume |β| of a form β ∈ Λp (Mα , E); iii) mα -dimensional measure < β∧, *β >E ≥ 0 at a fixed orientation on Mα .

Integrability of Multidimensional Differential Systems on Riemannian Manifolds

749

Owing to the conditions i)-iii) one can endow the spaces Λp (Mα , E), p ∈ Z+ , with the natural scalar product Z Z (20) < β, γ >(p) dµgα < β∧, ∗γ >E = (β, γ) := Mα

Mα

for any β, γ ∈ Λp (Mα , E). Subject to the scalar product (20) we can naturally construct the corresponding Hilbert space mα

k HΛ (Mα ) := ⊕ HΛ (Mα ) k=0

well suitable for our further consideration. Notice also here that the Hodge star *-operation satisfies the following easily checkable property: For any β, γ ∈ HΛk (Mα ), k = 0, mα , < β, γ >(k) =< *β, *γ >(mα −k) ,

(β, γ) = (*β, *γ),

(21)

that is, the Hodge operation * : HΛ (Mα ) → HΛ (Mα ) is an isometry and ′ its standard adjoint (·) with respect to the scalar product (20) operation ′ satisfies the condition (*) = (*)−1 . Denote by d′α the formally adjoint expression to the Cartan type connection mapping dα : HΛ (Mα ) → HΛ (Mα ) in the Hilbert space HΛ (Mα ). Here dα := i∗α dA , where dA :HΛ (M ) → HΛ (M ) is a suitable Cartan connection mapping and iα : Mα → M is the corresponding integral submanifold imbedding mapping, associated with a given multi-dimensional nonlinear integrable differential system on the Riemannian manifold M . Making use of these operations d′α and dα in HΛ (Mα ), one can naturally define [7,34,35] a generalized Laplace-Hodge operator ∆α : HΛ (Mα ) → HΛ (Mα ) as ∆α := d′α dα + dα d′α .

(22)

Take a form β ∈ HΛ (Mα ) satisfying the equality ∆α β = 0.

(23)

Such a form is called harmonic. One can also verify that a harmonic form β ∈ HΛ (Mα ) satisfies simultaneously the following two adjoint conditions: d′α β = 0, dα β = 0,

(24)

easily stemming from (22) and (24). It is not hard to check that the differential operation d∗α := *d′α (*)−1

(25)

750

A. K. Prykarpaysky & N. N. Bogolubov

in HΛ (Mα ) defines also a usual [26–28,30,33] Cartan type connection mapping in HΛ (Mα ). The corresponding dual to (6) co-chain d∗

d∗

d∗

d∗

α α α α E → Λ0 (Mα , E) → Λ1 (Mα , E) → ... → Λm (Mα , E) → 0

(26)

is, evidently, a de Rham complex too, as the property d∗α d∗α = 0 holds owing to the definition (25). k Denote further by HΛ(α) (Mα ), k = 0, mα , the cohomology groups k of dα -closed and by HΛ(α∗ ) (Mα ), k = 0, mα , the cohomology of d∗α k closed differential forms, respectively, and by HΛ(α k = 0, mα , ∗ α) (Mα ), the abelian groups of harmonic differential forms from the sub-spaces HΛk (Mα , E), k = 0, mα . Before formulating the next results, define the stank dard sub-spaces for harmonic forms HΛ(α ∗ α) (Mα ) and cohomology groups k k HΛ(α) (Mα ), HΛ(α (M ) for k = 0, m . ∗) α α Assume also that the Laplace0 Hodge operator (22) is elliptic in HΛ (Mα ). Now by reasonings similar to those in [7,30,34,35,41] one can formulate the following a slightly generalized de Rham-Hodge theorem. k Theorem 3.1. The groups of harmonic forms HΛ(α ∗ α) (Mα ), k = 0, mα , are, respectively, isomorphic to the cohomology groups (H k (Mα , R))|Σ| , k = 0, mα , where H k (Mα , R) is the k−th cohomology group of the manifold M α with real coefficients, Σ ⊂ Rp , p ∈ Z+ , |Σ| := card Σ, is a set of suitable “spectral” parameters labeling the linear space of independent d ∗α -closed 0 (Mα ) and, moreover, the direct sum decompositions 0-forms from HΛ(α) k k k HΛ(α ∗ α) (Mα ) ⊕ ∆HΛ (Mα ) = HΛ (Mα ) ′

k−1 k+1 k (Mα ) ⊕ dα HΛ (Mα ) = HΛ(α ∗ α) (Mα ) ⊕ dα HΛ

(27)

hold for any k = 0, mα . Another variant of the statement above was formulated in [10,11,23,41] and reads as the following generalized de Rham-Hodge theorem. k Theorem 3.2. The generalized cohomology groups HΛ(α) (Mα ), k = 0, mα , k are isomorphic, respectively, to the cohomology groups (H (Mα , R))|Σ| , k = 0, mα .

The next lemma [10,11,22,24,41] is fundamental for the proof of the above isomorphism Theorem 3.2. Lemma 3.1. There exist a set of differential (k + 1)-forms Z (k+1) [ϕ(0) (λ), dα ψ (k) ] ∈ Λk+1 (Mα , R), k = 0, mα − 1, and a set of kforms Z (k) [ϕ(0) (λ), ψ (k) ] ∈ Λk (Mα , R), k = 0, mα − 1, parametrized by a

Integrability of Multidimensional Differential Systems on Riemannian Manifolds

751

set Σ ∋ λ and semilinear in (ϕ(0) (λ), ψ (k) ) ∈ H0∗ × HΛk (Mα ), such that Z (k+1) [ϕ(0) (λ), dα ψ (k) ] = dZ (k) [ϕ(0) (λ), ψ (k) ]

(28)

for all k = 0, mα − 1 and λ ∈ Σ. Based now on Lemma 3.1, one can construct the cohomology group isomorphism claimed in Theorem 3.2 formulated above. Namely, following [10,11,13,41], let us take some singular simplicial [29,30,33–35] com(k) plex K(Mα ) of the manifold Mα and introduce linear mappings Bλ : k HΛ (Mα ) → Ck (Mα , R)), k = 0, mα − 1, λ ∈ Σ, where Ck (Mα , R), k = 0, mα − 1, are as before free abelian groups over the field R generated, respectively, by all k-chains of simplexes S (k) ∈ Ck (Mα , R), k = 0, mα − 1, from the singular simplicial complex K(Mα ), as Z X (k) (k) (k) Bλ (ψ ) := S Z (k) [ϕ(0) (λ), ψ (k) ] (29) S (k)

S (k) ∈Ck (Mα ,R))

with ψ (k) ∈ HΛk (Mα ), k = 0, mα − 1. The following theorem [10–12,22,41] based on mappings (29) holds. Theorem 3.3. The set of operations (29) parametrized by λ ∈ Σ realizes the isomorphism of cohomology groups formulated in Theorem 3.2. Assume now that the Riemannian compactified manifold M = Mα ×Rs , mα

dimM = s + dim Mα ∈ Z+ , and E := RN , where Mα ≃ × Mα,j , Mα,j := j=1

[0, Tj ) ⊂ R+ , j = 1, mα , and put dα =

mα X j=1

dtj ∧ Aj (t|∂), Aj (t|∂) := ∂/∂tj + Aj (t),

(30)

with Aj (t), j = 1, mα , being matrices parametrically dependent on t ∈ Mαmα . It is also assumed that operators Aj (t|∂) : HΛ0 (Mα ) → HΛ0 (Mα ), j = 1, mα , are commuting with each other. mα Take now such a fixed pair (ϕ(0) (λ), ψ (0) (µ)dt) ∈ H0∗ × HΛ(α) (Mα ), parametrized by elements (λ, µ) ∈ Σ × Σ, for which owing to both Theorem 3.3 and the Stokes theorem [26,27,30,35] the equality Z (m ) (m ) Ω (mα −1) [ϕ(0) (λ), ψ (0) (µ)dτ ] (31) Bλ α (ψ (0) (µ)dt) = S(t) α (m

∂S(t) α

)

(m )

holds, where S(t) α ∈ Cmα (Mα , R) is some fixed element parametrized by (s)

an arbitrarily chosen point t ∈ S(t) ⊂ Mα . Consider next the integral

752

A. K. Prykarpaysky & N. N. Bogolubov

expressions Ω(t) (λ, µ) : = Ω(t0 ) (λ, µ) : =

Z

(m

∂S(t) α

Z

)

(m ) ∂S(t α 0)

Ω (mα −1)) [ϕ(0) (λ), ψ (0) (µ)dτ ], Ω (mα −1) [ϕ(0) (λ), ψ (0) (µ)dτ ],

(32)

(m )

with a point t0 ∈ S(t0 )α ⊂ Mα , being taken fixed, λ, µ ∈ Σ, and interpret them as the corresponding kernels [15,16] of the integral invertible (ρ) (ρ) operators of Hilbert-Schmidt type Ω(t) , Ω(t0 ) : L2 (Σ; R) → L2 (Σ; R), where ρ is some finite Borel measure on the parameter set Σ. In par(m −1) (m ) ticular, we assumed above that the boundaries ∂S(t) α := σ(t) α and (m )

(m −1)

are taken homological to each other as t → t0 ∈ Mα . ∂S(t0 )α := σ(t0 )α Define now the expressions Ω± : ψ (0) (η) → ψ˜(0) (η)

(33)

−1 ψ˜(0) (η) : = ψ (0) (η) · Ω(t) Ω(t0 ) Z Z −1 dρ(ξ)ψ (0) (µ)Ω(t) (µ, ξ)Ω(t0 ) (ξ, η), dρ(µ) =

(34)

dα˜ ψ˜(0) (η) = 0

(35)

for any ψ (0) (η) ∈ HΛ0 (Mα ), η ∈ Σ, and some ψ˜(0) (η) ∈ HΛ0 (Mα ), where, by definition, for any η ∈ Σ,

Σ

Σ

motivated by the expression (31). Suppose now that the elements (34) are ones being related to some other Delsarte-Lions transformed cohomology 0 group HΛ( α) ˜ (Mα ), that is, the condition 0 holds for ψ˜(0) (η) ∈ HΛ(α) (Mα ), η ∈ Σ, and some new connection mapping

dα˜ :=

mα X j=1

˜ j (t|∂), dtj ∧ A

(36)

in HΛ (Mα ), where A˜j (t; ∂) := ∂/∂tj + A˜j (t), j = 1, mα , are parametrically dependent on t ∈ Mα . Put now ˜ j := Ω± Aj Ω −1 A ±

(37)

for each j = 1, mα , where Ω± : HΛ0 (Mα ) → HΛ0 (Mα ) are the corresponding Delsarte-Lions transmutation operators [8,9] corresponding to some ele(m −1) (m −1) ments S± (σ(t) α , σ(t0)α ) ∈ Cmα (Mα , R) naturally generated by homo(m )

(m −1)

logical to each other boundaries ∂S(t) α = σ(t) α

(m )

(m −1)

and ∂S(t0)α = σ(t0 )α

.

Integrability of Multidimensional Differential Systems on Riemannian Manifolds

753

Since all the differential expressions Aj : HΛ0 (Mα ) → HΛ0 (Mα ), j = 1, mα , were taken commuting, the same property also holds for the transformed operators (37), that is [A˜j , A˜k ] = 0, k, j = 0, mα . The latter is, evidently, equivalent owing to (37), to the general expression −1 dα˜ = Ω± dα Ω± .

(38)

For the conditions (38) and (35) to be satisfied, let us consider the expressions ˜(t) (λ, η), ˜ (mα ) (ψ˜(0) (η)dt) = S (mα ) Ω B λ (t)

(39)

based on (31) and related with the corresponding external differentiation (m ) (38), where S(t) α ∈ Cmα (M, R) and (λ, η) ∈ Σ × Σ. Assume further that there are also defined mappings ⊛ Ω± : ϕ(0) (λ) → ϕ˜(0) (λ)

(40)

⊛ 0 0 with Ω± : HΛ(α) (Mα ) → HΛ(α) (Mα ), being some operators associated (but not necessarily adjoint!) with the corresponding Delsarte-Lions transmutation operators Ω± : HΛ0 (Mα ) → HΛ0 (Mα ) and satisfying the standard ⊛ ∗ ⊛,−1 relationships A˜∗j := Ω± Aj Ω± , j = 1, mα . The proper Delsarte-Lions type operators Ω± : HΛ0 (Mα ) → HΛ0 (Mα ) are related with two different realizations of the action (34) under the necessary conditions ∗ (0) ˜ (λ) = 0, dα˜ ψ˜(0) (η) = 0, dα ˜ϕ

(41)

needed to be satisfied and meaning, evidently, that the embeddings 0 ˜(0) (η) ∈ H 0 (Mα ), η ∈ Σ, are ϕ˜(0) (λ) ∈ HΛ( α ˜ ∗ ) (Mα ), λ ∈ Σ, and ψ Λ(α) ˜ satisfied. Now we need to formulate a lemma very important for conditions (41) to hold. Lemma 3.2. The invariance property −1 (mα ) −1 Z (mα ) = Ω(t0 ) Ω(t) Z Ω(t) Ω(t0 )

(42)

holds for any t and t0 ∈ Mα . As a result of (42) and the symmetry invariance between cohomology 0 0 spaces HΛ(α) (Mα ) and HΛ( α) ˜ (Mα ) one obtains the pair of related mappings (0) ˜ −1 Ω ˜ ˜ ⊛,−1 Ω ˜⊛ , ψ (0) = ψ˜(0) Ω = ϕ˜(0) Ω (t) (t0 ) , ϕ (t) (t0 ) ⊛,−1 ⊛ −1 Ω(t0 ) , ψ˜(0) = ψ (0) Ω(t) Ω(t0 ) , ϕ˜(0) = ϕ(0) Ω(t)

(43)

754

A. K. Prykarpaysky & N. N. Bogolubov (ρ)

(ρ)

where the integral operator kernels in L2 (Σ; R) ⊗ L2 (Σ; R) are defined as Z ˜ (mα −2) [ϕ˜(0) (λ), ψ˜(0) (µ)dτ ], ˜ Ω (44) Ω(t) (λ, µ) := (m

σ(t) α

Z

˜ ⊛ (λ, µ) := Ω (t)

(m

σ(t) α

)

)

˜ Ω

(mα −2),⊺

[ϕ˜(0) (λ), ψ˜(0) (µ)dτ ]

for all (λ, µ) ∈ Σ×Σ, giving rise to proper Delsarte-Lions transmutation operators ensuring the pure differential nature of the transformed expressions (37). Note here also that owing to (42) and (43) the operator property −1 ˜(t ) Ω −1 Ω(t ) = 0 Ω(t0 ) Ω(t) Ω(t0 ) + Ω 0 0 (t)

(45)

˜(t) = −Ω(t ) . holds for any t and t0 ∈ Mα meaning that Ω 0; Similarly, one can now define additional three subspaces 0 (Mα ) : dα ψ (0) (µ) = 0, µ ∈ Σ}, H0 := {ψ (0) (µ) ∈ HΛ(α)

˜ 0 := {ψ˜(0) (µ) ∈ H0 (Mα ) : dα˜ ψ˜(0) (µ) = 0, µ ∈ Σ}, H Λ(e α)

˜ 0∗ H

(0)

:= {ϕ˜

(η) ∈

0 HΛ( α) ˜ (Mα )

:

∗ (0) dα ˜ (η) ˜ϕ

(46)

= 0, η ∈ Σ}

closed and dense in HΛ0 (Mα ), and construct the actions ⊛,−1 ⊛ −1 ⊛ Ω(t0 ) Ω± : ψ (0) 7→ ψ˜(0) := ψ (0) Ω(t) Ω(t0 ) , Ω± : ϕ(0) 7→ ϕ˜(0) := ϕ(0) Ω(t) (47) on arbitrary but fixed pairs of elements (ϕ(0) (λ), ψ (0 (µ)) ∈ H0∗ × H0 , parametrized by the set Σ, where by definition, one needs that all obtained 0 0 pairs (ϕ˜(0) (λ), ψ˜(0) (µ)), λ, µ ∈ Σ, belong to HΛ( α) ˜ (Mα ) × HΛ(α) ˜ (Mα ). Note also that the related operator property (45) can be compactly written as

˜(t) = Ω ˜(t ) Ω −1 Ω(t ) = −Ω(t ) Ω −1 Ω(t ) . Ω 0 0 0 0 (t) (t)

(48)

Construct now from the expressions (47) the operator kernels from the (ρ) (ρ) Hilbert space L2 (Σ; R) ⊗ L2 (Σ; R) Z Ω (mα −1) [ϕ(0) (λ), ψ (0) (µ)dτ ] Ω(t) (λ, µ)−Ω(t0 ) (λ, µ) = (m

∂S(t) α

−

Z

)

(mα ) 0)

∂S(t

= (m )

Z

Ω (mα −1) [ϕ(0) (λ), ψ (0) (µ)dτ ]

(m −1)

S(±)α (σ(t) α

(49)

dΩ (mα−1) [ϕ(0) (λ), ψ (0) (µ)dτ ] (mα−1) ) 0)

,σ(t

Integrability of Multidimensional Differential Systems on Riemannian Manifolds

Z

= (m

Z (mα ) [ϕ(0) (λ), ψ (0) (µ)dτ ],

(m −1)

)

755

S(±)α (σ(t) α

(mα−1) ) 0)

,σ(t

and, similarly, ⊛ ⊛ Ω(t) (λ, µ)−Ω(t (λ, µ) 0)

=

Z

(m

∂S(t) α

−

Z

)

(mα ) 0)

∂S(t

= (mα )

S±

(mα )

¯ (mα −1),⊺ [ϕ(0) (λ), ψ (0) (µ)dτ ] Ω

Z

(m −1)

Z

(mα−1) ) 0)

,σ(t

Z (mα−1),⊺ [ϕ(0) (λ), ψ (0) (µ)dτ ],

(m −1)

(σ(t) α

(50)

¯ (mα−1),⊺ [ϕ(0) (λ), ψ (0) (µ)dτ ] dΩ

(σ(t) α

= S±

¯ (mα −1),⊺ [ϕ(0) (λ), ψ (0) (µ)dτ ] Ω

(mα−1) ) 0)

,σ(t

where λ, µ ∈ Σ, and by definition, mα -dimensional surfaces (m −1) (m ) (m −1) (m −1) (m ) (m −1) S+ α (σ(t) α , σ(t0 )α ) and S− α (σ(t) α , σ(t0 )α ) ⊂ Cmα −1 (Mα ) are spanned smoothly without self-intersection between two homo(m ) (m −1) (m ) (m −1) ∈ := ∂S(t0 )α = ∂S(t) α and σ(t0 )α logical cycles σ(t) α (mα )

Cmα (Mα , R) in such a way that the boundary ∂(S+

(m −1) (m ) (m −1) ∪ S− α (σ(t) α , σ(t0 )α )) = ∅. Since the integral (ρ) (ρ) ⊛ : L2 (Σ; R) → L2 (Σ; R) are at a fixed Ω(t0 ) , Ω(t 0)

(m −1)

(σ(t) α

(m −1)

, σ(t0 )α

)

operator expressions

point t0 ∈ Mα , evidently constant and assumed to be invertible, for extending the actions given (47) on the whole Hilbert space HΛ0 (Mα ) × HΛ0 (Mα ) one can apply to them the classical constants variation approach, making use of the expressions (50). As a result, we easily obtain the Delsarte-Lions transmutation integral operator expressions Z ˜ ξ)Ω −1 (ξ, η) dρ(ξ)dρ(η)ψ(t; (51) Ω± = 1− (t0 ) Σ×Σ Z × Z (mα ) [ϕ(0) (η), ·], (mα )

S±

⊛ = 1− Ω±

Z

−1)

(m

,σ(t) α

−1)

)

⊛,−1 dρ(ξ)dρ(η)ϕ(t; ˜ η)Ω(t (ξ, η) 0) Σ×Σ Z × Z (mα ),⊺ [·, ψ (0) (ξ)dτ ]

(mα )

S±

(m

(σ(t) α

(m

(σ(t) α

−1)

(mα −1) ) 0)

,σ(t

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A. K. Prykarpaysky & N. N. Bogolubov

for fixed pairs (ϕ(0) (ξ), ψ (0) (η)) ∈ H0∗ × H0 , ξ, η ∈ Σ, and ˜ 0∗ × H ˜ 0 , λ, µ ∈ Σ, being bounded invertible inte(ϕ˜(0) (λ), ψ˜(0) (µ)) ∈ H ˜ ∗ . Applying the gral operators of Volterra type on the whole space H × H standard arguments, one can show also that transformed sets of opera⊛ ∗ ⊛,−1 −1 Ak Ω± , k = 1, mα , tors A˜j := Ω± Aj Ω± , j = 1, mα , and A˜∗k := Ω± prove to be, respectively, purely differential too. Thereby, one can formulate [23,40,41] the following final theorem.

Theorem 3.4. The expressions (51) are bounded invertible Delsarte-Lions transmutation integral operators of Volterra type onto HΛ0 (Mα ) × HΛ0 (Mα ), transforming, respectively, given commuting sets of differential expressions Aj , j = 1, mα , and their formally adjoint ones A∗k , k = 1, mα , into the −1 pure differential sets of expressions A˜j := Ω± Aj Ω± , j = 1, mα , and ⊛,−1 ⊛ ∗ ∗ ˜ Ak := Ω± Ak Ω± , k = 1, mα . Moreover, the suitably constructed closed 0 ˜ 0 ⊂ H 0 (Mα ) such that the operasubspaces H0 ⊂ HΛ(α) (Mα ) and H Λ(α) ˜ tors Ω and Ω ⊛ ∈ B(HΛ0 (Mα )) depend strongly on the topological struc0 0 ture of the generalized cohomology groups HΛ(α) (Mα ) and HΛ( α) ˜ (Mα ), being (mα )

parametrized by elements S±

(m −1)

(σ(t) α

(m −1)

, σ(t0 )α

) ∈ Cmα (Mα , R).

4. Conclusion The study done above presents some of the recent results devoted to the development of a generalized de Rham-Hodge theory [10,22,30,33–35,40,41] and related differential-geometric aspects of Chern characteristic classes, concerning special differential complexes with Cartan type connections, which give rise to effective analytical tools of studying multi-dimensional integrable nonlinear differential systems of M. Gromov type [3] on Riemannian manifolds. Some results on the structure of the Delsarte-Lions transmutation operators can be easily adapted for constructing effective transformations of Cartan-type connections for multidimensional integrable DaveyStewartson-type nonlinear differential system on a Riemannian manifold M , vanishing on a three-dimensional integral submanifold Mα ⊂ M . The results obtained can be used for studying a wide class of exact special solutions of such differential systems, having applications [4,14,31,32,35,37,40] at solving some problems of modern differential topology and mathematical physics.

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Acknowledgements The authors are cordially thankful to the Abdus Salam International Center for Theoretical Physics in Trieste, Italy, for the hospitality during their 2007 research scholarships. A.P. also appreciates interesting discussions of the results with Profs. Y. Kondratyev and Y. Kozicki during the Third International Conference on Infinite Dimensional Systems, held June 23– 28, 2007 in Kazimierz Dolny, Poland, and to Profs. O. C ¸ elebi, K. Zheltukhin, G. Barsegian, O. Zhdanov and V. Golubeva during the 2007 International ISAAC Congress, held August 13–18, 2007 in Ankara, Turkey. The last but not the least thanks belong to the referee of the article for many insightful and qualified comments which helped in an essential way in improving the exposition. References 1. R. Abraham, J. Marsden, Foundations of mechanics, Cummings, NY, USA, 1978, 806P. 2. V. I. Arnold, Mathematical methods of classical mechanics, Springer, NY, 1978, 472P. 3. M. Gromov, Partial differential relations, Springer, NY, 1986, 536P. 4. Y. A. Mitropolski, N. N. Bogoliubov, Jr., A. K. Prykarpatsky and V. H. Samoilenko, Integrable dynamical systems: Differential-geometric and spectral aspects, Nauka Dumka, Kiev, 1987, 296P. (in Russian). 5. N. N. Bogoliubov, Y. A. Prykarpatsky, A. M. Samoilenko and A. K. Pryakrpatsky, A generalized de Rham-Hodge theory of multidimensional Delsarte transformations of differential operators and its applications for nonlinear dynamic systems, Physics of Particles and Nuclei, 2005, 36, N1, P.111–121. 6. E. Cartan, Lecons sur invariants integraux, Hermann, Paris, 1971, 260P. 7. S. S. Chern, Complex manifolds, Chicago University Publ., USA, 1956, 240P. 8. J. Delsarte, Sur certaines transformations fonctionelles relative aux equations lineaires aux derives partielles du second ordre, C. R. Acad. Sci. Paris, 1938, 206, P.178–182. 9. J. Delsarte and J. Lions, Transmutations d’operateurs differentielles dans le domain complex, Comment. Math. Helv., 1957, 52, P.113–128. 10. I. V. Skrypnik, Periods of A-closed forms, Proceedings of the USSR Academy of Sciences, 1965, 160, N4, 1965, P.772–773 (in Russian). 11. I. V. Skrypnik, Harmonique fields with singularities, Ukr. Math. Journal, 1965, 17, N4, P.130–133 (in Russian). 12. I. V. Skrypnik, The generalized de Rham theorem, Proc. UkrSSR Acad. Sci., 1965, 1, P.18–19 (in Ukrainian). 13. Y. B. Lopatynski, On harmonic fields on Riemannian manifolds, Ukr. Math. Journal, 1950, 2, P.56–60 (in Russian). 14. L. D. Faddeev and L. A. Takhtadjyan, Hamiltonian approach to soliton theory, Moscow, Nauka, 1986, 528P. (in Russian).

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15. Y. M. Berezansky, Eigenfunctions expansions related with selfadjoint operators, Kiev, Nauk. Dumka Publ., 1965, 798P. (in Russian). 16. F. A. Berezin and M. A. Shubin, Schrodinger equation, Moscow, the Moscow University Publisher, 1983, 470P. (in Russian). 17. A. L. Bukhgeim, Volterra equations and inverse problems, Moscow, Nauka, 1983, 208P. (in Russian). 18. V. B. Matveev and M. I. Salle, Darboux-Backlund transformations and applications, NY, Springer, 1993, 245P. 19. L. P. Nizhnik, Inverse scattering problems for hyperbolic equations, Kiev, Nauk. Dumka Publ., 1991, 232P. (in Russian). 20. I. C. Gokhberg and M. G. Krein, Theory of Volterra operators in Hilbert spaces and its applications, Moscow, Nauka, 1967, 506P. (in Russian). 21. Y. V. Mykytiuk, Factorization of Fredholmian operators, Mathematical Studii, Proceedings of Lviv Mathematical Society, 2003, 20, N2, P.185–199 (in Ukrainian). 22. A. M. Samoilenko, A. K. Prykarpatsky and V. G. Samoylenko, The structure of Darboux-type binary transformations and their applications in soliton theory, Ukr. Mat. Journal, 2003, 55, N12, 1704–1723 (in Ukrainian). 23. Y. A. Prykarpatsky, A. M. Samoilenko and A. K. Prykarpatsky, The multidimensional Delsarte transmutation operators, their differential-geometric structure and applications, Part 1, Opuscula Mathematica, 2003, 23, P.71–80. 24. Y. A. Prykarpatsky, A. M. Samoilenko and A. K. Prykarpatsky, The de Rham-Hodge-Skrypnik theory of Delsarte transmutation operators in multidimension and its applications. Reports on Mathem. Physics, 2005, 55, N3, P.351–363. 25. J. Golenia, Y. A. Prykarpatsky, A. M. Samoilenko and A. K. Prykarpatsky, The general differential-geometric structure of multidimensional Delsarte transmutation operators in parametric functional spaces and their applications in soliton theory, Part 2, Opuscula Mathematica, 2004, 24, P.71–83. 26. C. Godbillon, Geometrie differentielle et mechanique analytique, Paris, Hermann, 1969, 188P. 27. Sternberg S., Lectures on differential geometry, Prentice Hall, USA, 1956, 410P. 28. A. S. Mishchenko and A. T. Fomenko, Introduction to differential geometry and topology, Moscow University Publ., 1983, 439P. 29. S. Kobayashi and K. Nomizu, Foundations of differential geometry, John Wiley and Sons, NY, v.1, 1963, 344P.; v.2, 1969, 357P. 30. R. Teleman, Elemente de topologie si varietati diferentiabile, Bucuresti Publ., Romania, 1964, 390P. 31. J. D. Moore, Lectures on Seiberg-Witten invariants, 2nd ed., Springer, 2001, 160P. 32. S. Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geometry, 1983, 18, P.279–315 33. G. de Rham, Varietes differentielles, Hermann, Paris, 1955, 249P. 34. G. de Rham, Sur la theorie des formes differentielles harmoniques, Ann. Univ. Grenoble, 1946, 22, P.135–152.

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35. F. Warner, Foundations of differential manifolds and Lie groups, Academic Press, NY, 1971, 346P. 36. S. P. Novikov, Topology, Institute of Computer Reserch Publ., Moscow, 2002 (in Russian). 37. A. M. Samoilenko and Y. A. Prykarpatsky, Algebraic-analytic aspects of completely integrable dynamical systems and their perturbations, Kyiv, NAS, Inst. Mathem. Publisher, v.41, 2002, 245P. (in Ukrainian). 38. D. L. Blackmore, Y. A. Prykarpatsky and R. V. Samulyak, The integrability of Lie-invariant geometric objects generated by ideals in Garssmann algebras, J. Nonlinear Math. Phys., 1998, 5, P.54–67. 39. A. K. Prykarpatsky and I. V. Mykytiuk, Algebraic integrability of nonlinear dynamical systems on manifolds: Classical and quantum aspects, Kluwer Acad. Publishers, the Netherlands, 1998, 553P. 40. O. Y. Hentosh, M. M. Prytula and A. K. Prykarpatsky, Differential-geometric integrability fundamentals of nonlinear dynamical systems on functional menifolds, 2nd revised ed., Lviv University Publisher, Lviv, Ukraine, 2006, 408P. 41. A. M. Samoilenko, Y. A. Prykarpatsky and A. K. Prykarpatsky, The spectral and differential-geometric aspects of a generalized de Rham-Hodge theory related with Delsarte transmutation operators in multidimension and its applications to spectral and soliton problems, Nonlinear Analysis, 2006, 65, P.395–432 42. V. Zakharov, S. Manakov, S. Novikov, L. Pitaevsky, Theory of solitons: The inverse scattering problem, Moscow, Nauka Publisher, 1980, 320P. (in Russian). 43. V. E. Zakharov, Integrable systems in multidimensional spaces, Lect. Notes in Phys., 1982, 153, P.190–216.

760

INTRODUCTORY BACKGROUND TO MODERN QUANTUM MATHEMATICS WITH APPLICATION TO NONLINEAR DYNAMICAL SYSTEMS A. K. PRYKARPATSKY Department of Nonlinear Mathematical Analysis, Institute for Applied Problems of Mechanics and Mathematics of NAS, Lviv, 79060, Ukraine & Department of Applied Mathematics, The AGH University of Science and Technology, Krakow, 30059, Poland E-mail: [email protected] J. GOLENIA Department of Applied Mathematics, The AGH University of Science and Technology, Krakow, 30059, Poland E-mail: [email protected] N. N. BOGOLUBOV, Jr. V. A. Steklov Mathematical Institute of RAS, Moscow, 117946 Russian Federation & Abdus Salam International Center for Theoretical Physics, Trieste, Italy E-mail: nikolai [email protected] U. TANERI˙ Department of Applied Mathematics and Computer Science, Eastern Mediterranean University EMU, Famagusta, North Cyprus ¨ & Institute of Graduate Studies, Kyrenia American University GAU, Kyrenia, North Cyprus E-mail: [email protected] Introductory background to a new mathematical physics discipline—Quantum Mathematics—is discussed and analyzed both from historical and analytical points of view. The magical properties of the second quantization method, invented by Fock in 1934, are demonstrated, and an impressive application to

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the nonlinear dynamical systems theory is considered. Keywords: Quantum mathematics; Creation and annihilation operators; Nonlinear dynamical systems;Representation.

1. Introduction There is a broad and inclusive view of modern mathematical physics by many mathematicians and mathematical physicists. During the last century, modern mathematical physics evolved within at least four components which illustrate [19] the development of the mathematics and quantum physics synergy: 1) the use of ideas from mathematics in shedding new light on the existing principles of quantum physics, either from a conceptual or from a quantitive point of view; 2) the use of ideas from mathematics in discovering new “laws of quantum physics”; 3) the use of ideas from quantum physics in shedding new light on existing mathematical structures; 4) the use of ideas from quantum physics in discovering new domains in mathematics. Each one of these topics plays some role in understanding the modern mathematical physics. However, our success in directions 2) and 4) is certainly more modest than our success in directions 1) and 3). In some cases it is difficult to draw a clear-cut distinction between these two sets. In fact, we are lucky when it is possible to make progress in directions 2) and 4); so much so that when we achieve a major progress, historians like to speak of a revolution. In any case, many of mathematical physicists strive to understand within their research efforts these deep and lofty goals. There are many situations however, when mathematical physicists’ research efforts are directed toward one other more mundane aspect: 5) the use of ideas from quantum physics and mathematics to benefit “economic competitiveness”. Here too, one might subdivide this aspect into conceptual understanding on one hand (such as the mathematical model of Black and Sholes for pricing of derivative securities in financial markets) and invention on the other: the creation of new algorithms or materials (e.g. quantum computers) which might revolutionize technology or change our way of life. As in the first four cases, the boundary between these domains is not sharp, and it remains open to views and interpretations. This fifth string can be characterized as “applied” mathematical physics. We will restrict our anal-

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A. K. Prykarpatsky, J. Golenia, N. N. Bogolubov & U. Taneri

ysis to the first four strands characterizing modern quantum physics and mathematics aspects; it is believed that that most of the profound applied directions arise after earlier fundamental quantum physics and mathematics progress. We have passed through an extraordinary 35-year period of development of modern fundamental mathematics and quantum physics. Much of this development has been derived from one subject to understand the other. Not only the concepts from diverse fields have been united: statistical physics, quantum field theory and functional integration; gauge theory and geometry; index theory and knot invariants, etc., but also, new phenomena have been recognized and new areas have emerged whose significance for both mathematics and modern quantum physics is only partially understood: for example, non-commutative geometry, super-analysis, mirror symmetry, new topological invariants of manifolds, and the general notion of geometric quantization. There is no doubt that, over the past 35 years, the ideas from quantum physics have led to far greater inventions of new mathematics than the ideas from mathematics have in discoveries of laws of quantum physics. Recognition of this underlines the opportunities for future progress in the opposite direction: a new understanding of the quantum nature of the world is certainly our expectation! Great publicity and recognition has been attached to the progress made in modern geometry, representation theory, and deformation theory due to this interaction. We will not also dwell here on the substantial progress in analysis and probability theory, which unfortunately is more difficult to understand because of its delicate dependence on subtle notions of continuity. On the other hand, there are deep differences between pure mathematics and modern quantum physics fundamentals. They have evolved from different cultures and they each have a distinctive set of values of their own, suited for their different realms of universality. But both subjects are strongly based on intuition, some natural and some acquired, which form our understanding. Quantum physics describes the natural micro-world. Hence, physicists appeal to observation in order to verify the validity of a physical theory. And, although much of mathematics arises from the natural world, mathematics has no analogous testing grounds - mathematicians appeal to their own set of values, namely mathematical proof, to justify validity of a mathematical theory. In mathematical physics, when announcing results of a mathematical nature, it is necessary to claim a theorem when the proof meets the mathematical community standards for a proof; other-

Quantum Mathematics with Application to Nonlinear Dynamical Systems

763

wise, it is necessary to make a conjecture with a detailed outline for support. Most of physics, on the other hand, has completely different standards. There is no question that the interaction between modern mathematics and quantum physics will change radically during this running century. We do hope however, that this evolution will preserve the positive experience of being a mathematician, a pure physicist, or a mathematical physicist, so that it remains attractive to the brightest and gifted young students today and tomorrow. It is instructive to look at the beginning of the 20th century and trace the way mathematics has been exerting influence on modern and classical quantum physics, and next observe the way the modern quantum physics is nowadays exerting so impressive influence on modern mathematics. With the latter, application of modern quantum mathematics to studying nonlinear dynamical systems in functional spaces will for example be a significant topic of our present work. We will begin with a brief history of quantum mathematics: The beginning of the 20th century: • P. A. M. Dirac — first realized and used in quantum physics the fact that the commutator operation Da : A ∋ b −→ [a, b] ∈ A, where a ∈ A is fixed and b ∈ A, is a differentiation of an associative algebra A; moreover, he first constructed a spinor matrix realization of the Poincar´e symmetry group P(1, 3) and invented the famous Dirac δfunction [8] (1920-1926); • J. von Neumann — first applied the spectral theory of self-adjoint operators in Hilbert spaces to explain the radiation spectra of atoms and the stability of the related matter [30] (1926); • V. Fock — first introduced the notion of many-particle Hilbert space, Fock space, and introduced the related creation and annihilation operators acting in it [13] (1932); • H. Weyl — first understood the fundamental role of the notion of symmetry in physics and developed a physics-oriented group theory; moreover he showed the importance of different representations of classical matrix groups for physics and studied the unitary representations of the Heisenberg-Weyl group related with creation and annihilation operators in Fock space [33] (1931). The end of the 20th century: New developments are due to

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A. K. Prykarpatsky, J. Golenia, N. N. Bogolubov & U. Taneri

• L. Faddeev with co-workers — quantum inverse spectral theory transform [10] (1978); • V. Drinfeld, S. Donaldson, E. Witten — quantum groups and algebras, quantum topology, quantum super-analysis [21,22,34] (1982–1994); • Y. Manin, R. Feynman — quantum information theory [11,12,28] (1980–1986); • P. Shor, E. Deutsch, L. Grover and others — quantum computer algorithms [18,20,32] (1985–1997). As one can observe, many exciting and highly important mathematical achievements were strictly motivated by the impressive and deep influence of quantum physics ideas and ways of thinking, leading nowadays to an altogether new scientific field often called quantum mathematics. Following this quantum mathematical way of thinking, we will demonstrate below that a wide class of strictly nonlinear dynamical systems in functional spaces can be treated as a natural object in specially constructed Fock spaces in which the corresponding evolution flows are completely linearized. Thereby, the powerful machinery of classical mathematical tools can be applied to studying the analytical properties of exact solutions to suitably well posed Cauchy problems. 2. Mathematical preliminaries: Fock space and its realizations Let Φ be a separable Hilbert space, F be a topological real linear space and A := {A(ϕ) : ϕ ∈ F } a family of commuting self-adjoint operators in Φ (i.e. these operators commute in the sense of their resolutions of the identity). Consider the Gelfand rigging [2] of the Hilbert space Φ, i.e., a chain D ⊂ Φ+ ⊂ Φ ⊂ Φ− ⊂ D

′

(1)

in which Φ+ and Φ− are further Hilbert spaces, and the inclusions are dense and continuous, i.e. Φ+ is topologically (densely and continuously) and quasi-nucleously (the inclusion operator i : Φ+ −→ Φ is of the HilbertSchmidt type) embedded into Φ, Φ− is the dual of Φ+ with respect to the scalar product < ., . >Φ in Φ, and D is a separable projective limit of Hilbert spaces, topologically embedded into Φ+ . Then, the following structural theorem [2,3] holds: Theorem 2.1. Assume that the family of operators A satisfies the following conditions:

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a) D ⊂ DomA(ϕ), ϕ ∈ F, and the closure of the operator A(ϕ) ↑ D coincides with A(ϕ) for any ϕ ∈ F , that is A(ϕ) ↑ D = A(ϕ) in Φ; b) the Range A(ϕ) ↑ D ⊂ Φ+ for any ϕ ∈ F ; c) for every f ∈ D the mapping F ∋ ϕ −→ A(ϕ)f ∈ Φ+ is linear and continuous; T d) there exists a strong cyclic (vacuum) vector |Ωi ∈ ϕ∈F DomA(ϕ), Qn such that the set of all vectors |Ωi, j=1 A(ϕj )|Ωi, n ∈ Z+ , is total in Φ+ (i.e. their linear hull is dense in Φ+ ). Then there exists a probability measure µ on (F ′ , Cσ (F ′ )), where F ′ is the dual of F and Cσ (F ′ ) is the σ−algebra generated by cylinder sets in F ′ such that, for µ−almost every η ∈ F ′ there is a generalized joint eigenvector ω(η) ∈ Φ− of the family A, corresponding to the joint eigenvalue η ∈ F ′ , that is < ω(η), A(ϕ)f >Φ = η(ϕ) < ω(η), f >Φ

(2)

with η(ϕ) ∈ R denoting the pairing between F and F ′ . The mapping Φ+ ∋ f −→< ω(η), f >Φ := fˆ(η) ∈ C

(3)

ˆ F f (η) := f(η)

(4)

for any η ∈ F ′ can be continuously extended to a unitary surjective operator (µ) F : Φ −→ L2 (F ′ ; C), where ′

for any η ∈ F is a generalized Fourier transform corresponding to the family A. Moreover, the image of the operator A(ϕ), ϕ ∈ F ′ , under the F− mapping is the operator of multiplication by the function F ′ ∋ η → η(ϕ) ∈ C. We assume additionally that the main Hilbert space Φ possesses the standard Fock space (bose)-structure [4,6,31], that is, Φ = ⊕n∈Z+ Φ⊗n (s) ,

(5)

where subspaces Φ⊗n (s) , n ∈ Z+ , are the symmetrized tensor products of a Hilbert space H := L2 (Rm ; C). If a vector g := (g0 , g1 , ..., gn , ...) ∈ Φ, its norm is !1/2 X 2 kgkΦ := kgn kn , (6) n∈Z+

Φ⊗n (s)

where gn ∈ ≃ L2,(s) ((R ) ; C) and k ... kn is the corresponding norm ⊗n in Φ(s) for all n ∈ Z+ . Note here that concerning the rigging structure (1), m n

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A. K. Prykarpatsky, J. Golenia, N. N. Bogolubov & U. Taneri

there holds the corresponding rigging for the Hilbert spaces Φ⊗n (s) , n ∈ Z+ , that is, ⊗n ⊗n n ⊂ Φ⊗n D(s) (s),+ ⊂ Φ(s) ⊂ Φ(s),−

(7)

with some suitably chosen dense and separable topological spaces of symn metric functions D(s) , n ∈ Z+ . Concerning expansion (5) we obtain by means of projective and inductive limits [2,3,6] the quasi-nucleous rigging of the Fock space Φ in the form (1): ′

D ⊂ Φ+ ⊂ Φ ⊂ Φ− ⊂ D . Consider now any vector |(α)n i ∈ Φ⊗n (s), n ∈ Z+ , which can be written [2,4,26] in the following canonical Dirac ket-form: |(α)n i := |α1 , α2 , ..., αn i,

(8)

1 X |α1 , α2 , ..., αn i := √ |ασ(1) i ⊗ |ασ(2) i...|ασ(n) i n! σ∈Sn

(9)

where, by definition,

m and |αj i ∈ Φ⊗1 (s) (R ; C) := H for any fixed j ∈ Z+ . The corresponding scalar product of base vectors as (9) is given as

h(β)n |(α)n i := hβn , βn−1 , ..., β2 , β1 |α1 , α2 , ..., αn−1 , αn i P = σ∈Sn hβ1 |ασ(1) i...hβn |ασ(n) i := per{hβi |αj i : i, j = 1, n},

(10)

where “per” denotes the permanent of matrix and h.|.i is the corresponding product in the Hilbert space H. Based now on representation (8) one can ⊗(n+1) for any |αi ∈ H by define an operator a+ (α) : Φ⊗n (s) −→ Φ(s) a+ (α)|α1 , α2 , ..., αn i := |α, α1 , α2 , ..., αn i,

(11)

which is called the “creation” operator in the Fock space Φ. Its adjoint ⊗(n+1) −→ Φ⊗n operator a(β) := (a+ (β))∗ : Φ(s) (s) with respect to the Fock space Φ (5) for any |βi ∈ H, called the “annihilation” operator, acts as X a(β)|α1 , ..., αn+1 i := hβ, αj i|α1 , ..., αj−1 , α ˆ j , αj+1 , ..., αn+1 i, (12) σ∈Sn

where the “hat” over a vector denotes that it should be omitted from the sequence. It is easy to check that the commutator relationship [a+ (α), a(β)] = hα, βi

(13)

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holds for any vectors |αi ∈ H and |βi ∈ H. Expression (13), owing to the rigged structure ( 1), can be naturally extended to the general case, when vectors |αi and |βi ∈ H− , conserving its form. In particular, taking |αi := |α(x)i = √12π eihλ,xi ∈ H− := L2,− (Rm ; C) for any x ∈ Rm , one easily gets from (13) that [a+ (x), a(y)] = δ(x − y),

(14)

where we put, by definition, a+ (x) := a+ (α(x)) and a(y) := a(α(y)) for all x, y ∈ Rm and denoted by δ(·) the classical Dirac delta-function. The construction above makes it possible to observe easily that there exists a unique vacuum vector |Ωi ∈ H+ , such that for any x ∈ Rm a(x)|Ωi = 0, and the set of vectors n Y

j=1

!

a+ (xj ) |Ωi ∈ Φ⊗n (s)

(15)

(16)

is total in Φ⊗n (s) , that is, their linear integral hull over the dual functional ˆ ⊗n is dense in the Hilbert space Φ⊗n for every n ∈ Z+ . This means spaces Φ (s)

(s)

that for any vector g ∈ Φ the representation Z gˆn (x1 , ..., xn )a+ (x1 )a+ (x2 )...a+ (xn )|Ωi g = ⊕n∈Z+

(17)

(Rm )n

ˆ ⊗n for all n ∈ Z+ , with holds with the Fourier type coefficients gˆn ∈ Φ (s) ˆ ⊗1 := H ≃ L2 (Rm ; C). The latter is naturally endowed with the Gelfand Φ (s) type quasi-nucleous rigging dual to H+ ⊂ H ⊂ H− ,

(18)

making it possible to construct a quasi-nucleous rigging of the dual Fock ˆ ⊗n . Thereby, chain (18) generates the dual Fock space ˆ := ⊕n∈Z+ Φ space Φ (s) quasi-nucleous rigging ˆ⊂Φ ˆ+ ⊂ Φ ˆ ⊂Φ ˆ− ⊂ D ˆ′ D

(19)

ˆ where D ˆ ≃ D, easily with respect to the central Fock type Hilbert space Φ, following from (1) and (18). Construct now the self-adjoint operator a+ (x)a(x) := ρ(x) : Φ → Φ,

(20)

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A. K. Prykarpatsky, J. Golenia, N. N. Bogolubov & U. Taneri

called the density operator at a point x ∈ Rm , satisfying the commutation properties [ρ(x), ρ(y)] = 0, [ρ(x), a(y)] = −a(y)δ(x − y),

(21)

[ρ(x), a+ (y)] = a+ (y)δ(x − y)

for all y ∈ Rm . R Now, if one is to construct the self-adjoint family A := ρ(x)ϕ(x)dx : ϕ ∈ F of linear operators in the Fock space Φ, where Rm F := S(Rm ; R) is the Schwartz functional space, one can derive, making use of Theorem 2.1, that there exists the generalized Fourier transform (4), such that Z ⊕ (µ) Φη dµ(η) (22) Φ(H) = L2 (S ′ ; C) ≃ ′

S′

for some Hilbert space sets Φη , η ∈ F , and a suitable measure µ on S ′ , with respect to which the corresponding joint eigenvector ω(η) ∈ Φ+ for any η ∈ F ′ generates the Fourier transformed family Aˆ = {η(ϕ) ∈ R : ϕ ∈ S}. Moreover, if dim Φη = 1 for all η ∈ F, the Fourier transformed eigenvector ′ ω ˆ (η) := Ω(η) = 1 for all η ∈ F . Now we will consider the family of self-adjoint operators A as generating a unitary family U := {U (ϕ) : ϕ ∈ F } = exp(iA), where for any ρ(ϕ) ∈ A, ϕ ∈ F , the operator U (ϕ) := exp[iρ(ϕ)]

(23)

is unitary, satisfying the abelian commutation condition U (ϕ1 )U (ϕ2 ) = U (ϕ1 + ϕ2 )

(24)

for any ϕ1 , ϕ2 ∈ F . Since, in general, the unitary family U = exp(iA) is defined in some Hilbert space Φ, not necessarily being of Fock type, the important problem of describing its Hilbertian cyclic representation spaces arises, within which the factorization Z ρ(ϕ) = a+ (x)a(x)ϕ(x)dx (25) Rm

jointly with relationships (21) hold for any ϕ ∈ F . This problem can be treated using mathematical tools devised both within the representation theory of C ∗ -algebras [9] and the Gelfand–Vilenkin [14] approach. Below we will describe the main features of the Gelfand–Vilenkin formalism, being

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much more suitable for the task, providing a reasonably unified framework of constructing the corresponding representations. Definition 2.1. Let F be a locally convex topological vector space, F0 ⊂ F be a finite dimensional subspace of F . The space F 0 ⊆ F ′ defined by F 0 := {ξ ∈ F ′ : ξ|F0 = 0} ,

(26)

is called the annihilator of F0 . The quotient space F ′0 := F ′ /F 0 may be identified with F0′ ⊂ F ′ , the adjoint space of F0 . Definition 2.2. Let A ⊆ F ′ ; then the subset  (A) XF 0 := ξ ∈ F ′ : ξ + F 0 ⊂ A

(27)

is called the cylinder set with base A and generating subspace F 0 .

Definition 2.3. Let n = dim F0 = dim F0′ = dim F ′0 . One says that a cylinder set X (A) has a Borel base if A is Borel when regarded as a subset of Rn . The family of cylinder sets with Borel base forms an algebra of sets. Definition 2.4. The measurable sets in F ′ are the elements of the σ− algebra generated by the cylinder sets with Borel base. Definition 2.5. A cylindrical measure in F ′ is a real-valued σ−preadditive function µ defined on the algebra of cylinder sets with Borel base ′ and `satisfying thePconditions 0 ≤ µ(X) ≤ 1 for any X, µ(F ) = 1 and = j∈Z+ µ(Xj ), if all sets Xj ⊂ F ′ , j ∈ Z+ , have a comµ j∈Z+ Xj mon generating subspace F0 ⊂ F . Definition 2.6. A cylindrical measure µ satisfies the commutativity condition if and only if for any bounded continuous function α : Rn −→ R of n ∈ Z+ real variables, the function Z α(η(ϕ1 ), η(ϕ2 ), ..., η(ϕn ))dµ(η) (28) α[ϕ1 , ϕ2 , ..., ϕn ] := F′

is sequentially continuous in ϕj ∈ F , j = 1, m. (It is well known [14,15] that in countably normed spaces the properties of sequential and ordinary continuity are equivalent.)

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A. K. Prykarpatsky, J. Golenia, N. N. Bogolubov & U. Taneri

Definition 2.7. A cylindrical measure µ is countably additive if for any ` cylinder set X = j∈Z+ Xj , which is the union of countably many mutually P disjoints cylinder sets Xj ⊂ F ′ , j ∈ Z+ , µ(X) = j∈Z+ µ(Xj ). The following propositions hold.

Proposition 2.1. A countably additive cylindrical measure µ can be extended to a countably additive measure on the σ−algebra generated by the cylinder sets with Borel base. Such a measure will also be called a cylindrical measure. Proposition 2.2. Let F be a nuclear space. Then any cylindrical measure µ on F ′ , satisfying the continuity condition, is countably additive. Definition 2.8. Let µ be a cylindrical measure in F ′ . The Fourier transform of µ is the nonlinear functional Z exp[iη(ϕ)]dµ(η). (29) L(ϕ) := F′

Definition 2.9. The nonlinear functional L : F −→ C on F , defined by (29), is called positive definite, if for all fj ∈ F and λj ∈ C, j = 1, n, the condition n X ¯j L(fk − fj )λk ≥ 0 λ (30) j,k=1

holds for any n ∈ Z+ .

Proposition 2.3. The functional L : F −→ C on F , defined by (29), is the Fourier transform of a cylindrical measure on F ′ , if and only if it is positive definite, sequentially continuous and satisfies the condition L(0) = 1. Suppose now that we have a continuous unitary representation of the unitary family U in a Hilbert space Φ with a cyclic vector |Ωi ∈ Φ. Then we can put L(ϕ) := hΩ|U (ϕ)|Ωi

(31)

for any ϕ ∈ F := S, the Schwartz space on Rm , and observe that functional (31) is continuous on F owing to the continuity of the representation. Therefore, this functional is the generalized Fourier transform of a cylindrical measure µ on S ′ : Z exp[iη(ϕ)]dµ(η). (32) hΩ|U (ϕ)|Ωi = S′

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From the spectral point of view, based on Theorem 2.1, there is an (µ) isomorphism between the Hilbert spaces Φ and L2 (S ′ ; C), defined by |Ωi −→ Ω(η) = 1 and U (ϕ)|Ωi −→ exp[iη(ϕ)] and next extended by linearity upon the whole Hilbert space Φ. In the the non-cyclic case there exists a finite or countably infinite family (µ ) of measures {µk : k ∈ Z+ } on S ′ , with Φ ≃ ⊕k∈Z+ L2 k (S ′ ; C) and the uni(µ ) tary operator U (ϕ) : Φ −→ Φ for any ϕ ∈ S ′ corresponds in all L2 k (S ′ ; C), k ∈ Z+ , to exp[iη(ϕ)]. This means that there exists a single cylindrical measure µ on S ′ and a µ−measurable field of Hilbert spaces Φη on S ′ , such that Z ⊕ Φ≃ Φη dµ(η), (33) S′

with U (ϕ) : Φ −→ Φ, corresponding [14] to the operator of multiplication by exp[iη(ϕ)] for any ϕ ∈ S and η ∈ S ′ . Thereby, having constructed the nonlinear functional (29) in an exact analytical form, one can retrieve the representation of the unitary family U in the corresponding Hilbert space Φ of the Fock type, making use of the suitable factorization (25) as follows: Φ = ⊕n∈Z+ Φn , where ) ( Y + a (xj )|Ωi , (34) Φn = span fn ∈L2,s ((Rm )n ;C)

j=1,n

for all n ∈ Z+ . The cyclic vector |Ωi ∈ Φ can be, in particular, obtained as the ground state vector of some unbounded self-adjoint positive definite Hamilton operator H : Φ −→ Φ, commuting with the self-adjoint particles number operator Z N := ρ(x)dx, (35) Rm

that is [H, N] = 0. Moreover, the conditions H|Ωi = 0

(36)

and inf hg, Hgi = hΩ|H|Ωi = 0

g∈domH

(37)

hold for the operator H : Φ −→ Φ, where dom H denotes the domain of definition of H. To find the functional (31), which is called the generating Bogolubov type functional for moment distribution functions Fn (x1 , x2 , ..., xn ) := hΩ| : ρ(x1 )ρ(x2 )...ρ(xn ) : |Ωi,

(38)

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A. K. Prykarpatsky, J. Golenia, N. N. Bogolubov & U. Taneri

where xj ∈ Rm , j = 1, n, and the normal ordering operation : · : is defined as ! j n X Y ρ(xj ) − δ(xj − xk ) , (39) : ρ(x1 )ρ(x2 )...ρ(xn ) : = j=1

k=1

it is convenient to choose the Hamilton operator H : Φ −→ Φ in the [5,15,16] algebraic form Z 1 H := K + (x)ρ−1 (x)K(x)dx + V (ρ), (40) 2 Rm which is equivalent in the Hilbert space Φ to the positive definite operator expression Z 1 (K + (x) − A(x; ρ))ρ−1 (x)(K(x) − A(x; ρ))dx, (41) H := 2 Rm

where A(x; ρ) : Φ → Φ, x ∈ Rm , is some specially chosen linear selfadjoint operator. The “potential” operator V (ρ) : Φ −→ Φ is, in general, a polynomial (or analytical) functional of the density operator ρ(x) : Φ −→ Φ and the operator is given as K(x) := ∇x ρ(x)/2 + iJ(x),

(42)

where the self-adjoint “current” operator J(x) : Φ −→ Φ can be defined (but non-uniquely) from the equality ∂ρ/∂t =

1 [H, ρ(x)] = − < ∇x · J(x) >, i

(43)

holding for all x ∈ Rm . Such an operator J(x) : Φ −→ Φ, x ∈ Rm can exist owing to the commutation condition [H, N] = 0, giving rise to the continuity relationship (43), if taking into account that supports supp ρ of the density operator ρ(x) : Φ −→ Φ, x ∈ Rm , can be chosen arbitrarily owing to the independence of (43) on the potential operator V (ρ) : Φ −→ Φ, but its strict dependence on the corresponding representation (33). Denote also that representation (41 ) holds only under the condition that there exists such a self-adjoint operator A(x; ρ) : Φ −→ Φ, x ∈ Rm , that K(x)|Ωi = A(x; ρ)|Ωi

(44)

for all ground states |Ωi ∈ Φ, correspond to suitably chosen potential operators V (ρ) : Φ −→ Φ. The self-adjointness of the operator A(x; ρ) : Φ −→ Φ, x ∈ Rm , can be stated following schemes from works [5,16], under the additional condition

Quantum Mathematics with Application to Nonlinear Dynamical Systems

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of the existence of such a linear anti-unitary mapping T : Φ −→ Φ that the invariance conditions T ρ(x)T −1 = ρ(x),

T J(x) T −1 = −J(x),

T |Ωi = |Ωi

(45)

hold for any x ∈ Rm . Therefore, owing to conditions (45), the expressions K ∗ (x)|Ωi = A(x; ρ)|Ωi = K(x)|Ωi

(46)

hold for any x ∈ Rm , giving rise to the self-adjointness of the operator A(x; ρ) : Φ −→ Φ, x ∈ Rm . Based now on the construction above one easily deduces from expression (43) that the generating Bogolubov type functional (31) obeys for all x ∈ Rm the functional-differential equation   1 δ 1 δL(ϕ) L(ϕ), (47) = A x; [∇x − i∇x ϕ] 2i δϕ(x) i δϕ whose solutions should satisfy the Fourier transform representation (32). In particular, a wide class of special so-called Poissonian white noise type solutions to the functional-differential equation (47) was obtained in [5,16] by means of functional-operator methods in the generalized form    Z   1 δ {exp[iϕ(x)] − 1}dx , (48) exp ρ¯ L(ϕ) = exp A i δϕ Rm where ρ¯ := hΩ|ρ|Ωi ∈ R+ is a Poisson distribution density parameter. Consider now the case that the basic Fock space Φ = ⊗sj=1 Φ(j) , where (j) Φ , j = 1, s, are Fock spaces corresponding to the different types of independent cyclic vectors |Ωj i ∈ Φ(j) , j = 1, s. This, in particular, means that the suitably constructed creation and annihilation operators aj (x), a+ k (y) : Φ −→ Φ, j, k = 1, s, satisfy the commutation relations [aj (x), ak (y)] = 0, [aj (x), a+ k (y)] = δjk δ(x − y)

(49)

for any x, y ∈ Rm . Definition 2.10. A vector |ui ∈ Φ, x ∈ Rm , is called coherent with respect to a mapping u ∈ L2 (Rm ; Rs ) := M, if it satisfies the eigenfunction condition aj (x)|ui = uj (x)|ui for each j = 1, s and all x ∈ Rm .

(50)

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A. K. Prykarpatsky, J. Golenia, N. N. Bogolubov & U. Taneri

It is easy to check that the coherent vectors |ui ∈ Φ exist. Indeed, the vector expression |ui := exp{(u, a+ )}|Ωi,

(51)

where (., .) is the standard scalar product in the Hilbert space M , satisfies the defining condition (50), and moreover, the norm 1 kukΦ := hu|ui1/2 = exp( kuk2 ) < ∞, 2

(52)

since u ∈ M and its norm kuk := (u, u)1/2 is bounded. 3. The Fock space embedding method, nonlinear dynamical systems and their complete linearization Consider any function u ∈ M := L2 (Rm ; Rs ) and observe that the Fock space embedding mapping ξ : M ∋ u −→ |ui ∈ Φ,

(53)

defined by means of the coherent vector expression (51) realizes a smooth isomorphism between Hilbert spaces M and Φ. The inverse mapping ξ −1 : Φ −→ M is given by the exact expression u(x) = hΩ|a(x)|ui,

(54)

holding for almost all x ∈ Rm . Owing to condition (52), one finds from (54) that the corresponding function u ∈ M . In the Hilbert space M , let’s now define a nonlinear dynamical system (which can, in general, be non-autonomous) in partial derivatives du/dt = K[u],

(55)

where t ∈ R+ is the corresponding evolution parameter, [u] := (t, x; u, ux , uxx , ..., urx ), r ∈ Z+ , and a mapping K : M −→ T (M ) is Frechet smooth. Assume also that the Cauchy problem u|t=+0 = u0

(56)

is solvable for any u0 ∈ M in an interval [0, T ) ⊂ R1+ for some T > 0. Thereby, the smooth evolution mapping is defined Tt : M ∋ u0 −→ u(t|u0 ) ∈ M, for all t ∈ [0, T ).

(57)

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It is now natural to consider the commutative diagram ξ

M −→ Φ Tt ↓ ↓ Tt

(58)

ξ

M −→ Φ,

where the mapping Tt : Φ −→ Φ, t ∈ [0, T ), is defined from the conjugation relationship ξ ◦ Tt = Tt . ◦ ξ

(59)

Now take a coherent vector |u0 i ∈ Φ, corresponding to u0 ∈ M, and construct the vector |ui := Tt · |u0 i

(60)

for all t ∈ [0, T ). Since vector (60) is, by construction, coherent, that is aj (x)|ui := uj (x, t|u0 )|ui

(61)

for each j = 1, s, t ∈ [0, T ) and almost all x ∈ Rm , owing to the smoothness of the mapping ξ : M −→ Φ with respect to the corresponding norms in the Hilbert spaces M and Φ, we derive that coherent vector (60) is differentiable with respect to the evolution parameter t ∈ [0, T ) . Thus, one can easily find [25,26] that d ˆ + , a]|ui, |ui = K[a dt

(62)

|ui|t=+0 = |u0 i

(63)

where

ˆ + , a] : Φ −→ Φ is defined by the exact analytical exand a mapping K[a pression ˆ + , a] := (a+ , K[a]). K[a

(64)

As a result of the consideration above we obtain the following theorem. Theorem 3.1. Any smooth nonlinear dynamical system (55) in Hilbert space M := L2 (Rm ; Rs ) is representable by means of the Fock space embedding isomorphism ξ : M −→ Φ in the completely linear form (62).

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A. K. Prykarpatsky, J. Golenia, N. N. Bogolubov & U. Taneri

We now make some comments concerning the solution to the linear equation (62) under the Cauchy condition (63). Since any vector |ui ∈ Φ has the series representation R L (n) (1) (1) 1 f (x , ..., xn1 ; |ui = P (n1 !...ns !)1/2 (Rm )n n1 ...ns 1 s n= j=1 nj ∈Z+ (65) Q  nj (s) Q (s) (2) (2) (j) + (j) x1 , ..., xn2 ; ...; x1 , ..., xns ) sj=1 dx a (x ) |Ωi, j k=1 k k Ps where for any n = j=1 nj ∈ Z+ , ∈ fn(n) 1 ...ns

and

s O j=1

L2,s ((Rm )nj ; C) ≃ L2,s (Rmn1 × ... × Rmns ; C),

kuk2Φ =

n=

X

Ps

j=1 nj ∈Z+

k2 = exp(kuk2 ). kfn(n) 1 n2 ...ns 2

(66)

(67)

By substituting (65) into equation (62), reduces (62) to an infinite recurrent set of linear evolution equations in partial derivatives on coefficient functions (66). The latter can often be solved [25] step by step analytically in exact form, thereby, making it possible to obtain, owing to representation (54), the exact solution u ∈ M to the Cauchy problem (56) for our nonlinear dynamical system in partial derivatives (55). Remark 3.1. Concerning some applications of nonlinear dynamical systems like (53) in mathematical physics problems, it is very important to construct their so-called conservation laws or smooth invariant functionals γ : M −→ R on M . Making use of the quantum mathematics technique described above one can suggest an effective algorithm for constructing these conservation laws in exact form. Indeed, consider a vector |γi ∈ Φ, satisfying the linear equation

∂ ˆ ∗ [a+ , a]|γi = 0. |γi + K ∂t Then the following result [25] holds.

(68)

Proposition 3.1. The functional γ := hu|γi

(69)

is a conservation law for dynamical system (53), that is, dγ/dt|K = 0 along any orbit of the evolution mapping (57).

(70)

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4. Conclusion Within the scope of this work we have described the main mathematical preliminaries and properties of the quantum mathematics techniques suitable for analytical studying of the important linearization problem for a wide class of nonlinear dynamical systems in partial derivatives in Hilbert spaces. This problem was analyzed in much detail using the Gelfand-Vilenkin representation theory [14] of infinite dimensional groups and the GoldinMenikoff-Sharp theory [15–17] of generating Bogolubov type functionals, classifying these representations. The related problem of constructing Fock type space representations and retrieving their creation-annihilation generating structure still needs a deeper investigation within the approach devised. Here we mention only that some aspects of this problem within the so-called Poissonian white noise analysis were studied in a series of works [1,2,24,27], based on some generalizations of the Delsarte type characters technique. It is also necessary to mention the related results obtained in [23,25,26], devoted to the application of the Fock space embedding method to finding conservation laws and the so called recursion operators for the well known Korteweg-de Vries type nonlinear dynamical systems. We plan to devote our next investigations to concerning some important applications of the methods devised in the work to concrete dynamical systems. Acknowledgements Two of the authors (N.B. and A.P.) are cordially thankful to the Abdus Salam International Center for Theoretical Physics in Trieste, Italy, for the hospitality during their 2007 research scholarships. A.P. is also thankful to Profs. Y. Kondratyev and Y. Kozicki for interesting discussions of the results during the Third International Conference on Infinite Dimensional Systems, held June 23–28, 2007 in Kazimierz Dolny, Poland, and to Profs. O. C ¸ elebi, K. Zheltukhin, G. Barsegian, O. Zhdanov and V. Golubeva during the 2007 International ISAAC Congress, held August 13–18, 2007 in Ankara, Turkey. The last but not least thanks belong to the referee of the article for many insightful and qualified comments which helped in an essentialway in improving the exposition. References 1. S. Albeverio, Y. G. Kondratiev and L. Streit, How to generalize white noise analysis to non-Gaussian measures, Preprint Bi Bo S, Bielefeld, 1992. 2. Y. M. Berezansky, A generalization of white noise analysis by means of theory of hypergroups, Reports on Math. Phys., 38, N.3 (1996), pp. 289–300.

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3. Y. M. Berezansky and Y. G. Kondratiev, Spectral methods in infinite dimensional analysis, v.1 and 2, Kluwer, 1995. 4. N. N. Bogolubov and N. N. Bogolubov, Jr., Introduction into quantum statistical mechanics. World Scientific, NJ, 1986. 5. N. N. Bogoliubov, Jr., A. K. Prykarpatsky, Quantum method of generating Bogolubov functionals in statistical physics: Current Lie algebras, their representations and functional equations, Physics of Elementary Particles and Atomic Nucleus, v.17, N.4 (1986), pp. 791–827 (in Russian). 6. F. A. Berezin, The second quantization method, Nauka Publisher, Moscow, 1986 (in Russian). 7. N. N. Bogoliubov, Collected works, v.2, Naukova Dumka, Kiev, 1960 (in Russian). 8. P. A. M. Dirac, The principles of quantum mechanics, Oxford University Press, 1932. 9. J. Dixmier, C ∗ -algebras, Amsterdam, North-Holland, 1982. 10. L. D. Faddeev, E. K. Sklyanin, Quantum mechanical approach to completely integrable field theories, Proceed. of the USSR Academy of Sciences (DAN), 243 (1978), pp. 1430–1433 (in Russian). 11. R. Feynman, Quantum mechanical computers, Found. Physics, 16 (1986), pp. 507–531. 12. R. Feynman, Simulating physics with computers. Intern. Journal of Theor. Physics, 21 (1982), pp. 467–488. 13. V. A. Fock, Konfigurationsraum und zweite Quantelung, Zeischrift Phys., Bd. 75 (1932), pp. 622–647. 14. I. Gelfand and N. Vilenkin, Generalized functions, 4, Academic Press, New York, 1964. 15. G. A. Goldin, Nonrelativistic current algebras as unitary representations of groups, Journal of Mathem. Physics, 12(3), 1971, pp. 462–487. 16. G. A. Goldin, J. Grodnik, R. T. Powers and D. Sharp, Nonrelativistic current algebra in the N/V limit, J. Math. Phys., 15, (1974), pp. 88–100. 17. G. A. Goldin, R. Menikoff and F. H. Sharp, Diffeomorphism groups, gauge groups, and quantum theory. Phys. Rev. Lett. 51 (1983), pp. 2246–2249. 18. L. K. Grover, Quantum mechanics helps in searching for a needle in a haystack, Phys. Rev. Lett., 79 (1997), pp. 325–328. 19. A. Jaffe and F. Quinn, Theoretical mathematics: Toward a cultural synthesis of mathematics and theoretical physics, Bull. Amer. Math. Soc., 29 (1993), pp. 1–13 Zeischrift Phys., Bd. 75 (1932), pp. 622–647. 20. D. Deutsch, Quantum theory, the Church-Turing principle and the universal quantum computer, Proc. Roy. Soc. London, A400 (1985), pp. 97–117. 21. V. G. Drinfeld, Quantum groups, Proceed. of the Int. Congress of Mathematicians, 1986, pp. 798–820. 22. S. K. Donaldson, An application of gauge theory to four dimensional topology, J. Diff. Geom., 17 (1982), pp. 279–315. 23. K. Kowalski and W.-H. Steeb, Symmetries and first integrals for nonlinear dynamical systems: Hilbert space approach, I and II, Progress of Theoretical Physics, 85, N.4 (1991), pp. 713–722 and 85, N.4 (1991), pp. 975–983.

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24. Y. G. Kondratiev, L. Streit, W. Westerkamp and J.-A. Yan, Generalized functions in infinite dimensional analysis, II, AS preprint, 1995. 25. K. Kowalski, Methods of Hilbert spaces in the theory of nonlinear dynamical systems, World Scientific, 1994. 26. K. Kowalski and W.-H. Steeb, Nonlinear dynamical systems and Carleman linearization, World Scientific, 1991. 27. E. W. Lytvynov, A. L. Rebenko and G. V. Shchepaniuk, Wick calculus on spaces of generalized functions compound Poisson white noise, Reports on Math. Phys., 39, N.2 (1997), pp. 219–247. 28. Y. I. Manin, Computable and uncomputable, Moscow, Sov. Radio, 1980 (in Russian). 29. Y. I. Manin, Classical computation, quantum computation and P. Shor’s factoraizing algorithm, Seminaire N. Bourbaki, 1998–1999, exp. N.862, p. 375–404. 30. J. von Neumann, Mathematische Grundlagen der Quanten Mechanik, J. Springer, Berlin, 1932. 31. A. K. Prykarpatsky, U. Taneri and N. N. Bogolubov, Jr., Quantum field theory and application to quantum nonlinear optics, World Scientific, NY, 2002. 32. P. W. Shor, Polynomial time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM Journ. Comput., 26, N.5 (1997), pp. 1484–1509. 33. H. Weyl, The theory of groups and quantum mechanics, Dover, New York, 1931. 34. E. Witten, Nonabelian bosonization in two dimensions, Commun. Mathem. Physics, 92 (1984), pp. 455–472.

780

A. K. Prykarpatsky, J. Golenia, N. N. Bogolubov & U. Taneri

Session 17

General Session

SESSION EDITOR A. O. C ¸ elebi

˙ Yeditepe University, Istanbul, Turkey

783

COHEN p-NUCLEAR MULTILINEAR MAPPINGS DAHMANE ACHOUR Laboratoire de Math´ ematiques Pures et Appliqu´ ees, Universit´ e de M’sila, 28000 M’sila, Algeria, E-mail: [email protected] We define the concept of Cohen p-nuclear multilinear operators. Some properties and composition theorems are shown. Keywords: Absolutely p-summing m-linear operator, Cohen strongly psumming, Cohen p-nuclear multilinear operator.

1. Introduction and Notation In a series of papers [1,2,5,8,9] the authors have developed the theory of summing multilinear operators, generalizing the nice behavior of linear psumming operators to the multilinear setting. Among other things, we introduced the concept of multilinear operators of type p-nuclear. In the linear case the concept of Cohen p-nuclear operators (1 ≤ p ≤ ∞) was introduced by Cohen in [4] and generalized to Cohen (p, q)-nuclear (1 ≤ q ≤ ∞) by Apiola in [3]. A linear operator T between two Banach spaces X, Y is Cohen p-nuclear for (1 < p < ∞) if there is a positive constant C such that for all n ∈ N; x1 , ..., xn ∈ X and y1∗ , ..., yn∗ ∈ Y ∗ we have



(hT (xi ) , yi∗ i)1≤i≤n n ≤ C sup k(x∗ (xi ))kln sup k(yi∗ (y))kln∗ . l1

x∗ ∈BX ∗

p

y∈BY

p

The smallest constant C, which is denoted by np (T ), such that the above inequality holds, is called the Cohen p-nuclear norm on the space Np (X, Y ) of all Cohen p-nuclear operators from X into Y which is a Banach space. For p = 1 and p = ∞ we have N1 (X, Y ) = π1 (X, Y ) (the Banach space of all 1-summing operators), and N∞ (X, Y ) = D∞ (X, Y ) (the Banach space of all strongly ∞-summing operators). In this paper, we introduce a multilinear generalization of the notion of Cohen p-nuclear linear operators. Some properties and composition theorems are shown.

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D. Achour

This paper is organized as follows: In Section 1, we give some basic definitions and properties. We introduce in Section 2, a multilinear version of Cohen p-nuclear operators studied by Cohen for which the resulting vector space is a Banach space. We prove, if T verifies a domination theorem then T is p-nuclear we asked if the statement of this theorem characterizes p-nuclear operators. In Section 3, some composition theorems are obtained. Now, we fix the notation used in this paper. Let m ∈ N and X1 , ..., Xm , Y be Banach spaces over the real numbers, we denote by L (X1 , ..., Xm ; Y ) the Banach space of all continuous m-linear mappings from X1 × ... × Xm into Y , under the norm kT k =

sup kT (x1 , ..., xm )k

xk ∈BXk

where BXk denotes the closed unit ball of Xk centred at 0. The vector space of all bounded linear operators from X into Y will be noted by B(X, Y ). Let now X be a Banach space and 1 ≤ p ≤ ∞. We denote by lpn (X) the space of all sequences (xi )1≤i≤n in X with the norm



(xi )1≤i≤n n

lp (X)

and by lpn

ω

=(

n P

i=1

p

1

kxi k ) p

(X) the space of all sequences (xi )1≤i≤n in X with the norm k(xn )klnp

ω (X)

=

sup (

n P

kξkX ∗ =1 i=1

p

1

|hxi , ξi| ) p

where X ∗ denotes the dual (topological) of X. We know (see [6]) that lp (X) = lpω (X) for some 1 ≤ p < ∞ iff dim (X) is finite. If p = ∞, we have ω l∞ (X) = l∞ (X). We begin by presenting the several classes of multilinear mappings related to the concept of absolutely summing operators. • Consider m in N. An m-linear operator T ∈ L (X1 , ..., Xm ; Y ) is Cohen strongly p- summing (1 < p ≤ ∞), if and only if there exists a constant C > 0 and a Radon probability measure µ on BY ∗∗ such that for all (x1 , ..., xm ) ∈ X 1 × ... × X m , we have Z m 1 Y

j


p∗

x ( T x1 , ..., xm , y ∗ ≤ C |y ∗ (y ∗∗ )| dµ) p∗ (1) j=1

BY ∗∗

Again the class of all Cohen strongly p-summing m-linear operators from X1 × ... × Xm into Y , which is denoted by Dpm (X1 , ..., Xm ; Y ) is a Banach space with the norm dm p (T ) which is the smallest constant C such

Cohen p-Nuclear Multilinear Mappings

785

that the inequality (1) holds. For p = 1, we have D1m (X1 , ..., Xm , Y ) = L(X1 , ..., Xm ; Y ). • One will say that an m-linear operator T ∈ L (X1 , ..., Xm ; Y ) is absolutely p-summing (1 ≤ p < ∞) if there is a constant C > 0 such that for any n ∈ N and (xji )1≤i≤n ⊂ Xj (1 ≤ j ≤ m), we have m n

Y X


p 1

j

T x1i , ..., xm

)p ≤ C . (2) (

(xi )1≤i≤n n ω i j=1

i=1

lp

(Xj )

The space of all absolutely p-summing m-linear mappings from X1 × ... × Xm into Y will be denoted by Lm p (X1 , ..., Xm ; Y ) , and the infimum of the C for which the inequality (2) always holds defines a norm, kT kp for Lm p (X1 , ..., Xm ; Y ) . The Cohen strongly p-summing mappings was introduced by the work of Achour-Mezrag [1]. The multilinear mappings of absolutely summing type are motivated by Pietsch in [9] and more recently in [2,7,10]. 2. General Properties We introduce the following generalization of the concept p-nuclear operators defined in [4], also we give some properties concerning this class of operators. Definition 2.1. An m-linear operator T : X1 × ... × Xm −→ Y is Cohen p-nuclear (1 < p < ∞), if and only if there is a constant C > 0 such that for any xj1 , ..., xjn ∈ Xj , (j = 1, ..., m), and any y1∗ , ..., yn∗ ∈ Y ∗ , we have n m

X

Y


j T x1i , ..., xm sup k(yi∗ (y))kln∗ (3) , yi∗ ≤ C

(xi ) n ω i p lp (Xj ) y∈BY i=1

j=1

Again the class of all Cohen p-nuclear m-linear operators from X1 ×...×Xm into Y , which is denoted by Npm (X1 , ..., Xm ; Y ) is a Banach space with the norm nm p (T ) which is the smallest constant C such that the inequality (3) holds. It is clear that, for every T ∈ Npm (X1 , ..., Xm ; Y ), T is continuous and kT k ≤ nm p (T ) .

Example 2.1. Let K be a compact Hausdorff space, let µ be a positive regular Borel measure on K and let 1 ≤ p < ∞. Each g ∈ Lp (µ) defines an m m-linear multiplication operator Tg ∈ L( C(K); Lp (µ)), Tg (f 1 , ..., f m ) = 1 m g · f · ... · f . This map is Cohen p-nuclear and nm p (Tg ) = kgkLp (µ) . Proposition 2.1. a) For p = 1, we have N1m (X1 , ..., Xm ; Y ) = π1m (X1 , ..., Xm ; Y ) . m m b) For p = ∞, we have N∞ (X1 , ..., Xm ; Y ) = D∞ (X1 , ..., Xm ; Y ) .

786

D. Achour

Proof. (a) Let T ∈ N1m (X1 , ..., Xm ; Y ) , then

m n

P
∗  Q

j 1 m sup k(yi∗ (y))kln T xi , ..., xi , yi ≤ nm

(xi ) n ω 1 (T ) ∞ l1 (Xj ) y∈BY j=1 i=1

m Q

j kyi∗ k ≤ nm

(xi ) n ω 1 (T ) l1

j=1

(Xj )

On the other hand, we have

( n ) n X

X


∗ 
1 m 1 m ∗

T xi , ..., xi = sup T xi , ..., xi , yi , sup kyi k ≤ 1 . i=1

i=1

This implies

m n

Y X




j m

T x1i , ..., xm

≤ n (T )

(xi ) n ω i 1 j=1

i=1

l1

(Xj )

.

Then T is absolutely 1-summing and kT k1 ≤ nm 1 (T ). Conversely, let T be a absolutely 1-summing m-linear operator, we have n n P



∗  1 m ≤ sup ky ∗ k P T x1 , ..., xm , y T x , ..., x i i i i i i i i=1 i=1

m Q

j sup kyi∗ k ≤ kT k1

(xi ) n ω l1 (Xj ) i j=1

m Q

j ≤ kT k1 sup k(yi∗ (y))kln .

(xi ) n ω l1

j=1

(Xj ) y∈BY

∞

This T is 1-nuclear m-linear operator and nm 1 (T ) ≤ kT k1 . (b) It is obvious, and this completes our proof.

Proposition 2.2. Consider m ∈ N and T ∈ L(X1 , ..., Xm ; Y ). The operator T is Cohen p-nuclear if and only if, for all n ∈ N and all v ∈ B(l pn , Y ∗ ), we have m n

X Y


j 1 m kvk . T xi , ..., xi , v (ei ) ≤ C

(xi ) n ω lp (Xj ) i=1

j=1

Proof. Let v : lp∗ −→ X be a linear operator such that v (ei ) = xi (namely, ∞ P ej ⊗ xj , ej denotes the unit vector basis of lp ). We can use kvk = v= 1

k(xn )klω (X) , as we wanted to prove. p

In the next theorem we shall present an integral condition which is sufficient to guarantee that an multilinear operator T belong to Npm (X1 , ..., Xm ; Y ) .

Cohen p-Nuclear Multilinear Mappings

787

Theorem 2.1. Let T ∈ L (X1 , ..., Xm ; Y ) and 1 < p < ∞. Suppose there exists a constant C ≥ 0 and positive Radon probability measures µj
∈ C(BXj∗ )∗ (1 ≤ j ≤ m) and λ ∈ C(BY ∗∗ )∗ such that for all x1 , ..., xm ∈ X1 × .... × Xm and y ∗ ∈ Y ∗ , we have m Y


 T x1 , ..., xm , y ∗ ≤ C kxj kLp (B

X ∗ ,µj ) j

j=1

ky ∗ kLp∗ (BY ∗∗ , λ) .

(4)

Then T is Cohen p-nuclear and nm p (T ) ≤ C.


∈ X1 × .... × Xm and yi∗ ∈ Y ∗ by (4), then Proof. Let x1i , ..., xm i m Y


T x1 , ..., xm , y ∗ ≤ C kxj kLp (B i i i

X ∗ ,µj ) j

j=1

ky ∗ kLp∗ (BY ∗∗ , λ) ,

for all 1 ≤ i ≤ n, and n

n

P 

∗  P 1 m T x1 , ..., xm , y ∗ , y T x , ..., x i i i i ≤ i i i=1 i=1 m n Q P

j kyi∗ kLp∗ (BY ∗∗ , λ) ). ( ≤C

xi Lp (BX ∗ ,µj )

i=1 j=1

j

We can use H¨ older’s inequality in order to write n P


∗  1 m , y T x , ..., x i i i i=1 n Q m n P 1 1 P ∗ p ≤ C( kxj kLp (B ∗ ,µj ) ) p ( ky ∗ kLp∗ (BY ∗∗ , λ) p ) p∗ X j i=1 j=1Z i=1 Z n m P n Q 1 1 P p∗ j ≤C ( |hxi , xj∗ i|p dµj (xj∗ )) p ( |yi∗ (y ∗∗ )| dλ (y ∗∗ )) p∗ ≤C

j=1 i=1 m Q

BX ∗ j n P

sup (

j=1x∗ j ∈BX ∗ i=1 j

i=1

j∗

|x

1 (xji )|p ) p

sup (

n P

y∈BY i=1

BY ∗∗ ∗

|yi∗ (y)|p

1

) p∗

Therefore T is p-nuclear and nm p (T ) ≤ C, as we wanted to prove. Question 2.1. Does T ∈ L (X1 , ...., Xm ; Y ) Cohen p-nuclear imply there exist Radon probability measures µj ∈ C(BXj∗ )∗ (1 ≤ j ≤ m) and λ ∈
C(BY ∗∗ )∗ such that for all x1 , ..., xm ∈ X1 × .... × Xm and y ∗ ∈ Y ∗ , we have (4)? 3. Composition Theorems In this section, we give some composition theorems. At first, it is easy to verify the following result.

788

D. Achour

Proposition 3.1. Let T ∈ L(X1 , ..., Xm ; Y ), w ∈ L (Y, Z) and uj ∈ L (Ej , Xj ) (1 ≤ j ≤ m). (i) If T is Cohen p-nuclear, then w ◦ T is Cohen p-nuclear and m nm p (w ◦ T ) ≤ kwk np (T ). (ii) If T is Cohen p-nuclear, then T ◦ (u1 , ..., um ) is Cohen p-nuclear m Q m and nm kuj k . p (T ◦ (u1 , ..., um )) ≤ np (T ) j=1

Theorem 3.1. Let m ∈ N and X1 , ..., Xm , Y , Z be Banach spaces. Let u : Y → Z be a strongly p-summing operator and let S : Y1 × ... × Ym −→ Y be absolutely p-summing. Then, the operator T = uS : X1 ×....×Xm −→ Z, ∗ is p-nuclear and nm p (T ) ≤ kSkp πp∗ (u ). Proof. Let (xji ) ⊂ Xj and (zi∗ ) ⊂ Z ∗ . We have n

n


∗ P
 P S x1 , ...xm , u∗ (z ∗ ) , zi ≤ T x1i , ...xm i i i i

i=1

i=1 n P

n


1 P ∗

S x1 , ..., xm p ) p ( ku∗ (z ∗ )kp ) p1∗ i i i i=1 i=1

n m P Q

j p∗ 1 ( ku∗ (zi∗ )k ) p∗ ≤ kSkp

(xi ) n ω

≤(

lp

j=1

(Xj ) i=1

since u∗ is p∗ -summing. Then

n

P
∗  1 m T xi , ...xi , zi i=1

m Q

j ≤ kSkp πp∗ (u∗ )

(xi ) n ω j=1

lp

sup (

n P

(Xj ) z∈BZ ∗∗ i=1

p∗

1

|hzi∗ , z ∗∗ i| ) p∗ ,

∗ and thus T is p-nuclear and nm p (T ) ≤ kSkp πp∗ (u ).

Theorem 3.2. Let X1 , ..., Xm , Y1 , ..., Ym ; Z be Banach spaces, with dim (Z ∗ ) < ∞. Let, for every 1 ≤ j ≤ m, vj : Xj → Yj be a p-summing operator and let R : Y1 × .... × Ym −→ Z be absolutely p-summing. Then the operator T = R (v1 , ..., vm ) : X1 × ... × Xm −→ Z is p-nuclear and nm p (T ) ≤ kRkp

m Y

j=1

πp (vj ) .

Cohen p-Nuclear Multilinear Mappings

789

Proof. Let x1 , ..., xm ∈ X1 × ... × Xm and z ∗ ∈ Z ∗ . We can use H¨ older’s inequality and (2) to obtain n P



∗  m 1 R v1 xi , ..., vm (xi ) , zi i=1 n n

p 1 P ∗ P

) p ( kz ∗ kp ) p1∗ ≤ ( R v1 x1i , ..., vm (xm i ) i i=1 i=1

n m P Q

p∗ 1 j ≤ kRkp

(vj (xi ))1≤i≤n n,ω ( kzi∗ k ) p∗ . lp

j=1

i=1

n

p 1

P On the other hand, (vj (xji ))1≤i≤n ln,ω ≤ ( vj (xji ) ) p . Because vj is p

i=1

p-summing and dim (Z ∗ ) < ∞, we have n



P R v1 x1 , ..., vm (xm ) , z ∗ i i i i=1

m m Q Q

j k(zi∗ )kln∗ω (Z ∗ ) , πp (vj ) ≤ kRkp

(xi ) n ω j=1

j=1

lp

(Xj )

and hence T is p-nuclear, and nm p (T ) ≤ kRkp

m Q

p

πp (vj ).

j=1

References 1. D. Achour and L. Mezrag, On the Cohen strongly p-summing multilinear operators, J. Math. Anal. Appl. 327 (2007), 550–563. 2. R. Alencar and M. C. Matos, Some classes of multilinear mappings between Banach spaces, Pub. Dep. An. Mat. Univ. Complut. Madrid 12 (1989). 3. H. Apiola, Duality between spaces of p-summable sequences, (p, q)-summing operators and characterizations of nuclearity, Math. Ann. 219 (1976), 53–64. 4. J. S. Cohen, Absolutely p-summing, p-nuclear operators and their conjugates, Math. Ann. 201 (1973), 177–200. 5. V. Dimant, Strongly p-summing multilinear operators, J. Math. Anal. Appl. 278 (2003), 182–193. 6. J. Diestel, H. Jarchow, A. Tonge, Absolutely summing operators, Cambridge University Press, 1995. 7. S. Geiss, Ideale multilinearer Abbildungen, Diplomarbeit, 1984. 8. D. P´erez-Garc`ıa, I. Villanueva, Multiple summing operators on Banach spaces, J. Math. Anal. Appl. 285 (2003), 86–96. 9. A. Pietsch, Ideals of multilinear functionals: Designs of a theory, Proceedings of the Second International Conference on Operator Algebras, Ideals, and their Applications in Theoretical Physics, Leipzig, Teubner-Texte, 1983, 185– 199. 10. B. Schneider, On absolutely p-summing and related multilinear mappings, Wissenchaftliche Zeitschrift der Brandenburger Landeshochschule 35 (1991), 105–117.

790

NUMERICAL MODELING OF A SYSTEM OF MUTUAL REACTION-DIFFUSION TYPE MERSAID ARIPOV∗ National University of Uzbekistan, Uzbekistan E-mail: [email protected] ABDUGAPPAR KHYDAROV National University of Uzbekistan, Uzbekistan The methods of numerical modeling of nonlinear processes of mutual diffusionreaction systems are offered. For modeling of diffusion-reaction systems, a degenerate-type parabolic system equations are used. The properties of weak solutions of initial-boundary value problems for the nonlinear reaction-diffusion systems in the presence of convective transfer, the volumetric absorption or source using a self-similar approach are investigated.

1. Introduction We consider n components of a gas reacting with each other in the course of R independent reactions at constant temperature. The system of equations of local balance of mass in case of n components ui (t, x) (i = 1, n) in nonlinear media at the presence of convective transfer may be written in the form [1-2]
∂ui s3−i i , ∇ui − div(vi (t)ui ) + εupi i uq3−i = ∇ u3−i ∂t
N t > 0, x ∈ R ; i = 1, 2 .

(1)

The different properties of solutions of system (1) such as blow-up, localization of solution, asymptotical behavior of self similar solution are studied. The numerical modeling aspect of a solution of system (1) is considered in [5]. They notice that some of their results required strong justifications. Below, the method of investigation of initial value problem for (1) based ∗ Corresponding

author.

Numerical Modeling of a System of Mutual Reaction-Diffusion Type

791

on the method of nonlinear decomposition [3-4] is offered. This method is valid for n > 1. For simplicity we consider the case n = 2. 2. Construction of the self-similar system The system of self-similar equations  
1 dfi qi N −1 s3−i dfi 1−N d − fi = 0 ξ f3−i + ψi fipi f3−i ϕi ξ +ξ 2 dξ dξ dξ

corresponding to equation (1) is constructed as follows: ui (t, x) = u ¯i (t)ωi (τ, x),

ωi (τ, x) = fi (ξ),

|ξ| =

N  1/2 Rt P xi − 0 vi (t)dt i=1

τ 1/2 (t)

,

(2)

where the functions u¯i (t) = di (T ± t)−αi are solutions of the system d¯ ui i ¯q3−i = ε¯ upi i u dt and τ = ds11 where

(T ± t)1−α1 s1 , α1 s1 − 1

h i p1 1−p i i , α1 s2 − 1 > 0, α1 s2 = α2 s1 , αi 3−i di = αq3−i −s

ϕi = ds11 d3−i3−i , ψi =

αi ϕi , αi si − 1

αi = (1 − p3−i + qi )p−1 , p = q1 q2 − (1 − p1 )(1 − p2 ),

i = 1, 2.

After substituting 1

fi (ξ) = f¯i (ξ)yi (η), η = − ln(a − ξ 2 ), f¯i (ξ) = Ai (a − ξ 2 ) si in (2), we have −s

−1 ′ y3−i ][yi′ + ai1 yi ] [yi′ + ai1 yi ]′ + [ai2 + ai6 y3−i3−i + ai3 a3−i,1 + ai3 y3−i −s

q −s3−i

i + ai4 yi y3−i3−i + ai5 yipi y3−i

= 0,

(3)

792

M. Aripov & A. Khydarov

where ai1 = −

1 , si

ai2 = 1 + ai1 + ai3 = 21 s3−i ,

N , 2aeη − 2

−s

ai4 = − 41 ψi A3−i3−i (aeη − 1)−1 , q −s3−i

i ai5 = 14 ψi Aipi −1 A3−i

−s

ai6 = 14 ϕi A3−i3−i ,

e−li η , a − e−η

qi pi − 1 + , i = 1, 2, s3−i si 2 − si = , 2s2i

li = 1 + bi1

−s

bi2 = − 4s1i ϕi A3−i3−i , bi3 =

pi −1 qi −s3−i 1 . A3−i 4a ψi Ai

3. Main results Theorem 3.1. For the existence of a solution to system (3) in the form yi (η) = yi0 + o(1), η → +∞, i = 1, 2,

0 < yi0 < +∞,

(4)

it is necessary that the numbers yi0 (i = 1, 2) be a solution of the following corresponding nonlinear algebraic system: −s

q −s3−i

i 1. li = 0, bi1 + bi2 z3−i3−i + bi3 zipi −1 z3−i

= 0,

2. l1 = 0, l2 > 0, b11 + b12 z2−s2 + b13 z1p1 −1 z2q1 −s2 = 0, b21 + b22 z1−s1 = 0, 3. l1 > 0, l2 = 0, b11 + b12 z2−s2 = 0, b21 + b22 z1−s1 + b23 z1q2 −s1 z2p2 −1 = 0, −s

4. li > 0, bi1 + bi2 z3−i3−i = 0. 4. Conclusion Primary investigations of the qualitative properties of system (1) allowed us to carry out a numerical experience depending on different values of numerical parameters. To reach this goal, the asymptotical solutions were used as initial approximation. For the linearization of system (1), Newton,

Numerical Modeling of a System of Mutual Reaction-Diffusion Type

793

Piker and a special method were used. The results of numerical experiments showed the effectivity of suggested approach. Asymptotes of different type solutions to the system (2) allowed also modeling of mutual reactiondiffusion processes in visualization form with animations which had the possibility of observing the evolution of the studied problem. Blow-up analysis of solutions for different values of parameters was performed too. References 1. M. Holodnyok, A. Klich, M. Kubichek, M. Marec, Methods of Analysis of Dynamical Models; Moscow, Mir, 1991, 365pp. 2. A. A. Samarskii, V. A. Galaktionov, S. P. Kurduomov, A. P. Mikhajlov, Blowup in Parabolic Equations; Berlin, New York, Walter de Gruyter, 1995, 535pp. 3. M. Aripov, Approximate Self Similar Approach to Solving of Quasilinear Parabolic Equations: Experimentation, Modeling and Computation in Flow, Turbulence and Combustion; John Wiley and Sons, 1997, vol. 2, 19–25. 4. M. Aripov, Asymptotics of the Solution of the Non-Newton Polytropical Filtration Equation; ZAMM, vol. 80 (2000), suppl. 3, 767–768. 5. S. P. Kurduomov, E. S. Kurkina, O. V. Telkovskaya. Blow-up in TwoComponent Media; Mathematical Modeling, 1989, vol. 5.

794

l2 (b) ON COMPLEMENTED SUBSPACES OF E0l2 (a) × E∞

E. KARAPINAR ˙ Izmir University of Economics, Departments of Mathematics ˙ Izmir, 35330, Turkey E-mail: [email protected] It is known that a complemented subspace of a finite-type nuclear K¨ othe space E0l2 (a) has a basis due to Mityagin [2]. By using the results of Prada [4] and Zakharyuta [6], it is obtained that any stable complemented subspace l2 (b) is basic, where ai or bi tends to infinity. of E0l2 (a) × E∞ Keywords: Complemented Subspaces, K¨ othe Spaces.

1. Introduction Let A = (ain )i∈I,n∈N be a matrix of real numbers such that 0 ≤ ain ≤ ain+1 . K¨ othe space K lp (A) defined by the matrix A is a Fr´echet space of all sequences ξ = (ξi ) such that !1/p X p p |ξ|n := (ain ) |ξi | < ∞, ∀n ∈ N i∈I

with the topology, generated by the system of semi-norms {|ξ|n = kξklp : n ∈ N}. In the case X = K l2 (A), Y = K l2 (B), where A = (aip ), B = ˆ l2 Y are naturally isomorphic, (bjp ), i, j ∈ N, the spaces X × Y and X ⊗ respectively, to the spaces K(C) and K(D), where ck,n equals to ai,n if k = 2i − 1 and bi,n if k = 2i, and also D = (dνp ), dνp = aip bjp , ν = (i, j) ∈ N2 = N × N. Let ω+ be the set of all sequences with positive terms. Here we consider two important particular cases: For a, b ∈ ω+ , 1 l2 (b) := K l2 (exp (qbi )), E0l2 (a) := K l2 (exp (− ai )) and E∞ p

whose elements are called finite- and infinite-type power series, respectively. For a ∈ ω+ , a subspace Eνl2 (˜ a) of Eνl2 (a) is called basic if a ˜ = (aik ), where

l2 On Complemented Subspaces of E0l2 (a) × E∞ (b)

795

ν = 0, ∞. The sequence a = (an ) will be called stable if a satisfies the condition < ∞. This is equivalent to saying that Eνl2 (a) isomorphic to limn sup aa2n n 2 l2 (Eν (a)) , where ν = 0, ∞. Let X and Y be locally convex spaces. A linear operator T : X → Y is called a near isomorphism if T (X) is closed in Y and an open map from X onto T (X), dim(T ) < ∞, co dim(T ) < ∞. The locally convex spaces X and Y are said to be nearly isomorphic, denoted by X ≈ Y if there is a near isomorphism T from X onto Y . Note that if X and Y are isomorphic, we shall write X ≃ Y . Definition 1.1. Let X and Y be locally convex spaces. A linear operator T : X −→ Y will be called compact if there exists a neighborhood U in X such that its image T (U ) is precompact in Y . We shall say that an ordered pair of locally convex spaces (X, Y ) satisfies condition K, and write (X, Y ) ∈ K, if every linear continuous operator T : X −→ Y is compact. Proposition 1.1. ([6]) Let X, Y be locally convex spaces and (X, Y ) ∈ K. Then (X0 , Y0 ) ∈ K for every subspace X0 that is topologically complemented in X and any subspace Y0 of Y . Proof. Let T : X0 −→ Y0 be an arbitrary linear continuous operator. By the assumption, there exists a subspace X1 in X such that  X1 ⊕ X2 . Let T0 x if x ∈ X0 , T : X −→ Y be the linear continuous operator T (x) = 0 if x ∈ X1 . Since (X, Y ) ∈ K, this operator is compact and therefore T0 is compact too. Hence (X0 , Y0 ) ∈ K. Proposition 1.2. ([4]) A complemented subspace of X1 ×X2 , where X1 , X2 are Fr´echet spaces such that (X1 , X2 ) ∈ K, is isomorphic to a subspace of X1 × X2 of the form L1 × L2 , where L1 (respectively L2 ) is a complemented subspace of X1 (respectively X2 ). Nuclearity of the classical K¨ othe spaces K lp (A) is equivalent to the Grothendieck-Pietsch condition [3]: ∀p ∃q > p and (ξn ) ∈ l1 with apn ≤ ξn aqn , n ∈ N . Mityagin [2] proved that a complemented subspace of finite type of a nuclear K¨ othe space E0l2 (a) has a basis, and asked about the existence of basis for a complemented subspace of infinite type of a nuclear K¨ othe

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l2 space E∞ (b). In this short paper, we observe that any stable complemented l2 subspace of E0l2 (a) × E∞ (b) is basic, where ai or bi tends to infinity.

2. Main l2 Theorem 2.1. Let L be a complemented subspace of E0l2 (a)×E∞ (b), where 2 a, b ∈ ω+ . If L ≃ L and ai → ∞ (or bi → ∞), then L is isomorphic to a l2 basic subspace of E0l2 (a)×E∞ (b) of the form L1 ×L2 , where L1 , respectively l2 L2 , is a complemented subspace of E0l2 (a), respectively E∞ (b). l2 Proof. Let X1 = E0l2 (˜ (˜ a). By Proposition 1.2, if L is a a) and X2 = E∞ complemented subspace of X1 × X2 , then there exists L1 , L2 such that Lν is a complemented subspace of Xν , where ν = 1, 2. Due to Mityagin [2], we know that any complemented subspace has a basis for the finite-type power series, i.e, it is of the same type with the a), where a ˜ := (ain ). original space, L1 ≃ E0l2 (˜ Since ai → ∞, we have (X1 , X2 ) ∈ K. Combining with Proposition 1.1, we get (L1 , L22 ) ∈ K, (L21 , L2 ) ∈ K . Let T : L → L2 be an isomorphism such   T T 11 12 that T = [Tij ] where Tij : Lj → L2i . By Douady Lemma [1], T = 0 T22 is a near isomorphism. In view of the result of Zahariuta [6, Prop. 4], we obtain that T11 : L1 → 2 L1 and T22 : L2 → L22 are near isomorphisms, that is, L ≈ L2 if and only if L1 ≈ L21 and L2 ≈ L22 . l2 ˜ (b), By Wagner [5], if L2 ≈ L22 then L2 has a basis. Hence L2 = E∞ where ˜b := (bjk ).

ˆ l2 l 2 = l 2 . The following result is due to the fact that l2 ⊗ l2 ˆ l2 E∞ Corollary 2.1. Let L be a complemented subspace of E0l2 (a)⊗ (b), 2 where a, b ∈ ω+ . If L ≃ L and ai → ∞ (or bi → ∞), then L is isol2 ˆ l2 E∞ ˆ l2 L2 , where morphic to basic subspace of E0l2 (a)⊗ (b) of the form L1 ⊗ l2 L1 (respectively L2 ), is a complemented subspace of E0 (a), (respectively l2 E∞ (b)).

Acknowledgements The author thanks Prof. Dr. V. Zakharyuta for his helpful discussions and endless support.

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References 1. A. Douady, Un espace de Banach dont le group lineaire n’est pas convexe, Indag. Math. 25 (1965), 27–29. 2. B. S. Mityagin, The equivalence of bases in Hilbert scales, Studia Math. 37 (1970), 111–137 (in Russian). 3. A. Pietsch, Nukleare lokalkonvexe R¨ aume, Berlin, 1965. 4. J. Prada, On idempotent operators on Fr´echet spaces, Arch. Math. 43 (1984), 179–182. 5. M. J. Wagner, Stable complemented subspaces of (s) have basis, Seminar Lecture, A-G Funktionalanalysis D¨ usseldolf/Wuppertal, 1985. 6. V. P. Zahariuta, On the isomorphism of Cartesian products of locally convex spaces, Studia Math. 46 (1973), 201–221.

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ASYMPTOTIC FORMULAS FOR EIGENVALUES AND EIGENFUNCTIONS OF A NONSELFADJOINT STURM-LIOUVILLE OPERATOR KHANLAR R. MAMEDOV† and HAMZA MENKEN∗ Mersin University, Faculty of Science and Letters, C ¸ iftlikk¨ oy Campus, 33343, Mersin, Turkey † E-mail: [email protected], ∗ E-mail: [email protected] We consider the nonselfadjoint Sturm-Liouville operator with periodic and antiperiodic boundary conditions. We obtain new accurate asymptotic formulas for the eigenvalues and eigenfunctions of the boundary value problem when the potential q(x) ∈ C (4) [0, 1] is a complex-valued function satisfying the condition q(0) = q(1). Keywords: Nonselfadjoint Sturm-Liouville operator, periodic and antiperiodic boundary conditions, accurate asymptotic formula.

1. Introduction We consider the nonselfadjoint differential operator generated by the differential equation ℓ(y) ≡ y ′′ + q(x)y = λy, 0 < x < 1,

(1)

and the boundary conditions y(0) = (−1)θ y(1), y ′ (0) = (−1)θ y ′ (1),

(2)

where θ = 1, 2, and q(x) is a complex-valued function in the space C (4) [0, 1] satisfying theR condition q(0) = q(1). Without loss of generality, we can 1 assume that 0 q(x)dx = 0. Under this condition, we get a new asymptotic form of the solution of the equation (1) with accuracy of the term O(λ−7 ). Using this form of the solution, accurate asymptotic estimates for eigenvalues and eigenfunctions of the boundary value problems (1), (2) are obtained, and it is shown that all but finitely many eigenvalues are simple. The main goal of this paper is to obtain the asymptotic formulas for

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eigenvalues and eigenfunctions. The asymptotic formulas for the eigenfunctions of the boundary problems (1), (2) have important role in obtaining spectral expansion formulas. In [1,2], using the asymptotic formulas in the case of q(0) 6= q(1), the basis property in Lp (0, 1) (1 < p < ∞ ) were proved. We note that eigenvalues and eigenfunction of a differential operator under differing conditions have been investigated by many authors for various purposes. In [3], the relation between the order of accuracy of the asymptotic form of Sturm-Liouville operator and the smoothness of the coefficient q(x) was investigated, and the asymptotic formulas were obtained when q(x) ∈ W2n (0, 1) was an real-valued function. The asymptotic formulas for eigenvalues and eigenfunctions of a differential operator order n were given in [4]. The accurate asymptotic estimates of the eigenvalues of the classical Sturm-Liouville boundary value problem were constructed in [5], and in the general case, they were obtained in [6]. First, we investigate the asymptotic estimates of the boundary value problem (1), (2), when θ = 1. 2. Asymptotic formulas for eigenvalues of the antiperiodic problem Theorem 2.1. All eigenvalues of the boundary-value problem (1), (2) when θ = 1, beyond a certain point, are simple and form two infinite sequences λk,1 , λk,2 , k = n0 , n0 + 1, · · ·, where n0 is a positive integer and R1 q ′ (1) − q ′ (0) − 0 q 2 (x)dx 1 2 + O( 3 ), (3) λk,1 = − [(2k + 1)π] + [2(2k + 1)π]2 k R1 q ′ (1) − q ′ (0) + 0 q 2 (x)dx 1 2 λk,2 = − [(2k + 1)π] − + O( 3 ). (4) [2(2k + 1)π]2 k Proof. Consider equation (1) or y ′′ + q(x)y + µ2 y = 0,

(5) √ √ iϕ/2 for −π < ϕ ≤ π. It is well-known (see [3,4]) where µ = −λ and re that the eigenvalues of the boundary problem (1), (2) are asymptotically located in pairs and the relation   p 1 µk,j = −λk,j = (2k + 1)π + O (k = n0 , n0 + 1, · · ·, j = 1, 2), k 1/2

holds for all µk,1 , µk,2 ∈ Q = {µ : Re µ ≥ 0, |Im µ| ≤ co }, where co > 0. Let S0 − ico ≡ T , where S0 = µ : 0 ≤ arg µ ≤ π2 and Q ⊂ T . According to

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[3,4], for µ in the region T of the complex plane, the equation (5) has two linearly independent solutions ϕ1 (x, µ), ϕ2 (x, µ) satisfying the relations # " 6 X um (x) 1 + O( 7 ) , j = 1, 2, ϕj (x, µ) = eµωj x (2ωj µ)m µ m=0 " # 6 X um (x) + 2u′m−1 (x) 1 ′ µωj x ϕj (x, µ) = µωj e u0 (x) + + O( 7 ) , j = 1, 2, (2ωj µ)m µ m=1

where ω1 = −ω2 = i, u0 (x) ≡ 1, and Z x ℓ (um−1 (t)) dt, m = 1, 2, 3, 4, 5, 6. um (x) = − 0

It follows that

1 ϕj (0, µ) = 1 + O( 7 ), µ  Z 1 1 1 µωj ′ ′ 1− ϕj (1, µ) = e [q (1) − q (0) + q 2 (t)dt] + [q ′′ (1) 4 (2ωj µ)3 (2ω j µ) 0 3 1 5 [q ′′′ (1)−q ′′′ (0) −q ′′ (0) + q 2 (1)− q 2 (0)−q(0)q(1)] − 2 2 (2ωj µ)5 +7q(1)q ′ (1) − 5q(0)q ′ (0) − q(0)q ′ (1) − q(1)q ′ (0) Z 1 Z 1 Z 1 2 + (q(1) − q(0)) q 2 (t)dt + 2 q 3 (t)dt − q ′ (t)dt] 0

0

0

1 [q (4) (1) − q (4) (0) + 9q(1)q ′′ (1) − 7q(0)q ′′ (0) + (2ωj µ)6 11 9 −q(0)q ′′ (1) − q(1)q ′′ (0) + q ′2 (1) − q ′2 (0) − q ′ (0)q ′ (1) 2 2 15 3 7 3 5 3 + q (1) − q (0) − q(0)q 2 (1) − q(1)q 2 (0) 2 2 2 2 Z 1  2 Z 1 1 1 2 ′ ′ 2 q (t)dt ] + O( 7 ) , + (q (1) − q (0)) q (t)dt + 2 µ 0 0  ′ 2q(0) 2q (0) 1 ϕ′j (0, µ) = µωj 1 − + − [2q ′′ (0) + 2q 2 (0)] (2ωj µ)2 (2ωj µ)3 (2ωj µ)4 ′′′ 1 1 [2q (0) + 8q(0)q ′ (0)] − [2q (4) (0) + (2ωj µ)5 (2ωj µ)6  1 +10q ′2 (0) + 12q(0)q ′′ (0) + 4q 3 (0)] + O( 7 ) , µ

ϕ′j (1, µ)

= µωj e

µωj

 Z 1 q(0) + q(1) 1 ′ ′ 1− + [q (1) + q (0) − q 2 (t)dt] (2ωj µ)2 (2ωj µ)3 0

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1 3 3 [q ′′ (1) + q ′′ (0) + q 2 (1) + q 2 (0) − q(0)q(1)] 4 (2ωj µ) 2 2 1 [q ′′′ (1) + q ′′′ (0) + 5q(1)q ′ (1) + 5q(0)q ′ (0) − q(0)q ′ (1) + (2ωj µ)5 Z 1 Z 1 Z 1 −q(1)q ′ (0)+(q(1)+q(0)) q 2 (t)dt−2 q 3 (t)dt+ q ′2 (t)dt] −

0

0

0

13 9 1 [q (4) (1) + q (4) (0) + q ′2 (1) + q ′2 (0) − q ′ (0)q ′ (1) − (2ωj µ)6 2 2 7 +7q(1)q ′′ (1) + 7q(0)q ′′ (0) − q(0)q ′′ (1) − q(1)q ′′ (0) + q 3 (1) 2 7 3 3 3 2 2 + q (0) − q(0)q (1) − q(1)q (0) 2 2 2 Z 1  2 Z 1 1 1 q 2 (t)dt ] + O( 7 ) . + (q ′ (1) + q ′ (0)) q 2 (t)dt − 2 µ 0 0

Let us substitute all these expressions into the characteristic determinant U (ϕ ) U (ϕ ) ∆(µ) = 1 1 1 2 , U2 (ϕ1 ) U2 (ϕ2 )

where U1 (y) = y(1) + y(0), U2 (y) = y ′ (1) + y ′ (0). By elementary transformations, we obtain the relation  Z 1 2q(0) 1 1 1 eiµ 2iµ − 1− − [2q ′′ (0)− q 2 (1) ∆(µ) = e q 2 (t)dt− iµ (2iµ)2 (2iµ)3 0 (2iµ)4 2 3 1 + q 2 (0) + q(0)q(1)] − [q(1)q ′ (1) − q(0)q ′ (1) − q(0)q ′ (0) 2 (2iµ)5 Z 1 Z 1 Z 1 + q(1)q ′ (0) − 2q(0) q 2 (t)dt + 2 q 3 (t)dt − q ′2 (t)dt] 0

0

0

1 21 1 [2q (4) (0) + q ′2 (1) + q ′2 (0) − q ′ (0)q ′ (1) − q(1)q ′′ (1) − 6 (2iµ) 2 2 + 11q(0)q ′′ (0) + q(0)q ′′ (1) + q(1)q ′′ (0) − 2q 3 (1) + 3q 3 (0) Z 1  2 1 1 2 2 q (t)dt ] + O( 7 ) + 3q(0)q (1) − 2 µ 0  2q(0) 1 1 + 2eiµ 1 − − [2q ′′ (0) + 2q 2 (0)]− [2q (4) (0) 2 4 (2iµ) (2iµ) (2iµ)6  1 ′′ ′2 3 +12q(0)q (0) + 10q (0) + 4q (0)] + O( 7 ) µ

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K. R. Mamedov & H. Menken

 Z 1 1 2q(0) 1 1 + 1− + [2q ′′ (0) − q 2 (1) q 2 (t)dt − (2iµ)2 (2iµ)3 0 (2iµ)4 2 1 3 [q(1)q ′ (1) − q(0)q ′ (0) − q(0)q ′ (1) + q 2 (0) + q(0)q(1)] + 2 (2iµ)5 Z 1 Z 1 Z 1 + q(1)q ′ (0) − 2q(0) q 2 (t)dt + 2 q 3 (t)dt − q ′2 (t)dt] 0

0

0

1 1 21 − [2q (4) (0) + q ′2 (1) + q ′2 (0) − q ′ (0)q ′ (1) − q(1)q ′′ (1) 6 (2iµ) 2 2

+ 11q(0)q ′′ (0) + q(0)q ′′ (1) + q(1)q ′′ (0) − 2q 3 (1) + 3q 3 (0) Z 1 2  1 1 2 2 + 3q(0)q (1) − q (t)dt ] + O( 7 ) , 2 µ 0

(6)

for µ ∈ T sufficiently large in absolute value. Let b(µ) be the coefficient of e2iµ in (6). It can be easily seen that the relation Z 1 1 1 1 2q(0) + [2q ′′ (0) − q 2 (1) q 2 (t)dt + b−1 (µ) = 1 + (2iµ)2 (2iµ)3 0 (2ωj µ)4 2 1 11 [q(1)q ′ (1) − q(0)q ′ (0) + q 2 (0) + q(0)q(1)] + 2 (2iµ)5 Z 1 Z 1 −q(0)q ′ (1) + q(1)q ′ (0) + 2q(0) q 2 (t)dt + 2 q 3 (t)dt Z

0

1

0

1 21 1 − q (t)dt] + [2q (4) (0) + q ′2 (1) + q ′2 (0) 6 (2iµ) 2 2 0 ′ ′ ′′ ′′ −q (0)q (1) − q(1)q (1) + 19q(0)q (0) + q(0)q ′′ (1) ′2

+q(1)q ′′ (0) − 2q 3 (1) + 17q 3 (0) + q(0)q 2 (1) Z 1 2 1 2 2 +4q(1)q (0) + q (t)dt ] + O( 7 ) µ 0

(7)

holds for µ ∈ T sufficiently large in absolute value. Thus, for µ ∈ T sufficiently large in absolute value, the equation ∆(µ) = 0 is equivalent to the equation −(iµ)−1 b−1 (µ)∆(µ)eiµ = 0.

(8)

Using (6), (7) and the relation q(1) = q(0), by the equation (8) we obtain R1 q ′ (1) − q ′ (0) − 0 q 2 (t)dt 1 iµ + O( 4 ), (9) e +1= (2iµ)3 µ R1 q ′ (1) − q ′ (0) + 0 q 2 (t)dt 1 iµ e +1=− + O( 4 ). (10) 3 (2iµ) µ

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By Rouche’s theorem, we have asymptotic expressions for the roots µk,1 and µk,2 , k = n0 , n0 + 1, · · · , of the equations (9 ) and (10), respectively, where n0 is a positive integer R1 q ′ (1) − q ′ (0) − 0 q 2 (t)dt 1 2 + O( 3 ), (11) µk,1 = − [(2k + 1)π] + [2(2k + 1)π]2 k R 1 q ′ (1) − q ′ (0) + 0 q 2 (t)dt 1 2 µk,2 = − [(2k + 1)π] − + O( 3 ). (12) (4kπ)2 k Note that µk,1 and µk,2 are simple roots of the equations (9) and (10), respectively. From the relations (11), (12) and the relations λk,1 = − µ2k,1 , λk,2 = − µ2k,2 , we obtain the formulas (3)-(4) and observe that these eigenvalues are simple. The theorem is proved. 3. Asymptotic formulas for eigenfunctions of the antiperiodic problem Theorem 3.1. The eigenfunctions of the boundary value problem (1), (2) when θ = 1, corresponding the eigenvalues above, are of the form 1 yk,1 (x) = sin(2k + 1)πx + O( ), k 1 yk,2 (x) = cos(2k + 1)πx + O( ). k

(13) (14)

Proof. Let us calculate U1 (ϕ1 (x, µk,1 )) and U1 (ϕ2 (x, µk,1 )). Since R1 q ′ (1) − q ′ (0) − 0 q 2 (t)dt 1 iµk,1 + O( 4 ), e = −1 + 3 (2iµk,1 ) µk,1 we have U1 (ϕ1 (x, µk,1 )) = ϕ1 (1, µk,1 ) + ϕ1 (0, µk,1 ) R1   q ′ (1) − q ′ (0) + 0 q 2 (t)dt 1 iµk,1 1− =e + O( ) (2iµk,1 )3 µ4k,1   1 + 1 + O( 4 ) µk,1 =

1 2 (q ′ (1) − q ′ (0)) + O( 4 ). 3 (2iµk,1 ) µk,1

Similarly, U1 (ϕ2 (x, µk,1 )) = ϕ2 (1, µk,1 ) + ϕ2 (0, µk,1 )

(15)

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=−

2 (q ′ (1) − q ′ (0)) 1 + O( 4 ). (2iµk,1 )3 µk,1

Since U1 (ϕj (x, µk,1 )) 6= 0, j = 1, 2, we seek the eigenfunction yk,1 (x) corresponding to the eigenvalue λk,1 in the form ϕ1 (x, µk,1 ) (2iµk,1 )2 ϕ2 (x, µk,1 ) . (16) yk,1 (x) = − 4 [q ′ (1) − q ′ (0)] U1 (ϕ1 (x, µk,1 )) U1 (ϕ2 (x, µk,1 ))

To normalized the eigenfunction yk,1 (x) we assume that q ′ (1) − q ′ (0) 6= 0. From the equalities   3 X uv (x) 1 ϕj (x, µk,1 ) = eµk,1 ωj x 1 + + O( ) , j = 1, 2 (2wj µk,1 )v µ4k,1 v=1

and the formulas (15), (16) we obtain

yk,1 (x) = cos µk,1 x + O(

1 ). µk,1

Therefore, the eigenfunction yk,1 (x) satisfies the asymptotic formula (13). In a similar way, we obtain U2 (ϕ1 (x, µk,2 )) = ϕ′1 (1, µk,2 ) + ϕ′1 (0, µk,2 ) R1  q(1) + q(0) q ′ (1) + q ′ (0) − 0 q 2 (t)dt iµk,2 + = iµk,2 e 1− (2iµk,2 )2 (2iµk,2 )3    2q(0) 2q ′ (0) 1 1 + +O( ) +O( 4 ) +iµk,2 1− µk,2 (2iµk,2 )2 (2iµk,2 )3 µ4k,1 =−

1 q ′ (1) − q ′ (0) + O( 3 ), (2iµk,1 )2 µk,1

and U2 (ϕ2 (x, µk,2 )) = −

q ′ (1) − q ′ (0) 1 + O( 3 ). 2 (2iµk,1 ) µk,1

Since U2 (ϕj (x, µk,1 )) 6= 0, j = 1, 2, we can seek the eigenfunction yk,2 (x) corresponding to the eigenvalues λk,2 in the form ϕ1 (x, µk,2 ) (2iµk,2 )2 ϕ2 (x, µk,2 ) yk,2 (x) = − . 2i [q ′ (1) − q ′ (0)] U2 (ϕ1 (x, µk,2 )) U2 (ϕ2 (x, µk,2 )) Thus, we obtain

yk,2 (x) = sin µk,2 x + O( This completes the proof of the theorem.

1 ). µk,2

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4. Asymptotic formulas for eigenvalues and eigenfunctions of the periodic problem In a similar way, the following results are obtained for the boundary problem (1), (2) when θ = 2. Theorem 4.1. All eigenvalues of the boundary value problem (1), (2) when θ = 2, beyond a certain point, are simple and form two infinite sequence λk,1 , λk,2 , k = n0 , n0 + 1, · · · , where n0 is a positive integer and R1 q ′ (1) − q ′ (0) + 0 q 2 (t)dt 1 2 + O( 3 ), λk,1 = −(2kπ) − 2 (4kπ) k R1 2 ′ ′ q (1) − q (0) − 0 q (t)dt 1 + O( 3 ). λk,2 = −(2kπ)2 + (4kπ)2 k The corresponding eigenfunctions are of the form 1 yk,1 (x) = sin 2kπx + O( ), k 1 yk,2 (x) = cos 2kπx + O( ). k References 1. N. B. Kerimov and K. R. Mamedov, On the Riesz basis property of the root functions in certain regular boundary value problems, Math. Notes V. 64, N.4, pp. 483–487 (1998). 2. V. M. Kurbanov, A theorem on equivalent bases for a differential operator, Dokl. Akad. Nauk V. 406, N.1, pp. 17–20 (2006). 3. V. A Marchenko, Sturm-Liouville Operators and Applications, Birkhauser Verlag, 1986. 4. M. A. Naimark, Linear Differential Operators, Part I, Frederick Ungar Pub. Co., New York, 1967. 5. V. A. Chernyatin, Higher-order spectral asymptotics for the Sturm-Liouville operator, Differ. Equ. V. 38, N.2, pp. 206–215 (2002). 6. V. A. Vinokurov and V. A. Sadovnichii, Izv. RAN. Ser. Mat. V. 64, N.2, pp. 47–108 (2000).

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ON THE MULTISUBLINEAR OPERATORS ` LAHCENE MEZRAG Laboratoire de Mathmatiques ´ Pures et Appliqu´ ees, Universit´ e de M’sila, 28000 M’sila, Algeria E-mail: [email protected] KHALIL SAADI Laboratoire de Math´ ematiques Pures et Appliqu´ ees, Universit´ e de M’sila, 28000 M’sila, Algeria E-mail: kh [email protected] We introduce and study the class of multisublinear operators. We give an approach to a Hahn-Banach theorem for this category like the one given by Ioffe in the linear case. Keywords: Banach lattices, Cohen strongly p-summing multilinear operators, Hahn-Banach theorem, multisublinear operators.

1. Introduction Let X be a Banach space and Y be a Banach lattice. Let T : X −→ Y be a sublinear operator, namely, subadditive and positively homogeneous. We denote by ∇T the set of all linear operators u : X −→ Y such that u(x) ≤ T (x) for all x in X. We know (see [1]) that ∇T is not empty if Y is a complete Banach lattice and T (x) = sup {u(x) : u ∈ ∇T }, moreover, the supremum is attained. If Y is simply a Banach lattice then ∇T is empty in general (see [12]). In this paper we introduce and study the category of m-sublinear or multisublinear operators and the relation with m-linear or multilinear operators. We try to find a relation between multisublinear and multilinear like the above equality. This is the first original motivation of our work. In the case of sublinear operators, to have T (x) = sup {u(x) : u ∈ ∇T }, we have used the Hahn-Banach theorem (see [11]): Let X0 be a subspace of X and u : X0 −→ Y be a linear operator such that u ≤ T , then u extends to a linear operator u e : X −→ Y (Y is a complete Banach lattice) such that u e ≤ T . For this, we attempt to give a

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multilinear version of Hahn-Banach theorem. The celebrated Hahn-Banach theorem for linear functionals has been generalized in many different directions. In particular, for linear and bilinear operators. In our knowledge, the first who has studied this problem is A. Grothendieck in his thesis [7]. He has shown that every bilinear integral form can be extended. In [9] Hayden has posed the following conjecture: Let X, Y be two Banach spaces and let X0 , Y0 be two subspaces respectively. Consider u ∈ B(X0 , Y0 ; R), the space of all continuous bilinear forms on X0 ×Y0 such that kuk ≤ C kxk kyk for an absolute constant C. Then, there is u e ∈ B(X, Y ; R) such that u e/X0 ×Y0 = u and ke uk ≤ C kxk kyk. He proved this conjecture in the case of Hilbert spaces and showed in [10] that it is not characterize Hilbert spaces. The problem of the extension of bilinear and multilinear operators has been studied by many authors. Contrary to the case of continuous linear operators, multilinear bounded forms on Banach spaces do not extend from subspaces to the whole spaces (see [4, p. 10]). A number of papers [2,3,6] among so many others have considered this problem from a subspace X0 to a larger X by regarding the manner of the embedding of X0 into X. But there is several positive answers for particular problems. Let m ∈ N and X1 , ..., Xm be real or complex Banach spaces and Y be a complete Banach lattice. Let Ej be a subspace of Xj for 1 ≤ j ≤ m and T be a continuous m-sublinear operator from X1 × ... × Xm into Y . If u0 is a continuous m-linear operator from E1 ×...×Em into Y such that u0 ≤ T . Does thereexist a continuous m-linear operator u from X1 × ... × Xm into Y such that u/E1 × ... × Em = u0 and u ≤ T ? This question remains partially open, it depends on u. We study this problem for some particular cases. If T (x1 , ..., xm ) = T1 (x1 )...Tm (xm ), where Tj is a symmetrical sublinear operator, then this problem turned out to be true for some particular m-linear operators u. Such operators T cover a large gamut of m-sublinear operators. If u(x1 , ..., xm ) = u1 (x1 )...um (xm ) such that uj ≤ Tj , then the extension holds. Our approach consists of studying these types of problems. 2. The multisublinear operators In this section we give some terminology concerning Banach lattices. These spaces are well-known. For more details, the reader can consult the references [13,14]. But for our convenience we recall some facts. We introduce also the class of multisublinear operators. Firstly, we start by recalling the abstract definition of Banach lattices. A real Banach lattice (resp. a real complete Banach lattice) X is an ordered vector space equipped with a lattice (resp. a complete lattice) structure and a Banach space norm satisfying

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the following compatibility condition: ∀x, y ∈ X,

|x| ≤ |y| =⇒ kxk ≤ kyk,

where (|x| = sup {x, −x}. Note that this implies obviously that for any x ∈ X the elements x and |x| have the same norm. We denote by X + = {x ∈ X : x ≥ 0}. An element x of X is positive if x ∈ X + . Let X1 , ..., Xm be Banach lattices then the product space X1 × ... × Xm is also a vector lattice equipped with the natural order x ≤ y ⇐⇒ x1 ≤ y 1 , ..., xm ≤ y m for all x, y ∈ X1 × ... × Xm , 
sup {x, y} = sup x1 , y 1 , ..., sup {xm , y m } .  + We denote by (X1 × ... × Xm ) the set (x1 , ..., xm ) : xj ∈ Xj+ . To render this presentation possible, we assume that X is a Banach lattice of equivalent classes of measurable functions of a measure space (Ω, µ) with the usual order, because we use the multiplication in X. We continue this section by defining the notion of multisublinear operators. It appeared firstly in [5] for bisublinear operators. A mapping T from a Banach space X into a Banach lattice Y is said to be sublinear if for all x, y in X and λ in R+ we have T (λx) = λT (x) and T (x+y) ≤ T (x)+T (y). Consider m in N. Let X1 , ..., Xm , Y be Banach spaces over the real numbers. We will denote by L (X1 , ..., Xm ; Y ) the space of all continuous m-linear operators from X1 × ... × Xm into Y . Definition 2.1. Let X1 , ..., Xm be Banach spaces and Y be a Banach lattice. A mapping T from X1 × ... × Xm into Y which is sublinear in each variable separately is called m-sublinear or multisublinear. In ther words, T is multisublinear means that it is positively homogeneous and subadditive in every component. Note that the sum of two multisublinear operators is a multisublinear operator and the multiplication by a positive number is also a multisublinear operator. Let us denote by SL(X1 , ..., Xm ; Y ) = {m-sublinear mappings T : X1 × ... × Xm −→ Y }. If X1 = ... = Xm = X, we write simply SL(m X, Y ). We equip SL(X1 , ..., Xm ; Y ) with the natural order induced by Y T1 ≤ T2 ⇐⇒ T1 (x) ≤ T2 (x),

∀x ∈ X1 × ... × Xm ,

(1.1)

∇T = {u ∈ L(X1 , ..., Xm ; Y ) : u ≤ T (i.e., ∀x ∈ X1 ×...×Xm, u(x) ≤ T (x))}.

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For a particular T , the set ∇T is not empty by Theorem 2.1 below if Y is a complete Banach lattice. As a consequence, for all x = (x1 , ..., xm ) in X1 × ... × Xm and j in {1, ..., m}, we have u ∈ ∇T ⇐⇒ −T (x1 , ..., −xj , ..., xm ) ≤ u(x) ≤ T (x).

(1.2)

Remark 2.1. We do not know if ∇T exists if T is an arbitrary multisublinear operator. Definition 2.2. We say now that an m-sublinear operator T is: (a) symmetrical with respect to j if T (x) = T (x1 , ..., −xj , ..., xm ) for all x in X1 × ... × Xm . (b) positive if T (x) ≥ 0 for all x in X1 × ... × Xm . The symmetry (for one j) implies the positivity; the converse is false. If T is symmetrical with respect one j, then u ∈ ∇T ⇐⇒ |u(x)| ≤ T (x),

∀x ∈ X,

T (x1 , ..., λxj , ..., xm ) = |λ| T (x),

for all λ.

Let T be an m-sublinear operator from X1 × ... × Xm into a Banach lattice Y . The operator T is continuous, if and only if, there is a positive constant C such that for all x ∈ X1 × ... × Xm , kT (x)k ≤ C x1 ... kxm k. In this case we say that T is bounded and we put

kT k = sup{kT (x)k : x1 BX = 1, ..., kxm kBXm = 1}. 1

We will denote by SB(X1 , ..., Xm ; Y ) the set of all bounded multisublinear operators from X1 × ... × Xm into Y .

Example 2.1. 1- If u is multilinear then |u| is multisublinear.

2- The operator T (x1 , ..., xm ) = x1 ... kxm k is multisublinear.
T : lm × ... × lm −→ l1 defined by T ( x1n n , ..., (xm n )n ) = 13- Consider ( xn ...xm n )n . The operator T is m-sublinear and continuous with kT k ≤ 1. Indeed, by using a general form of H¨ older’s inequality, we obtain ∞ P x1n ...xm n ≤

n=1



∞ P x1n m

n=1

 m1

...



∞ P

n=1

m

|xm n|

 m1

.

4- Let us now define the notion of m-quasilinear operators. An operator T from X1 × ... × Xm into a Banach lattice Y is called to be m-quasilinear

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if for all x = (x1 , ..., xm ) in X1 × ... × Xm , y j ∈ Xj (1 ≤ j ≤ m) and λ in R, we have (i) T (x1 , ..., λxj , ..., xm ) = |λ| |T (x)| for all 1 ≤ j ≤ m, (ii) T (x1 , ..., xj + y j , ..., xm ) ≤ |T (x)| + T (x1 , ..., y j , ..., xm ) .

Note that in general the sum of two m-quasilinear operators is not an m-quasilinear operator but the multiplication by a scalar is m-quasilinear. If we put ϕ (x) = |T (x)| then ϕ is a symmetrical m-sublinear operator. If T is m-sublinear and symmetrical for one j, then T is m-quasilinear. Remark 2.2. Let m in N. Let X1 , ..., Xm , E1 , ..., Em , Y, Z be Banach spaces such that Y, Z are Banach lattices. (a) Consider T in SL(X1 , ..., Xm ; Y ) and u in L (Y ; Z) (assume that u is positive, i.e., u (y) ≥ 0 if y ∈ Y + ). Then, u ◦ T ∈ SL(X1 , ..., Xm ; Z). (b) Consider uj in L (Ej , Xj ) (1 ≤ j ≤ m) and T in SL(X1 , ..., Xm ; Y ). Then, T ◦ (u1 , ..., un ) ∈ SL(E1 , ..., Em ; Y ). (c) ∀T ∈ SL(X1 , ..., Xm ; Y ) and ∀λ ∈ R, we have λT (x) ≤ T (x1 , ..., λxj , ..., xm ) for all j. As an immediate consequence, T ≤ u (u ∈ L(X1 , ..., Xm ; Y )) ⇒ T = u. Indeed, we have for all x in X
1 × ... × Xm that T ≤ u =⇒ T (x) ≤ u (x) =⇒ T x1 , ..., −xj , ..., xm
≤ u(x1 , ..., −xj , ..., xm ) =⇒ (by (1.2) −T (x) ≤ T x1 , ..., −xj , ...., xm ≤ −u (x) =⇒ T (x) ≥ u (x). 3. Extension of Hahn-Banach theorem for the multisublinear operators

The main objective in this section is to study a particular version of HahnBanach theorem for this category of operators. We establish under some suppositions the relation between multilinear and multisublinear operators. The problem of extension of a multilinear operator u defined on a subspace E1 × ... × Em of X1 × ... × Xm (Ej ⊂ Xj ) such that u ≤ T where T ∈ SL(X1 , ..., Xm ; Y ) is not true in general as we have seen above. In this section, we study an extension for the multisublinear of the form

(2.1) T x1 , ..., xm = T1 x1 ...Tm (xm ) . The following theorem generalizes the Hahn-Banach theorem in the case of the multisublinear operators. It seems simple, but its applications are very useful. Theorem 3.1. Let X1 , ..., Xm be Banach spaces and let Y be a complete Banach lattice. Consider T in SL(X1 , ..., Xm ; Y ) of the form (2.1), where

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Tj : Xj → Y is a symmetrical bounded sublinear operator. Then, for all x0 in X1 × ... × Xm there is ux0 ∈ ∇T such that T (x0 ) = ux0 (x0 ), i.e., the supremum is attained, T (x) = sup {u(x) : u ∈ ∇T }. 1 Proof. Let x0 = (x10 , ..., xm 0 ) be in X1 × ...× Xm
. Consider v1x
0 : G = Rx0m× 1 m m ... × Rx0 → Y defined by vx0 λ1 x0 , ..., λm x0 = λ1 T1 x0 ...λm Tm (x0 ) for all (λj )1≤j≤m ⊂ R. The operator vx0 is multilinear on G. We have by

Remark 2.2 (c), for any j, that λj Tj (xj0 ) ≤ Tj (λj xj0 ). Thus, this implies vx0 ≤ T on G. For vj : Rxj0 → Y defined by vj (λxj0 ) = λj Tj (xj0 ) and Tj : Xj → Y , we have vj ≤ Tj on Xj . By Hahn-Banach theorem (see [11]) applied to sublinear operators, there is a linear extension of vj noted uj such that uj ≤ Tj on Xj and u
j /Xj = vj .
Since Tj is positive (Tj is symmetrical)
for all j we get |uj x
j | ≤ Tj xj for all xj ∈ Xj . Hence, we m have |u1 x1 ...um (xm ) | ≤ T1 x1 ...Tm (x
), ∀x ∈mX1 × ... × Xm . 1 Furthermore if we put u (x) = u1 x ...um (x ), we obtain |u (x)| ≤ T (x), ∀x ∈ X1 × ... × Xm . Consequently u is multilinear and u (x) ≤ T (x) for all x ∈ X1 × ... × Xm and hence u/G = v1 ...vm . This concludes the proof
1 1 m m x (x , ..., x ) = T because u(x0 ) = u(x10 , ..., xm ) = v ...T 1 x0 m (x0 ) = 0 0 0 0 T (x0 ).

Remark 3.1. We have made some assumptions on T in Theorem 3.1. Without these assumptions, we have no answer in general. Corollary 3.1. Let X1 , ..., Xm be Banach spaces and Y be a complete Banach lattice. Let T be in SB(X1 , ..., Xm ; Y ) such that T (x1 , ..., xm ) = T1 (x1 )...Tm (xm ), where Tj : Xj → Y is a symmetrical bounded sublinear operator. Then, the following properties are equivalent. (i) The operator T is bounded. (ii) For every u in ∇T , u is bounded. Proof. The first implication is obvious. Concerning the second, we use Banach-Steinhauss theorem for multilinear operators (we are grateful to V. Dimant and I. Villanueva for their help concerning Banach-Steinhauss theorem). As we have seen there is no Hahn-Banach theorem, also no uniform continuity and no open mapping theorem, but there is closed graph theorem and a Banach-Steinhaus theorem. Corollary 3.2. With the same conditions as in the above corollary, we have for all x = (x1 , ..., xm ) in X1 × ... × Xm

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(i) (ii)

kT (x)k

=

kT k

=

sup ku(x)k,

u∈∇T

sup kuk .

u∈∇T

Proof. (i) Let x = (x1 , ..., xm ) be in X1 × ... × Xm . There exists ux ∈ ∇T such that kT (x)k = kux (x)k ≤ sup ku(x)k . This is one direction. u∈∇T

For the other direction, we have for u ∈ ∇T , m −u(x) = u(−x1 , ..., T (−x1 , ..., xm )
x ) ≤m 1 ≤ T1 −x
...Tm (x ) (T1 is symmetrical) ≤ T1 x1 ...Tm (xm ) ≤ T (x). This implies that |u(x)| ≤ T (x) and hence ku(x)k ≤ kT (x)k. Thus sup ku(x)k ≤ kT (x)k. u∈∇T

(ii) This follows immediately from (i). Remark 3.2. In general, we have kuk ≤ 2 kT k . Indeed, u(x) ≤ T (x) implies that −u(x1 , ..., xm ) ≤ T (−x1 , ..., xm ) and hence |u(x)| ≤ sup T (x), T (−x1 , ..., xm ) 1 m ≤ sup |T (x)| , ..., x , T (−x ) 1 m ≤ |T (x)| + T (−x , ..., x ) .

Corollary

3.3. Let T ∈ SB(X1 , ..., Xm ; K) (K = R or C) be given by T (x) = x1 ... kxm k. Then (i) ∇T = {u ∈ L(X1 , ..., Xm ; K) : kuk ≤ 1} = BL(X1 ,...,Xm ;K) . (ii) kx1 k... kxm k = sup{ |u(x)| : u ∈ BL(X1 ,...,Xm ;K) } and kT k = sup{ kuk : u ∈ BL(X1 ,...,Xm ;K) } = 1. Proof. (i) is easy and (ii) is clear from Corollary 3.2.

Corollary 3.4. Consider x0 ∈ X1 × ... × Xm . Then there exists u ∈ 1 2 m 2 L(X1 , ..., Xm ; K) such that kuk = kx10 k... kxm 0 k and u (x0 ) = kx0 k ... kx0 k .
Proof. Let 1 ≤ j ≤ m. We define Tj on Xj by Tj xj = kxj0 kkxj k, and uj on Rxj0 by uj (λj xj0 ) = λj kxj0 k2 . We have uj (λj xj0 ) ≤ kxj0 k kλj xj0 k = Tj (λj xj0 ), ∀j, 1 ≤ j ≤ m. Then u0 = u1 ...um ≤ T = T1 ...Tm on G = Rx10 ×...×Rxm 0 . By Theorem 3.1, there exists u ∈ L(X1 , ..., Xm ; K) such that u (x) ≤ T (x), ∀x ∈ X1 × ... × Xm and u/G = u0 . Hence kuk = kx10 k... kxm 0 k 1 2 m 2 because ku0 k = kx10 k... kxm k and u (x ) = u (x ) = kx k ... kx k . 0 0 0 0 0 0

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References 1. D. Achour and L. Mezrag. Little Grothendieck’s theorem for sublinear operators. J. Math. Anal. Appl. 296 (2004), 541-552. 2. D. Carando. Extendable polynomials on Banach spaces. J. Math. Anal. Appl. 223 (1999), 359-372. 3. J. M. F. Castillo, R Garc´ıa and J. A. Jaramillo. Extension of bilinear forms on Banach spaces. Proc. Amer. Math. Soc. 129 (2001), 3647-3656. 4. A. Defant, K. Floret. Tensor Norms and Operator Ideals. North-Holland, 1993. 5. K. Donner. Extension of Positive Operators and Korovkin Theorems. Lecture Notes in Mathematics, 90. 6. P. Calindo, D. Garc´ıa, M. Maestre and J. Mujica. Extension of multilinear mappings on Banach spaces. Studia Math. 108 (1994), 55-76. 7. A. Grothendieck. Produits tensoriels topologiques et espaces nucleaires. Memoirs Amer. Math. Soc. 16 (1955). 8. L. Grafakos and T. Tao. Multilinear interpolation between adjoint operators. J. Funct. Anal. 199 (2003), 379-385. 9. T. L. Hayden. A conjecture on the extension of bilinear functionals. Amer. Math. Monthly 74 (1967), 1108-1109. 10. T. L. Hayden. The extension of bilinear functionals. Pacific J. Math. 22 (1967), 99-108. 11. A. Ioffe. A new proof of the equivalence of the Hahn-Banach extension and the least upper bound properties. Proc. Amer. Math. Soc. 82 (1981), 385-389. 12. Y. E. Linke. Linear operators without subdifferentials. Sibirsk. Mat. Zh. 32 (1991), 219-221. 13. J. Lindenstrauss and L. Tzafriri. Classical Banach Spaces, I and II. SpringerVerlag, Berlin, 1996. 14. P. Meyer-Nieberg. Banach Lattices. Springer-Verlag, Berlin, Heidelberg, NewYork, 1991.

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PROBABILISTIC MODULAR SPACES K. NOUROUZI Department of Mathematics, K. N. Toosi University of Technology, P.O. Box 15875-4416, Tehran, Iran E-mail: [email protected] The notion of a probabilistic modular space is given, and then some results in convexity is examined in such spaces. Keywords: Modular; Probabilistic modular; Convexity.

1. Introduction On a real linear space X, a real functional ρ is called a modular on X if it satisfies the following conditions: 1. ρ(x) = 0 iff x = 0, 2. ρ(x) = ρ(−x), 3. ρ(αx + βy) ≤ ρ(x) + ρ(y), for all x, y ∈ X and α, β ≥ 0, α + β = 1. The theory of modulars was initiated by Nakano [4] and generalized by Musielak and Orlicz [3]. For more details, the reader is referred to [1] and [2]. The main purpose of this note is to introduce the concept of probabilistic modular spaces. In fact, following the notion of a probabilistic norm [5], this is carried out, and then we give some results in convexity. Throughout this note, X denotes a real vector space and ∆ stands for the set of all nondecreasing functions f : R → R+ 0 satisfying inf t∈R f (t) = 0 and supt∈R f (t) = 1. The symbol ∧ means the minimum. Definition 1.1. A pair (X, µ) is called a probabilistic modular space (briefly, P-modular space) if µ is a mapping from X into ∆ (for x ∈ X, the function µ(x) is denoted by µx , and µx (t) is the value µx at t ∈ R) satisfying the following conditions: 1. µx (0) = 0, 2. µx (t) = 1 for all t > 0 iff x = 0, 3. µ−x (t) = µx (t), 4. µαx+βy (s + t) ≥ µx (s) ∧ µy (t) for all x, y ∈ X, and α, β, s, t ∈ R+ 0

Probabilistic Modular Spaces

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with α + β = 1. If we replace 4 by t s µαx+βy (s + t) ≥ µx ( ) ∧ µy ( ), α β for all x, y ∈ X, and α, β, s, t ∈ R0 + , α + β = 1, we say that (X, µ) is convex. We will also say (X, µ) is β-homogeneous, where β ∈ (0, 1], if µαx (t) = µx ( for every x ∈ X, t > 0, and α ∈ R.

t ), |α|β

Example 1.1. Suppose that ρ is a modular on X. Define ( 0, t≤0 µx (t) = t , r t > 0, t+ρ(x) for all x ∈ X. Then (X, µ) is a P-modular space. Example 1.2. Suppose that ρ is a modular on X. Define  0, t ≤ ρ(x) µx (t) = 1, t > ρ(x), for all x ∈ X. Then (X, µ) is a P-modular space. Example 1.3. Define ρ : R → R by  0, x = 0 ρ(x) = 1, x 6= 0. Then it is evident that (R, ρ) is a modular space. If we define µ as µx (t) =

t , t + ρ(x)

then (R, µ) is a non β-homogeneous P-modular space for every β ∈ (0, 1] t t since t+1 6= t+2 β for all β ∈ (0, 1]. Definition 1.2. Let (X, µ) be a P-modular space. • A sequence (xn ) in X is said to be µ-convergent to a point x ∈ X and µ denoted by xn −→ x if for every t > 0 and r ∈ (0, 1), there exists a positive integer k such that µxn −x (t) > 1 − r for all n ≥ k. • The µ-closure of a subset E of X is denoted by E and defined as the set of all x ∈ X such that there is a sequence (xn ) of elements of E such that xn → x. The subset E is µ-dense in X if E = X.

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• For x ∈ X, t > 0, and 0 < r < 1, the µ-ball centered at x with radius r is defined by B(x, r, t) = {y ∈ X : µx−y (t) > 1 − r}. • An element x ∈ E is called a µ-interior point of E if there are r ∈ (0, 1) and t > 0 such that B(x, r, t) ⊆ E. We say that E is µ-open in X if each element of E is a µ-interior point. According to the notions above, we have the folowing elementary results. Lemma 1.1. The µ-limit of any µ-convergent sequence is unique. Proof. Let (X, µ) be a P-modular space and (pn ) be a sequence in X. If µ µ pn −→ p and pn −→ q, then

t t µ 21 (p−q) (t) = µ 12 (pn −p)+ 21 (q−pn ) (t) ≥ µ(pn −p) ( ) ∧ µ(pn −q) ( ) 2 2 for all t > 0. Thus t t µ 12 (p−q) (t) ≥ lim (µ(pn −p) ( ) ∧ µ(pn −q) ( )) = 1, n→∞ 2 2 for all t > 0. This implies that p = q.

Lemma 1.2. The operations + and · in the β-homogeneous P-modular space (X, µ) are continuous. µ

R

µ

Proof. Let ǫ > 0 and r ∈ (0, 1). Suppose xn −→ x, yn −→ y, and αn −→ α, where x, y, xn , yn ∈ X and α, αn ∈ R, for all n. We may assume αn 6= α and αn 6= 0 for all n. Then there exists k ∈ N such that for every n ≥ k, we have |αn | < |α| + 1,     ǫ ǫ µxn −x 2(β+1) ∧ µ > 1 − r, yn −y 2 |αn |β 2(β+1) and

µx



ǫ 2(β+1) 2 |αn − α|β



> µxn −x



ǫ 2(β+1) 2 (|α| + 1)β



> 1 − r.

Putting γ = µ(αn xn +yn )−(αx−y) (ǫ), we obtain       ǫ ǫ ∧ µ γ ≥ µxn −x 22(β+1)ǫ |αn |β ∧ µx 22(β+1) |α y −y β+1 β n 2 n −α| >

1 − r,

for all n ≥ k. This completes the proof.

Probabilistic Modular Spaces

817

2. Convex sets in P-modular spaces In this section, we aim to give some elementary results in convexity. By a convex set in a vector space X, it is understood a subset of X containing the segment joining every two points of it. Theorem 2.1. An open ball in a convex P-modular space (X, µ) is convex. Proof. Let B(x, r, t) be an arbitrary open ball in (X, µ). If y, z ∈ B(x, r, t) and λ ∈ (0, 1), then µx−(λy−(1−λ)z) (t)

= ≥

=

µλ(x−y)+(1−λ)(x−z) (λt  λ)t)  + (1 −
(1−λ)t λt µ(x−y) λ ∧ µ(x−z) (1−λ) µ(x−y) (t) ∧ µ(x−z) (t) > 1 − r.

Hence λy + (1 − λ)z ∈ B(x, r, t). Theorem 2.2. Suppose that A is a convex subset of a convex βhomogeneous P-modular space (X, µ). Suppose further that a ∈ A, and x is aµ-interior point of A. Then every element of T = {λx + (1 − λ)a; λ ∈ (0, 1)} is a µ-interior point of A. Proof. If q ∈ T, then q = λx + (1 − λ)a for some λ ∈ (0, 1). There exist r0 ∈ (0, 1) and t0 > 0 such that B(x, r0 , t0 ) ⊆ A. We have B(q, r0 , λβ t0 ) ⊆ A. Indeed, if y ∈ B(q, r0 , λβ t0 ), then µλ−1 (y−q) (t0 ) = µy−q (λβ t0 ) > 1 − r0 , which implies that µx+λ−1 (y−q)−x (t0 ) > 1 − r0 . Then u = x + λ−1 (y − q) ∈ B(x, r0 , t0 ), that is, y = a + λ(u − a) ∈ A. Theorem 2.3. Suppose that A is a convex subset of a convex βhomogeneous P-modular space (X, µ). If x is a µ-interior point of A, and y belongs to the µ-closure of A, then all points of T = {λy + (1 − λ)x; λ ∈ (0, 1)}, are in the µ-interior of A.

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Proof. Let x be an µ-interior point of A, and y a µ-closure point of A. Then µ there exists a sequence (an ) of element of A such that an −→ y. If z ∈ T, then z = (1 − λ)x + λy, for some λ ∈ (0, 1). Define zn = (1 − λ)x + λan . There exist r0 ∈ (0, 1) and t0 > 0, such that B(x, r0 , t0 ) ⊆ A. By the same argument presented in the proof of the preceeding theorem, we obtain µ B(zn , r0 , λβ t0 ) ⊆ A, for all n. Since zn −→ z and zn ∈ B(zn , r0 , λβ t0 ) for all n ≥ n0 , we have z ∈ B(zn0 , r0 , λβ t0 ), that is, z ∈ A. Corollary 2.1. If the µ-interior of a convex set A is a nonempty, then 1) the µ-closure of µ-interior of A is equal to A, 2) A and A have the same µ-interior. Proof. 1) It is clear that µ − intA ⊆ A. Conversely, let y ∈ A and x ∈ µ − intA. By Theorem 2.3, T = {λx + (1 − λ)y; λ ∈ (0, 1)} ⊆ µ − intA. Now if λn ∈ (0, 1) with λn → 0, then {λn x + (1 − λn )} is a sequence in T , which is µ-convergent to y as λn → 0. Hence y ∈ µ − intA. 2) The inclusion µ−intA ⊆ A is trivial. Conversely, choose x ∈ µ−intA. If y ∈ µ − intA, then T ⊆ µ − intA. Let zλ = (1 − λ)−1 (y − λx) where µ λ ∈ (0, 1). Then zλ −→ y as λ → 0. Thus zλ ∈ A for some λ. Thus y = λx + (1 − λ)zλ ∈ µ − intA and therefore µ − intA ⊆ intA. Acknowledgments The author would like to thank the Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran, for supporting this research (Grant no. 86470033). References 1. W. M. Kozlowski, Modular Function Spaces, Marcel Dekker, 1988. 2. J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, 1983. 3. J. Musielak and W. Orlicz, On modular spaces, Studia Math. 18 (1959), 49–65. 4. H. Nakano, Modular Semi-Ordered Spaces, Tokyo, Japan, 1959. 5. B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North Holland, New York, Amsterdam, Oxford, 1983.

819

AN APPLICATION OF EXTREMAL POINTS IN DUAL SPACES S. REZAPOUR Department of Mathematics, Azarbaidjan University of Tarbiat Moallem, Azarshahr, Tabriz, Iran E-mail: [email protected] By using extremal points of dual spaces, we shall improve an incorrect result in the well-known book of Ivan Singer [3]. Keywords: ε-approximation, extremal point.

1. Introduction Let X be a normed space and W a subspace of X. A point w0 ∈ W is said to be an ε-approximation for x ∈ X if kx − w0 k ≤ d(x, W ) + ε. For x ∈ X, put PW,ε (x) = {y ∈ W : kx − yk ≤ d(x, W ) + ε}. Note that PW,ε (x) is a non-empty, closed, bounded and convex subset of X, and when ε = 0, PW (x) = PW,ε (x) is the set of all best approximations of x in W . A subspace W is called k-semi-Chebyshev subspace whenever for every x ∈ X, there do not exist more than k linearly independent elements in PW (x). But PW (x) may be empty. Throughout, (S, µ) is a positive σ-finite measure space. There are some works about ε-approximation, for example, [1,2]. Also, there are some mistakes in [3] (see, for example, [4]). Let x ∈ X and r > 0. We say that a set A ⊆ X supports the cell S(x, r) = {y ∈ X : kx − yk ≤ r} whenever d(x, A) = r. Let B ⊆ X be a closed convex subset. We say that D ⊆ B is extremal if (a) D is closed and convex, (b) x, y ∈ B, λx + (1 − λ)y ∈ D and 0 < λ < 1 imply x, y ∈ D. An extremal subset of B consisting of a single point is called an extremal point of B. We shall denote by E(B) the set of all T extremal points of B. It is known that E(C) = E(B) C, whenever C is an extremal subset of B. Let x ∈ X, f ∈ X ∗ and α be a real number. We say that H = {y ∈ X : Re f (x − y) ≥ α} is extremal whenever f ∈ E(SX ∗ ). Here we list some known results which we will need in the main results.

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Lemma 1.1 ([2]; Lemma 1.2). Let W be a closed subspace of a normed linear space X, x ∈ X and ε > 0 be given. Then M ⊆ PW,ε (x) if and only if there exists f ∈ X ∗ such that kf k = 1, f |W = 0 and f (x − y) ≥ kx − yk − ε for all y ∈ M . Lemma 1.2 ([3]; Lemma 1.13; p. 83). Let X = L1 (S, µ) and f ∈ X ∗ . Then, f ∈ E(SX ∗ ) if and only if there exists h ∈ L∞ (S, µ) such that |h| = 1, µ − a.e. on S and Z f (g) = gh dµ, f or all g ∈ X. (1) S

Lemma 1.3 ([3]; Theorem 1.19; p. 85). Let G be a subspace of the normed space X = L1 (S, µ), f ∈ X and g0 ∈ G. Then, g0 ∈ PG (f ) if and only if for each g ∈ G there exists f g ∈ L∞ (S, µ) such that |f g | = 1, µ − a.e. on S, Z (2) Re (g0 − g)f g dµ ≥ 0, S Z Z |f − g0 | dµ. (3) Re (f − g0 )f g dµ = S

S

Finally, we will show that there exists a logical error in the next lemma. Lemma 1.4 ([3]; Theorem 4.3; p. 133). Let W be a subspace of X = L1 (S, µ) and 0 ≤ k < ∞. Then the following are equivalent: (1) W is a k-semi-Chebyshev subspace. (2) There do not exist a measurable subset A of S with µ(A) > 0, a function f0 ∈ L∞ (S, µ) and k + 1 linearly independent elements g0 , g1 , · · · , gk in W such that Z

kf0 k∞ = 1,

f0 hdµ S k X i=0

= 0, f or all h ∈ W,

|gi | = 6 0, µ − a.e. on A,

f0 = ± sign gi , µ − a.e. on S\ker(gi ), (i = 0, 1, · · · , k), gi = 0, µ − a.e. on S\A, (i = 0, 1, · · · , k).

(4) (5) (6) (7) (8)

2. Main Results Proposition 2.1. Let W be a subspace of a normed space (X, k.k), x ∈ X, ε > 0 be given, and w0 ∈ PW,ε (x). Then, there exists f ∈ X ∗ such that

An Application of Extremal Points in Dual Spaces

821

f ∈ E(SX ∗ ), Re f (w0 ) ≥ 0 and Re f (x − w0 ) ≥ kx − w0 k − ε. Proof. Put Mεx,w0 = {f ∈ X ∗ : kf k = 1, Re f (x − w0 ) ≥ kx − w0 k − ε}. Note that Mεx,w0 is convex and σ(X ∗ , X)-compact, where σ(X ∗ , X) is the weak topology. Now by using Lemma 1.1, choose f1 ∈ Mεx,w0 such that f1 (w0 ) = 0. Now, define ψ : X ∗ −→ C by ψ(f ) = f (w0 ). Then, ψ is linear and continuous for σ(X ∗ , X). Thus, ψ(Mεx,w0 ) is convex and compact. Since ψ(f1 ) = f1 (w0 ) = 0, so 0 ∈ ψ(Mεx,w0 ). Now, suppose that ∆ = {z ∈ C : Re z ≥ 0}, and ξ ⊆ ∆ be a support line of ψ(Mεx,w0 ) parallel to the boundary of ∆. Then, ξ, whence also ∆, contains an extremal point ξ0 of ψ(Mεx,w0 ). Thus, by using a virtue of the T Krein-Milman theorem, there exists f0 ∈ E(Mεx,w0 ψ −1 (ξ0 )). Therefore, f0 ∈ E(Mεx,w0 ), and so f0 ∈ E(SX ∗ ). But, f0 (w0 ) = ψ(f0 ) = ξ0 ∈ ∆, hence Re f0 (w0 ) ≥ 0. Since f0 ∈ Mεx,w0 , Re f0 (x − w0 ) ≥ kx − w0 k − ε. Proposition 2.2. Let W be a subspace of a normed space (X, k.k), x ∈ X, w0 ∈ W , f ∈ X ∗ and ε > 0 be given. Then the following are equivalent: (i) f ∈ E(Mεx,w0 ) and Re f (w0 ) ≥ 0. (ii) H = {y ∈ X : Re f (x − y) ≥ kx − w0 k − ε} is extremal, supports the cell S(x, kx − w0 k − ε), Re f (w0 ) ≥ 0 and 0, w0 ∈ H. Proof. (i)⇒ (ii). It is clear that H is extremal and w0 ∈ H. If y ∈ H, then kx − yk ≥ Re f (x − y) ≥ kx − w0 k − ε. Thus, d(x, H) = infy∈H kx − yk = kx − w0 k − ε, that is, H supports the cell S(x, kx − w0 k − ε). Since Re f (w0 ) ≥ 0, Re f (x) ≥ Re f (x − w0 ) ≥ kx − w0 k − ε). Hence, 0 ∈ H. (ii)⇒ (i). Since H is extremal, f ∈ E(SX ∗ ). Theorem 2.1. Let W be a subspace of a normed space (X, k.k), x ∈ X, ε > 0 be given. Then, w0 ∈ PW,ε (x) if and only if for each w ∈ W , there exists f w ∈ E(SX ∗ ) such that Re f w (w0 − w) ≥ 0 and

Re f w (x − w0 ) ≥ kx − w0 k − ε.

Proof. First suppose that w0 ∈ PW,ε (x) and w ∈ W . Then, w0 − w ∈ PW,ε (x − w), so by Proposition 2.1, there exists f w ∈ E(SX ∗ ) such that Re f w (w0 −w) ≥ 0 and Re f w (x−w0 ) ≥ kx−w0 k−ε. For the converse part, let w ∈ W be arbitrary. Then, we have kx − w0 k − ε ≤ Re f w (x − w0 ) = Re f w (x − w) + Re f w (w − w0 ) ≤ Re f w (x − w) ≤ kx − wk. Therefore, w0 ∈ PW,ε (x).

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Corollary 2.1. Let W be a subspace of a normed space (X, k.k), x ∈ X, and ε > 0 be given. Then w0 ∈ PW,ε (x) if and only if for each w ∈ W , Hw = {y ∈ X : Re f w (x − y) ≥ kx − w0 k − ε} is extremal, supports the cell S(x, kx − w0 k − ε), and w0 , w ∈ H, where f w is the functional obtained in Theorem 2.1. We can obtain the following result from Lemma 1.2 and Theorem 2.1. Corollary 2.2. Let G be a subspace of the normed space X = L1 (S, µ), f ∈ X, g0 ∈ G and ε > 0 be given. Then, g0 ∈ PG,ε (f ) if and only if for each g ∈ G there exists f g ∈ L∞ (S, µ) such that |f g | = 1, µ − a.e. on S, Z Re (g0 − g)f g dµ ≥ 0, (9) S Z Z Re (f − g0 )f g dµ ≥ |f − g0 | dµ − ε. (10) S

S

Now, we show that there is Ran error in the Lemma 1.4. In fact, by the second part of the lemma, S f0 gi dµ = 0 (i = 0, 1, · · · , k). Hence, R R R 0 = S f0 gi dµ = S\ker(gi ) f0 gi dµ = ± S\ker(gi ) (sign gi )gi dµ. Thus R S |gi | dµ = 0, and so gi = 0 (i = 0, 1, · · · , k). Therefore, there do not exist linearly independent elements in W that satisfy conditions (4)-(6). This shows that the second statement is always true whereas the first statement may be incorrect. This logical error shows that the result of [3] is incorrect. Now, by using Lemma 1.3, we can improve the result as follows. Proposition 2.3. Let G be a subspace of X = L1 (S, µ) and 0 ≤ k < ∞. Then the following are equivalent: (1) G is a k-semi-Chebyshev subspace. (2) There do not exist f ∈ X, k + 1 linearly independent elements g0 , g1 , · · · , gk in G such that the following statement holds: For each g ∈ G and i = 0, 1, · · · , k there exists fig ∈ L∞ (S, µ) such that |fig | = 1, µ − a.e. on S,

(11)

(gi − g)fig dµ ≥ 0, (i = 0, 1, · · · , k), S Z Z Re (f − gi )fig dµ = |f − gi | dµ (i = 0, 1, · · · , k).

(12)

Re

S

Z

(13)

S

Proof. (1)⇒(2). First suppose that G is a k-semi-Chebyshev subspace. If there exist f ∈ X, k + 1 linearly independent elements g0 , g1 , · · · , gk in G

An Application of Extremal Points in Dual Spaces

823

such that the following statement holds: For each g ∈ G and i = 0, 1, · · · , k there exists fig ∈ L∞ (S, µ) such that |fig | = 1, µ − a.e. on S, Z (14) Re (gi − g)fig dµ ≥ 0, (i = 0, 1, · · · , k), S Z Z |f − gi | dµ (i = 0, 1, · · · , k). (15) Re (f − gi )fig dµ = S

S

Then we have Z Z |f − gi | dµ = Re (f − gi )fig dµ S ZS Z g = Re (f − g)fi dµ + Re (g − gi )fig dµ Z S ZS g ≤ Re (f − g)fi dµ ≤ | (f − g)fig dµ| Z S Z S g |(f − g)| dµ |(f − g)fi | dµ = ≤ S

S

for all g ∈ G. Thus, gi ∈ PG (f ) for all i = 0, 1, · · · , k. Therefore, G is not a k-semi-Chebyshev subspace. This contradiction shows that (2) holds. (2)⇒(1). If G is not a k-semi-Chebyshev subspace, then there exist f ∈ X and at least k + 1 linearly independent elements g0 , g1 , · · · , gk in G such that gi ∈ PG (f ) for all i = 0, 1, · · · , k. Since gi ∈ PG (f ), by Lemma 1.3, for each g ∈ G and i = 0, 1, · · · , k there exists fig ∈ L∞ (S, µ) such that |fig | = 1, µ − a.e. on S, Z (16) Re (gi − g)fig dµ ≥ 0, S Z Z |f − gi | dµ. (17) Re (f − gi )fig dµ = S

S

This contradiction completes the proof.

References 1. S. Rezapour, ε-weakly Chebyshev subspaces of Banach spaces, Anal. Theory Appl. 19 (2003), 130–135. 2. S. Rezapour, Weak compactness of the set of ε-extensions, Bull. Iranian Math. Soc. 30 (2004), 13–20. 3. I. Singer, Best approximation in normed linear spaces by elements of linear subspaces, Springer-Verlag, Berlin, 1970. 4. X. Xu, A result on best proximity pair of two sets, J. Approx. Theory 54 (1988), 322–325.

824

ON A PROBLEM FOR A PARABOLIC-HYPERBOLIC EQUATION WITH A NONSMOOTH LINE OF TYPE CHANGING A. K. URINOV∗ and I. U. KHAYDAROV Ferghana State University, Ferghana, 712000, Uzbekistan ∗ E-mail: [email protected] The uniqueness of a boundary value problem for a parabolic-hyperbolic equation with a nonsmooth line of type changing will be proved. Keywords: Nonlocal problem; parabolic-hyperbolic equation; energy integral; integral equation.

1. Formulation of the problem Consider the equation  uxx − uy − λ21 u, (x, y) ∈ Ω0 , 0 = Lλ u ≡ uxx sign y + uyy sign x − λ22 u sign (x + y) , (x, y) ∈ Ωj ∪ Ωj∗ ,

where λ1 , λ2 are given complex numbers (j = 1, 2), and let Ω0 = {(x, y) : 0 < x < 1, 0 < y < 1}, Ω1 = {(x, y) : −y < x < 1 + y, (−1/2) < y < 0} , Ω2 = {(x, y) : −x < y < 1 + x, (−1/2) < x < 0}, where Ω1∗ , Ω2∗ domains, which are symmetric with respect to straight line x + y = 0 to domains Ω1 , Ω2 , respectively. Let O(0, 0), A(1, 0), B(0, 1), A∗ (0, −1), B ∗ (−1, 0), C(1/2, −1/2), D(−1/2, 1/2), A0 (1, 1), OA(OB ∗ ), OB(OA∗ ), OC(OD), AA0 are segments of straight lines y = 0, x = 0, x + y = 0, x = 1, respectively; Ω = Ω0 ∪ Ω1 ∪ Ω2 ∪ Ω1∗ ∪ Ω2∗ ∪ OA ∪ OB ∪ OC ∪ OD. Boundary-value problems for equation Lλ u = 0 in various subdomains of Ω studied by many authors [1,2,3]. In work [4] for the first time an inner-boundary problem for the equation Lλ u = 0 in the domain Ω was formulated, which was called problem F1 , and also the unique solvability theorem for this problem was stated. In the present work, problem F1 is investigated in detail. Problem F1 . Find a function u(x, y) such that

Parabolic-Hyperbolic Equation with a Nonsmooth Line of Type Changing

825

1) u is a regular solution of the equation Lλ u = 0 in the domains Ω0 , Ω1 , Ω2 , Ω1∗ , Ω2∗ ;
2) u (x, y) ∈ C Ω ∩ C 1 ((Ω ∪ OA∗ ∪ OB ∗ ) \ (OC ∪ OD)); 3) u satisfies u(x, y) = ϕ(y), (x, y) ∈ AA0 ;

(1)

u (0, y) = f1 (y) , (0, y) ∈ A∗ O; ux (0, y) − ux (0, −y) = f2 (y) , (0, y) ∈ A∗ O;

(2)

u (x, 0) = g1 (x) , (x, 0) ∈ B ∗ O; uy (x, 0) − uy (−x, 0) = g2 (x) , (x, 0) ∈ B ∗ O.

(3)

Here ϕ(y), fj (y), gj (x) (j = 1, 2) are given functions; moreover ϕ(y) ∈ C 1 [0, 1]; f1 (t), g1 (t) ∈ C [−1, 0]∩C 2 (−1, 0), f1 (0) = g1 (0); f2 (t) , g2 (t) ∈ C 1 (−1, 0), and they can have singularities of order less than 1 as t → 0 and t → −1. 2. Main functional relations obtained from hyperbolic subdomains Let u(x, y) be a solution of the problem F1 . Suppose that functions ux (x, 0) , ux (0, y) , uy (x, 0) , uy (0, y) are integrable and continuously differentiable in the intervals (−1, 0) , (1, 0) . Then function u (x, y) can be found by Riemann’s formula as a solution of Cauchy problem for the equation Lλ u = 0 in the domains Ωj and Ωj∗ (j = 1, 2) [5]: 1 [u (x + y, 0) + u (x − y, 0)] 2   p Z λ2 y x+y J1 λ2 (x − ξ)2 − y 2 p u (ξ, 0) dξ + 2 x−y (x − ξ)2 − y 2 Z i ih 1 x+y h p + J0 λ2 (x − ξ)2 − y 2 uη (ξ, η)|η=0 dξ 2 x−y

u (x, y) =

(4)

for (x, y) ∈ Ω1 ∪ Ω2∗ ,

1 [u (0, y + x) + u (0, y − x)] 2   p Z λ2 x y+x J1 λ2 (y − η)2 − x2 p u (0, η) dη + 2 y−x (y − η)2 − x2 Z i ih 1 y+x h p + J0 λ2 (y − η)2 − x2 uξ (ξ, η)|ξ=0 dη 2 y−x

u (x, y) =

(5)

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A. K. Urinov & I. U. Khaydarov

for (x, y) ∈ Ω2 ∪ Ω1∗ , where Js (x) is the Bessel function of the first kind of s-th order. Denote τ1 (x) = u (x, 0) , 0 ≤ x ≤ 1; τ2∗ (y) = u (0, y) , −1 ≤ y ≤ 0;

ν1 (x) = uy (x, 0) , 0 < x < 1; (6) ν2∗ (y) = ux (0, y) , −1 < y < 0.

From the statement of the problem F1 , it follows that lim

x→−y+0

u (x, y) =

lim

x→−y−0

u (x, y) , 0 ≤ x ≤ 1/2

(7)

Substituting (4), (5) into (7) and taking (6) into account, after some evaluations we obtain Z x h p i x ∂ J0 λ2 t (t−x) dt = γ (x) (8) [τ1 (x)−τ2∗ (−x)]+ [τ1 (t)−τ2∗ (−t)] t ∂x 0 i h p Rx for 0 ≤ x ≤ 1, where γ (x) = 0 [ν1 (t) + ν2∗ (−t)] J0 λ2 t (t − x) dt. Transforming equation (8) according to the work [6, p. 59], we find Z x h p i ∂ τ1 (x) − τ2∗ (−x) = γ (x) − γ (t) J0 λ2 x (x − t) , 0 ≤ x ≤ 1. ∂t 0 Substituting presentation of the function γ (x) and taking Z x h p h p i∂ h p i i J0 λ2 x(x−t) dt = J0 λ2 ξ(ξ−x) −J0 [λ2 (x−ξ)] , J0 λ2 ξ(ξ−t) ∂t ξ

into account, we have τ1 (x) −

τ2∗ (−x)

=

Z

x

0

[ν1 (t) + ν2∗ (−t)] J0 [λ2 (x − t)] dt, 0 ≤ x ≤ 1.

Taking (2) and (6) into account, from the last equality we get Z x τ1 (x) = f1 (−x) + [ν1 (t) + ν2 (t) + f2 (−t)] J0 [λ2 (x − t)] dt.

(9)

0

Similarly using formulas (4), (5) and condition (3) we obtain Z x τ2 (x) = g1 (−x) + [ν1 (t) + ν2 (t) + g2 (−t)] J0 [λ2 (x − t)] dt

(10)

0

for 0 ≤ x ≤ 1, where τ2 (y) = u (0, y) , 0 ≤ y ≤ 1; ν2 (y) = ux (0, y) , 0 < y < 1. From (9) and (10) follows that τ2 (x) = τ1 (x) + Φ0 (x) , 0 ≤ x ≤ 1, (11) Z x where Φ0 (x) = g1 (−x) − f1 (x) + [g2 (−t) − f2 (−t)] J0 [λ2 (x − t)] dt. 0

Further, transforming equation (9) in view of ν1 (x) + ν2 (x) we get

0,λ2 ν1 (x) + ν2 (x) = −f2 (−x) + C0x [τ1 (x) − f1 (−x)] , 0 < x < 1,

(12)

Parabolic-Hyperbolic Equation with a Nonsmooth Line of Type Changing

Z

827

x

J1 [λ2 (x − t)] p (t) dt. λ2 (x − t) Equalities (11) and (12) are main functional relations among τ1 (x), τ2 (x), ν1 (x) and ν2 (x), obtained from hyperbolic subdomains of Ω. 0,λ2 [p (x)] ≡ p′ (x) + λ22 where C0x

0

3. The uniqueness of solution Lemma 3.1. If u(x, y) is a solution of homogeneous problem F1 and υ(x, y) = eδ(x+y) u(x, y), ∀δ ∈ \{0}, then the equality ZZ h i
Re λ21 − δ 2 − δ |υ (x, y)|2 + |υx (x, y)|2 dxdy +

1 2

Z

Ω0

1

0

2

|υ (x, 1)| dx+ +Re

Z

Z

1

0 1

0

Z 1  1 2 2 |T1 (x)| dx |T1′ (x)| dx + Reλ21 − δ 2 − 2 0

0,λ2 e2δx τ¯1 (x) C0x [τ1 (x)] dx ≡ 0

(13)

is valid, where T1 (x) = υ (x, 0), 0 ≤ x ≤ 1, and τ¯1 (x) is the conjugate function of τ1 (x). Proof. Let u (x, y) be a solution of the problem F1 at ϕ (y) ≡ fj (y) ≡ ≡ gj (x) ≡ 0 (j = 1, 2). Then equalities uxx − uy − λ22 u = 0,

(x, y) ∈ Ω0 ,

τ2 (x) = τ1 (x) , 0 ≤ x ≤ 1; 0,λ2 ν2 (x) = −ν1 (x) + C0x [τ1 (x)] ,

0 0}, 0 < β = const < 1/2. References 1. T. D. Dzhuraev, A. Sopuev, M. Mamazhonov, Boundary-value problems for parabolic-hyperbolic equations, Tashkent, Fan, 1986, 220 pp. 2. K. B. Sabytov, To the theory of mixed parabolic-hyperbolic type equations with spectral parameter, Differentsialnije uravnenija, 1988, V. XXIV, 117–126. 3. E. T. Karimov, Boundary-value problems for parabolic-hyperbolic type equations with spectral parameter, Ph.D. Thesis, Tashkent, 2006. 4. A. K. Urinov, I. U. Khaydarov, On a problem for parabolic-hyperbolic type equation with non-smooth line of type change, Abstracts, 6th international ISAAC congress, Ankara, Turkey, 2007, p. 105. 5. A. N. Tikhonov, A. A. Samarskij, Equations of mathematical physics, Moscow, Nauka, 1977, 737 pp. 6. I. N. Vekua, New methods of solving for elliptic equations, Moscow, Gostekhizdat, 1948, 296 pp. 7. M. S. Salakhitdinov, A. K. Urinov, Boundary-value problems for mixed type problems with spectral parameter, Tashkent, Fan, 1997, 168 pp. 8. A. K. Urinov, I. T. Tojiboev, On a uniqueness of solution of non-local problem, Proceedings of international conference “Modern problems of differential equations, theory of operators and space technologies”, Almaty, 2006, 120–121. 9. A. K. Urinov, K. T. Karimov, On the unique solvability of a problem for mixed type equation with two singular coefficients, Proceedings of international conference “Modern problems of differential equations, theory of operators and space technologies”, Almaty, 2006, 119–120.

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Addenda to Session 06

Dispersive Equations

SESSION EDITORS V. Georgiev M. Reissig

Universit`a di Pisa, Pisa, Italy Technische Universit¨at, Freiberg, Germany

835

ASYMPTOTIC BEHAVIOR AND REGULARITY FOR NONLINEAR DISSIPATIVE WAVE EQUATIONS IN Rn G. TODOROVA University of Tennessee, Knoxville, TN, USA E-mail: [email protected] ˘ D. UGURLU ˙ Abant Izzet Baysal University, Bolu, Turkey E-mail: ugurlu [email protected] B. YORDANOV University of Tennessee, Knoxville, TN, USA E-mail: [email protected] We discuss several results on asymptotic behavior and regularity for wave equations with nonlinear damping utt − ∆u + |ut |m−1 ut = 0 in R+ × Rn , where m > 1. In particular, we show that ∇u ∈ Lm+1 (I × Rn ) for all initial data (u, ut )|t=0 ∈ H 1 (Rn ) × L2 (Rn ) when m ≤ (n + 2)/n and I ⊂ R+ is a compact interval. We also consider global well-posedness in Sobolev spaces H k (Rn ) × H k−1 (Rn ) for k > 2. The strength of nonlinear damping is critical when m = n/(n − 2) and n ≥ 3, which makes classical techniques ineffective. We outline the proof of global well-posedness for m = 3 and n = 3 (critical) under the additional condition that (u, ut )|t=0 are spherically symmetric. Finally, we present scattering results for spherically symmetric global solutions. Keywords: Wave equation, nonlinear damping, regularity, asymptotic behavior.

1. Introduction The asymptotic behavior and regularity of solutions to wave equations with nonlinear damping (m > 1) utt − ∆u + |ut |m−1 ut = 0,

(t, x) ∈ R+ × Rn ,

(1)

are not only challenging questions but also starting points for understanding basic mechanisms in nonlinear hyperbolic equations, such as scatteringdamping [18–20] and smoothing-focussing [3,9,14]. A classical result of Lions and Strauss [15] states that (1) is globally well-posed for all initial

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data u(0, x) = u0 (x), k

n

with (u0 , u1 ) ∈ H (R ) × H Moreover, if

ut (0, x) = u1 (x),

k−1

u0 ∈ H 2 (Rn ),

x ∈ Rn ,

(2)

n

(R ) and 1 ≤ k ≤ 2; see also [9,14,22].

u1 ∈ H 1 (Rn ) ∩ L2m (Rn ),

problem (1)–(2) has a unique global solution u with the following properties: (a) u ∈ L∞ (R+ , H 2 (Rn )),

ut ∈ L∞ (R+ , H 1 (Rn )); Z tZ |us |m+1 dxds, t ≥ t0 , (b) E1 (u, t) = E1 (u, t0 ) − t0 Z 1 X (|∇α ut |2 + |∇α ∇u|2 )dx; where Ek (u, t) = 2 |α|=k−1

(c) E1 (u, t) ≤ E1 (u, t0 ),

E2 (u, t) ≤ E2 (u, t0 ),

t ≥ t0 ;

(d) supp(u, ut ) ⊂ {|x| ≤ t + R} if supp(u0 , u1 ) ⊂ {|x| ≤ R}. Hence, E1 (u, t) and E2 (u, t) are non-increasing functions of t. A natural question is whether these norms decay to zero or not as t → ∞. The answers are known in several cases described in Section 2. Another interesting problem is suggested by identity (b) which implies ut ∈ Lm+1 (R+ × Rn ) even for data (u0 , u1 ) ∈ H 1 (Rn ) × L2 (Rn ). Section 3 shows that the gain of regularity in ∇u is similar if m ≤ (n + 2)/n: ∇u ∈ Lm+1 (I × Rn ) for every compact I ⊂ R+ . The final Section 4 deals with the question Ek (u, t) < ∞ at all t ∈ R+ for k > 2 and arbitrarily large Ek (u, 0). It is expected, based on the invariant scaling of (1), that either m < ∞ and n = 1, 2, or m ≤ n/(n − 2) and n ≥ 3 will be a sufficient condition. However, the critical cases m = n/(n − 2), n ≥ 3 remain open as they are difficult to study by current perturbation techniques. This paper outlines authors contribution25 which covers m = 3, n = 3, and spherically symmetric data in H k (R3 ) × H k−1 (R3 ), k ≥ 3. 2. Asymptotic behavior of energy The long time behavior of solutions to the Cauchy problem (1)–(2) is not well-understood yet. We should mention that the boundary value problem x ∈ Ω, where Ω ⊂ Rn is a bounded domain, is quite different. Nakao [21] and Haraux [7] have found polynomial decay rates of E1 (u, t) under the Dirichlet boundary condition u = 0 on ∂Ω. The current state of this problem and its generalizations for localized damping and source is presented

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in [13]. A surprising similarity with the Cauchy problem is that no decay estimates are known for E2 (u, t) and higher-order norms. There are three types of results about the asymptotic behavior of (1)– (2) with data (u0 , u1 ) ∈ H 1 (Rn ) × L2 (Rn ): Decay estimates for E1 (u, t), lower bounds on E1 (u, t), and E1 (u − u+ , t) → 0 as t → ∞ for a finiteenergy solution u+ of the wave equation, i.e., scattering in the energy space. Clearly, the existence of scattering states implies that the energy approaches a non-zero constant. Below is a short review of significant results. A sufficient condition for scattering, due to Motai and Mochizuki [20], is m > 1 + 2/(n − 1), n ≥ 2. In such cases, E1 (u, t) can not decay as t → ∞. The latter result is generalized to certain exterior domains in n = 3 in [17]. The strongest decay estimate for problem (1)–(2) has long been logarithmic. For 1 < m < 1 + 2/n and 0 < µ < 2/(m − 1), Mochizuki and Motai [19] have established E1 (u, t) ≤ Cµ {ln(2 + t)}−µ ,

(3)

where Cµ depends on µ and (u0 , u1 ). The gap between conditions for scattering and energy decay means that the asymptotic behavior of E1 (u, t) is unknown if 1 + 2/n ≤ m ≤ 1 + 2/(n − 1). Estimate (3) is very weak but it may be sharp in the critical cases n ≥ 2, m = 1 + 2/(n − 1). Kubo [12] has found such modified asymptotic profiles in n = 2, m = 3. The logarithmic decay rate of energy is unlikely to be sharp, however, if 1 < m < 1 + 2/n. Then the invariant scaling and nonlinear diffusion phenomenon for (1) strongly suggest polynomial rates depending on n and m. To explain the first argument, we consider a change of variables u(t, x) 7→ uλ (t, x) = λ(2−m)/(m−1) u(λ(t − t0 ), λ(x − x0 )),

(4)

with λ > 0 and (t0 , x0 ) ∈ R × Rn . Equation (1) is invariant under these transformations. Since {uλ (t, x)} resembles a delta family concentrating at (t0 , x0 ), as λ → ∞, it can be used for testing conjectures about the asymptotic behavior and regularity of u. Changes may occur at critical m where the energy is λ-invariant. We readily calculate Ek (uλ , t) = λ2/(m−1)+2(k−1)−n Ek (u, λ(t − t0 ))

(5)

and set t = 1, t0 = 0, and k = 1 to obtain E1 (u, λ) = λn−2/(m−1) E1 (uλ , 1). Thus the behavior of u(λ, x) as λ → ∞ is determined by the relative smoothness of uλ (1, x) compared to uλ (0, x); any non-trivial smoothing estimate of uλ (1, x) will imply a decay estimate of u(λ, x). The former type

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of results is very plausible since the nonlinear damping has a regularizing effect on solutions. Notice that always E1 (uλ , 1) → ∞ as λ → ∞, so the energy of u is not expected to decay when m ≥ 1 + 2/n. To present the second argument in favor of polynomial decay rates, we assume that (1) exhibits the nonlinear diffusion phenomenon. Here the relevant correspondence is ut ≈ v, where v solves the fast diffusion equation (v m )t − ∆v = 0,

(t, x) ∈ R+ × Rn .

Every positive solution v has a polynomial asymptotic profile v(t, x) ∼ t−nγ (α + β|x|2 /t2mγ )−1/(m−1) ,

(6)

with α, β > 0, and γ = (n − m(n − 2))−1 ; see [4] and the references therein. The wave equation with a nonlinear damping (1) is formally transformed into the fast diffusion equation if utt is neglected, the remaining terms are differentiated with respect to t, and ut is replaced by v. Such manipulations cannot be justified although they are valid if the damping is linear; see [16]. Theorem 2.1. Let u be a solution of (1)–(2) with compactly supported (u0 , u1 ) ∈ H 2 (Rn ) × H 1 (Rn ). If n ≥ 3 and 1 < m ≤ (n + 2)/(n + 1), then E1 (u, t) ≤ Ca t−a ,

t → ∞,

where a > 0 depends only on m and n, while Ca depends on a and (u0 , u1 ). Unfortunately, the above decay rate [26] is implicit. Here the the lack of control on kukL2 is the main difficulty. This is also an essential difference with the wave equation in a bounded domain or the Klein-Gordon equation utt − ∆u + u + |ut |m−1 ut = 0. The proof of Theorem 2.1 uses “parabolic” effects coming from the presence of damping in (1). A crucial idea is to introduce weights similar to the asymptotic profile of fast diffusion (6). 3. Smoothing effects The regularizing effect of nonlinear damping on solutions of (1)–(2) has already been studied. Examples are constructed of piecewise C 2 data with jump discontinuities which are partially smoothed during the evolution when m > 1 + 2/(n − 1); see [9] (radial data) and [14] (general data). We consider the complementary range 1 < m ≤ 1 + 2/n and general data (u0 , u1 ) ∈ H 1 (Rn ) × L2 (Rn ). Recall that ut ∈ Lm+1 (R+ × Rn ), so ut is more regular than u1 for almost all t > 0; see Section 1 (b). The partial smoothing of ut is a significant difference with the linear wave equation.

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A natural question is whether the regularizing effect extends to ∇u. The answer is affirmative, if the nonlinear damping is relatively weak. Theorem 3.1. Assume that 1 < m ≤ 1 + 2/n. For every compact interval I ⊂ R+ , the solution u of (1) satisfies ³ ´ 1/(m+1) m/(m+1) k∇ukLm+1 (I×Rn ) ≤ C E1 (u, 0) + E1 (u, 0) , where C depends only on m, n, and I. We begin with a simple estimate for the wave operator ¤ = ∂t2 − ∆. Lemma 3.1. Assume that 1 < p ≤ q < ∞ and 1/p − 1/q = 1/(n + 1). There exists a C depending on p, q, and n, such that k∇ukLq (Rn+1 ) ≤ Ck∂t ukLq (Rn+1 ) + Ck¤ukLp (Rn+1 ) for every compactly supported distribution u ∈ E ′ (Rn+1 ) which has finite norms on the right side. Proof of Theorem 3.1. Let I ⊂ J be compact intervals in R+ and χ ∈ C0∞ (R+ ) be a cut-off function satisfying ½ 1, if t ∈ I, χ(t) = 0, if t ∈ R \ J. We denote the first two derivatives of χ by χ′ and χ′′ . Applying Lemma 3.1 to χu, where u is a solution of (1), we have k∇x (χu)kLq (Rn+1 ) ≤ Ck∂t (χu)kLq (Rn+1 ) + Ck¤(χu)kLp (Rn+1 ) . Since ¤(χu) = −χ|ut |m−1 ut + 2χ′ ut + χ′′ u and ∂t (χu) = χut + χ′ u, the initial estimate becomes kχ∇x ukLq (Rn+1 ) ≤ C(kχut kLq (Rn+1 ) + kχ′ ukLq (Rn+1 ) ) + Ckχ|ut |m kLp (Rn+1 )

(7)

′

′′

+ C(kχ ut kLp (Rn+1 ) + kχ ukLp (Rn+1 ) ). The strongest regularization occurs when q = m + 1,

p = (m + 1)(n + 1)/(m + n + 2), q

(8) n

as the nonlinear damping requires q ≤ m + 1 for ut ∈ L (R+ × R ). Notice also that m ≤ 1 + 2/n implies p ≤ (m + 1)/m. Hence, the most singular term in (7) is estimated by H¨older’s inequality: m/(m+1)

kχ|ut |m kLp (Rn+1 ) ≤ Ckut km Lm+1 (J×Rn ) ≤ CE1

(u, 0).

(9)

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We can assume that u(t, ·) is compactly supported, since the estimates are local in time and the equation has a finite speed of propagation. The other terms in (7) are trivially bounded by E1 (u, 0). Recall that the integration takes place on a compact set {(t, x) : t ∈ J, |x| ≤ t + R} and p ≤ (m + 1)/m < 2, where p and q are defined in (8). We begin with the norms involving ut on the right side of (7): kχut kLq (Rn+1 ) + kχ′ ut kLp (Rn+1 ) ≤ kut kLm+1 (J×Rn ) + Ckut kL2 (J×Rn ) 1/(m+1)

≤ CE1

1/2

(u, 0) + CE1 (u, 0). (10)

The Sobolev embedding H 1 (Rn+1 ) ֒→ Lq (Rn+1 ) for q ≤ 2 + 2/n implies kχ′ ukLq (Rn+1 ) ≤ C(kut kL2 (J×Rn ) + k∇ukL2 (J×Rn ) ) 1/2

≤ CE1 (u, 0).

(11)

Finally, we use H 1 (Rn+1 ) ֒→ L2 (Rn+1 ) and p < 2 : 1/2

kχ′′ ukLp (Rn+1 ) ≤ k∇ukL2 (J×Rn ) ≤ CE1 (u, 0).

(12)

Substituting estimates (9), (10), (11), and (12) into (7), we obtain ³ ´ 1/2 1/(m+1) m/(m+1) k∇x ukLm+1 (I×Rn ) ≤ C E1 (u, 0) + E1 (u, 0) + E1 (u, 0) . 1/2

Since 1/(m + 1) ≤ 1/2 ≤ m/(m + 1), we can drop E1 (u, 0) from the last estimate. The proof of Theorem 3.1 is complete. Proof of Lemma 3.1. Define the Fourier transform F as Z e−i(τ t+ξx) u(t, x)dxdt, (τ, ξ) ∈ Rn+1 , Fu(τ, ξ) = Rn+1

and denote its inverse by F −1 . From F(∂t u) = iτ Fu and F(∇u) = iξFu, we have F(¤u) = (−τ 2 + |ξ|2 )Fu. To express ∇u in terms of ∂t u and ¤u, we rewrite this identity as |ξ|Fu = |τ |Fu + (|ξ| + |τ |)−1 F(¤u) and apply F −1 to both sides: ¡ ¢ ¡ ¢ ∇u = F −1 ξ|ξ|−1 sign(τ )F(∂t u) +iF −1 ξ|ξ|−1 (|ξ| + |τ |)−1 F(¤u) . (13)

Here sign(τ ) = τ /|τ | for τ 6= 0. It is well known [5] that ξ/|ξ| and sign(τ ) are Lq (Rn+1 ) Fourier multipliers, while (|ξ| + |τ |)−1 is an Lp (Rn+1 ) − Lq (Rn+1 ) Fourier multiplier if 1/p − 1/q = 1/(n + 1) and 1 < q < ∞. Using (13), k∇ukLq (Rn+1 ) ≤ kF −1 (ξ|ξ|−1 sign(τ )F(∂t u))kLq (Rn+1 ) + kF −1 (ξ|ξ|−1 (|ξ| + |τ |)−1 F(¤u))kLq (Rn+1 ) ≤ Ck∂t ukLq (Rn+1 ) + Ck¤ukLp (Rn+1 ) .

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4. Regularity and scattering for critical damping The global well-posedness of (1) is a highly nontrivial problem in Sobolev spaces H k (Rn ) × H k−1 (Rn ) with k > 2 since the monotonicity of |ut |m−1 ut no longer implies a priori estimates. Standard techniques treat the nonlinear damping as a perturbation and restrict its growth m ≤ n/(n − 2) if n ≥ 3; this is implied by scaling laws (4), (5) and E2 (u, t) ≤ E2 (u, 0): E2 (uλ , t) = λ2/(m−1)+2−n E2 (u, λt) ≤ E2 (u, 0),

λ → ∞,

for m ≥ n/(n − 2), n ≥ 3. Hence the existence of concentrating solutions (increasing norms of order k > 2) cannot be excluded for large m. There is some possibility to handle the critical case m = n/(n − 2) by improved classical techniques similar to the nonlinear wave and Schrodinger equations; see [1,6,23] for the former and [2,24] for the latter. So far the global wellposedness of (1) is established along these lines in a special case: m = 3, n = 3, and radial data [25]. We present the main ideas and analytic tools. Our result concerns mainly large data, as the global existence and asymptotic behavior are well-understood for small data; see [8,11,17]. Let us define the Sobolev spaces of spherically symmetric functions k Hrad (R3 ) = {u ∈ H k (R3 ) : u is a function of |x|},

for k ≥ 1. These are invariant under the evolution determined by (1)–(2). We write Du = (∇u, ut ) for the space-time derivative of u. The following is partial answer to the question about global well-posedness [25]. 3 2 Theorem 4.1. Let m = 3, n = 3, and (u0 , u1 ) ∈ Hrad (R3 ) × Hrad (R3 ). Then problem (1)–(2) admits a unique global solution u, such that 3−|α|

Dα u ∈ C(R+ , Hrad (R3 )),

|α| ≤ 3.

Moreover, u satisfies the estimates X Z t kDα u(s)k2L∞ ds ≤ C1 (u0 , u1 ), |α|≤2

t ≥ 0,

0

X

E1 (Dα u, t) ≤ C2 (u0 , u1 ),

t ≥ 0,

|α|≤2

for implicit constants Ck (u0 , u1 ), k = 1, 2, which are finite whenever the norm ku0 kH 3 + ku1 kH 2 is finite. Remark 4.1. We rely on the “forbidden” L1t L2x −L2t L∞ x Strichartz estimate for the wave equation in R × R3 ; see [10]. The estimate is valid only for

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G. Todorova, D. U˘ gurlu & B. Yordanov

spherically symmetric solutions which explains the condition for spherically symmetric data. Proof of Theorem 4.1 (outline). Recall a useful fact: for |α| ≤ 1, Z t E1 (Dα u, t) + (2|α| + 1) kus Dα us k2L2 ds = E1 (Dα u, 0). 0

Step 1. Let u be a local solution of (1)–(2), such that E3 (u, t) < ∞ for t ∈ [0, T∗ ). The equation for Dα u, |α| = 2, involves non-dissipative terms: X ¤Dα u + 3u2t Dα ut + cβ,γ ut Dβ ut Dγ ut = 0. (14) β+γ=α

From E2 (u, t) ≤ E2 (u, 0), the L2 -norms of these terms are bounded by Z t (15) kDα u(s)k2L∞ ds, |α| ≤ 2. 0

Step 2. We can handle L2t L∞ x norms by means of µZ t ¶1/2 Z t 1/2 k¤u(s)kL2 ds, ku(s)k2L∞ ds ≤ CE1 (u, 0) + C 0

0

which is an endpoint Strichartz estimate. If |α| = 1, the inequality leads to µZ t ¶1/2 Z t 1/2 kDα u(s)k2L∞ ds ≤ CE2 (u, 0) + C ku2s (s)Dα us (s)kL2 ds. 0

0

L2t L∞ x

It is not trivial to bound the right side by the We need a simple “non-concentration” lemma.

norm on the left side.

Lemma 4.1. Assume that (u0 , u1 ) ∈ H 3 (R3 ) × H 2 (R3 ) and |α| = 1. Let u be the solution of problem (1)–(2) extended on a maximal interval [0, T∗ ). For every ǫ > 0 there exists δ ∈ (0, T0 ) such that Z t sup (kus (s)k4L4 + kus (s)Dα us (s)k2L2 ) ds < ǫ. t∈[T0 ,T∗ )

t−δ

This result follows from kus (s)k4L4 + kus (s)Dα us (s)k2L2 ∈ L1 ([0, T∗ )). Applying Lemma 4.1, we find C > 0 and δ = δ(u0 , u1 , α) > 0, such that µZ t ¶1/2 1/2 1/2 kDα u(s)k2L∞ ds ≤ CE2 (u, 0)(1 + CE2 (u, 0))1+t/δ (16) 0

for all t ∈ [0, T∗ ). The estimate for |α| = 2 in (15) is similar. Step 3. To complete the proof of E3 (u, t) < ∞, we combine the standard energy estimate from (14) and the L2t L∞ x -estimate (16) with |α| = 1, 2.

Asymptotic Behavior and Regularity for Wave Equations in Rn

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The solutions constructed in Theorem 4.1 are asymptotically free, since the nonlinear term u3t is supercritical for scattering theory in R3 . 3 2 Theorem 4.2. Let m = 3, n = 3, and (u0 , u1 ) ∈ Hrad (R3 ) × Hrad (R3 ). The global solution u of problem (1)–(2) is asymptotically free:

kDα u(t) − Dα u+ (t)k2 → 0,

t → ∞,

for 1 ≤ |α| ≤ 3, where u+ is a solution of ¤u+ = 0 with initial data 2 1 Du+ (x, 0) ∈ Hrad (R3 ) × Hrad (R3 ).

Proof (outline). We use the estimates in Theorem 4.1 and H¨older’s inequality to show a sufficient condition for scattering: Z ∞ kDα u3t (t)k2 dt < ∞, 0 ≤ |α| ≤ 2. 0

References 1. H. Bahouri, P. G´erard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math. 121 (1999), 131–175. 2. J. Bourgain, Global wellposedness of defocusing critical nonlinear Schr¨ odinger equation in the radial case, J. Amer. Math. Soc. 12 (1999), 145–171. 3. R. Carles, J. Rauch, Focusing of Spherical Nonlinear Pulses in R1+3 , Proc. Amer. Math. Soc. 130 (2002), 791–804. 4. J. A. Carrillo, J. L. V´ azquez, Fine asymptotics for fast diffusion equations, Comm. Partial Differential Equations 28 (2003), 1023–1056. 5. J. Duoandikoetxea, Fourier Analysis, transl. and rev. D. Cruz-Uribe, Amer. Math. Soc., Providence, 2000. 6. M. Grillakis, Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity, Ann. of Math. 132 (1990), 485–509. 7. A. Haraux, Comportement ` a l’infini pour une ´equation d’ondes non lin´eaire dissipative, C. R. Acad. Sci. Paris S´er. A-B 287 (1978), A507–A509. 8. L. H¨ ormander, Lectures on Nonlinear Wave Equations, Math. Appl., vol. 26, Springer, Berlin, 1997. 9. J.-L. Joly, G. M´etivier, J. Rauch, Nonlinear hyperbolic smoothing at a focal point, Michigan Math. J. 47 (2000), 285–312. 10. S. Klainerman, M. Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46 (1993), 1221–1268. 11. S. Klainerman, G. Ponce, Global, small amplitude solutions to nonlinear evolution equations, Comm. Pure Appl. Math. 63 (1983), 133–141. 12. H. Kubo, Large-time behavior of solutions to semilinear wave equations with dissipative terms, preprint, 2006. 13. I. Lasiecka, D. Toundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms—an intrinsic approach, preprint, 2005.

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14. J. Liang, Nonlinear hyperbolic smoothing at a focal point, preprint, 2004. 15. J.-L. Lions, W. A. Strauss, Some non-linear evolution equations, Bull. Soc. Math. France 93 (1965), 43–96. 16. A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci. 12 (1976/77), 169–189. 17. T. Matsuyama, Asymptotics for the nonlinear dissipative wave equation, Trans. Amer. Math. Soc. 355 (2003), 865–899. 18. K. Mochizuki, Decay and asymptotics for wave equations with dissipative terms, Lecture Notes in Phys., vol. 39, Springer, 1975, pp. 486–490. 19. K. Mochizuki, T. Motai, On energy decay-nondecay problems for wave equations with nonlinear dissipative term in RN , J. Math. Soc. Japan 47 (1995), 405–421. 20. T. Motai, K. Mochizuki, On asymptotic behaviors for wave equation with a nonlinear dissipative term in RN , Hokkaido Math. J. 25 (1996), 119–135. 21. M. Nakao, Asymptotic stability of the bounded or almost periodic solution of the wave equation with nonlinear dissipative term, J. Math. Anal. Appl. 58 (1977), 336–343. 22. J. Serrin, G. Todorova, E. Vitillaro, Existence for a nonlinear wave equation with damping and source terms, Differential Integral Equations 16 (2003), 13–50. 23. J. Shatah, M. Struwe, Regularity results for nonlinear wave equations, Ann. of Math. 138 (1993), 503–518. 24. T. Tao, M. Visan, X. Zhang, Global well-posedness and scattering for the defocusing mass-critical nonlinear Schr¨ odinger equation for radial data in high dimensions, Duke Math. J. 140 (2007), 165–202. 25. G. Todorova, D. U˘ gurlu, B. Yordanov, Regularity and scattering for the wave equation with a critical nonlinear damping, preprint, 2007. 26. G. Todorova, B. Yordanov, The energy decay problem for wave equations with nonlinear dissipative terms in Rn , Indiana Univ. Math. J. 56 (2007), 389–416.

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THE BLOW-UP BOUNDARY FOR A SYSTEM OF SEMILINEAR WAVE EQUATIONS HIROSHI UESAKA Department of Mathematics, College of Science and Technology, Nihon University, Tokyo, 101-8308, Chiyodaku Kanda Surugadai 1-8, Japan E-mail: [email protected] We shall show the existence of a blow-up boundary of blow-up solutions for a system of semilinear wave equations. The blow-up boundary is proved to be Lipschitz continuous. It is well-known for a single semilinear wave equation that Caffarelli and Friedman investigated the blow-up boundary of blow-up solutions and proved various remarkable results. Keywords: Blow-up boundary, system of semilinear wave equations.

1. Introduction Let u be a real-valued solution for some wave equation in R3 × [0, ∞) and let u blow-up in finite time. The set Γ is defined by Γ = ∂{u = ∞} ∩ {t > 0} and it is given by Γ : t = ϕ(x). Then we call u a pointwise blow-up solution. The set of (x, ϕ(x)) is called the blow-up boundary of u. Caffarelli and Friedman investigated a blow-up boundary for a single semilinear wave equation and published their results in [2,3]. Later Alinhac summarized a part of their results in his book [1]. They studied the following Cauchy problem for a semilinear wave equation: ½ ¤u = F (u) = up in R3 × (0, T ), (1) u(x, 0) = f (x), ∂t u(x, 0) = g(x) in R3 , where ¤ = ∂t2 − △, T is a positive constant and p > 1. Caffarelli and Friedman proved that the blow-up boundary is a Lipschitz continuous surface and further it is a C 1 space-like one. Let u be a pointwise blow-up solution within (0, T ] for some nonlinear wave equation. T is considered in two ways as follows:

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H. Uesaka

1. T is determined from a pointwise blow-up solution given in advance (see [4,5,8]), or, 2. we can manage initial data for any fixed T given in advance so that a solution blows up pointwise within (0, T ] (see [1,2]). We consider the following Cauchy problem for a system in R3 × [0, T ) with 0 < T < ∞ and assign initial data to it, that is, we are interested in ¤1 u1 = F1 (u1 , u2 ), ¤2 u2 = F2 (u1 , u2 ) in R3 × (0, T ),

(2)

where ¤k = ∂t2 − c2k △, c1 ≥ c2 > 0, and ½

u1 (x, 0) = f1 (x), ∂t u1 (x, 0) = g1 (x) in R3 , u2 (x, 0) = f2 (x), ∂t u2 (x, 0) = g2 (x) in R3 ,

(3)

where fk (x) ∈ C 3 (R3 ), gk (x) ∈ C 2 (R3 ) and additional assumptions are supposed later. The function Fk = Fk (u1 , u2 ) (k = 1, 2) is given by Fk (u1 , u2 ) = ak up1k uq2k

or ak up1k + bk uq2k ,

where pk , qk > 1 and ak , bk > 0. Let Z Z ´ ∂³ t t uk,0 (x, t) = fk (x + ck tω)dω + gk (x + ck tω)dω, ∂t 4π |ω|=1 4π |ω|=1 Z £ 1 fk (x + ck tω) = 4π |ω|=1 ¤ + t (ck ω · ∇fk (x + ck tω) + gk (x + ck tω) dω, (4)

R where k = 1, 2 and x, ω ∈ R3 , |ω| = 1, and |ω|=1 · · · dω means the integral over the unit sphere. Then the problem (2)–(3) is converted as usual into the system of integral equations: uk (x, t) = uk,0 (x, t) +

Z Z 1 t (t − s)ds Fk (~u(x + ck (t − s)ω, s))dω, (5) 4π 0 |ω|=1

where ~u = (u1 , u2 ) ∈ R2 . Many results concerning (2)–(3) have been proved, for example, existence, regularity, blow-up and other properties of solutions. Here we do not give the proof of the existence of solutions. But we suppose the following. A.1.1. The Cauchy problem (2)–(3) has a local unique C 2 -solution ~u within (0, T ] which satisfies the integral equation (5).

The Blow-up Boundary for a System of Semilinear Wave Equations

847

2. Positive Solutions Let R be any fixed positive number. We denote a ball of radius R with the center at the origin by BR = {x ∈ R3 | |x| ≤ R}. Let T be any fixed positive constant. We formulate two assumptions in this section. These are essentially the same as those in [2]. A.2.1. Let X = x + c1 tω, where x ∈ BR , 0 ≤ t ≤ T and ω ∈ R3 , |ω| ≤ 1. We suppose (1) fk (x) ∈ C 3 (R3 ) and gk (x) ∈ C 2 (R3 ), (2) fk (X) ≥ 0 and gk (X) ≥ ck |∇fk (X)|. A backward characteristic cone for ¤k is defined and denoted by Ck (x, T ) = {(ξ, τ ) ∈ R3 × [0, ∞) | |x − ξ| ≤ ck (T − τ )}. We define Kk,R,T = ∪x∈BR Ck (x, T ). Theorem 2.1. Suppose A.1.1 and A.2.1. Let ~u be a solution of (2)–(3). Then uk (x, t) ≥ 0 for k = 1, 2 in the domain of existence of solutions ~u contained in K1,R,T . Theorem 2.1 is essential for proving other results. Not only the result of Theorem 2.1 but also the reasoning to prove the theorem are often applied to get statements which we shall introduce below. Proof. We only give an outline. The assumption A.2.1 shows that u0,k (x, t) ≥ 0 for k = 1, 2 (see (5)). Assume either u1 or u2 becomes negative at some point (x, t) ∈ K1,R,T . We take the backward characteristic cone Ck (x, t) and use the finite speed of propagation of solutions. By means of Keller’s argument [5,7,8] we are led to a contradiction if uk < 0. We remark that uk (x, t) > 0 holds if fk (X) > 0 in A.2.1. Moreover, we suppose the following. A.2.2. Let k = 1, 2. Let each λk be any fixed positive number. Put X = x + c1 tω, x ∈ BR , 0 ≤ t ≤ T . Suppose (1) fk (x) ∈ C 4 (BR+T ) and gk (x) ∈ C 3 (BR+T ), (2) gk (X) ≥ λk |∇fk (X)|, (3) c2k △fk (X) + Fk (f~(X)) ≥ (ck + λk )|∇gk (X)| + λk ck |∇2 fk (X)|, qP 3 ∂2 2 where |∇2 f (x)| = i,j=1 | ∂xi ∂xj f (x)| .

848

H. Uesaka

Theorem 2.2. Suppose that the assumptions A.1.1 and A.2.1-2 are satisfied. Let ~u = (u1 , u2 ) be a solution of (2)–(3). Then ∂t uk (x, t) ≥ λk |∇uk (x, t)| in K1,R,T .

(6)

Proof. We define Lk (x, t) = ∂t uk (x, t) + λk e · ∇uk (x, t), where e is any unit vector of R3 . Then Lk satisfies   ¤k Lk (x, t) = ∂1 Fk (~u)L1 (x, t) + ∂2 Fk (~u)L2 (x, t), Lk (x, 0) = gk (x) + λk e · ∇fk (x),  ∂t Lk (x, 0) = c2k △fk (x) + Fk (f~(X)) + λk e · ∇gk (x),

where ∂l Fk (s1 , s2 ) = ∂sl Fk (s1 , s2 ) for k, l = 1, 2. According to the same reasoning as in the proof of Theorem 2.1, the assumption A.2.2 assures that the above equation has a positive solution Lk . The desired result follows from Lk ≥ 0. 3. Blow-up Let ~u = (u1 , u2 ) be a solution of (2)–(3) with positive components. It is called a pointwise blow-up solution in BR if there exists a positive constant Tk (x) < ∞ depending on x ∈ BR such that lim

tրTk (x)

uk (x, t) = ∞ (k = 1, 2).

We shall show that there exists a sufficient condition for (2)–(3) to have a pointwise blow-up solution in BR . Lemma 3.1 (Keller [5]). Let f (s) = Asp + B, where A > 0, B ≥ 0 and p > 1. Let v be the solution of the ordinary differential equation ½ ′′ d v (t) = f (v(t)) (t > 0, ′ = dt ), (7) ′ v(0) = α, v (0) = β, where α, β ≥ 0. Then there exists some finite positive T such that lim v(t) = ∞.

tրT

(8)

Suppose the following. A.3.1. fk (X) ≥ α and gk (X) ≥ β + ck |∇fk (X)| for X = x + c1 tω, x ∈ BR , 0 ≤ t ≤ T. Then we have the following result. Theorem 3.1. Let ~u = (u1 , u2 ) be the C 2 -solution of (2)–(3). Let T be given in Lemma 3.1. Moreover, let us suppose that A.3.1 holds. If ~u keeps

The Blow-up Boundary for a System of Semilinear Wave Equations

849

to be a solution from the class C 2 till uk attains ∞, then there exists Tk (x) depending on x ∈ BR such that 0 < Tk (x) ≤ T and lim

tրTk (x)

uk (x, t) = ∞ for x ∈ BR .

Proof. We first put f (v) = F1 (v, α) in (7). Then Lemma 3.1 holds. Let w1 (x, t) = u1 (x, t) − v(t). We remark that F1 (u1 , u2 ) ≥ F1 (u1 , α) for t ≥ 0 because uk is monotone increasing in t. By the mean value theorem F1 (u1 , u2 ) − F1 (v, α) ≥ F1 (u1 , α) − F1 (v, α) = ∂1 F1 (u′1 , α)w1 ,

(9)

where u′1 is a point between u1 and v, and ∂1 F1 (u′1 , α) > 0 when u′1 , α > 0. We make the same computation for u2 . Then we have ½ ¤k wk = Fk (~u) − Fk (~v ) ≥ ∂k Fk (~u′ )wk , wk (x, 0) = fk (x) − α, ∂t wk (x, 0) = gk (x) − β, where ~u′ = (u′1 , α) or (α, u′2 ), and ~v = (v, α) or (α, v). According to the same argument as in the proof to Theorem 2.1, we can get uk (x, t) ≥ v(t) for x ∈ BR , t ≤ T . By Lemma 3.1, there exists Tk (x) such that 0 < Tk (x) ≤ T and limtրTk (x) uk (x, t) = ∞. Thus the proof is completed. 4. Blow-up Boundary Let c1 and c2 be the propagation speeds and c1 ≥ c2 . We shall prove that a blow-up boundary actually exists over BR and it is Lipschitz continuous with Lipschitz constant c−1 1 . First we define the influence domain (see [1,6]). Definition 4.1. Let κ be a positive constant. The open set Fκ is called an influence domain if it satisfies (1) Fκ ⊂ R3 × [0, ∞), (2) Cκ (x, t) ⊂ Fκ for (x, t) ∈ Fκ , where Cκ (x, t) = {(x′ , t′ ) ∈ R3 × [0, ∞) | |x − x′ | ≤ κ(t − t′ )}. Lemma 4.1. Let Fκ be an influence domain and let ψ(x) := sup{t, (x, t) ∈ Fκ }. Then the following is true: either ψ(x) ≡ ∞ or |ψ(x) − ψ(y)| ≤

1 |x − y|. κ

The proof is easy, so we omit it (see [1,6]). We assume the following.

850

H. Uesaka

A.4.1. Put λ1 = λ2 = c1 in A.2.2 and in Theorem 2.2. Let ~u = (u1 , u2 ) be a solution of (2)–(3) satisfying the assumptions A.1.1, A.2.1-2 and A.4.1. We suppose that there exists a positive constant T such that ~u(x, t) does not exist as a C 2 -solution for t > T . Let x be any fixed point in BR . Then uk (x, t) is monotone increasing in t. Moreover, we let {uk,n }, k = 1, 2 to be a sequence of successively approximating solutions to uk in K1,R,T . Here uk,0 is equal to (4), and the approximating solutions uk,n satisfy for n ≥ 1 in K1,R,T ½ ¤uk,n = Fk (~un−1 ), (10) uk,n (x, 0) = fk (x) and ∂t uk,n (x, 0) = gk (x), where ~un = (u1,n , u2,n ). Using (5), ~un is defined as an integral of ~un−1 . Then uk,n has the following properties in K1,R,T : (1) uk,1 ≤ uk,2 ≤ · · · ≤ uk,n ≤ · · · , (2) ∂t uk,n (x, t) ≥ c1 |∇uk,n (x, t)|, (3) uk,n (x, t) is monotone increasing in t. We can prove these properties easily by the expression of the integral form and by the same argument as those for proving the properties for uk . Let x ∈ BR . We define a subset Ωk ⊂ Kk,R,T by Ωk = {(x, t) | uk (x, t) = lim uk,n exists and is of C 2 }. n→∞

We set Ω = Ω1 ∩ Ω2 . We define ϕk (x) for x ∈ BR by ϕk (x) = sup{ t | (t, x) ∈ Ωk }, and ϕ(x) by ϕ(x) = min(ϕ1 (x), ϕ2 (x)). We shall show that Ω is the influence domain, ϕ1 (x) = ϕ2 (x), and limtրϕ(x) uk (x, t) = ∞. Theorem 4.1. Let ~u be the solution stated in the above paragraphs. Then ~u is a blow-up solution of class C 2 . The following holds: (1) ϕ(x) = ϕ1 (x) = ϕ2 (x), whence Ω = Ω1 = Ω2 , (2) |ϕ(x) − ϕ(y)| ≤ c−1 1 |x − y| for x, y ∈ BR , (3) limtրϕ(x) uk (x, t) = ∞ for k = 1, 2. Proof. Let x be any point in BR . We shall prove limtրϕ(x) uk (x, t) = ∞ for k = 1, 2. We take any t < ϕ(x) and fix it. We set m = (x, t). We define the backward characteristic cone Ck (m) by Ck (m) = { (x′ , t′ ) | |x − x′ | ≤ ck (t − t′ ), t′ ≥ 0 }. We suppose that limτ րϕ(x) uk (x, τ ) = A < ∞. Let m′ = (x′ , t′ ) be any point ∈ C1 (m). Then 0 ≤ uk (m′ ) ≤ A holds for k = 1, 2. We shall show

851

The Blow-up Boundary for a System of Semilinear Wave Equations

this inequality. By [mm′ ] we denote the segment connecting m and m′ . Let p be any point on [mm′ ], so p = m + s(m − m′ ) with 0 ≤ s ≤ 1. By the Taylor formula and (2) of the above property we have uk,n (m) = uk,n (m′ ) +

Z

1

{(t − t′ )∂t uk,n (p) + (x − x′ ) · ∇uk,n (p)} ds

0

≥ uk,n (m′ ),

(11) ′

| because ∂t uk,n (p) ≥ c1 |∇uk,n (p)| > |x−x t−t′ |∇uk,n (p)|. Thus we have shown ′ 0 ≤ uk (m ) ≤ A. We compare the first and second order derivatives of uk,n with W = M exp(Bt) and its derivative. By the same argument as in Lemma 2.3 from [2] or in the proof of Theorem 4.1 of Chapter 3 from [1], if we choose M and B suitably large we can show that the absolute value of these derivatives are less than some positive constant C, which depends on A, B and M , and is independent of n. By an argument similar to that in Satz 1 from [4] we can show that uk,n converges in Ck (m) to uk as n → ∞ and uk is of class C 2 in Ck (m). Thus both u1 and u2 exist together with their derivatives up to second order at (x, ϕ(x)). This is contradiction. There is a possibility that either u1 or u2 does not go to ∞ at x ∈ BR as t ր ϕ(x). But by means of the construction of the solution such a possibility is excluded. Then limtրϕ(x) uk (x, t) = ∞ for k = 1, 2. Consequently, we have ϕ1 = ϕ2 , Ω1 = Ω2 and C1 (m) ⊂ Ω. Then Ω is the influence domain and ϕ(x) is Lipschitz continuous with Lipschitz constant c−1 1 by Lemma 4.1. Thus the proof is complete.

5. Blow-up Rates In this section we study the order of infinity of uk (x, t) when t → ϕ(x). We cannot yet prove such results for different propagation speeds and for product type nonlinearity. For this reason we suppose the following. A.5.1. (1) c1 = c2 = 1, (2) Fk (~u) = ak up1k + bk uq2k , where ak , bk > 0 and pk , qk > 1. We put Gk (u1 ) = ak up1k and Hk (u2 ) = bk uq2k . Let ~u be the pointwise blow-up solution. The function Jk is defined in Ω by Jk (x, t) = ∂t2 uk (x, t) − Fk (~u) + mk ∂t uk (x, t) = △uk (x, t) + mk ∂t uk (x, t),

852

H. Uesaka

where mk is some positive large constant given afterward, and it satisfies  ¤Jk (x, t) = G′k (u1 )J1 (x, t) + Hk′ (u2 )J2 (x, t)    + G′′ (u )|∇u |2 + H ′′ (u )|∇u |2 , k

1

1

k

2

2

J (x, 0) = △fk (x) + mk gk ,    k ∂t Jk (x, 0) = △gk + mk (△fk + Fk (f~)).

By the same reasoning as in the proof of positivity of solutions (Theorem 2.1) we get the following lemma. Lemma 5.1. Suppose fk (x) > 0 and that the assumptions A.1.1, A.2.1-2 and A.5.1 are satisfied. Let mk be sufficiently large such that ∂t Jk (X, 0) ≥ |∇Jk (X, 0)| holds. Then we have Jk (x, t) ≥ 0 in Ω. Theorem 5.1. Let ~u be the solution stated in Lemma 3.1. Then there exists a suitable positive constant Ck such that uk (x, t) ≤

Ck ′ in Ω, (ϕ(x) − t)pk

where

p′k =

2 . pk − 1

(12)

Proof. Let k = 1. We have ∂t2 u1 + m1 ∂t u1 ≥ a1 up11 + b1 uq21 ≥ a1 up11 . By multiplying this inequality by ∂t u1 e2m1 t and by integrating it from t0 to t, where t0 < t < ϕ(x), we have Z t 2 ∂s (u1 (·, s)p1 +1 )ds + C2 ≥ C3 u1 (·, t)p1 +1 . (∂t u1 (·, t)) ≥ C1 t0

We choose a suitable constant C and we have ∂t u1 (x, t) ≥ Cu1 (x, t)

p1 +1 2

, C=

p

C3 .

(13)

We integrate both sides of (13) from 0 to t and get (12) for k = 1. We can carry out the same calculation for u2 and get (12) for k = 2. Remark 5.1. We do not have an estimate for u1 and u2 from below, but at least we have u1 (x, t) + u2 (x, t) ≥ where r′ =

2 r−1 ,

C , (ϕ(x) − t)r′

r = max{p1 , p2 , q1 , q2 }.

Remark 5.2. We can consider the same problem in Rd ×[0, T ) for d = 1, 2.

The Blow-up Boundary for a System of Semilinear Wave Equations

853

References 1. S. Alinhac, Blowup for Nonlinear Hyperbolic Equations, Birkh¨ auser, 1995. 2. L. A. Caffarelli and A. Friedman, The blow-up boundary for nonlinear wave equations, Trans. Amer. Math. Soc. 297 (1986), 223–241. 3. L. A. Caffarelli and A. Friedman, Differentiability of the blow-up curve for one dimensional nonlinear wave equations, Arch. Ration. Mech. Anal. 91 (1985), 83–98. 4. K. J¨ orgens, Das Anfangswertproblem im Grossen f¨ ur eine Klasse nichtlinearer Wellengleichungen, Math. Z. 77 (1961), 295–308. 5. J. B. Keller, On solutions of nonlinear wave equations, Comm. Pure Appl. Math. 10 (1957), 523–530. 6. H. Lindblad, Blow-up for solutions of ¤u = |u|p with small initial data, Comm. Partial Differential Equations 15 (1990), 757–821. 7. H. Uesaka, Oscillation or nonoscillation property for semilinear wave equations, J. Comput. Appl. Math. 164/165 (2004), 723–730. 8. H. Uesaka, Non-negative solutions of the Cauchy problem for semilinear wave equations and non-existence of global non-negative solutions, to appear, Advances in Analysis (Proceedings of 5th ISAAC Congress 2005), World Scientific.

854

H. Uesaka

855

AUTHOR INDEX Abdulayev, S. K., 593 Abreu Blaya, R., 167 Achour, D., 783 Ak¸ca, H., 635 Akhalaia, G., 303 Akperov, A. A., 593 ¨ 311 Aksoy, U., Akta¸s, M. F., 507 Almeida, A., 601 Andersson, A., 281 Aripov, M., 790 Ashyralyev, A., 645, 654, 663, 670, 679, 689, 698, 707 Avcı, M., 321 Aydın, A., 717 Ayele, T. G., 611 Barsegian, G. A., 3 Barza, I., 197 Bastos, M. A., 111 Begehr, H., 327 Berdyshev, A. S., 727 Bian, X., 175 Bitew, W. T., 611 Bogolubov, N. N., 743, 760 Bory Reyes, J., 167 B¨ ohme, C., 415

Dibl´ık, J., 515 Dovbush, P. V., 122 Drihem, D., 622 Dryga´s, P., 291 Erdo˘ gan, A. S., 654 Erg¨ un, E., 525 Evans, W. D., 33 Fardigola, L. V., 337 Fernandes, C. A., 111 Fonte, M., 434 Fujita, K., 567 Fujiwara, H., 574 Ger¸cek, O., 663 Ghisa, D., 197 Giorgadze, G. K., 347 Golberg, A., 218 Golenia, J., 760 Goncharov, A. P., 229 Guliyev, V. S., 132 Guseinov, G. S., 525 Hirosawa, F., 444 Hung, N. Q., 185 Jachmann, K., 454

Covachev, V., 635 Covacheva, Z., 635 C ¸ elebi, A. O., 311 Dadashova, I. B., 260 Dalla Riva, M., 208 Del Santo, D., 425 Demirci, E., 654

Kadowaki, M., 465 Karapınar, E., 794 Karas¨ ozen, B., 717 Karlovich, Y. I., 111 Karupu, O. W., 233 Kats, B. A., 241 Khaydarov, I. U., 824 Khydarov, A., 790

856

Author Index

Lanza de Cristoforis, M., 249 Luong, N. C., 185 Makatsaria, G., 303 Mamedkhanov, J. I., 260 Mamedov, K. R., 798 Manjavidze, N., 303 Martinez, M., 679 Mashiev, R. A., 321 Matsuura, T., 574 Menken, H., 798 Mert, R., 535 Mezrag, L., 806 Mochizuki, K., 476 Mohamad, S., 635 Moussai, M., 142 Nakazawa, H., 465 Nourouzi, K., 814 O˘ gra¸s, S., 321 ¨ Ozbekler, A., 545 ¨ Ozdemir, Y., 689 ¨ urk, E., 698 Ozt¨

Selmi, R., 368 Sj¨ ostrand, J., 45 Sommen, F., 167 Son, L. H., 185 S¨ ozen, Y., 707 Sugimoto, M., 486 Svoboda, Z., 515 Taneri, U., 760 Tiryaki, A., 507 Todorova, G., 835 Topal, F. G., 555 Uesaka, H., 845 U˘ gurlu, D., 835 Urinov, A. K., 824 Vaitekhovich, T. S., 327 Wang, J., 379 Wang, Y., 379 Watanabe, K., 465 Wirth, J., 495 Wu, Z., 132 Xu, Y., 387

Pastor, J., 679 Pe˜ na Pe˜ na, D., 167 Piskarev, S., 679 Plaksa, S. A., 268 Prykarpatsky, A. K., 743, 760 Qi, Q., 152 Qiao, Y., 175 Rajabov, N., 356 Rakhmatullaeva, N. A., 727 Ramazanov, M. D., 734 Reissig, M., 415, 454 Rezapour, S., 819 Ruzhansky, M., 486 R˚ uˇziˇckov´ a, M., 515 Saadi, K., 806 Saitoh, S., 574, 584 Sawano, Y., 574

Yamada, M., 584 Yang, H., 175 Yantır, A., 555 Yıldırım, C. Y., 75 Yordanov, B., 835 Y¨ uksel, U., 395 Zafer, A., 535 Zhang, Y., 152 Zhong, S., 404