Complex Analysis and Potential Theory: Proceedings of the Conference Satellite to ICM 2006 [illustrated edition] 9789812705983, 981-270-598-8

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Complex Analysis and Potential Theory

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Potential Theory Proceedings of the Conference Satellite to ICM 2006 Gebze Institute of Technology, Turkey 

8­14 September 2006

Editors

Tahir Aliyev Azeroglu Gebze  Institute cf Technology, Turkey

Promarz M. Tamrazov National Academy of Sciences,  Ukraine

Gebze  Institute of  Technology

World Scientific NEW JERSEY •  L O N D O N . SINGAPORE •  B E I J I N G • SHANGHAI • HONGKONG • TAIPEI  •  C H E N N A I

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Published by

World Scientific Publishing Co. Pte. Ltd.

5 Toh Tuck Link, Singapore 596224 USA ofice: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 CIK ofice: 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.

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COMPLEX ANALYSIS AND POTENTIAL THEORY Proceedings of the Conference Satellite to ICM 2006

Copyright 0 2007 by World Scientific Publishing Co. Pte. Ltd

All rights reserved. This book, orparts thereoJ niay 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.

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ISBN-13 978-981-270-598-3 ISBN-10 981-270-598-8

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Printed in Singapore by World Scientific Printers (S) Pte Ltd

V

PREFACE The satellite to ICM (2006) International Conference on Complex Analysis and Potential Theory was held at the Gebze Institute of Technology in Gebze, Turkey during the period 8-14 September, 2006. The Proceedings involves the most of presentations delivered at the Conference by the participants, among of which many of the top-notch mathematicians were present. Topics discussed include Grunsky inequalities and Moser's conjecture, speed of approximation to degenerate quasiconformal mappings, combinatorial Theorems of complex analysis, geometry of the general Beltrami equations, polyharmonic Dirichlet problem, isoperimetric inequalities for sums of reciprocal eigenvalues of the Laplacian, contour-solid theorems for finely meromorphic functions, residues on a Klein surface, functional analytic approach to the analysis of nonlinear boundary value problems, generalized quasiconformal mappings, properties of separately quasi-nearly subharmonic functions, implicit function theorem for Sobolev Mappings, The Martin boundary and the restricted mean value property for harmonic functions, approximate properties of the Bieberbach polynomials on the complex domains, asymptotic expansions of solutions of the heat equation with generalized functions initial data, Hausdorff operators, harmonic transfinite diameter and Chebyshev constants, analogues of the Mittag-Leffler and the Weierstrass theorems for harmonic differential forms on noncompact Riemannian manifolds, harmonic commutative Banach algebras and spatial potentials fields, parameter space of error functions and others. Besides, Prof. C. C. Yang proposed to include in the Proceedings a set of the articles devoted to so-called "open problems," i.e., the problems of great importance, but unsolved yet. This suggestion was approved by the scientific peers and accepted. In this connection, the Proceeding is composed of two Parts. The first one, Part A, involves the talks, which were presented and discussed at the Conference. The second one is devoted to the open problems completely. However, one can find some open problems in the first part as well, where they are given in passing with the main contents of the talks. The articles published in this Proceedings are oriented on the active researchers, who works in these areas directly and in the adjacent fields of mathematics and would like to update recent developments in the field, This book will be useful for the Ph.D. and M.S. students as well as researchers who just start only or continue their activity in this area of mathematics and its applications in engineering. We would like to thank all participants for their invaluable contributions. We acknowledge great efforts of our colleagues, Dr. Faik Mikailov, Tugba Akyel and

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others who made major contribution t o the organization of the meeting. Special thanks t o Prof. Alinur Buyukaksoy, Rector of Gebze Institute of Technology, who supported every stages of preparation and holding of the meeting, and t o the Scientific and the Technical Council of Turkey (TUBITAK) for the support of this conference. Tahir Aliyev Azerojjlu Department of Mathematics, Gebze Institute of Technology, Gebze, 41410 Kocaeli, Turkey

Promarz M. Tamrazov Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev, 01601, Ukraine

vii

PARTICIPANTS Olli Martio, University of Helsinki, Finland Vladimir Mazya, Ohio State University, USA Samuel Krushkal, Bar-Ilan University, Ramat Gan, Israel Hakan Hedenmalm, Royal Institute of Technology, Stockholm, Sweden Promarz M. Tamrazov, Institute of Mathematics of NAS, Ukraine Cliung Chun Yang, HKUST, Hong Kong Vladimir Miklyukov, Volgograd State University, Russia Reiner Kuhnau, Martin-Luther Universitat Halle-Wittenberg, Germany Bodo Dittmar, Martin-Luther-Universitat Halle-Wittenberg, Germany Bogdan Bojarski, Polish Academy of Sciences, Warsaw, Poland Heinrich Begehr, I. Math. Inst., F U Berlin, Germany Tahir Aliyev Azeroglu, Gebze Institute of Technology, Turkey Sergei Favorov, Kharkov, Ukraine Tatyana Shaposhnikova, Linkoping, Sweden Lev Aizenberg, Bar-Ilan University, Ramat-Gan, Israel Akif Gadjiev, Institute of Mathematics and Mechanics, Baku, Azerbaijan Massimo Lanza De Cristoforis, Universita Degli Studi di Padova, Italy Arturo Fernandez Arias, UNED, Madrid, Spain Yurii Zelinskii, Institute of Mathematics of NAS, Ukraine Anatoly Golberg, Bar-Ilan University, Ramat-Gan, Israel A. V.Pokrovskii, Institute of Mathematics of NAS, Ukraine Juhani Riihentaus, University of Joensuu, Joensuu, Finland Sergiy Plaksa, Institute of Mathematics of NAS, Ukraine Matti Vuorinen, University of Turku, Finland Oleg F. Gerus, Zhytomyr, Ukraine Shunsuke Morosawa, Kochi University, Japan Victor V. Starkov, Petrozavodsk State University, Russian Allami Benyaiche, Universiti! Ibn Tofail, Kenitra, Morocco Igor V. Zhuravlev, Volgograd, Russia Elijah Liflyand, Bar-Ilan University, Ramat-Gan, Israel Mubariz T. Karayev, Suleyman Demirel University, Isparta, Turkey Kunio Yoshino, Sophia University, Tokyo, Japan Eugenia Malinnikova, Trondheim, Norway Daniyal M.Israfilov, Balikesir University, Turkey Aydin Aytuna, Sabanci University, Turkey

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... 

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Aydin Aytuna, Sabanci University, Turkey Ana,toliy Pogoruy, Zhytomyr, Ukraine Andrey L. Targonskii, Institute of Mathematics of NAS, Ukraine Yasuyuki O h , Sophia University, Tokyo, Japan Bulent N. Ornek, Gebze Institute of Technology, Turkey Shahram Rezapour, Azarbaidjan University of Tarbiat Moallem, Iran Hiroshige Shiga, Tokyo, Japan Vyacheslav Zakharyuta, Istanbul, Turkey H. Turgay Kaptanoglu, Ankara, Turkey Olena Karupu, Institute of Mathematics of NAS, Ukraine Mehmet Acikgoz, University of Gaziantep, Turkey Yu.V. Vasil’eva, Institute of Mathematics of NAS, Ukraine Coskun Yakar, Gebze Institute of Technology, Kocaeli, Turkey Allaberen Ashyralyev, Fatih University, Istanbul, Turkey Ali Sirma, Gebze Institute of Technology, Gebze, Kocaeli, Turkey Mehmet Kucukaslan, Mersin University, Turkey Peter Tien-Yu Chern, Kaohsiung, Taiwan

1x

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CONTENTS Preface

V

vii

Participants

Part A

TALKS

Strengthened Moser’s Conjecture and Finsler Geometry of Grunsky Coefficients S. Krushkal Decompositions of Meromorphic Functions Over Small Functions Fields C.-C. Yang and P. La Speed of Approximation to Degenerate Quasiconformal Mappings and Stability Problems V. M. Miklyukov

3

zy

17

33

Grunsky Inequalities, Fredholm Eigenvalues, Reflection Coefficients R. Kuhnau

46

Sums of Reciprocal Eigenvalues B. Dittmar

54

Geometry of the General Beltrami Equations B. Bojarski

66

A Particular Polyharmonic Dirichlet Problem H. Begehr

84

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Finely Meromorphic Functions in Contour-Solid Problems T. Aliyeu Azeroilu and P. M. Tamrazou

116

A Generalized Schwartz Lemma at the Boundary T. Aliyev Azeroilu and B. N . Ornek

125

zyz zyxwvu z

X

Singular Perturbation Problems in Potential Theory and Applications M. Lanza de Gristoforis

131

Residues on a Klein Surface A . Ferna’ndez Arias and J . Pirez Alvarez

140

Combinatorial Theorems of Complex Analysis Yu. B.Zelinskii

145

Geometric Approach in the Theory of Generalized Quasiconformal Mappings 148

A . Golberg

Separately Quasi-Nearly Subharmoriic Functions J. Riihentaus

156

Harmonic Commutative Banach Algebras and Spatial Potential Fields S. A . Plaksa

166

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The Parameter Space of Error Functions of the Form a e-w2dw S. Morosawa

 s;

174

On Potential Theory Associated to a Coupled PDE A. Benyaiche

178

An Implicit Function Theorem for Sobolev Mappings I. V. Zhuravlev

187

A Relation Among Ramanujan’s Integral Formula, Shannon’s Sampling Theorem and Plana’s Summation Formula K. Yoshino

191

Asymptotic Expansions of the Solutions to the Heat Equations with Generalized Functions Initial Value K. Yoshino and Y . Oka

198

On the Existence of Harmonic Differential Forms with Prescribed Singularities E. Malinnikova

207

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Approximate Properties of the Bieberbach Polynomials on the Complex Domains

D.M. Israfilov

214

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Harmonic Transfinite Diameter and Chebyshev Constants N . Skiba and V. Zakharyuta

222

On Properties of Moduli of Smoothness of Conformal Mappings 0. W. Karupu

231

Strict Stability Criteria of Perturbed Systems with Respect to Unperturbed Systems in Terms of Initial Time Difference C. Yakar

239

Piecewise Continuous Riemann Boundary Value Problem on a Closed Jordan Rectifiable Curve Yu. V. Vasil’eva

249

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A Note on the Modified Crank-Nicholson Difference Schemes for Schrodinger Equation A. Ashyralyev and A. Sirma

Part B

256

OPEN PROBLEMS

Some Old (Unsolved) and New Problems and Conjectures on Functional Equations of Entire and Meromorphic Functions C.-C. Yang

275

An Open Problem on the Bohr Radius L. Aizenberg

279

Open Problems on Hausdorff Operators E. Laflyand

280

Author Index

287

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PART A

TALKS

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3

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STRENGTHENED MOSER’S CONJECTURE AND FINSLER GEOMETRY OF GRUNSKY COEFFICIENTS SAMUEL KRUSHKAL

Department of Mathematics, Bar-Ilan University,52900 Ramat-GanJsrael and Department of Mathematical Sciences, University of Memphis, Memphis, T N 38152, USA

T h e Grunsky and Teichmiiller norms ~ ( f and ) k(f) of a univalent2unction f in a finitely connected domain D 3 00 with quasiconformal extension t o @.  are related by ~ ( f 5) k(f). In 1985, Jurgen Moser conjectured that any univalent function in the disk A* =  ( 2 :  121  >  1) can be approximated locally uniformly by functions with ~ ( f 1) if and only if

5 1,

,,h% a,,z,x,I  

zyx zyx zyxwv

where a,,

are generated by

m,n=l

x = ( 2 , ) runs over the unit sphere S(12)of the Hilbert space 1’ with

w 1 1 ~ 1 =  1 ~C   lxnl 2 ,  1

and the principal branch of logarithmic function is chosen (cf.Gr).The quantity

{I  c 00

N(f)

:= sup

m,n=l

Jmn mnz,,z,l

: x = ( 2 , ) E  S(Z’)) 

is called the Grunsky norm o f f . We denote by C the class of univalent holomorphic functions f ( z ) = z bo b1z-l .. .  mapping A* into \  {0}, and by C ( k ) its subclass of f with kquasiconformal extensions to the unit disk =  1 .( 2 - m a j o r i z e do n D . T h e n we have:

(i) t h e canonical h o m e o m o r p h i s m F : B(0,R ) + R and t h e inverse homeom o r p h i s m F-' : R + B(0,R ) ) have t h e L u s i n N - p r o p e r t y if a n d only i f f : D + R (respectively, f-' : R + D ) have this property; ( i i ) if f E W;:(D),

(iii) i f f - ' DIED:

t h e n F E Wly:(B(O,R));

E Wlkr(fl) f o r a n u m b e r

J

(911922

-g:2)

4

Q, 

25

Q

5

Q 

dx1dx2 _  1 and

EO

(11)

Ih(z) ­ 21  5 C’ A(RJZ6,), and f o r every measurable set E

h = W,

C  B(0,R )

0

I.(

w-l

, 

Here

G

E B(O,R),

(12)

the following estimate holds

I’H2(h(E)) ­  ‘Hz(E)l5 C”A,(RJZS,).

A, ( E )

’majorized in D . In other words, there exists a function K ( z ) E  W’,2(D),for which

P ( Z )5 K ( z ) for a.e.

zyxw zyxw IC

E D.

Denote by Lf(R’) the set of the functions cp(z) which are as the Bessel potentials cp(X)

1

= 

w2

Gl(lZ ­ Y O  4 Y )  d YldY2, 

of the functions u E  L2(R2)with the Bessel kernel of the order 1 (see, e.g., Adams and H e d b e ~ - g ~Since ~ ) . D is a disk, there exists by Theorem 1.2.3 of34 for

zyxwvuts z z zyx zyxwvu 41

every function p(x) E W’>’(D) a function cp*(x) E L:(IR2) such that a.e. cp*(x)=  p(x). Thus, since K E W112(D),there is a function u E L 2 ( R 2 )such that

J

zyx zy zyxwvutsrq

K ( x ) =  G1(1x ­ y I ) u ( y ) d l ~ l d y z a.e. on w2

By Theorem 6.2.1,34the set

{x

E  D : lim

D.

K ( < )= co}

C-tx

has zero conformal capacity, and therefore the following statement holds. Theorem 4.1. If P i s W1>’-majorized,the set

P, = {z E D

: lim T+O

J

­ 

r2

P(y)dxn=ca}

B(x,r)

is of zero capacity. Problem I. Describe the class of functions p ( z ) : D majorized in D .

+ [ l , ~that ) are W1>’-

B. The class is not empty. We privide the conditions for characteristic p ( z ) which ensure that the quantities 6, ­­+  0 and, consequently, the class of mappings described by Theorem 3.1 is not empty. Namely, the following statement holds. Theorem 4.2. Let p(x) :  D + [l,co]be a continuous function of the class W’i2(D). Then the quantities S, defined b y

satisfy the condition

zyxw

lim S, = 0 ,--to3

Sketch of the proof. Let ( p ( z ) ,O(x))be the Lavrentiev’s characteristics in D, and let p : D ­+ [l,fco]  be continuous. For a given sequence Q , ­­+  00,  we find the sets I , c D and the functions P, involved in Theorem 3.1. Since p is continuous, the sets I , are closed. Suppose that for some Q , > 1, the set I , G D . Without loss of generality, we may assume that D \ I , is a domain.

42 

zyxwvutsr zyxwv

For every function K, we have

6: 5

J’ lVI(:,/2(z)I2 p ( z )dz1dz2

D

Let cp :  D \  I , be a function which is extremal for the condenser capacity (I,,aD; D ) , that is, cp = 0 on a D , cp = 1 on In and

J’

~ ~ c p ( zdzldzZ ) l ~   = cap (I,, a D ; D )

D\In

We choose K, of the form

This function has the desired properties and by the proved above,

Thus, if lim

n+m

Qb cap ( I n ,a D ; D ) = 0 ,

then 6, + 0. We show that the assumption p ( z )

An,,, = 

J’

In0 \In

zyxw

W ’,2 (D) implies (14).Indeed, let

1 ~ p ( z ) 1dzIdz2 2  1 so that the set In, lies strongly interior to D . Fix Q n > Qno. Consider the condenser ( D \ I n o , I,; D ) . The function

43

is admissible in the variational problem for the capacity of the condenser. Therefore,

This yields

n

zyxwvut zyxw z zyxwvuts zyxw

we get that Setting Qn, =  $Qn, + 00, and the theorem is proved.

the condition (14) holds. Thus, 6,

Problem 11. Describe the class of continuous functions p ( z ) : D which

+ 0

as 0

+ [l,co) for

liminf 6, = 0 . n+oo

zyxwvutsrq

References

1. I.N. PESIN:Mappings which are quasiconformal in the mean. ­  Dokl. Akad. Nauk SSSR 187, 740-742, 1969 (in Russian). 2. 0. LEHTO:Homeomorphisms with a given dilatation. ­  Proceedings of the Fifteenth Scandinavian Congress (Oslo, 1968), Springer-Verlag, 55-73, 1970. 3. 0. LEHTO:Remarks o n generalized Beltrami equations and conformal mappings. ­  Proceedings of the Romanian-Finnish Seminar on Teichmuller Spaces and Quasiconformal Mappings (Bragov, 1969), Pub1 House of the Acad. of the Socialist Republic of Romania, Bucharest, 203-214, 1971. and G.D. SUVOROV: O n the existence and uniqueness of quasi4. V.M. MIKLYUKOV conformal mappings with unbounded characteristics (in Russian). ­  Investigations in the theory of functions of a complex variable and its applications, Inst. Mat. Akad. Nauk Ukr., Kiev, 45-53, 1972. Existence and Uniqueness of Quasiconformal “in Mean” Mappings 5. V. I. KRUGLIKOV: (in Russian). ­ Metric questions of functions theory and mappings, n. 4, 123-147, 1973. General properties of quasiconformal mappings (in Russian). ­  6. P. P. BELINSKII: Novosibirsk, Nauka, Sibirskoe otdelenie, 1974. 7. G. DAVID:Solutions de l’e‘quation de Beltrami avec Ilplloo =  1. ­  Ann. Acad. Sci. Fenn. Ser. A I Math. 13, 25-70, 1988. Compactness properties of p-homeomorphisms. ­  Ann. Acad. Sci. Fenn. 8. P . TUKIA: Ser. A I Math. 16, 47-69, 1991. and J.A. JENKINS: O n solutions of the Beltrami equation. ­  J . Anal. 9. M.A. BRAKALOVA Math. 76, 67-92, 1998. 0 . MARTIO,T. SUGAWA and M. VUORINEN: O n the Degenerate 10. V. GUTLYANSKII, Beltrami Equation. Trans. Amer. Math. SOC.,v. 357, n. 3, 2005, 875-900. 11. 0. MARTIO,V.M. MIKLYUKOV: O n existence and uniqueness of degenerate Beltrami equation. ­  Complex Var. Theory Appl. 49, n. 7-9, 2004, 647-656., 12. V.M. MIKLYUKOV: Isothermac coordinates o n singular surfaces. ­  Mat. Sb. 195,n. 1, 2004, 69-88. Conformal representation. ­  Ca.mhridge at the University Press, 13. C . CARATH~ODORY: 1932.

zyxw zyxwvu

44

zyxwv zy zyxwvu zyxw

14. J. EELLSand B. FUGLEDE: Harmonic m a p s between R i e m a n n i a n polyhedra. ­  Cambridge Tracts in Mathematics 142, Cambridge Univ. Press, UK, 2001. 15. Yu.G. RESHETNYAK: Two-dimensional manifolds of bounded curvature, ­  Modern problems of math. Fundamental directions, v. 70, Itogi nauki i thechniki, VINITI, Moscow: 1989, 8-189. 16. T. TORO:Surfaces with generalized second fundamental f o r m in L2 are Lipschitz manifoZds. ­  J. Differential Geometry, v. 39, 1994, 65-101. 17. S. MULLER,V. S V E R ~ K On : surfaces of finite total curvature. ­  J. Differential Geometry, v. 42, 1995, 229-258. 18. I.M. GRUDSKII: Construction of i n n e r coordinates in composite R i e m a n n i a n surfaces. ”Differential, integral equations and complex analysis”, Elista, 1986, 30-45. 19. I.M. GRUDSKII: Christoffel ­ Schwarz f o r m u l a f o r polyhedral surfaces. ­  DAN SSSR, v. 307, n. 1, 1989, 15-17. Les conditions de monoge‘ne‘ite‘. ­  Act. sci. et ind. v. 329, 1936, 1-52. 20. D . E . MENCHOFF: 21. Yu.Yu. TROHIMCHUK: Continuous mappings and conditions of monogeneity. ­  GIFML, Moskow, 1963. 22. E.P. DOLZHENKO: O n ”erasure” of singularities of analytic functions. ­ Uspechi math. nauk, v. 18, n. 4, 1963, 135-142. 23. S.YA. KHAVINSON: On erasure of singularities. ­  Litov. math. sb., 111, 1, 1963, 271287. 24. G . DAVIDand P. MATTILA:Removable sets f o r Lipschitz harmonic functions in the plane. ­  Revista Matem. Iberoamericana, v. 16, n. 1, 2000, 137-215. S u r u n e classe de representations continue. ­  Math. sb., v. 42, 25. M.A. LAVRENTIEV: n. 4, 1935, 407-424. On distortion under quasiconformal mappings. ­  DAN SSSR, v. 91, 26. P.P. BELINSKII: n. 5, 1953, 997-998. 27. P.P. BELINSKII: On measure area under quasiconformal mappings .  ­  DAN SSSR, v. 121, n. 1, 1958, 16-17. Solution of extremal problems of the quasiconformal mappings theory 28. P . P . BELINSKII: with the variational method. ­  Sib. math. j., v. 1, n. 3, 1960, 303-330. 29. S.L. KRUSHKAL’: O n mappings E-quasiconformal in ”means”. ­  Sib. math. j., v. 8, n. 4, 1967, 798-806. 30. G . D . SUVOROV: Equistability of canformal mappings of closed domains. ­  Ukr. math. j., v. 20, n. 1, 1968, 78-84. On a class of quasiconformal mappings with invariant boundary 31. J. LAWRYNOWICZ: points, II, Applications and generalizations. ­  Ann. polon. math., v. 21, n. 3, 1969. 32. V.I. KRUGLIKOV, V.M. MIKLJUKOV: Stability theorems f o r B L - m a p p i n g s . ­  Dopov. AN USSR. ser. A, 1972, n. 5, 421-423 (in Ukr6inian); In sb. ”  Metr. vopr. teor. funct. i otobr.” , vyp. 111, ”Naukova Dumka” , Kiev, 1971, 55-70 (in Russian). 33. I.A. VOLYNEC: On distortion u n d e r B L - m a p p i n g s . ­  Sib. math. j., v. 18, n. 6, 1977, 1259-1270 (in Russian). 34. D.R. ADAMSand L.I. HEDBERG: f i n c t i o n Spaces and Potential Theory. ­  SpringerVerlag, Berlin ­ Heidelberg ­ New York etc., 1996. 35. V.M. MIKLYUKOV: Conformal mapping of nonregular surface and its application. ­  Volgograd State University Press, Volgograd, 2005 (in Russian). 36. H.A. SCHWARZ:Conforme Abbildung der Oberflache eines Tetraeders auf die OberfEache einer Kugel. ­  J reine angew. Math., v. 70, 1869, 121-136. 37. REINERKUHNAU:Trianguliere Riemannsche Mannigfaltigkeiten mit ganz-linear Bezugssubstitutionen und quasikonforme Abbildungen mat stuckweise kpnstanter k o m plexer Dilatation. ­  Mathematische Nachrichten, Band 46, Heft 1-6, 1970, 243 -261.

z zyxwvu 45

38. WALTER GRESKY: Konforme Abbildung der Oberflache eines rektangularen Hexaeders auf die Kugeloberflache. ­  Inaugural ­ Dissertation zur Erlangung der Doktorwurde der Hohen Philosophischen Fakultat der Universitat Leipzig, Weida, i. Thur. 1928, 1-74.

46

zyxwvutsrqp

zyxwv zyx

GRUNSKY INEQUALITIES, FREDHOLM EIGENVALUES, REFLECTION COEFFICIENTS REINER KUHNAU

F B Mathematik und Informatik Martin-Luther- Universitat Halle- Wittenberg 0-06099 Halle-Saale, Germany [email protected]

We give the exact domain of variability of a fixed Grunsky functional in the class of all hydrodynamically normalized schlicht conformal mappings of the exterior of the unit disk for which the unit circle transforms onto a quasicircle with a given Fredholm eigenvalue.

Keywords: Univalent functions, Grunsky coefficients, quasiconformal map, Fredholm eigenvalues.

zyx zyx zy

1. Introduction

Today a central role in Geometric Function Theory plays the Grunsky functional. We will restrict ourself here to the simply-connected case. Then we consider in the complex plane the class of all schlicht conformal mappings w = f ( z ) of IzI > 1 with the hydrodynamical normalization

The corresponding Grunsky coefficients a k l are defined by the development

and with a given fixed system of complex numbers the Grunsky functional Gf is defined as

Gs = 

C", k I-1

zyx 1,

1xk

l2 > 0)

zyxwvutsrqpon

­ 

11.

, xn ( n 2

~ 1 ,. .  .

aklxkxl 1Zkl2

ck=l7

The starting point of the theory was the famous

zyx

. 

(3)

Theorem 0 [Gr], [PI. For every n and all fixed z1, ,x, the exact domain of variability of the functional Gf is the closed unit disk IGfI 5 1

' 

(4)

z zyxwv zyxwvu 47 

This theorem solves the extremal problem IGfI  + max if we fix the constants x k and vary the mappings f ( z ) .

It arises the following complementary question: What is the solution of the extremal problem lGfl + max if we conversely fix the mapping f ( z ) E  C and vary the systems xk? Surprisingly, the maximal value for 1GfI is not always again 1. We have rnax,, IGfI  . ~ An easy consequence of the Expansion Theorem is the following formula for the sum of all reciprocals3

zy zyxwvu 57

Theorem 5.(Dittmar 2002) Let f be a conformal mapping from the unit disk U1 onto the domain D with the area A, then it holds

zyxwvut zyxwvut

where G ( z ,()  denotes Green's function of the unit disk. From this theorem it follows a formula'l in terms of the coefficients of the Taylor series for the conformal mapping which is similar to a formula for the torsional rigidity given in [25, p.3301 Theorem 6.(Hantke 2006) For the eigenvalues of the fixed membrane problem holds 0 0 .

0 3 0 3

k=l m=l 1=1

m=l l=1

zyxwvut k=2

where the coefficients A, B, C , D , E are known and 00

0 3 0 3

m=l n=l

n=O

It is worth pointing out that in some cases it is possible to calculate (10) exactly.'' Examples 1. Disk Let D1 be the unit disk then holds

zyx

zyxwvutsr

­  03 8 _7r2 _ _5   + c n = 1 4(2n+4)(4n+4)(2n+2) ­ 48 32 For the unit disk see also [3, p.511.

4

C;l*=,,

' 

2. Cardioid

f ( z ) = z + z 2 / 2 maps the unit disk onto a cardioid. It follows from the formula above C;, = &x2 ­ &.  Similar results are given in" for the image of the

$ 

+ 

unit disk by f n ( z ) = z ;zn,n = 3,4, ...  .  For the cardioid follows in the same matter also the value ll7r/48 for the torsional rigidity" .  Starting with (10) it follows3 Corollary 1. Let f be the conformal mapping from the unit disk U1 onto the domain D with the area A =  If'(z)12dA_  0. Then

zyx zyxwvu z

where n is either a positive integer or +m, and D is the conformal image of the unit disk U1 under the conformal mapping f with If'(0)l = 1, with strict inequality unless D equals the unit disk. 2 . 2 . Free membrane

In the free membrane case the situation is more complicated. Encouraged by a conjecture of Kornhauser and Stakgold it was proven by Szego 1954 that of all simply connected domains of a given area, the circle yields the maximum value of p2. Szego and Weinberger noticed that the same proof causes the disk to minimize

zyxw zyxwvu

zyxwv 59 

­1+ ­   1 P2

P3

of all simply connected domains of a given area. Generalizations are given by C. Bandle 1972, T. Gasser and J. Hersch 19681’7 and N. Nadirashvili 1997.” In order t o derive a formula similar to (10) we introduce a special symmetric function which will be the kernel in an integral equation for the eigenfunctions of Problem (2).5 Lemma 3. Let N f ( z , C ) be the following symmetric function depending on an univalent conformal map f

zyxwvutsrq zyxwvuts

zyxw zyx

where

sulIf‘(z)12dAZ 2 . Let the following conditions are satisfied (i) for a given 70 E  A, limsup C+z,

f (01 = 0(P1(IZ ­ "iol)),

­ 

CED

E  (aD\Q),

+ 70, ( 3 )

(ii) for all

VY E  A f \ {YO} limsup 14") 

­ f(O1 = O($((I. ­ "iol)), 

z E (aD\Q>, z

­+ 

70.

C+z,

(4) Then 4("(dJio)by setting

Then we have

Lo

ii(t)= 

S,(t

­  s ) ~ ( sda, )

V t E clI",

zy

zyxwvutsr zy zyxwv zyx zyxwv zyxw z 133

and

/ 

+ i v ( t )+ 

dU

-(t)

1

= - - p ( t )

2

8UO

.h. 

+ 

an i

ui(t). DtS,(t

b't E

s ) q ( s )do,

­ 

dIi,

u"(t) . D&(t ­ s ) p ( s ) dos

where Dt denotes the gradient with respect to the t-variable. By plugging (4)-(6) into the boundary conditions of (2), one obtains a system of two integral equations in the unknowns 7,  p, which depends on the parameter E €10, EO[. We rewrite such a system into the abstract form

Mk, 17, PI = 0 , 

(7)

where M is a suitable nonlinear map of 10, ~O[xC~,"(diTz)  x  Co>"(dlI")to Co@(dlIi) x  C0>"(aII0). In many instances M may be seen t o be the restriction of some regular map (still denoted M ) of ]  ­ E O , E O [ X C ~ , "x (Co@(dIo) ~IZ) to Co>"(dIi) x  Co>"(dIo), and equation M[O,0, p] = 0 corresponds to the integral equation relative to problem (I) for 6. Under suitable conditions on B", Bi, we may be able to solve locally around ( O , O , p ) equation ( 7 ) , and to prove that the set of solutions of (7) is the graph of a nonlinear operator E H ( E [ E ] , R [of E ]] )­ E O , E O [ to C0,"(dIi) x  C0>"(dIo) for a possibly smaller € 0 . If B", Biare real analytic, one would expect that (I?[.], I?[.]) is also real analytic. Then by setting

zyxwvu

~ ( tt ) ,E

Lo

S,(t

L

s ) R [ E ] (dos s ) + 

­ 

S,(t ­ S E ) E [ E ]dos (S)

V t E c l A ( ~ ) ,(8)

one obtains a solution of (2) which converges in the C1)u-norm on the compact subsets of clIo \ (0) t o U as E tends to 0. Once the family of solutions { u ( E ,.))tE]O,Eo[ is established, two questions appear as natural.

4

0 be a bounded open subset of I['\ (0) such that 0 cl0. What can be said on the map 10, E O [ ~E ++  U ( F , t),clfi E C',"(clfi) around E = 0? (jj) What can be said on the map 10, E O [ E~ H € [ E ] E J D u ( Et)I2 , d t E  R around E = O? (j) Let

SA(.,

Problems of this type with linear boundary conditions have long been investigated with the methods of Asymptotic Analysis, which aims at giving complete asymptotic expansions of the solutions in terms of the parameter E. It is perhaps difficult to provide a complete list of the contributions. Here, we mention the work of Kozlov, Maz'ya and Movchan Ill],Maz'ya, Nazarov and Plamenewskii [19],

134

Kuhnau [la], Movchan [20], Ozawa [22], Ward and Keller [27]. For nonlinear problems on domains with small holes far less seems t o be known, we mention the results which concern the existence of a limiting value of the solutions or of their energy integral as the holes degenerate to points, as those of J. Ball [l],Sivaloganathan, Syector and Tilakraj [23], and the literature of homogenization theory. We also mention the computation of the expansions in the case of quasilinear equations of Ward, Henshaw and Keller [25], Ward and Keller [26], Titcombe and Ward [24]. The goal of asymptotic analysis for problem (2) would be to write asymptotic expansions for the maps in (j), (jj). Thus for example an expansion of the form

zy zyxwvu zyxwvutsrqponmlk

for suitable coefficients a j . Our goal is instead to represent E E]O,to[  by means of

U ( E , .)lclfi

and I [ €for ]

(a) real analytic maps defined on a whole neighborhood of t = 0; (b) possibly singular at 6 = 0, but known functions of E (such as t p l , logt, etc. . . . ). 

zyxw zyxwvu zyx

We observe that our approach does have its advantages. Indeed, if for example we can prove that there exists a real analytic real valued function 3(.) defined in a whole neighborhood of 0 such that F[t]= & I t ] for E ~ ] O , t o [ ,then we know that an asymptotic expansion such as (9) for all T would necessarily generate a convergent series Cj”=,a j & , and that the sum of such a series would be €[E] for E > 0. Now let fi be as in (j). Under conditions in which R[.])is real analytic, the map Ufi of ] ­ € 0 , E n [ to C1,a(clfi) defined by

(&?[.I,

U,[t] E

Luo

Sn(t ­ s ) R [ E ] (do, s)

+ 

Sn(t ­ SE)E[E](S) dg,

V t E clfi , 

LUi

for all t E] ­ t O , t O [ offers a real analytic continuation for the map E u(~,.)~~,fi, which is defined only for E E ] O , E ~ [Thus . in this case the answer for question (j) is that

In order to analyze question (jj), we write

135

and we note that

zyxwv zyxwv zy zyxwv

Now we note that for n > 2, we have Sn(e(t ­ s))  = (2 ­ n)s,r,(c)S,(t ­ s),  and that S2(t(t­ s)) = T ~ ( E ­t  )& (t ­ s ) . Hence, we can prove that there exist two real analytic operators Fl,F2 of ]  ­ € 0 , €01 to R such that

&[O]

= 0. Next we note that

v " ( t ) ' D&(t

­ S E ) E [ € ] ( S )do,

1

dot.

Hence, one can prove that there exists a real analytic operator such that

By (12) and (14) we conclude that €[el = (&[el

+ E ~ [ +  ETn[~]Fz[e] ])

z z

F3 of ] ­ €0,€01 t o R

€]O,eo[, 

zy (15)

an equality which answers question (jj). We note that our conclusions (lo), (15) are relative to the various simplifications we have made so far to carry on the elementary presentation of this introduction, such as that the solutions of (1) and of  ( 2 ) can be represented as simple layers, and that those boundary value problems are solavable exactly when the corresponding integral equations are solvable, and that one can solve locally equation (7) and obtain an implicitly defined operator ( E ( . ]&[.I). , All such circumstances are not always present. Hence, under different circumstances, formulas (lo), (15) may have a different form. This is the case for example when both B', Bi correspond t o the linear Dirichlet boundary conditions, as shown in 1131, 1161, [171, [187.

136

zyxwvu zyxwvu zyx zy zyx zyxwv zyxw

One could extend the ideas exposed above in order t o consider also pertubations of 812, 81"by representing aili, dJI" by global parametrizations say 42, 4" defined for example on the unit sphere dBn of Rn, and analyze the corresponding singular perturbation problem in the complex of variables ( E , 42, gY) considered as a point x  (C1>"(dl"))",as done for linear problems in the Banach space R x (C'~"(811i))" in [16] ­ [18]and for problems related to the Riemann mapping in [13], [14]. In this case the difficulties would increase. Thus for example the analysis of an equation such as (7) is more complicated and the presentation is necessarily longer, and the corresponding treatment is not illustrated here. 2. A concrete case: the case of the nonlinear Robin boundary conditions

The material of this section is entirely based on the paper [15], where the special case Bi ( c , t ~ - ' , u ( t )s(t)) , =  -s(t), B" ( t , u ( t )g(t)) , =  ­  G"(t,u(t)) has been considered. Here Go is a continuous map of 81"x R to iW. We first have the following Theorem, which asserts the existence and local uniqueness of the family of solutions U ( E , .).  Theorem 1. Let a E]O,l[. Let I i , JIo be bounded open connected subsets of Rn of class Let Rn \  clIi, Rn \ clI" be connected. Let 0 E I i , 0 E I". Let Go E Co(dII"x  R) be such that the operator T G of ~ Co~"(dl")to Co(dJIo)defined by

g(t)

T G ~ [ WE] (G~")( t , v ( t ) )

V t E  dII",

z

VW E  Co'"(dJIo)

map Co>"(dl")real analytically to itself and map bounded sets of Co~"(dIIo) to bounded sets of Co~"(dIIo). Assume that there exists a solution fi C1."(cllIo) of problem

AU = 0 in I", ( t )= G"(t,u ( t ) )V t E 81°, such that

Then the following statements hold. (i) There exists E' E]O,co[ and a family { ~ ( c , . ) ) ~ ~ l Oof , ~ ,solutions [ u(E;) E C'>"(clB(E)) of problem

{ ---=O

A(€), on E W , =(t) = G"(t,~ ( t 'dt ) ) E  dI",

Au=O

in

(3)

such that lim,,ou(e, .),clfi = fi,,,fi(.) in C'@(clfi) for all bounded open subsets

fi of H" \ (0)

such that 0 $ clfi.

zyxwv zyxwv zyxw zyxwvuts zy zyxw 137

(ii) If {cj}jtw is a sequence of 10, -too[ converging to 0 and if {uj}jEw is a sequence of functions such that Cl@(Clh(€j)), (3) for E = ~j ,  uj = U in C’,cy(clfi)for all bounded open subsets fi of 1”\ (0) such that 0 clfi ,  uj E

uj solves

limj+,

4

then there exists j o E

N such that uj(.)= u ( q ,.)  for all j 2 j o .

Then we answer questions (j), (jj) of section 1 by means of the following. Theorem 2. Let the assumptions of Theorem 2 hold. Then the following statements hold. such that 0 4 clfi. Then there exists E” E]O,.5’[ and a real analytic operator U, of ]  ­  E ” , E ” [ to C1,a(clfi) such that clfi C A(€)for all E E] ­ E ” , E ” [ and such that

(i) Let

fi be a bounded open subset of IIo \ (0)

Moreover, U,[O] = 2LIClfi. (ii) There exists a real analytic operator 3 of ]  ­ d’,E ” [ to

s, 

3[€] = 

ID@(€,t)I2d t

lR such that

v.5 €10, ?[. 

6)

sI0IDtG(t)12d t .

Moreover, 3 [ 0 ]=  Acknowledgments

The author is indebted to Prof. B. Dittmar and to Prof. E. Wegert for pointing out a number of references concerning the existence of solutions for problems (l), (l),and to Prof. A.B. Movchan, and to Prof. J. Sivaloganathan, and to Prof. M.J. Ward, for pointing out a number of references on nonlinear singular perturbation problems on domains with small holes. References 1. J.M. Ball, Discontinuous equilibrium solutions a n d cavitation in nonlinear elasticity, Philos. Trans. Roy. SOC.London Ser. A, 306, (1982), 557-611. 2. H. Begehr and G.N. Hile, Nonlinear R i e m a n n boundary value problems f o r a nonlinear elliptic s y s t e m in the plane, Math. Z., 179, (1982), 241-261. 3. H. Begehr and G.C. Hsiao, Nonlinear boundary value problems f o r a class of elliptic systems, Komplexe Analysis und ihre Anwendung auf partielle Differentialgleichungen, Martin-Luther-Univeristat, Halle-Wittenberg, (1980), 90-102. 4. H. Begehr and G.C. Hsiao, Nonlinear boundary value problems of Riemann-Hilbert type, Contemporary Mathematics, 11, (1982), 139-153. 5. T. Carleman, Uber e i n e nichtlineare Randwertaufgabe bei der Gleichung Au = 0, Math. Z., 9 , (1921), 35-43.

138

zyxwv zyxwv zyxw zyxw zyxwv

6. M.A. Efendiev, H. Schmitz and W. Wendland, O n s o m e nonlinear potential problems, Electron. J. Differential Equations, 1999, (1999), 1-17 .  7. K. Klingelhofer, Modified Hammerstein integral equations and nonlinear harmonic boundary value problems, J. Math. Anal. Appl. 28, (1969), 77-87. 8. K. Klingelhofer, Nonlinear harmonic boundary value problems. I, Arch. Rational Mech. Anal. 31, (1968)/(1969), 364-371. 9. K. Klingelhofer, Nonlinear harmonic boundary value problems. II. Modified H a m m e r stein integral equations, J. Math. Anal. Appl. 2 5 , (1969), 592-606. 10. K. Klingelhofer, Uber nichtlineare Randwertaufgaben der Potentialtheorie, Mitt. Math. Sem. Giessen Heft 76, (1967), 1-70. 11. V.A. Kozlov, V.G. Maz’ya and A.B. Movchan, Asymptotic analysis of fields in multistructures, Oxford Mathematical Monographs, The Clarendon Press, Oxford university Press, New York, 1999. 12. R. Kiihnau, Die Kapazitut dunner Kondensatoren, Math. Nachr., 203, (1999), 125130. 13. M. Lanza de Cristoforis, Asymptotic behaviour of the conformal representation of a Jordan d o m a i n with a small hole, and relative capacity, in ‘Complex Analysis and Dynamical Systems’, edited by M. Agranovsky, L. Karp, D. Shoikhet, and L. Zalcman, Contemp. Math., 364, (2004) pp. 155-167. 14. M. Lanza de Cristoforis, Asymptotic behaviour of the conformal representation of a Jordan d o m a i n with a small hole in Schauder spaces, Computat. Methods Funct. Theory, 2, 2002, pp. 1-27. 15. M. Lanza de Cristoforis, Asymptotic behaviour of the solutions of a nonlinear Robin problem f o r the Laplace operator in a d o m a i n with a small hole. A functional analytic approach, submitted, 2006. 16. M. Lanza de Cristoforis, Asymptotic behaviour of the solutions of t h e Dirichlet problem f o r the Laplace operator in a domain with a small hole. A functional analytic approach, submitted, 2004. 17. M. Lanza de Cristoforis, A singular domain perturbation problem f o r the Poisson equation, submitted, 2005. 18. M. Lanza de Cristoforis, A singular perturbation Dirichlet boundary value problem f o r harmonic functions o n a domain with a small hole, Proceedings of the 12th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications, Tokyo July 27-31 2004, edited by H. Kazama, M. Morimoto, C. Yang, Kyushu University Press, (2005), 205-212. 19. V.G. Mazya, S.A. Nazarov and B.A. Plamenewskii, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, I, 11, (translation of the original in German published by Akademie Verlag 1991), Oper. Theory Adv. Appl., 111,112, Birkhauser Verlag, Basel, 2000. 20. A.B. Movchan, Contributions of V.G. Maz’ya t o analysis of singularly perturbed boundary value problems, The Maz’ya anniversary collection, 1 (Rostock, 1998), Oper. Theory Adv. Appl., 109, Birkhauser, Basel, 1999, pp. 201-212. 21. K. Nakamori and Y. Suyama, O n a nonlinear boundary problem f o r the equations Au = 0 and Au = f(x,y ) (Esperanto) Mem. Fac. Sci. Kyusyu Univ. A,, 5 , (1950), 99-106. 22. S. Ozawa, Electrostatic capacity and eigenvalues of the Laplacian, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 30, (1983), 53-62 .  23. J. Sivaloganathan, S.J. Spector and V. Tilakraj, T h e convergence of regularized minimizers for cavitation problems in nonlinear elasticity, SIAM J. Appl. Math., 66, (ZOOS), 736-757.

zyxwv

zyxwvut zy zyx zyx 139

24. M.S. Titcombe, M.J. Ward, S u m m i n g logarithmic expansions f o r elliptic equations in multiply-connected d o m a i n s with small holes, Canad. Appl. Math. Quart., 7,(1999), 313-343. 25. M.J. Ward, W. Henshaw, and J . Keller, S u m m i n g logarithmic expansions f o r singularly perturbed eigenvalue problems, SIAM J. Appl. Math., 53, (1993), pp. 799-828. 26. M.J. Ward, J . Keller, Nonlinear eigenvalue problems u n d e r strong localized perturbations w i t h applications t o chemical reactors, Stud. Appl. Math., 8 5 , (1991), 1-28. 27. M.J. Ward and J.B. Keller, Strong localized perturbations of eigenvalue problems, SIAM J . Appl. Math., 53 (1993), 770-798.

140

zyxwvutsrq

zyxw

RESIDUES O N A KLEIN SURFACE ARTURO FERNANDEZ ARIAS

Dpto. de Matematicas Fundamentales Facultad de Ciencias UNED C/Senda del Rey s / n Madrid 28040 Spain [email protected] JAVIER PEREZ ALVAREZ

Dpto. de Matemciticas Fundamentales Facultad de Ciencias U N E D C/Senda del Rey s / n Madrid 28040 Spain [email protected]

zyxwvu

T h e reconstruction of a Riemann surface starting from the meromorphic function field K , comes from Dedekind and Weber who developed an algebraic function theory in one variable over an algebraically closed field k . Alling and Greenleaf present a counterpart t o this approach starting from a real algebraic curve. From this point of view, the residues theorem is a classical result which depends strongly on the algebraically closed character of the base field. In this paper, via the complex double, we translate this fact to the case where we start from a function field in one variable over R.

zyxwvu zyxwvutsrq zyxwv z

Keywords: Klein surface, valuation ring, residue.

1. Klein surface of a real function field

Let K be a field and A be a subring of K ; we shall say that A is a valuation ring of K if for each II: E  K then x E  A or x-l E  A; it holds that A is a local ring: if it has two different maximal ideals p l and p a , then, taking a E p1\ p2 and b E  p ~ \ p l , we have a / b 4 A and b / a 4 A

Definition 1. Let R be the field of real numbers and K be a finite extension of transcendence degree 1 over R in which -1 is not a square. We shall call Klein surface of the field K , the set S of valuations rings of K which contain R. In the same way, the set S, of valuation rings of K [ i ]which contain C shall be called the double cover of S. We shall denote a point P of S by A, when we intend to indicate the ring that it represents, denoting then by p or m, its only maximal ideal. Let us fix an element z E K \ R, let n be the degree of the extension R(z) 9K and let us denote with 0 the integral clausure of R[z] in K , that is, the subring

zyxw

zyxwvu zyx zyxwvu zyxwv zyxw 141

of K formed by the elements which verify a monic polynomial with coefficients in

RbI.

For each element X  = XI 

+ i X 2 in K[i], let us write X = XI 

­ i X 2 .

Proposition 1. 0 i s a noetherian ring.

Proof. We shall see that 0 is a free R[z]-module of rank n. If t is a primitive element of the extension R ( z ) L) K , we can suppose, multiplying the minimal polynomial o f t over R[z] by an adecuate element of R[z], that t E 0. In this way, if y E 0, we can write y  = 

+ a l t + a2t2 + ... + an-1tn-',

ai E  R(z)

If (ai):=' are the R(z)-isomorphisms of K in the minimal normal algebraic extension L of K over R(z), we obtain the n following equations: a i ( y ) = a0

+ a lai(t)+ a m ( t ) 2 + ... + a n-lai(t)n-l

(1 5 i

 0 for any 5 #  z . Here & ( z ) = {C E  C :  v( 0,3c, > 0 s.t.

z

zyxwvuts zy (4 ­ 1) I f ( z ) l 5 C, exp(HK(z)

+ E I z I ) ,

+ i y E C n )

( v z = z 

Ezp(Cn : K ) denotes the space of entire functions which satisfiy the estimate (4-1). Following theorem gives a characterization of Fourier-Bore1 image of analytic functionals. Theorem 2 (Polya-Martineau ["I) Fourier-Borel transform is a topological isomorphism fiom H ' ( K ) to Ezp(C" : K ) .

4-2 Avanissian-Gay transform

zyx

Now we introduce Avanissian-Gay transform G T ( w )of analytic functional T .  It is called z-transform in digital signal analysis. In2 V. Avanissian and R. Gay introduced Avanissian-Gay transform for analytic functionals with compact carrier. Later, ins M. Morimoto and K. Yoshino extended Avanissian-Gay transform for analytic functional with non-compact carrier. If K is contained in {t E C : J l r n t (  0, 3CE,,l > 0 s.t.

Theorem 3 Suppose that holomorphic function f ( z ) satisfies (5-1). If 0 5 k then Ramanujan's Integral formula always valids in the following sense.

 0 s.t. I   I  5 CllcpII~,~  ,  (Vcp  E  S~,A(J@)).

Remark : For the details of Si(EXd),we refer the reader to2 and.5

The following Theorem 3 is Main result in this section. Theorem 3. Let U ( z ,t ) E  Cm(Rd x  (0, m)) satisfy the following conditions :  (i) (& ­ A) ~ ( zt ), =    o ,  (ii) VE  >  o ,  3 ~ 2, 0 ,  X , 2 0 s.t. l ~ ( x , t ) 5 I C,t-NceEJ+I,  (0 ' . Harmonic differential forms on Riemannian manifolds can be considered as a generalization of analytic functions of one complex variable or analytic differentials on Riemann surfaces3 . In this note we proof versions of the Mittag-Leffler and Weierstrass theorems for harmonic forms. Construction of harmonic forms with prescribed singularities turns out to be a classical problem, it was discussed for example in the book by B.Rodin and L.Sario4 . We will give another approach to this problem and prove a version of the MittagLeffler theorem in Section 3 . In Section 2 we formulate necessary and sufficient conditions for separation of singularities (in the sense of N.Aronszajn5). In several complex variable separation of singularities corresponds to the additive Cousin problem. This scheme of proofing Mittag-Leffler theorem was applied to harmonic functions on manifolds by T.Bagby and P.Blanchet' , and by P.Gauthier7 ; and to harmonic differential forms on open subsets of R" by the author* . In Sections 2 and 3 the results of Ref. 8 are generalized to forms on Riemannian manifolds. We give a new proof of Lemma 1 based on the Hodge decomposition. Further we formulate the result on separation on singularities for a given form (see Theorem a ) , the last corollary repeats our result for forms on

zyxwvu

208

open subspace of Rn. Finally, Theorem 2 is used to obtain a version of the MittagLeffler theorem. In the last section a generalization of the Weierstrass theorem is given. In particular, we construct a harmonic form that vanishes on a prescribed discrete set of points, the form can be chosen to be both exact and coexact. One of the main tools we employ for solving both problems is approximation of harmonic forms by elementary ones (as in the classical theorem by Runge), we adjust the methods used in Ref.9. We use also an abstract result on simultaneous approximation and interpolation due to H.Yamabe.lo For harmonic and analytic functions simultaneous approximation and interpolation can be found in the works by L.Walsh and his coauthors.

zyxwv zyxwvuts zyxw zyxwv

2. Separation of singularities for harmonic differential forms

In this article M denotes a smooth (of class C") connected oriented noncompact Riemannian manifold of dimension n 2 3. We use the Hodge-de Rham decompositions for forms (and currents) on M T = HIT

+ H2T + H T

and refer the reader to de Rham's book' .  Lemma 1. Let R be an open subset of M and let 1 5 p there exists p E C p l ( R ) such that 6dp = ha.

zy

5 n. For any Q  E  CF(R)

Proof. Let { K j } j be an exhaustion of R by compact subsets such that R \ Kj has no connected components relatively compact in R. We take I/Jj E D(R) such that 0 5 $ j 5 1and I / J j = 1on a neighborhood of K j , I/Jo = 0. Then 1 = C3.($j-$j-l) =  C j$ j , where $ j E D(R) and $ j = 0 on a neighborhood of Kj-1. We define t c j = Hl(q5jcu). Then ~j E C F ( M ) and tcj is exact. Further, 6Kj

= d H l ( 4 j a ) = 6($jf2

H ( $ j Q ) )= 6($jCX).

­  Hz($jCC) ­ 

Now we consider ~j in a neighborhood of Kj-1 where 4j = 0. We claim that ~j can be uniformly on KjP1 approximated by exact in R forms that are coclosed on R. First, suppose that R is relatively compact in M . We will show that ~j can be approximated uniformly on Kj-1 by forms of the form H l ( c ) , where e n   = 0.  Let T be a current that is orthogonal to all such forms with T C Kj-1. So we have

for all c whose support does not intersect R. (We used that T n c =  0  to move H1,H2 above, see details in the book by de Rham.') Clearly H Z ( T ) is a harmonic form off Kj-1. Last identity implies that HZ(T) vanishes with all it's derivatives at each point of M \  But M \  0 intersects all components of M \ Kj-1, thus H 2 ( T )= 0 on M \ Kj-1. So we have

a.

zyxwv zyx z zyx zy zyxwvu 209 

In general, let R = U l R l and each 01 is relatively compact in 0 , 0 2 1 cc RI+I. As we have seen ~j can be approximated on Kj-l by uj,l = H l ( c j , l )where (cj,l)nn,) =  0.We then approximate uj,l on 01 by a form H l ( c j , 2 ) with ( c ~ J n  ) 0 2 = 0. And continue the procedure. We get a sequence of forms uj,l that are exact and coclosed on R1. This sequence can be done convergent uniformly on compact subsets of R . Then it follows from the de Rham theorems that the limit form wj is exact and coclosed on R. It provides an'approximation to ~j on Kj-1,

zj(&j

The series w =  ­  wj)is uniformly absolutely convergent on compact subsets of R , for each 1 the sum C j , l ( ~­  j w j ) is a harmonic form on a neighborhood of KLand we have

j=1

j=1

on a neighborhood of Kl. Thus Sw = ha. Our last step is to show that w is an exact form on R. For any finite p-cycle in R we get

r

We applied dominated convergence theorem, see ( l ) ,and then used that forms /c.j and w j are exact in 0. Then, by the de Rham theorem, w =  d/3 for some

P

E

cpl(fl).

zyx

We denote by H q ( 0 )the singular homology spaces of an open subset R of M (we consider smooth singular q-chains with real coefficients, see textbook 2 for details). There is a natural mapping sq : ~

~n R2) ( + Hq(R1) 0 ~ @  H q ( o 2 ) ,

defined by s,(r) = (r,-F). Let R be an open subset of M . We denote by Hq(R) the (de Rham) q-cohomology spaces, Hq(R) is the factor space of all (smooth) closed q-forms on R over the subspace of exact q-forms. De Rham cohomology spaces are dual to singular homology spaces. The dual operator to the operator sq is defined as

aq : Hq(Rl)e Hq(R2)+ H q ( R 1 n R2), where a,([wl], [wz]) = [(wl -w2)1a2"a2], here we mean that the difference w1 -w2 is restricted to the intersection of the open sets and then the equivalence class is taken, we omit the restriction sign in what follows. It is easy to see that ker(s,) = (0) if and only if aq is surjective. Now, using Lemma 1, we can prove the theorem on separation of singularities. Theorem 2. Let R1,Rz be two open subsets of M , and R =  01 n 0 2 . Then a

210

zyxwvu zyxwvuts zyx zy zyx zyxwvut zyxwvu zyxwvut

harmonic q-form w E C r ( 0 )  can be written as 21 = w1 ­ 212, where v j E  C r ( 0 j )are harmonic q-forms if and only if [w]E Im(a,) and [*w]E Im(anpq).

Proof. The necessity of the condition is clear, if w = 211 ­ v2, then clearly [w]=  a,([vl], [m]) and = an-q([wl,[w]). Suppose now that [w]= a q ( [ v l ][w~]). , We have w = u1 ­ 212 d h in R. Moreover, using Lemma 1, we can find f j such that 6dfj = 6wj in R j then w j = uj ­ dfj is both closed and coclosed and w = w1 ­ w2 dg. Now [*(w1 ­  W Z ) ]= U ~ - ~ ( [ * W [*wz]) ~ ] , and *dg = *w ­ * ( w 1 - w2) E 1m(unuq).It is enough to show that dg can be written as the difference of two forms each harmonic on the corresponding domain R j . We can always write g = g1 ­ g2, where gj E Coo(Rj), then *dg = *dgl ­ *dg2. Since *dg E Im(anPq),we have *dg = 171 ­ 172 for some forms ql and 172 closed in 01 and 0 2 respectively. (Really we have *dg = s1 ­ s2 dr = (s1 d r l ) ­ (s2 d r a ) , where T = r1 ­ r2 is a Coo-decomposition.) Now we consider

I.*[

+ 

+ 

+ 

K = {

*171 ­  dg1 *172 ­ 492

on on

+ 

+ 

01 0 2

Clearly, r; is well-defined on R1 U 0 2 . Using Lemma I once again, we find r E C p l such that 6 d r = 6 ~Then . dg = (dgl + d r ) ­ (dg2 d r ) is the desired decomposition.

+ 

Corollary 3. A harmonic q-form w E  C r ( R ) can be written as 'u  = u1 ­ 212, where v j E C r ( R j ) are harmonic q-forms if and only if there exist q-forms ul,u2 such that u = u1 ­ u2,uj E  C r ( 0 j ) ,  and

are exact forms on

zyxwvut zyxwv

01 u R2.

Proof. The corollary follows immediately from the theorem and the exactness of the Mayer-Vietoris exact sequence of the pair: ... H P ( R ~ CB ) H P ( R ~ )aP, H P ( R ~n 0,) 

% ~

u 

p + l ( o ~0,) .... 

We use that Im(u,) = Ker(dP) and apply this for p = q , n ­ q. The statement of the corollary does not depend on the choice of the decomposition 'u  = u1 ­ u2. Detailed discussion of the mapping d p can be found in the book by R.Bott and L.Tu."

Similar to the result' for harmonic forms on the open subsets of n , we formulate necessary and sufficient condition for separation of singularities, using mappings sp: Corollary 4. Let R l , R 2 be two open subsets of M , and R = 01 n 0 2 . Then every harmonic q-form 'u  E  C r ( R ) can be written as 'u  = w1 ­ 212, where uj E C r ( R j ) are harmonic q-forms if and only if ker sq = (0) and ker s , - ~ = (0).

zy zyxwvu 211

3. Mittag

zyxwvu zyxwv zyxwvu

Leffler theorem for harmonic differential forms

­ 

As we have seen earlier,8 even for the case of the Euclidean space there are some obstacles for construction of harmonic forms with prescribed (massive) singularities. On the manifold we will divide the problem into two parts. First, given a compact set e and a form u harmonic on w \ e for some neighborhood of w of el  we want to construct a form v harmonic on M \ e that has same singularities on el  i.e. such that v ­ u has harmonic extension to a neighborhood of e. This problem can be reformulated as separation of singularities problem. Let 01 = M \ e, 0 2 = w . We have a harmonic form on 01 n 0 2 and want to decompose it into the sum of two forms, one harmonic in 01 and another harmonic in 0 2 . As we have seen in the previous section it can be done if and only if sq

: % ( w

zyx

\ e ) + H q ( M \ e ) @ H P ( W )

has trivial kernel for q = p , n ­ p . For example, a sufficient condition is

\ e ) ­+ 

j,+l %+1(M

%+l(M)

is surjective for q = p , n ­ p . (To see it one can write the exact homology sequence of the pair (111\ e, w ) . ) The last condition is satisfied if e is a subset of a coordinate chat of M . Another part of the classical Mittag Leffler theorem addresses the question of sewing together a sequence of singularities to one harmonic form. Here the situation is exactly the same as in the theory of functions of one complex variable. Theorem 5. Let e = U j e j be the union of a sequence of compact pairwise disjoint subsets of M . Assume that each compact subset of M intersects only finitely many sets of the sequence. Let hj be a sequence of p-forms such that hj is defined and harmonic on M \ ej. Then there exists a q-form h harmonic on M \ e such that h ­ hj has harmonic extension in a neighborhood of ej. ~ 

zyxwvu zyxwvut

Proof. Let oj be a relatively compact neighborhood of e j such that aj is a sequence of pairwise disjoint subsets of M and each compact subset of M intersects only finitely many sets of the sequence. Further for each j we chose an open set w j such that ej c w j CC a j . We define v on w = U j w j by u = hj on w j . We want to show that v can be written as the difference of a form harmonic on M \ e and a form harmonic on w . Using the corollary proved after Theorem 2, for each j we get hj = uj ­ r j , where uj E  Cw(M \ e j ) , rj E C m ( w j ) and aj = 

d u j on M d r j on w j

\ ej

and ,8j = 

d * uj on M d * rj on w j

\ ej

is exact and coexact on M . Moreover, we can choose uj with compact support in a j . Then v = C juj ­  r . and both (u  = aj and ,l3  = pj are exact. To see it, we 3 3 apply the de Rham theorem, all periods of a and /? vanish.

c.

zyxwvutsrqp

212 

zyxwv zyxwvut z zyxwvu zyxw

4. Weierstrass's theorem for harmonic differential forms

We will show that there exists a harmonic form that interpolates given sequence of germs {uj} at a discrete set of points { p j } on M . We say that a smooth differential form 4 has zero of order m at point p if in some coordinate chart all coefficients of q5 vanish with all their partial derivatives up to order m. The following result is simple after it is formulated [Yamabelo]Let Y be a dense linear subspace of the normed linear space X . Then for any L1, ..., L , E  X * , any x E X and every E >  0 there exists y E  Y such that 1/11: ­ yI(   0, then there exists a q-form U exact and coexact on M and such that Iu ­ UI "(K) of forms exact and coexact in a neighborhood of K . We equip this space with the uniform norm. Applying the methods we used in Section 1, we see that q-forms that are exact and coexact on M are dense in H,C,e(K).Thus the statement of the lemma follows from the claim formulated above.

Theoem 7. Let { p j } j be a discrete sequence of points in M and for each j let u j be a q-form harmonic near p j . Then there exists a harmonic q-form u on M such that u ­ u j has zero of order mj at p j for each j . Proof. There exists an exhaustion of M by compact subsets, M = U z 1 K j such that M \ Kj has no relatively compact in connected components. For simplicity (and Poincark's lemma without loss of generality) we may assume that p j E K: \  implies that u1 is exact and coexact near p l . Then, applying Lemma ??,  we get a q-form U1 that is exact and coexact in M and such that Ul ­ u1 has zero of order ml at p l . Suppose that we have constructed a q-forms Uj-1 on M that is exact and coexact, and such that Uj-1 ­ ul has zero of order ml at pl for 1 = 1,...,j ­ 1. Since p j $2 Kj-1 we have d j = ( p j , KjPl) > 0. Let Ej = KjP1U B ( p j , r j ) , where rj  0) in the class of all functions analytic in

0, f ’ ( 2 0 ) = 1. In the literature there are results concerning the region of exponents p which lead to the univalence of ‘pp.In this work we study the approximation of ‘ p p by the extremal polynomials defined below. In fact this problem is a particular case of a more general one, formulated in [14, pp. 318-319, problems 1, 21 and is important for the approximately construction of the conformal mappings. Let 11, be the class of all polynomials p , of degree at most n satisfying the conditions p , (zo) = 0, p k ( Z O ) =  1. Then we can prove that the integral 1)  pb ­  p i I :p(G) ( p >  1) is minimized in 11, by an unique polynomial T , , ~ . We call (see a l ~ o : ) ~ these , ~ extremal polynomials T , , ~ the generalized Bieberbach polynomials for the pair (G,zo). In case of p = 2 they are the usual Bieberbach polynomials T,, n = 1 , 2 , .... The approximation problems for 972 = cp  in closed domains with various boundary conditions, where approximation is conducted by the usual Bieberbach polynomials were intensively studied in2>4>7-11)13 .  Similar problems for ‘ p p ( p > 1) using the generalized Bieberbach polynomials were investigated in1>516 (see also,*).In the above cited works the rate of convergence to zero of the quantity ( 2 0 ) = 

zyxwvu zyxwvutsr

II  PP

­ r n , p

l l ~  

as n tends to 00 , has been estimated by means of the geometric properties of G. One of the interesting problem in this direction is the problem connected with a conjecture due to S. N. Mergelyan, who in” showed that the Bieberbach polynomials satisfy

for every E > 0, whenever L is a smooth Jordan curve and stated it as a conjecture that the exponent $  ­ E in (I) could be replaced by 1 ­ E . In7 it has been possible for us to obtain some improvement of the above cited Mergelyan’s estimation (1).For its formulation, it is necessary to give some definitions as follows. We denote by L p ( L ) and EP (G) the set of all measurable complex valued functions such that 1  f Ip is Lebesgue integrable with respect to arclength, and the Smirnov class of analytic functions in G respectively. For a weight function w given on L , and p > 1 we also set LP(L,W)

:= {

f E  L1 ( L ) :I  f

y

w E L1 ( L ) } ,

EP ( G , w ) := { f E  El ( G ) : f E  L p ( L , w ) } .

216

zyxwvutsr zyxwvuts zyxwvu zyxwv zyxwv zyxwvu zyx zyxw zyxwvuts

Let g E  L p (T, w ) and let gh(W) be the mean value function for g defined as: l

g h ( w ) := 

lh h

O 1. Now, applying Theorem

we have

2 (in case of r := ~ p o we ) conclude tha.t

Choosing the number po sufficiently close to 1 we finally get

with c = c ( E ) . The proof of the following theorem is similar to that of Theorem 11 from3 .  Theorem 3 Let G be a domain with a smooth boundary L and let p

> 1. Then

with a constant c > 0 . The approximation properties of the polynomials L,(G) are given in the following lemmas.

T : , ~ ,n

= 1 , 2 , ..., in the space

Lemma 1 Let G be a domain with a smooth boundary L and let p

for every

E

> 0 and with c = c ( E ) > 0 .

> 1. Then

zy

Proof. For the polynomials qn,p ( z ) best approximating y ~ bin L,(G) we set

Qn,p (2)

.i 

:=  q n , p ( t )d t , a

tn,p (2)

:= Q n , p

(2)

+ [I

­  qn,p

( z o ) ]( 2 ­ ZO) . 

zyxwv zyx zyxwvuts 219 

Then t,,p (zo) = 0 , tL,P( 2 0 ) = 1 and hence by Theorem 3 we obtain

This relation by the inequality

If (Z0)l  5 c Ilf  IILp(G) , 

zy

which holds for every analytic function f with IlfllLPcG,