Semigroups of Bounded Operators and Second-Order Elliptic and Parabolic Partial Differential Equations [1 ed.] 9780367206291, 9780429262593

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Semigroups of Bounded Operators and Second-Order Elliptic and Parabolic Partial Differential Equations

Monographs and Research Notes in Mathematics Series Editors: John A. Burns, Thomas J. Tucker, Miklos Bona, Michael Ruzhansky

About the Series This series is designed to capture new developments and summarize what is known over the entire field of mathematics, both pure and applied. It will include a broad range of monographs and research notes on current and developing topics that will appeal to academics, graduate students, and practitioners. Interdisciplinary books appealing not only to the mathematical community, but also to engineers, physicists, and computer scientists are encouraged. This series will maintain the highest editorial standards, publishing well-developed monographs as well as research notes on new topics that are final, but not yet refined into a formal monograph. The notes are meant to be a rapid means of publication for current material where the style of exposition reflects a developing topic. Summable Spaces and Their Duals, Matrix Transformations and Geometric Properties Feyzi Basar, Hemen Dutta Spectral Geometry of Partial Differential Operators (Open Access) Michael Ruzhansky, Makhmud Sadybekov, Durvudkhan Suragan Linear Groups: The Accent on Infinite Dimensionality Martyn Russel Dixon, Leonard A. Kurdachenko, Igor Yakov Subbotin Morrey Spaces: Introduction and Applications to Integral Operators and PDE’s, Volume I Yoshihiro Sawano, Giuseppe Di Fazio, Denny Ivanal Hakim Morrey Spaces: Introduction and Applications to Integral Operators and PDE’s, Volume II Yoshihiro Sawano, Giuseppe Di Fazio, Denny Ivanal Hakim Tools for Infinite Dimensional Analysis Jeremy J. Becnel Semigroups of Bounded Operators and Second-Order Elliptic and Parabolic Partial Differential Equations Luca Lorenzi, Abdelaziz Rhandi For more information about this series please visit: https://www.crcpress.com/Chapman--HallCRCMonographs-and-Research-Notes-in-Mathematics/book-series/CRCMONRESNOT

Semigroups of Bounded Operators and Second-Order Elliptic and Parabolic Partial Differential Equations

Luca Lorenzi

University of Parma, Italy

Abdelaziz Rhandi

University of Salerno, Italy

First edition published 2021 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN © 2021 Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www. copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data ISBN: 9780367206291(hbk) ISBN: 9780429262593 (ebk)

v

To Adriana, Alfredo, Asma, Ursula and Rainer.

Contents

Symbol Description

xi

Preface

xiii

Introduction

xv

1 Function Spaces 1.1 Spaces of (H¨ older) Continuous Functions . . . . . . . . . . . . . . . . . 1.1.1 Functions defined on the boundary of a smooth open set . . . . . 1.2 Anisotropic and Parabolic Spaces of H¨older Continuous Functions . . . 1.2.1 Anisotropic spaces of functions defined on the boundary of a set 1.3 Lp - and Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Besov Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I

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Semigroups of Bounded Operators

2 Strongly Continuous Semigroups 2.1 Definitions and Basic Properties . . 2.2 The Infinitesimal Generator . . . . 2.3 The Hille-Yosida, Lumer-Phillips and 2.4 Nonhomogeneous Cauchy Problems 2.5 Notes and Remarks . . . . . . . . . 2.6 Exercises . . . . . . . . . . . . . . .

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Parabolic Equations

4 Elliptic and Parabolic Maximum Principles 4.1 The Parabolic Maximum Principles . . . . . 4.1.1 Parabolic weak maximum principle . . 4.1.2 The strong maximum principle . . . . 4.2 Elliptic Maximum Principles . . . . . . . . . 4.3 Notes . . . . . . . . . . . . . . . . . . . . . . 4.4 Exercises . . . . . . . . . . . . . . . . . . . .

1 1 7 8 14 15 26 34

35

. . . . . . . . . . . . . . . . . . . . . . . . . . . . Trotter-Kato Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Analytic Semigroups 3.1 Prelude . . . . . . . . . . . . . . . . . . . . . 3.2 Sectorial Operators and Analytic Semigroups 3.3 Interpolation Spaces . . . . . . . . . . . . . . 3.4 Nonhomogeneous Cauchy Problems . . . . . 3.5 Notes . . . . . . . . . . . . . . . . . . . . . . 3.6 Exercises . . . . . . . . . . . . . . . . . . . .

II

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85 85 85 92 97 104 104

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Contents

5 Prelude to Parabolic Equations: The Heat Equation and the GaussWeierstrass Semigroup in Cb ( d ) 107 5.1 The Homogeneous Heat Equation in Rd . Classical Solutions: Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.2 The Gauss-Weierstrass Semigroup . . . . . . . . . . . . . . . . . . . . . . . 111 5.2.1 Estimates of the spatial derivatives of T (t)f . . . . . . . . . . . . . . 114 5.3 Two Equivalent Characterizations of H¨older Spaces . . . . . . . . . . . . . 120 5.4 Optimal Schauder Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

R

R

6 Parabolic Equations in d 6.1 The Continuity Method . . . . . . . . . . . . . . 6.2 A priori Estimates . . . . . . . . . . . . . . . . . 6.2.1 Solving problem (6.0.1) . . . . . . . . . . 6.2.2 Interior Schauder estimates for solutions domains: Part I . . . . . . . . . . . . . . . 6.3 More on the Cauchy Problem (6.0.1) . . . . . . 6.4 The Semigroup Associated with the Operator A 6.4.1 Interior Schauder estimates for solutions domains: Part II . . . . . . . . . . . . . . 6.5 Higher-Order Regularity Results . . . . . . . . . 6.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Exercises . . . . . . . . . . . . . . . . . . . . . .

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7 Parabolic Equations in Rd+ with Dirichlet Boundary Conditions 7.1 Technical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 An Auxiliary Boundary Value Problem . . . . . . . . . . . . . . . 7.3 Proof of Theorem 7.0.2 and a Corollary . . . . . . . . . . . . . . . 7.4 More on the Cauchy Problem (7.0.1) . . . . . . . . . . . . . . . . 7.5 The Associated Semigroup . . . . . . . . . . . . . . . . . . . . . . 7.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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137 138 139 145 146 157 163 170 171 174 174 175 176 184 190 192 193 197 197

R

8 Parabolic Equations in d+ with More General Boundary Conditions 199 8.1 A Priori Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 8.2 Proof of Theorem 8.0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 8.3 Interior Schauder Estimates for Solutions to Parabolic Equations in Domains: Part III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 8.4 More on the Cauchy Problem (8.0.1) . . . . . . . . . . . . . . . . . . . . . 224 8.5 The Associated Semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 8.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 9 Parabolic Equations in Bounded Smooth Domains Ω 229 9.1 Optimal Schauder Estimates for Solutions to Problems (9.0.1) and (9.0.2) . 230 9.2 Interior Schauder Estimates for Solutions to Parabolic equations in Domains: Part IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 9.3 More on the Cauchy Problems (9.0.1) and (9.0.2) . . . . . . . . . . . . . . 243 9.4 The Associated Semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 9.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

Contents

III

ix

Elliptic Equations

247

10 Elliptic Equations in Rd 10.1 Solutions in H¨ older Spaces . . . . . . . . . . . . . . . . . 10.1.1 The Laplace equation . . . . . . . . . . . . . . . . 10.1.2 More general elliptic operators . . . . . . . . . . . 10.1.3 Further regularity results and interior estimates . . 10.2 Solutions in Lp (Rd ; C) (p ∈ (1, ∞)) . . . . . . . . . . . . 10.2.1 The Calder´ on-Zygmund inequality . . . . . . . . . 10.2.2 The Laplace equation . . . . . . . . . . . . . . . . 10.2.3 More general elliptic operators . . . . . . . . . . . 10.2.4 Further regularity results and interior Lp -estimates 10.3 Solutions in L∞ (Rd ; C) and in Cb (Rd ; C) . . . . . . . . . 10.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 Elliptic Equations in Rd+ with Homogeneous tions 11.1 Solutions in H¨ older Spaces . . . . . . . . . . 11.1.1 Further regularity results . . . . . . . 11.2 Solutions in Sobolev Spaces . . . . . . . . . . 11.2.1 Further regularity results . . . . . . . 11.3 Solutions in L∞ (Rd+ ; C) and in Cb (Rd+ ; C) . . 11.4 Exercises . . . . . . . . . . . . . . . . . . . .

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249 251 251 252 254 256 256 266 269 279 286 290

Dirichlet Boundary Condi. . . . . .

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291 292 294 296 304 309 314

12 Elliptic Equations in Rd+ with General Boundary Conditions 12.1 The C α -Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.1 Further regularity . . . . . . . . . . . . . . . . . . . . . 12.2 Elliptic Equations in Lp (Rd+ ; C) . . . . . . . . . . . . . . . . . 12.2.1 Further regularity results . . . . . . . . . . . . . . . . . 12.3 Solutions in L∞ (Rd+ ; C) and in Cb (Rd+ ; C) . . . . . . . . . . . . 12.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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315 316 319 321 332 334 338

13 Elliptic Equations on Smooth Domains Ω 13.1 Elliptic Equations in C α (Ω; C) . . . . . . . . . . . 13.1.1 Further regularity results . . . . . . . . . . 13.2 Elliptic Equations in Lp (Ω; C) . . . . . . . . . . . 13.2.1 Further regularity results . . . . . . . . . . 13.3 Solutions in L∞ (Ω; C), in C(Ω; C) and in Cb (Ω; C) 13.4 Exercises . . . . . . . . . . . . . . . . . . . . . . .

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339 339 348 351 360 364 368

14 Elliptic Operators and Analytic Semigroups 14.1 The Semigroup in Cb (Rd ; C) . . . . . . . . . 14.2 The Semigroups in Cb (Rd+ ; C) . . . . . . . . 14.2.1 Proof of Theorems 7.4.1 and 7.4.3 . . 14.3 The Semigroups in Cb (Ω; C) . . . . . . . . . 14.4 Exercises . . . . . . . . . . . . . . . . . . . .

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x

Contents

15 Kernel Estimates 15.1 Dunford-Pettis Criterion and Ultracontractivity . . . . . . 15.2 Gaussian Estimates for Second-Order Elliptic Operators Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 15.3 Integral Representation for the Semigroups in Chapters 6, 7 15.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IV

. . . . . . . . . with Dirichlet . . . . . . . . . and 9. . . . . . . . . . . . . . . . . . . . . . .

Appendices

A Basic Notions of Functional Analysis in Banach A.1 Bounded and Closed Linear Operators . . . . . A.2 Vector Valued Riemann Integral . . . . . . . . . A.3 Holomorphic functions . . . . . . . . . . . . . . A.4 Spectrum and Resolvent . . . . . . . . . . . . . A.5 A Few Basic Notions from Interpolation Theory A.5.1 Marcinkiewicz’s Interpolation Theorem . A.6 Exercises . . . . . . . . . . . . . . . . . . . . . .

397 397 400 412 413 414

415 Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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417 417 418 422 424 426 437 440

B Smooth Domains and Extension Operators B.1 Partition of Unity . . . . . . . . . . . . . . . . . . . . . . . . B.2 Smooth Domains . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Traces of Functions in Sobolev Spaces . . . . . . . . . . . . . B.4 Extension Operators . . . . . . . . . . . . . . . . . . . . . . . B.4.1 Extending functions defined on open sets . . . . . . . B.4.2 Extending functions defined on the boundary of a set

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441 441 443 450 454 454 460

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Bibliography

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Index

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Symbol Description

xi

Symbol Description Functions

Sets set of all positive natural numbers set of all real numbers set of all complex numbers set of all real d-tuples subset of Rd of d-tuples x = (x1 , . . . , xd ) with xd > 0 Rd−1 := (−∞, T ) × Rd−1 T d R+,T := (−∞, T ) × Rd+ Cd set of all complex d-tuples AbB given two subsets of Rd with B open, it means that A is contained in B Y ,→ X given two Banach spaces, it means that Y is continuously embedded into X B(x, r) open disk in Rd with centre at x and radius r > 0 C(x, r) open cube in Rd with centre at x and radius r > 0 Cr (t0 , x0 ) the cylinder (t0 − r2 , t0 ) × B(x0 , r) F closure of F ∂F boundary of F L(X, Y ) set of all the bounded linear operators from X to Y L(X) := L(X, X) X0 dual space of the Banach space X N R C Rd Rd+

Matrix and linear algebra detB determinant of the matrix B diag(λ1 , . . . , λd ) diagonal matrix whose entries on the main diagonal are λ1 , . . . , λd ej j-th vector of the canonical basis of Rd λmin (A) minimum eigenvalue of the matrix A λmax (A) maximum eigenvalue of the matrix A Tr(B) the trace of the d × d matrix B, Pd i.e., Tr(B) = i=1 bii hx, yi Euclidean inner product between the vectors x, y ∈ Rd x·y := hx, yi ||Q|| Euclidean norm of the d × d matrix Q = (qij ), namely ||Q||2 = Pd 2 i,j=1 qij .

u+

positive part of the real-valued function u : Ω → R, i.e., u+ (x) = max{u(x), 0} for every x ∈ Ω − u negative part of the real-valued function u : Ω → R, i.e., u− (x) = min{u(x), 0} for every x ∈ Ω fo odd extension with respect to the last variable of the function f : Rd+ → R fe even extension with respect to the last variable of the function f : Rd+ → R f trivial extension to Rd of the function f : Ω → R χA characteristic function of the set A, i.e., the function equal to 1 in A and null elsewhere 1l := χRd ||f ||∞ sup-norm of f : Ω ⊂ Rd → R, i.e., ||f ||∞ := supΩ |f | r f Di1 ,...,ir f derivative ∂xi ∂,...,∂x of the funci 1

Dα f

∇f Dk f |Dk f | ||Dk f ||∞

r

tion f : Ω ⊂ Rd → R |α| f derivative ∂xα1∂,...,∂x αd of the funcd tion f : Ω ⊂ R → R, where α = (α1 , . . . , αd ) gradient of f vector consisting of all the k-th order derivatives of f , for k ∈ N Euclidean norm of Dk f for k ∈ N the by ||Dk f ||2∞ = P norm αdefined 2 |α|=k ||D f ||∞

||Dk f ||Lp (Ω;K) the Lp -norm of the function |Dk u| over Ω div f divergence of f Jac g Jacobian matrix of the vectorvalued function g Γ The Euler function or the Poisson kernel Operators D(A) ρ(A) σ(A)

domain of the linear operator A resolvent set of the linear operator A spectrum of the linear operator A

xii I

a+ a− a∨b a∧b [x] |α| δij

Symbol Description identity operator (in a Banach dist(x, F ) distance of the point x from the space X) set F , i.e., dist(x, F ) = inf y∈F |x− y| Miscellanea dist(F, G) distance of the set F from the set G, i.e., the number inf x∈F d(x, G) := max{a, 0} dx Lebesgue measure in Rd := min{a, 0} d−1 dH (d − 1)-dimensional Hausdorff maximum between a and b measure on Rd minimum between a and b dσ(y) := dHd−1 integer part of x ∈ R real part of λ ∈ C length of the multi-index α, i.e., Re λ Im λ imaginary part of λ ∈ C |α| = α1 + . . . + αd Kronecker delta, i.e., δij = 1 if ωd surface measure of the unit ball in i = j and δij = 0 otherwise Rd

Preface

This book originated from the notes of the 20th edition of the Internet Seminar, an online course addressed to Ph.D. students and young researchers in mathematical analysis, organized once per year. The subject of the 20th edition of the Internet Seminar were partial differential equations (PDEs) and their connections with the theory of semigroups of bounded operators. The interest shown by participants which amounted in more than 250 subscriptions from many countries in the world (Algeria, Austria, Brazil, Chile, Colombia, Ethiopia, France, Germany, Hungary, India, Iran, Italy, Kazazhstan, Morocco, New Zeland, Pakistan, Polonia, Russia, Senegal, Slovenia, Switzerland, The Netherlands, Tunisia, United Kingdom, United States, Vietnam), motivated us in using the notes as a starting point for a more general project. The purpose of this book is to present a unified approach to the classical theory of elliptic and parabolic equations, to introduce the theory of semigroups of bounded operators and show how analytic semigroups are naturally associated with second-order elliptic and parabolic equations and that the theory of semigroups provides powerful tools to study existence, uniqueness and regularity properties of solutions to elliptic and parabolic equations in Rd , in Rd+ or in bounded and sufficiently smooth domains of Rd . Our idea was to write a book which could serve as a reference for researchers working in the field of elliptic and parabolic equations and at the same time could help young researchers to enter in the world of partial differential equations and semigroups. The relevance of partial differential equations is easy to explain due to their numerous applications. Indeed, many mathematical models from applied sciences (chemistry, physics, engineering, etc.) lead to partial differential equations. On the other hand, the theory of semigroups furnishes a powerful tool to study partial differential equations. One key idea which moved the researchers to introduce the concept of semigroup of bounded operators is that a partial differential equation of parabolic type can be written as an abstract ordinary differential equation associated with a linear (unbounded) operator A. Based on this basic but really relevant remark, the theory of semigroups has grown up. Having in mind the very classic theory of systems of ordinary differential equations, the main point of the theory of semigroups, was to find the right properties of the operator A which allows to extend the concept of the exponential of a matrix. In the parabolic case, we confine ourselves to classical solutions, i.e., solutions which are once continuously differentiable with respect to time and twice continuously differentiable with respect to the spatial variables. On the contrary, when dealing with elliptic equations, we consider both classical solutions and solutions in Sobolev spaces. Both for elliptic and parabolic equations on bounded (smooth) domains of Rd , we consider Dirichlet, Neumann and more general first-order boundary conditions. In the first part of the book, we introduce the theory of semigroup of bounded linear operators, pointing the attention to strongly continuous semigroups and, mainly, analytic semigroups. We also devote a chapter to classical maximum principle for both parabolic and elliptic equations, subject to suitable boundary conditions. Such principles will be the main tool to prove the uniqueness of the classical solutions to parabolic Cauchy problem

xiii

xiv

Preface

and elliptic equations, where by classical solutions to elliptic equations, we mean that we are interested in solutions which are twice continuously differentiable in their domain. In the second part of the book, we deal with parabolic Cauchy problems in Rd , in Rd+ and in sufficiently smooth open subsets of Rd . We prove optimal Schauder estimates, interior estimates and further regularity results for the solutions to such problems. The analysis begins considering the heat equation in the whole Rd . Indeed, this is the easiest case, since an explicit integral representation formula for the homogenenous Cauchy problem associated with the heat equation is available. This greatly simplifies the analysis. Moreover, it helps to make clear the connections between partial differential equations and semigroups. In the third part of the book, we consider elliptic equations in Rd , in Rd+ and in sufficiently smooth open subsets of Rd . Also in this case, we show optimal Schauder results, interior estimates and further regularity results. As already mentioned, we consider both classical solutions and solutions in Sobolev spaces. We partly use these last results to show that the semigroups associated with the Cauchy problems in the second part of the book are analytic. The last part of the book, which consists of two appendices, collects some tools from functional analysis and a series of extension operators, thoroughly used in the book.

Acknowledgments The authors wish to thank D. Addona, F. Gregorio, A. Lunardi, E.M. Mangino, G. Metafune, C. Tacelli, H. Vogt and J. Voigt for useful discussions and suggestions during the preparation of this book. They also thank the staff of Taylor and Francis for their help and useful suggestions they supplied during the preparation of this book.

Parma and Salerno, August 2020

Luca Lorenzi Abdlaziz Rhandi

Introduction

Elliptic and parabolic equations with bounded coefficients have been widely studied in the literature both in Rd and in its open subsets, in the last century, and nowadays they are well understood. In this book, we present most of the significant results of the theory of partial differential equations of parabolic and elliptic type, showing how these two classes of PDE’s are strongly connected each other. Some regularity results for solutions to elliptic equations are obtained as a byproduct of the corresponding results proved for parabolic Cauchy problems. This is in the spirit of the theory of semigroups of bounded operators, which makes clear the connections between the eigenvalue problem λu − Au = f , when A is a suitable closed linear operator, and the abstract Cauchy problem u(0) = f for the abstract differential equation u0 = Au. We devote two chapters to develop the theory of semigroups of bounded operators, describing the main properties of uniformly continuous semigroups and mainly focussing on analytic semigroups, based on the fact that these latter semigroup are strictly connected with second-order elliptic and parabolic problems. The book is intended to be self contained. For this reason, we prove almost all the results that the present. Each chapter of the book (and also Appendix A) contains some exercises, which are aimed at helping the reader to better understand the topics of the book. We present here below in details the structure of the book.

Overview Aimed at making the book as much self contained as possible we recall in Chapter 1 all the functions spaces that we use in this book and give the proofs of classical results concerning interpolation inequalities in spaces of H¨older continuous functions, Lp - and Sobolev spaces as well as some important properties of Besov spaces that we need to define traces of functions which belong to Sobolev spaces over smooth domains. The rest of the book is divided into three parts.

PART I: Semigroups of bounded operators In this part, we introduce the concept of semigroup of bounded linear operators on Banach spaces. As we will explain, semigroups of bounded linear operators arise naturally in the study of linear parabolic equations. The idea which leads to the definition of semigroups of bounded operators is to generalize the concept of exponential of a matrix, which is usually introduced in a course of Calculus to analyze systems of ordinary differential equations. The easiest generalization of the exponential of a matrix is the exponential P∞ n of a bounded linear operator A acting on a Banach space X. Indeed, the series n=0 tn! An converges in the space L(X) of bounded linear operators over X and the convergence is locally uniformly in R with respect to the variable property allows to define the exponential of the P∞t. This tn n operator A by setting etA = A for every t ∈ R. This one-parameter family of n=0 n! xv

xvi

Introduction

bounded operators enjoys the so-called semigroup property, i.e., etA ◦esA = e(t+s)A for every s, t ≥ 0 (actually, this property holds true for every s, t ∈ R and the family {etA : t ∈ R} is a group). Moreover, the function t 7→ etA is continuous on R with values on L(X). These semigroups are the so-called uniformly continuous (semi)groups. Quite often, in the applications, one has to deal with linear unbounded operators, so that the above definition of semigroup via a series is useless. One of the reason is that the domain of the powers An becomes smaller and smaller as n increases. So, a different strategy should be adopted and this leads to the definition of strongly continuous (or C0 -) and analytic semigroups. Chapter 2 is a detailed study of strongly continuous semigroups. A strongly continuous semigroup is a semigroup (i.e., a one-parameter family of operators {T (t) : t ≥ 0} in L(X) such that T (t) ◦ T (s) = T (t + s) for every t, s ≥ 0 and T (0) is the identity operator), which satisfies the continuity property limt→0 T (t)x = x for every x ∈ X. Here, we prove the famous Hille-Yosida, Lumer-Phillips and Trotter-Kato theorems. Hille-Yosida theorem characterizes the infinitesimal generators of strongly continuous semigroups, where the generator of a strongly continuous semigroup is the operator A defined by Ax = limt→0 t−1 (T (t)x − x) for every x ∈ X such that the previous limit exists. We also provide conditions to insure the existence and uniqueness of strict solutions to the nonhomogeneous Cauchy problem ( u0 (t) = Au(t) + f (t), t ∈ [0, T ], (0.0.1) u(0) = x, when A is the generator of the infinitesimal generator of the C0 -semigroup {T (t)}. As in the classical case of linear ordinary differential equations, the candidate to be the strict solution to problem (0.0.1) is the function u given by variation-of-constants formula, i.e., Z

t

T (t − s)f (s) ds,

u(t) = T (t)x +

t ∈ [0, T ].

0

By strict solution, we mean a solution which is continuous in [0, T ] with values in D(A) (endowed with the graph norm) and continuously differentiable in [0, T ] with values in X. Due to the fact that strongly continuous semigroups, in general, do not improve the smoothness of the datum to which they are applied to, it turns out that the right assumptions to guarantee the existence of a strict solution to the above Cauchy problem are x ∈ D(A) and f continuous in [0, T ] with values in D(A) or f continuously differentiable in [0, T ] with values in X. Chapter 3 deals with the study of semigroups of bounded linear operators which are analytic but not necessary strongly continuous. This class of semigroups appears naturally when studying parabolic problems in several function spaces. The idea which leads to the definition of such semigroups comes from observing that the series that defines uniformly continuous semigroups can be replaced by a suitable integral over a closed curve which surrounds the spectrum of operator A. In the case of unbounded operators, the spectrum is an unbounded set in general, so that the closed curve should be replaced by an unbounded curve and, to make the integral converge, suitable conditions on the spectrum of A and on the behaviour at infinity of the resolvent function λ 7→ R(λ, A) should be assumed. This leads to the definition of analytic semigroup via the Dunford integral and the operators for which this procedure works are the so-called sectorial operators. After proving some relevant properties of analytic semigroups, we introduce and give several characterizations of the interpolation spaces DA (α, ∞) and DA (1 + α, ∞). Such spaces play a central role in the existence of strict and classical solutions to nonhomogeneous Cauchy problems under

Introduction

xvii

weaker assumptions on the initial value and the nonhomogeneous term. Here, by a classical solution to problem (0.0.1), we mean a function u which is continuous in [0, T ] with values in X, continuous in (0, T ] with values in D(A) and continuously differentiable in (0, T ] with values in X. Comparing the analysis of strongly continuous and analytic semigroups it comes clear how these latter semigroups enjoy better regularity properties, since when applied to a datum x of the underlying Banach space X, they immediately regularize the datum. In fact, T (t)x belongs to the intersection of the domains of all the powers of the associated sectorial operator. This fact allows to weaken the assumptions on x and f such that the Cauchy problem (0.0.1) admits a strict solution u, when A is a sectorial operator.

PART II: Parabolic equations In this part, we study the existence, uniqueness and regularity properties of solutions to linear parabolic problems in the whole Rd , or in Rd+ and in bounded smooth domains of Rd , subject to Dirichlet and more general boundary conditions (including Neumann boundary conditions). Chapter 4 is dedicated to the parabolic and elliptic weak and strong maximum principles. These are fundamental tools for proving uniqueness of solutions to parabolic problems as well as elliptic boundary value problems in the setting of continuous functions. The idea which leads to weak maximum principle is easy and we explain it in the elliptic case: a function of class C 2 which admits a maximum at some point x0 of an open set Ω has gradient which vanishes at that point and the Hessian matrix at x0 is negative definite. Hence, if A is a second-order elliptic operator then (Au)(x0 ) ≤ c(x0 )u(x0 ), where c is the potential term of the operator A (see the forthcoming formula (0.0.3)). From this remark, it turns out that, if u is twice continuously differentiable on Ω, bounded and continuous on Ω, λu − Au ≤ 0 in Ω for λ > c0 = supx∈Ω c(x) and u is nonpositive on ∂Ω (if Ω 6= Rd ) then u ≤ 0 on Ω. In view of parabolic and elliptic problems with first-order boundary conditions, we also show that if x0 ∈ ∂Ω is a point where the smooth (enough) function u attains its ∂u (x0 ) is strictly positive for every outward direction η. Clearly, this maximum point, then ∂η property can be extended to smooth enough functions u : (0, T ] × Ω → R. The strong maximum principle for elliptic equations shows that, if u attains a maximum at some point of the open set Ω and u is bounded and continuous in Ω then u is constant in Ω. A similar result holds true in the parabolic case and it shows that, if u admits its maximum value at some point (t0 , x0 ) ∈ (0, T ] × Ω, then u is constant in [0, t0 ] × Ω. Chapter 5 is devoted to the study of existence, uniqueness and regularity properties of the classical solution to the nonhomogeneous Cauchy problem ( Dt u(t, x) = Au(t, x) + g(t, x), t ∈ (0, T ], x ∈ Rd , (0.0.2) u(0, x) = f (x), x ∈ Rd , when f ∈ Cb (Rd ), g ∈ Cb ([0, T ] × Rd ) and A is the Laplacian. Here, we explain how to get the explicit solution T (·)f to (0.0.2) with g ≡ 0 via the Gauss-Weierstrass semigroup {T (t)} in Cb (Rd ). We prove optimal uniform estimates for the spatial derivatives up to the third-order of T (t)f and characterize the spaces of H¨older and continuous functions on Rd in terms of the Gauss-Weierstrass semigroup {T (t)}. These properties allow to obtain optimal Schauder estimates for the solution to problem (0.0.2), where optimality means that, if f ∈ Cb2+α (Rd ) (i.e., u is bounded and twice continuously differentiable in Rd , with bounded derivatives, and the second-order derivatives are α-H¨older continuous in Rd ) and α/2,α g ∈ Cb ((0, T ) × Rd ) (i.e., g is α H¨older continuous with respect to the parabolic distance

xviii

Introduction

over Rd+1 ), then the Cauchy problem (0.0.2) admits a unique classical solution u such that all the terms which appear in the differential equation have the same regularity as the function g (i.e., the second-order spatial derivatives and the time derivative of u belong to α/2,α Cb ((0, T ) × Rd )). We stress that the conditions g ∈ Cb ([0, T ] × Rd ) is not enough to guarantee the existence of a classical solution to problem (0.0.2). The results of this chapter are crucial for the analysis of more general parabolic equations on Rd , Rd+ and on bounded and smooth domains Ω of Rd . In Chapter 6, we extend the results obtained in Chapter 5 for the Laplace operator to the more general second-order elliptic operator A, defined on smooth functions ψ : Rd → R, by Aψ(x) =

d X

qij (x)Dij ψ(x) +

i,j=1

d X

x ∈ Rd .

bi (x)Di ψ(x) + c(x)ψ(x),

(0.0.3)

i=1

In this situation, an explicit formula for the solution to the Cauchy problem is not available, in general. This prevents us from using the same arguments as in Chapter 5. We overcome this difficulty, using the continuity method, which requires to prove some a priori estimates for classical solutions to the Cauchy problem (0.0.2), and the results in Chapter 5. Thanks to this machinery, we prove that there exists a unique classical solution u to Problem (0.0.2) 1+α/2,2+α which belongs to Cb ((0, T ) × Rd ) and satisfies the estimate ||u||C 1+α/2,2+α ((0,T )×Rd ) ≤ C(||f ||C 2+α (Rd ) + ||g||C α/2,α ((0,T )×Rd ) ), b

b

b

α/2,α

provided that f ∈ Cb2+α (Rd ) and g ∈ Cb ((0, T ) × Rd ). This result serves also a starting point to prove interior Schauder estimates for solutions of parabolic problems in domains and to show that the more the coefficients of the operator A and the data f and g are smooth, the more the solution u to problem (0.0.2) is itself smooth. Finally, using all the tools so far obtained, we show that the classical solution to the homogeneous parabolic problem (0.0.2) defines a semigroup {T (t)} of bounded linear operators on Cb (Rd ) and prove optimal estimates for the spatial derivatives of the function T (t)f , when f belongs to Cb (Rd ) and to subspaces of this space. Chapter 7 is devoted to the study of the existence, uniqueness and optimal regularity of solutions to the Cauchy-Dirichlet problem  t ∈ (0, T ], x ∈ Rd+ ,   Dt u(t, x) = Au(t, x) + g(t, x), u(t, x0 , 0) = ψ(t, x0 ), t ∈ (0, T ], x0 ∈ Rd−1 , (0.0.4)   u(0, x) = f (x), x ∈ Rd+ , on Rd+ = Rd−1 ×R+ . More precisely, under suitable conditions on the coefficients of operator A (which are now defined on Rd+ ), we show that Problem (0.0.4) admits a unique classical 1+α/2,2+α solution u ∈ Cb ((0, T ) × Rd+ ) and it satisfies the estimate ||u||C 1+α/2,2+α ((0,T )×Rd ) ≤ C(||f ||C 2+α (Rd ) + ||g||C α/2,α ((0,T )×Rd ) + ||ψ||C 1+α/2,2+α ((0,T )×Rd−1 ) ), b

+

b

+

α/2,α

b

+

b

1+α/2,2+α

provided that f ∈ Cb2+α (Rd+ ), g ∈ Cb ((0, T ) × Rd+ ), ψ ∈ Cb ((0, T ) × Rd−1 ) the due compatibility conditions f (·, 0) ≡ ψ(0, ·), Dt ψ(0, ·) ≡ Af (·, 0) + g(0, ·, 0) on Rd−1 are satisfied. The argument used to prove this result is much trickier than that in Chapter 6, due to the boundary and compatibility conditions. Then, we state further regularity results for the solution to problem (0.0.4) and conclude the chapter by introducing the semigroup

Introduction

xix

of bounded operators associated with operator A with homogeneous Dirichlet boundary conditions on Cb (Rd+ ). In Chapter 8, we study the existence, uniqueness and optimal regularity of solutions to the Cauchy problem, with general boundary conditions on Rd+ ,  t ∈ (0, T ], x ∈ Rd+ ,   Dt u(t, x) = Au(t, x) + g(t, x), Bu(t, x0 , 0) = ψ(t, x0 ), t ∈ (0, T ], x0 ∈ Rd−1 , (0.0.5)   d u(0, x) = f (x), x ∈ R+ , ∂ . Under suitable conditions on the coefficients of operators A on the ∂η functions a and η, we prove the existence and uniqueness of the classical solution u to 1+α/2,2+α Problem (0.0.5), which also belongs to Cb ((0, T ) × Rd+ ) and satisfies the estimate where B = aI +

||u||C 1+α/2,2+α ((0,T )×Rd ) ≤ C(||f ||C 2+α (Rd ) + ||g||C α/2,α ((0,T )×Rd ) + ||ψ||C (1+α)/2,1+α ((0,T )×Rd−1 ) ), b

+

+

b

b

+

b

α/2,α

provided that f ∈ Cb2+α (Rd+ ), g ∈ Cb

((0, T ) × Rd+ ) and ψ ∈ C (1+α)/2,1+α ((0, T ) × Rd−1 ) ∂f satisfy the due compatibility conditions af (·, 0)+ (·, 0) ≡ ψ(0, ·) on Rd−1 . As a byproduct, ∂η we show some interior Schauder estimates up to a portion of ∂Rd+ satisfied by the solutions to the equation Dt u = Au + g in (0, T ] × Rd+ . Also in this case, the more the coefficients of the operators A, B and the data f , g and ψ are smooth, the more the solution to the Cauchy problem (0.0.5) is itself smooth. The last part of this chapter deals with the semigroups of bounded linear operators associated to operator A, subject to the above mixed boundary conditions on Cb (Rd+ ) and Cb (Rd+ ). In Chapter 9, we complete the analysis of parabolic Cauchy problems. Here, we study the existence, uniqueness and optimal regularity of solutions to the following Cauchy problems with Dirichlet and also with general boundary conditions on a bounded and smooth domain Ω of Rd :  t ∈ [0, T ], x ∈ Ω,   Dt u(t, x) = Au(t, x) + g(t, x), Bu(t, x) = h(t, x), t ∈ [0, T ], x ∈ ∂Ω, (0.0.6)   u(0, x) = f (x), x ∈ Ω,    Dt u(t, x) = Au(t, x) + g(t, x), u(t, x) = h(t, x),   u(0, x) = f (x),

t ∈ [0, T ], t ∈ [0, T ],

x ∈ Ω, x ∈ ∂Ω,

(0.0.7)

x ∈ Ω.

We also prove that, the more the coefficients of the operators A, B, and the data f , g and h are smooth, the more the solutions to the Cauchy problems (0.0.6) and (0.0.7) are themselves smooth. Using optimal Schauder estimates and the results in Chapter 8, we prove some interior Schauder estimates up to a portion of ∂Ω, satisfied by the solutions to the equation Dt u = Au + g in (0, T ] × Ω, where Ω is an open connected set of class C 2+α (which could be unbounded). As we did in the previous chapters, we conclude by introducing the semigroups of bounded linear operators associated to the homogeneous Cauchy problems (with g ≡ 0) (0.0.6) and (0.0.7) on Cb (Ω) and C(Ω). In all these chapters, we deal with real-valued functions but the results presented can be easily extended to complex-valued functions. Indeed, since the coefficients of the operators A and B are real-valued, if the data f , g, h and ψ are complex-valued, then it suffices to

xx

Introduction

split the problem into two problems: the first one has as data the real parts of the data f , g, h and ψ the second one has the imaginary parts of f , g, h and ψ as data. We stress that, even if we consider autonomous elliptic and parabolic operators (to exploit the connections with the theory of semigroups), the proofs that we present can be adapted to cover also the case when the operators are nonautonomous, i.e., their coefficients depend also on time.

PART III: Elliptic equations In this part we study existence, uniqueness and regularity properties of solutions to the elliptic equation λu − Au = f (0.0.8) in Rd , Rd+ and in smooth bounded domains of Rd subject to Dirichlet and mixed boundary conditions. We study the above problems both in Lp -spaces and in the space of bounded and continuous functions. In Chapter 10, we first solve equation (0.0.8) in spaces of H¨older continuous functions over Rd . Using the Gauss-Weierstrass semigroup, it can be easily seen (with the help of the results in Chapter 5) that, for each λ ∈ C with positive real part, there exists a unique solution u ∈ Cb2+α (Rd ; C) of the equation λu − ∆u = f and it satisfies the estimate ||u||C 2+α (Rd ;C) ≤ C||f ||Cbα (Rd ;C)

(0.0.9)

b

for some positive constant C, independent of u and f . From the above result and the continuity method, one then deduces that there exists a unique solution u ∈ Cb2+α (Rd ; C) of (0.0.8) for every λ ∈ C with Reλ > supx∈Rd c(x), which, in addition, satisfies (0.0.9), provided that the coefficients of A are bounded and α-H¨older continuous functions on Rd , for some α ∈ (0, 1), and the matrix-valued function Q = (qij ) is uniformly elliptic. In the second part of the chapter, we study problem (0.0.8) when f ∈ Lp (Rd ; C) in the context of Sobolev spaces. More precisely, we show that there exists λp ∈ R such that, for every f ∈ Lp (Rd ; C) and λ ∈ C with Reλ > λp , there exists a unique solution u ∈ W 2,p (Rd ; C) of (0.0.8), which, in addition, satisfies the estimate p |λ|||u||Lp (Rd ;C) + |λ||||∇u|||Lp (Rd ;C) + ||D2 u||Lp (Rd ;C) ≤ C||f ||Lp (Rd ;C) , provided that the diffusion coefficients are uniformly continuous and bounded functions, the other coefficients belong to L∞ (Rd ) and the uniform ellipticity property is satisfied. The main ingredients are the celebrated Calder´on-Zygmund inequality (that allows to estimate the second-order derivatives of a function u ∈ W 2,p (Rd ; C) from above in terms of the Lp (Rd ; C)-norms of u and its Laplacian), the continuity method and the a priori estimate p |λ|||u||Lp (Rd ;C) + |λ||||∇u|||Lp (Rd ;C) + ||D2 u||Lp (Rd ;C) ≤ C||λu − Au||Lp (Rd ;C) , which is satisfied by every u ∈ W 2,p (Rd ; C), every λ ∈ C with Re λ > λp and some positive constant C, independent of u and λ. In the last part of the chapter, we study the solvability of the equation (0.0.8) in L∞ (Rd ; C) and in Cb (Rd ; C). Here, the main ingredient is the following a priori estimate |λ|||u||∞ +

p d |λ||||∇u|||∞ + |λ| 2p sup ||D2 u||Lp (B(x0 ,|λ|−1/2 );C) ≤ Cp ||λu − Au||∞ x0 ∈Rd

2,p satisfied by every u ∈ Cb1+α (Rd ; C) ∩ Wloc (Rd ; C), for every p ∈ [1, ∞) and α ∈ (0, 1), such

Introduction

xxi

that Au ∈ L∞ (Rd ; C) (resp. Au ∈ Cb (Rd ; C)), every λ ∈ C, with sufficiently large real part, and some positive constant Cp , independent of u and λ. Chapter 11 is devoted to the boundary value problem u ≡ g on ∂Rd+ for the elliptic equation (0.0.8) in Rd+ when g belongs to Cb2+α (Rd−1 ; C) or to W 2,p (Rd−1 ; C). We show that, if f ∈ Cbα (Rd+ ; C) (resp. Lp (Rd+ ; C)) and the coefficients of the operator A satisfies appropriate and natural conditions, then there exists a unique classical solution u ∈ Cb2+α (Rd+ ; C) (resp. W 2,p (Rd+ ; C)), which, in addition, satisfies the estimate ||u||X ≤ C(||f ||Y + ||g||Z ) for all λ ∈ C with sufficiently large real part, and a positive constant C, which is independent of u and λ. Here, X = Cb2+α (Rd+ ; C),Y = Cbα (Rd+ ; C) and Z = Cb2+α (Rd−1 ; C) (resp. X = W 2,p (Rd+ ; C), Y = Lp (Rd+ ; C) and Z = W 2,p (Rd−1 ; C)). In the particular case when g ≡ 0, the solution u ∈ W 2,p (Rd+ ; C) ∩ W01,p (Rd+ ; C) satisfies the estimate p |λ|||u||Lp (Rd+ ;C) + |λ||||∇u|||Lp (Rd+ ;C) + ||D2 u||Lp (Rd+ ;C) ≤ Cp ||f ||Lp (Rd+ ;C) for every λ ∈ C with Reλ ≥ λp and a positive constant Cp , independent of u and λ. We also deal with weak solutions to the above boundary value problem when g ≡ 0. We conclude the chapter by analyzing the boundary value problem in L∞ (Rd+ ; C) and Cb (Rd+ ; C) in the case g ≡ 0. In Chapter 12, we solve the boundary value problem consisting of equation (0.0.8) in Rd+ and the first-order boundary condition Bu ≡ g on ∂Rd+ , in the context of spaces of H¨older continuous functions, when g ∈ Cb1+α (Rd−1 ; C) and in Lp -spaces when g ∈ B 1−1/p,p (Rd−1 ; C). In the first case, we show the existence and uniqueness of the solution to the above boundary value problem when A = ∆ and use the method of continuity to deduce the existence and uniqueness of the classical solution u ∈ Cb2+α (Rd+ ; C) satisfying the estimate ||u||C 2+α (Rd ;C) ≤ C(||f ||Cbα (Rd+ ;C) + ||g||C 1+α (Rd−1 ;C) ), under appropriate and nat+ b b ural conditions on the coefficients of operators A and B. In the Lp -setting, we first prove the a priori estimate p |λ|||u||Lp (Rd+ ;C) + |λ||||∇u|||Lp (Rd+ ;C) + ||D2 u||Lp (Rd+ ;C) p ≤Cp (||λu − Au||Lp (Rd+ ;C) + |λ|||g0 ||Lp (Rd+ ;C) + |||∇g0 |||Lp (Rd+ ;C) ) (0.0.10) satisfied by every function u ∈ W 2,p (Rd+ ; C) and some positive constant Cp , independent of u and g0 , where g0 is an arbitrary extension in W 1,p (Rd+ ; C) of the function Bu. The solvability of the Neumann-boundary value problem associated with the Laplacian and estimate (0.0.10) permit us to obtain the existence and uniqueness of the solution u ∈ W 2,p (Rd+ ; C) to the above boundary value problem. Based on these results we analyze the previous boundary value problem in the L∞ - and Cb -settings. In Chapter 13, we analyze the boundary value problems ( ( λu − Au = f, in Ω, λu − Au = f, and u = g, on ∂Ω Bu = g,

in Ω, on ∂Ω,

when Ω is a bounded and sufficiently smooth domain of Rd . Here, through local charts and suitable partitions of the unity, we transform both boundary value problems into two problems: the first one set in Rd and the second one in Rd+ . To these problems we can apply results obtained in Chapters 10 to 12. Chapter 14 is dedicated to prove that the semigroups introduced in Chapters 6 to 9 are analytic. We also characterize the interpolation spaces and prove some of the regularity results for solutions to parabolic problems in Rd+ and in bounded smooth domains of Rd , which we stated in the second part of the book. We use the semigroup approach to prove

xxii

Introduction

these results, to make still more evident the connection between PDE’s and semigroup of bounded operators. Chapter 15 concludes the third part. Here, we consider the elliptic operator A0 in divergence form, defined on smooth functions ψ : Ω → C by A0 ψ = −

d X

Dj (qij Di ψ) +

i,j=1

d X (bi Di ψ − Di (ci ψ)) + c0 ψ, i=1

subject to Dirichlet boundary conditions on an arbitrary open subset Ω of Rd and under general conditions on its coefficients. Using form methods, we prove that a suitable realization A0 of operator A0 in L2 (Ω; C) generates an analytic semigroup {T0 (t)} on L2 (Ω; C) which interpolates in the Lp -spaces over Ω for p ∈ [1, ∞)∪{∞}. Applying the Beurling-Deny and the Dunford-Pettis criteria, we deduce that {T0 (t)} admits an integral representation of the form Z (T0 (t)f )(x) = kt (x, y)f (y) dy, x ∈ Ω, f ∈ L2 (Ω), Ω

where kt ∈ L∞ (Ω × Ω) satisfies the estimate d

0 ≤ kt (x, y) ≤ M t− 2 eωt e−

|x−y|2 4ωt

,

a.e. (x, y) ∈ Ω × Ω, t > 0,

for some positive constants M and ω. Using these and the results of Chapter 14, we show that, under suitable assumptions on the coefficients of the operator A, considered in Chapters 6, 7 and 9, the semigroups constructed in those chapters actually admit an integral kernel, which satisfies the above estimate.

PART IV: Appendices In the last part of the book, we first collect (in Appendix A) some basic notions of functional analysis, operator theory in Banach spaces and interpolation theory. We do not intend to be exhaustive, just provide those results which are really used in the book, referring the reader to well recognized monographs for further details and a much more complete overview. Again, for the sake of completeness, we prove almost all the results that we present, we just skip the proof of the simplest ones. Appendix B is devoted to introducing two important tools used in this book. The first one is the definition of (bounded) domains of class C α for α > 0. In literature, there are seemingly different definitions of this concept. We prove that all these definitions actually coincide. We also prove some relevant properties of smooth domains. The second tool that we introduce, in the second part of the chapter, are the extension operators. We first address the problem of extending a function defined on the closure of an open set to the whole Rd via a linear bounded extension operator which preserves the smoothness of the function. In defining these extension operators, the smoothness of the domain Ω plays a crucial role. The starting (and easiest) case is when Ω = Rd+ . To cover the general case, we take advantage of a suitable partition of the unity, which allows us to go back to the case Ω = Rd+ . The extension operators that we define map C α (Ω; K) into C α (Rd ; K) (resp. W k,p (Ω; K) into W k,p (Rd ; K)). Here, K denotes both the set of real numbers and the set of complex numbers. In the last part of the chapter, we deal with the problem of extending a function, which belongs to the Besov space B k−1/p,p (∂Ω; K) for some p ∈ (1, ∞), when Ω is Rd+ or a bounded open set of class C k , with a function which belongs to W k,p (Ω; K), i.e., we show that B k−1/p,p (∂Ω; K) is the space of traces of functions in W k,p (Ω; K). Again the strategy consists in first considering the easiest case Ω = Rd+ and then addressing the general case, taking advantage of a suitable partition of the unity.

Chapter 1 Function Spaces

In this chapter, we introduce the main function spaces that we use in this book and collect some of their remarkable properties that will be extensively used. In Section 1.1, we begin by considering spaces of (H¨ older) continuous functions defined on a domain Ω of Rd and prove some very important interpolation estimates. Then, in Section 1.2 we introduce anisotropic and parabolic spaces of H¨ older continuous functions, which appear naturally in the analysis of optimal regularity for classical solutions to parabolic equations (see Chapters 5 to 9). Section 1.3 is devoted to introducing the main results of the classical theory of Lp - and Sobolev spaces that we use in this book. Finally, in Section 1.4 we introduce the Besov spaces which are intrinsically related to the theory of traces of functions which belongs to Sobolev spaces over (sufficiently smooth) domains. Aimed at making the book as much self contained as possible, we prove all the results that we present in this chapter. Notation. Throughout the chapter, Ω will denote a domain (i.e., an open and connected subset of Rd (d ≥ 1)) or its closure. Moreover, K will denote both the sets R and C.

1.1

Spaces of (H¨ older) Continuous Functions

Definition 1.1.1 ˆ By Cb (Ω; K) we denote the set of all functions f : Ω → K which are bounded and continuous in Ω. It is a Banach space when endowed with the norm ||f ||∞ = sup |f (x)|,

f ∈ Cb (Ω; K),

x∈Ω

the so-called sup-norm.

ˆ By BU C(Ω; K) we denote the set of all the bounded and uniformly continuous functions f : Ω → K. It is a Banach space when endowed with the sup-norm. ˆ By C0 (Ω; K) we denote the set of all the functions f : Ω → K which vanish on the boundary of Ω. If Ω is unbounded, then sometimes we also require that f vanishes as |x| → ∞. C0 (Ω; K) is a Banach space when endowed with the sup-norm. ˆ For each α ∈ (0, 1), Cbα (Ω; K) is the subset of Cb (Ω; K) consisting of functions f : Ω → K such that [f ]Cbα (Ω;K) = sup

x,y∈Ω x6=y

|f (x) − f (y)| < ∞. |x − y|α

It is a Banach space when endowed with the norm ||f ||Cbα (Ω;K) = ||f ||∞ + [f ]Cbα (Ω;K) ,

f ∈ Cbα (Ω; K). 1

2

Function Spaces More generally, for α ∈ (0, ∞), Cbα (Ω; K) is the subset of Cb (Ω; K) of functions f : Ω → K which admit bounded derivatives up to the order [α] and their derivatives of order [α] are (α − [α])-H¨ older continuous in Ω (if α 6∈ N). It is a Banach space when endowed with the norm X X f ∈ Cbα (Ω; K). ||Dβ f ||∞ + [Dβ f ]C α−[α] (Ω;K) , ||f ||Cbα (Ω;K) = |β|≤[α]

|β|=[α]

b

ˆ By Lip(Ω; K) we denote the set of all functions f : Ω → K such that [f ]Lip(Ω;K) = sup x,y∈Ω x6=y

|f (x) − f (y)| < ∞. |x − y|

When, Ω is bounded, it is a Banach space when endowed with the norm ||f ||Lip(Ω;K) = ||f ||∞ + [f ]Lip(Ω;K) ,

f ∈ Lip(Ω; K).

ˆ For k ∈ N, BU C k (Ω; K) is the set of functions f : Ω → K which belong to BU C(Ω; K) together with their derivatives up to the k-th-order. ˆ For α ∈ (0, ∞), C0α (Ω; K) is the subset of Cbα (Ω; K) of functions which vanish on ∂Ω together with all their derivatives up to the order [α]. α (Ω; K) the set of all functions f : Ω → K ˆ For each α ∈ (0, ∞) \ N, we denote by Cloc α which belong to C (K; K) for each compact subset K of Ω. α We write Cb (Ω), BU C(Ω), C0α (Ω), Cbα (Ω) and Cloc (Ω) when we consider real-valued functions. When the subscript “b” is not needed, we will skip it. For m ≥ 2, we also use the notation Cb (Ω; Km ) for vector valued functions f : Ω → Km , whose components belong to α Cb (Ω). The subspaces BU C(Ω; Km ), C0α (Ω; Km ), Cbα (Ω; Km ) and Cloc (Ω; Km ).

Remark 1.1.2 If Ω is a convex domain or a domain of class C α and α ∈ (1, ∞) \ N (see Section B.2.1), then Cbα (Ω; K) can be, equivalently, characterized as the set of all α−[α] f : Ω → K which admits classical derivatives up to the order [α] and Dβ f ∈ Cb (Ω; K) for every multi-index β with length at most [α]. Moreover, the norm ||·||Cbα (Ω;K) is equivalent P β to the norm |β|≤[α] ||D f ||Cbα−[α] (Ω;K) . Indeed, it is immediate to check that this latter space continuously embeds into Cbα (Ω; K). On the other hand, as noticed in Remark B.4.3, α−[α] Cb1 (Ω; K) ,→ Cb (Ω; K), so that all the existing derivatives of f ∈ Cbα (Ω; K) belong to α−[α] Cb (Ω; K) and ||Dβ f ||C α−[α] (Ω;K) ≤ C||f ||Cbα (Ω;K) for some constant C independent of f b and every multi-index β with length at most [α]. Lemma 1.1.3 For each α ∈ (0, 1), a function f belongs to Cbα (Rd ; K) if and only if f is bounded and, for some (and, hence, all ) r > 0, |f (x) − f (y)| ≤ Cr |x − y|α for every x, y ∈ Rd , such that |x − y| ≤ r, and some positive constant Cr . Moreover, for every r > 0, the norm |f (x) − f (y)| , f ∈ Cbα (Rd ; K), (1.1.1) ||f ||∞ + sup |x − y|α x,y∈Rd 0 0, then for |x − y| > r we can estimate |f (x) − f (y)| ≤ 2||f ||∞ ≤ 2r−α ||f ||∞ |x − y|α . Hence, f ∈ Cbα (Rd ; K) and [f ]Cbα (Rd ;K) ≤ Cr ∨ (2r−α ||f ||∞ ). This estimate also shows that  the classical norm of Cbα (Rd ; K) and the norm in (1.1.1) are equivalent. In the following propositions, we prove very useful interpolation results. Proposition 1.1.4 For each α, θ, β ∈ [0, ∞), with θ < β < α, there exists a positive constant C = C(α, β, θ) such that β−θ

α−β

f ∈ Cbα (Rd ; K).

α−θ ||f ||C β (Rd ;K) ≤ C||f ||Cα−θ θ (Rd ;K) ||f ||C α (Rd ;K) , b

b

b

(1.1.2)

Proof In this book, we will use this proposition only when α ≤ 3. For the sake of simplicity, we limit ourselves to proving it only in these situations but we stress that the proof in the other cases is quite similar. Since it is rather long, we split the proof into several steps. More precisely, ˆ in Step 1, we consider the case α, β, θ ∈ [0, 1]; ˆ in Step 2, we consider the case α ∈ (1, 2), θ ∈ [0, 1) and β = 1; ˆ in Step 3, we complete the case α < 2; ˆ in Step 4, we consider the case α ∈ (2, 3], θ ∈ [0, 1] and β = 2; ˆ in Step 5, we consider the case α ∈ [2, 3], θ ∈ [0, 1] and β < 2; ˆ in Step 6, we complete the proof.

Note that it is enough to prove the assertion for real-valued functions. Indeed, if f is a complex-valued function, it suffices to consider its real and imaginary parts. We fix f ∈ Cbα (Rd ) and, throughout the proof, by c we denote a positive constant, which may vary from line to line, but it is independent of the functions that we consider. Step 1. We fix x, y ∈ Rd and observe that |f (x) − f (y)| ≤ [f ]Cbθ (Rd ) |x − y|θ ,

|f (x) − f (y)| ≤ ||f ||Cbα (Rd ) |x − y|α .

Hence, α−β

β−θ

|f (x) − f (y)| =|f (x) − f (y)| α−θ |f (x) − f (y)| α−θ β−θ   α−β   α−θ α−θ ≤ [f ]Cbθ (Rd ) |x − y|θ ||f ||Cbα (Rd ) |x − y|α α−β

β−θ

b

b

α−θ β ≤[f ]Cα−θ θ (Rd ) ||f ||C α (Rd ) |x − y| ,

and estimate (1.1.2) follows immediately. Step 2. Of course, to prove (1.1.2) with α ∈ (1, 2), θ ∈ [0, 1) and β = 1, we just need to show that 1−θ X  α−θ d α−1 α−θ ||∇f ||∞ ≤ c||f ||C θ (Rd ) [Dj f ]C α−1 (Rd ) . (1.1.3) b

b

j=1

4

Function Spaces

For this purpose, we fix x ∈ Rd , such that ∇f (x) 6= 0, and y ∈ Rd . Moreover, we set h(t) = f (x + ty) for every t > 0 and use the fundamental theorem of calculus to write Z

1 0

Z

0

1

(h0 (t) − h0 (0)) dt,

h (t) dt = h(0) + h (0) +

h(1) = h(0) +

0

0

i.e., Z

1

f (x + y) = f (x) + h∇f (x), yi +

h∇f (x + ty) − ∇f (x), yi dt 0

and then estimate |h∇f (x), yi| |f (x + y) − f (x)| ≤ + |y| |y| θ−1

≤||f ||Cbθ (Rd ) |y|

1 + α

1

Z

|∇f (x + ty) − ∇f (x)| dt 0

X d

 [Dj f ]C α−1 (Rd ) |y|α−1 . b

j=1

Now, we fix s > 0 and set y = s∇f (x) to get   d X θ−1 θ−1 α−1 α−1 |∇f (x)| ≤ c s ||f ||Cbθ (Rd ) |∇f (x)| +s |∇f (x)| [Dj f ]C α−1 (Rd ) . b

j=1

Minimizing with respect to s > 0, we get α−1 α−θ Cbθ (Rd )

|∇f (x)| ≤ c||f ||

X d

1−θ  α−θ

.

[Dj f ]

Cbα−1 (Rd )

j=1

Estimate (1.1.3) follows at once. Step 3. Based on Step 2, we can now prove estimate (1.1.2) with α < 2. We first consider the case θ ∈ [0, 1) and β ∈ (θ, 1). Using Steps 1 and 2, we estimate   β−θ 1−β 1−β β−θ α−1 1−θ 1−θ 1−θ 1−θ α−θ α−θ ||f ||C β (Rd ) ≤ c||f ||C1−θ θ (Rd ) ||f ||C 1 (Rd ) ≤ c||f ||C θ (Rd ) ||f ||C θ (Rd ) ||f ||C α (Rd ) b

b

b

b

b

b

and (1.1.2) follows. If θ ∈ [0, 1) and β ∈ (1, α), we apply Step 1, with (α, β, θ) = (α − 1, β − 1, 0), to each first-order derivative of f to infer that α−β

β−1

α−1 ||Dj f ||C β−1 (Rd ) ≤ c||Dj f ||∞ ||Dj f ||Cα−1 . α−1 (Rd ) b

b

From this estimate and Step 2 it follows easily that   α−β α−β β−1 β−1 α−1 1−θ α−1 α−1 α−θ α−θ ||f ||C β (Rd ) ≤ c||f ||Cα−1 ||f || ||f ||Cα−1 α α (Rd ) 1 (Rd ) ||f ||C α (Rd ) ≤ c ||f || θ d C (R ) C (Rd ) b

b

b

b

b

b

and (1.1.2) follows. Finally, if θ ∈ [1, 2), then we apply Step 1, with (α, θ, β) = (α − 1, θ − 1, β − 1) to each first-order derivative of f and (1.1.2) follows at once. Step 4. We now prove estimate (1.1.2) with α ∈ (2, 3], θ ∈ [0, 1] and β = 2. We use

Semigroups of Bounded Operators and Second-Order PDE’s

5

again Taylor formula with k(t) = f (x + ty) for each t > 0, with y ∈ Rd \ {0} and x ∈ Rd arbitrarily fixed, to get Z 1 1 k(1) = k(0) + k 0 (0) + k 00 (0) + (k 00 (t) − k 00 (0))(1 − t) dt, 2 0 i.e., 1 f (x + y) = f (x) + hf (x), yi + hD2 f (x)y, yi + 2

1

Z

h(D2 f (x + ty) − D2 f (x))y, yi(1 − t) dt.

0

Let λ(x) be any eigenvalue of the matrix D2 f (x) and let yλ,x be a corresponding unit eigenvector. Taking y = syλ,x (with s positive) in the previous formula, we infer that Z 1 2|f (x + y) − f (x)| 2|∇f (x)| + + 2 |D2 f (x + ty) − D2 f (x)|(1 − t) dt |λ(x)| ≤ |y|2 |y| 0   ≤c sθ−2 [f ]Cbθ (Rd ) + s−1 ||∇f ||∞ + sα−2 ||f ||Cbα (Rd ) . To estimate the term s−1 ||∇f ||∞ , we observe that (1.1.3) clearly holds true also with α = 2. Applying such estimate to the first-order derivative of f and, then, estimating ||D2 f ||∞ , using again (1.1.3) (α, θ, g) = (α − 1, 0, Dj f ), we deduce that 1−θ

1

α−2

1

1

2−θ α−1 2 ≤ c||f ||C2−θ ||f ||Cα−1 ||∇f ||∞ ≤ c||f ||C2−θ α (Rd ) θ (Rd ) ||D f ||∞ θ (Rd ) ||∇f ||∞ b


1−θ 2−θ

b

b

or, equivalently, α−1

1−θ

α−θ ||∇f ||∞ ≤ c||f ||Cα−θ θ (Rd ) ||f ||C α (Rd ) . b

b

Thus, α−1

1−θ

α−θ s−1 ||∇f ||∞ ≤cs−1 ||f ||Cα−θ θ (Rd ) ||f ||C α (Rd ) b

b

θ−2

=c(s

||f ||Cbθ (Rd ) )

α−1 α−θ

1−θ

(s2−α ||f ||Cbα (Rd ) ) α−θ

≤c(sθ−2 ||f ||Cbθ (Rd ) + s2−α ||f ||C 2+α (Rd ) ), b

where we have taken advantage of the Young inequality. From the previous two inequalities we conclude that |λ(x)| ≤ csθ−2 ||f ||Cbθ (Rd ) + cs2−α ||f ||Cbα (Rd ) . Minimizing with respect to s > 0 gives 2−θ

2−α

α−θ |λ(x)| ≤ c||f ||Cα−θ θ (Rd ) ||f ||C α (Rd ) . b

b

Now we are almost done. Indeed, since D2 f (x) is a symmetric matrix, there exists an orthogonal matrix Px such that D2 f (x) = Px Λ(x)PxT , where Λ(x) is a diagonal matrix whose entries are the eigenvalues of D2 f (x). Hence, 2

||D f (x)||

≤||Px ||||Λ(x)||||PxT ||

2

≤ ||Px ||

d X

|λi (x)| ≤ d

i=1

d X i=1

α−2

2−θ

α−θ |λi (x)| ≤ c||f ||Cα−θ θ (Rd ) ||f ||C α (Rd ) b

b

and (1.1.2) follows, taking the supremum with respect to x ∈ Rd . Step 5. The core of this step is the case α = 2. Once we prove that there exists a positive constant c such that 2−β β−θ 2−θ ||f ||C β (Rd ) ≤ c||f ||C2−θ (1.1.4) θ (Rd ) ||f ||C 2 (Rd ) , b

b

b

6

Function Spaces

we will be done. Indeed, from this estimate and Step 4, it follows easily that   β−θ 2−β 2−α 2−θ 2−θ α−θ α−θ ||f || ||f || ||f ||C β (Rd ) ≤ c||f ||C2−θ θ (Rd ) C α (Rd ) C θ (Rd ) b

b

b

b

from which (1.1.2) follows. So, let us prove estimate (1.1.4). We consider separately the cases β ∈ (0, 1) and β ∈ (1, 2) (note that the case β = 1 has been proved in Step 2). The case β < 1 is easier: it is enough to write (1.1.2) with α = 1 and replace ||f ||Cb1 (Rd ) with the estimate given by (1.1.4). For β > 1, we observe that . ||f ||β−1 ||f ||C β (Rd ) ≤ c||f ||2−β C 2 (Rd ) C 1 (Rd ) b

(1.1.5)

b

b

β−1 ||f ||C Indeed, ||f ||Cb1 (Rd ) ≤ ||f ||2−β 2 d ; moreover, applying (1.1.2) to Dj f with β being Cb1 (Rd ) b (R ) replaced with β − 1 and with θ = 0 and α = 1, we get β−1 2−β β−1 ||Dj f ||C β−1 (Rd ) ≤ c||Dj f ||2−β ∞ ||Dj f ||C 1 (Rd ) ≤ c||f ||C 1 (Rd ) ||f ||C 2 (Rd ) b

b

b

b

for each j = 1, . . . , d. Combining these two estimates, we obtain (1.1.5). Using (1.1.5) and (1.1.4) with β = 1 we get  2−β 1−θ 1 β−1 2−θ ||f ||C β (Rd ) ≤c||f ||2−β ||f ||β−1 ≤ c ||f ||C2−θ ||f ||C 2 (Rd ) θ (Rd ) ||f ||C 2 (Rd ) C 2 (Rd ) C 1 (Rd ) b

b

b

2−β 2−θ Cbθ (Rd )

=c||f ||

β−θ 2−θ Cb2 (Rd )

||f ||

b

b

b

,

which is (1.1.4), in this case. Step 6. Here, we complete the proof of (1.1.2). The case α ∈ (2, 3], β ∈ (2, α) and θ ∈ [0, 1] ∪ {2} follows almost straightforwardly from the previous steps. Indeed, applying Step 1, with (α, θ) = (α − 2, 0), to the second-order derivatives of f , we obtain α−β

β−2

α−2 ||f ||C β (Rd ) ≤ ||f ||Cα−2 2 (Rd ) ||f ||C α (Rd ) , b

b

b

which is (1.1.2) in the case θ = 2. Next, using Step 4, with (α, β, θ) = (α, 2, θ) we can estimate ||f ||Cbα (Rd ) and cover the other case. If α ∈ [2, 3] and θ ∈ (1, 2), then it suffices to observe that Steps 2, 3 and 5 show that α−β

β−θ

||g||Cα−θ , ||g||C β−1 (Rd ) ≤ c||g||Cα−θ α−1 θ−1 (Rd ) (Rd ) b

b

b

g ∈ Cbα−1 (Rd ).

Writing this estimate with g = Dj f (j = 1, . . . , d), (1.1.2) follows. Finally, if α ∈ (2, 3] and θ ∈ (2, α), then it suffices to apply Step 1, with (α, β, θ) = (α − 2, β − 2, θ − 2) to the second-order derivatives of f .  The previous proposition can be easily generalized to the case of smooth enough domains thanks to Proposition B.4.1. Corollary 1.1.5 Let Ω = Rd+ or let Ω be a bounded open set of class C α for some α ∈ (0, ∞). Then, for every θ, β ∈ [0, ∞) such that θ < β < α, there exists a positive constant C = C(α, β, θ, Ω) such that α−β

β−θ

α−θ ||f ||C β (Ω;K) ≤ C||f ||Cα−θ θ (Ω;K) ||f ||C α (Ω;K) , b

b

b

f ∈ Cbα (Ω; K).

(1.1.6)

Semigroups of Bounded Operators and Second-Order PDE’s

7

Proof We fix α, β, θ as in the statement, f ∈ Cbα (Ω; K) and consider the extension operator Eα introduced in Proposition B.4.1. Applying estimate (1.1.2) to the function Eα f , which belongs to Cbα (Rd ; K), we obtain α−β

β−θ

α−θ ||f ||C β (Ω;K) ≤||Eα f ||C β (Rd ;K) ≤ C||Eα f ||Cα−θ θ (Rd ;K) ||Eα f ||C α (Rd ;K) b

b

b

b

α−β α−θ L(Cbθ (Ω;K),Cbθ (Rd ;K))

≤C||Eα ||

β−θ α−θ

α−β

β−θ

α−θ ||Eα ||L(C α (Ω;K),C α (Rd ;K)) ||f ||Cα−θ θ (Ω;K) ||f ||C α (Ω;K) . b

b

b

b

Estimate (1.1.6) follows immediately.

1.1.1



Functions defined on the boundary of a smooth open set

In this subsection, Ω is a bounded open set of class C α for some α > 0 (see Definition B.2.1). We want to define the space C α (∂Ω; K). If α ∈ [0, 1), then we can give a straightforward definition which mimic the definition of the space C α (Ω; K) given in the first part of this section. If α ≥ 1, then the definition of the space C α (∂Ω; K) is somehow more complicate. To state it, according to Definition B.2.1, we introduce a covering {Ux : x ∈ ∂Ω} of ∂Ω such that for every open set Ux there exists a function ψx : Ux → B(0, r) of class C α which is invertible with inverse function which is of class C α as well. Since ∂Ω is a compact set, we can extract a subcovering {Uxj : j = 1, . . . , N } of ∂Ω. Associated with this covering of ∂Ω, we consider a partition of the unity {ηj : j = 1, . . . , N } (see Proposition B.1.1). Moreover, we denote by B 0 (0, r) the ball of Rd−1 centered at zero with radius r. Definition 1.1.6 The space C α (∂Ω; K) is the set of all functions f : ∂Ω → K such that (y, 0)) for every k = 1, . . . , N , the function vk : B 0 (0, r) → K, defined by wk (y) = (ηk u)(ψx−1 k α 0 0 for every y ∈ B (0, r), belongs to C (B (0, r); K). It is normed by setting ||u||C α (∂Ω;K) =

N X

||wk ||C α (B 0 (0,r);K) ,

u ∈ C α (∂Ω; K).

k=1

Remark 1.1.7 The previous definition is independent of the functions ψxj and ηj . Changing the functions ψj = ψxj and/or the functions ηj produces just a change in the norm of C α (∂Ω; K) in the sense that the norm introduced in Definition 1.1.6 is replaced by an equivalent norm. Indeed, suppose that {Ui∗ : i = 1, . . . , M } is a different covering of ∂Ω, ϕi : Ui∗ → ∗ B(0, r) are bijective functions of class C α with inverse of class C α and let {η1∗ , . . . , ηM } be a partition of the unity subordinated to this new covering. Fix j ∈ {1, . . . ,P M } and denote by Ij the set of all index i ∈ {1, . . . , N } such that Ui ∩ Uj∗ 6= ∅. Then, i∈Ij ηi = 1 on P P ∗ ∂Ω∩Uj∗ and, consequently, uηj∗ = i∈Ij uηj∗ ηi = i∈Ij ui,j . Note that ui,j ◦ϕ−1 j (·, 0) = (ηj ◦ ψi−1 (·, 0)wi )◦Θi,j on the open set Ω0i,j = PRd−1 ϕj (∂Ω∩Ui ∩Uj∗ ) = B 0 (0, r)∩PRd−1 ϕj (Ui ∩Uj∗ ), where PRd−1 x = x0 for every (x0 , xd ) ∈ Rd and Θi,j = ψi ◦ ϕ−1 j (·, 0) is a diffeomorphism of −1 α 0 ∗ class C on Ωi,j . Clearly, the function (ηj ◦ ψi (·, 0)wi ) ◦ Θi,j belongs to C α (Ω0i,j ; K) and ||(ηj∗ ◦ ψi−1 (·, 0)wi ) ◦ Θi,j ||C α (Ω0i,j ,K) ≤ C||wi ||C α (B 0 (0,r);K) for some positive constant C, independent of wi . Moreover, since the supports of ηi and ηj∗ are compact subsets of Uj and Uj∗ , respectively, the support of the function (ηj∗ ◦ψi−1 (·, 0)wi )◦Θi,j is compact in Ω0i,j . We can extend it in a trivial way to B 0 (0, r) and conclude that the so extended function ui,j ◦ ϕ−1 j (·, 0) belongs to C α (B 0 (0, r); K) and ||ui,j ◦ ϕ−1 j (·, 0)||C α (B 0 (0,r);K) ≤ C||wi ||C α (B 0 (0,r);K) , where the constant C is independent of u as all the other constants appearing in this remark, which

8

Function Spaces

P 0 are still denoted by C. Since wj∗ = i∈Ij ui,j ◦ ϕ−1 j (·, 0) in B (0, r), from the above estimate ∗ α 0 we conclude that wj belongs to C (B (0, r); K) and ||wj∗ ||C α (B 0 (0,r);K) ≤

X

||ui,j ◦ ϕ−1 j (·, 0)||C α (B 0 (0,r);K)

i∈Ij

≤C

X

||wi ||C α (B 0 (0,r);K) ≤ C||u||C α (∂Ω;K) .

i∈Ij

We thus conclude that M X

||wj∗ ||C α (B 0 (0,r);K) ≤ CM ||u||C α (∂Ω;K) .

j=1

In a completely similar way, we can show that e ||u||C α (∂Ω;K) ≤ C

M X

||wj∗ ||C α (B 0 (0,r);K) .

j=1

The equivalence of the norms follows. Remark 1.1.8 Clearly, if α ∈ (0, 1) then the definition of the space C α (∂Ω; C), mentioned at the very beginning of this subsection, coincides with Definition 1.1.6.

1.2

Anisotropic and Parabolic Spaces of H¨ older Continuous Functions

Definition 1.2.1 Let I ⊂ R and Ω ⊂ Rd be, respectively, an interval and a domain, or a closure of a domain, and let α, β ∈ (0, 1) and k ∈ N be fixed.

ˆ Cbα,0 (I ×Ω; K) denotes the space of all the bounded continuous functions f : I ×Ω → K such that the function f (·, x) is α-H¨ older continuous in I for each x ∈ Ω. It is a Banach space with the norm ||f ||C α,0 (I×Ω;K) = sup ||f (·, x)||C α (I;K) , b

x∈Ω

f ∈ Cbα,0 (I × Ω; K).

ˆ Cb0,β (I ×Ω; K) denotes the space of all the bounded continuous functions f : I ×Ω → K such that the function f (t, ·) is β-H¨ older continuous in Ω for each t ∈ I. It is a Banach space with the norm ||f ||C 0,β (I×Ω;K) = sup ||f (t, ·)||C β (Ω;K) , b

b

t∈I

f ∈ Cb0,β (I × Ω; K).

ˆ Cbα,β (I × Ω; K) = Cbα,0 (I × Ω; K) ∩ Cb0,β (I × Ω; K). It is a Banach space with the norm ||f ||C α,β (I×Ω;K) = ||f ||C 0,β (I×Ω;K) + sup[f (·, x)]C α (I;K) , b

b

x∈Ω

f ∈ Cbα,β (I × Ω; K)

Semigroups of Bounded Operators and Second-Order PDE’s

9

ˆ Cbα,1+β (I × Ω; K) denotes, for 2α 6= 1 + β, the space of all the functions f ∈ Cbα,0 (I × Ω; K) such that Dj f ∈ Cbα,β (I × Ω; K) for each j = 1, . . . , d. It is a Banach space with the norm ||f ||C α,1+β (I×Ω;K) = ||f ||C α,0 (I×Ω;K) +

d X

b

b

||Dj f ||C α,β (I×Ω;K) , b

f ∈ Cbα,1+β (I × Ω; K).

j=1

ˆ Cb1,2 (I × Ω; K) denotes the set of all the bounded functions f : I × Ω → K which are once-continuously differentiable with respect to time and twice continuously differentiable with respect to the spatial variables in I × Ω, with bounded derivatives. It is a Banach space when endowed with the norm ||f ||

Cb1,2 (I×Ω;K)

=||f ||∞ +

d X

||Dj f ||∞ +

j=1

d X

||Dij f ||∞ + ||Dt f ||∞

i,j=1

for each f ∈ Cb1,2 (I × Ω; K).

ˆ Cb1+α,2+β (I × Ω; K) denotes, for 2α 6= β, the space of all the bounded functions f : I × Ω → K which are once continuously differentiable with respect to the time variable and twice continuously differentiable with respect to the spatial variables in I × Ω with Dt f and Dij f in Cbα,β (I × Ω; K) for each i, j = 1, . . . , d. It is a Banach space with the norm ||f ||C 1+α,2+β (I×Ω;K) = ||f ||∞ +

d X

b

||Dj f ||∞ +

d X

||Dij f ||C α,β (I×Ω;K) +||Dt f ||C α,β (I×Ω;K) , b

b

i,j=1

j=1

for each f ∈ Cb1+α,2+β (I × Ω; K).

ˆ Cb0,k (I × Ω; K) denotes the space of all the bounded and continuous functions f : I × Ω → K which admit spatial derivatives up to the order k, which are bounded and continuous on I × Ω. It is a Banach space with the norm X ||Dxβ f ||∞ , f ∈ Cb0,k (I × Ω; K). ||f ||C 0,k (I×Ω;K) = b

|β|≤k

ˆ Cbk,0 (I × Ω; K) denotes the space of all the bounded and continuous functions f : I × Ω → K which admit time derivatives up to the order k, which are bounded and continuous on I × Ω. It is a Banach space with the norm ||f ||C k,0 (I×Ω;K) =

k X

b

||Dtj f ||∞ ,

f ∈ Cbk,0 (I × Ω; K).

j=0 (k+α)/2,k+α

ˆ Cb (I × Ω; K) denotes the space of all the bounded and continuous functions f : I × Ω → K such that the derivative Dtj Dxβ f and all the derivatives obtained from this one permuting the order of derivation, exist in the classical sense in I × Ω and are therein bounded for every j ∈ N and every multi-index β such that 2j + |β| ≤ k. (k+α−2j−|β|)/2,α Moreover, Dtj Dkβ f ∈ Cb (I × Ω; K) if 2j + |β| ∈ (k − 2, k]. It is a Banach space with the norm X ||f ||C (k+α)/2,k+α (I×Ω;K) = ||Dtj Dxβ f ||∞ b

2j+|β|≤k−2

10

Function Spaces X + ||Dtj Dxβ f ||C (k+α−2j−|β|)/2,α (I×Ω;K) , b

k−1≤2j+|β|≤k

where

X

||Dtj Dxβ f ||∞ is replaced by 0 if k − 2 < 0.

2j+|β|≤k−2 α,⠈ Cloc (I × Ω; K) is the space of all the functions f ∈ C α,β (J × K; K) for each J b I 1+α,2+β and K b Ω. Similarly, Cloc (I × Ω; K) is the space of all the functions f ∈ 1+α,2+β C (J × K; K) for each J b I and K b Ω. A similar definition is given for the k+α

space Cloc2

,k+α

(I × Ω; K).

When the subscript “b” is not needed, we will skip it. Moreover, we simply write Cbα,0 (I ×Ω) instead of Cbα,0 (I × Ω; R). We apply the same convention to the other spaces introduced in this definition. Remark 1.2.2 Note that, if a function f : I × Ω → K, I and Ω being as in Definition 1.2.1 is continuous and admits, for some j ∈ {1, . . . , d}, the derivatives Dj f , Dt f and Dt Dj f , which are continuous in I × Ω, then also the derivative Dj Dt f exists in I × Ω and Dt Dj f = Dj Dt f . Indeed, fix x0 ∈ Ω and r > 0 such that the point x0 + hej belongs to Ω for every h ∈ [−r, r]. Then, Z f (t, x0 + hej ) = f (t, x0 ) +

h

Dj f (t, x0 + rej ) dr,

t ∈ I.

0

Clearly, we can differentiate under the integral sign with respect to t and obtain that Dt f (t, x0 + hej ) − Dt f (t, x0 ) 1 = h h

Z

h

Dt Dj f (t, x0 + rej ) dr,

t ∈ I.

0

Letting h tend to 0, we conclude that the derivative Dj Dt f exists at (t, x0 ) for each t ∈ I and it coincides with Dt Dj f (t, x0 ). α/2,α

Remark 1.2.3 As it is easily checked, Cb (I × Ω; K) can be also characterized as the space of all bounded functions f : I × Ω → K p which are α-H¨older continuous with respect to the parabolic distance d((t1 , x1 ), (t2 , x2 )) = |t2 − t1 | + |x2 − x1 |2 for each t1 , t2 ∈ I and α/2,α (k+α)/2,k+α x1 , x2 ∈ Ω. This is the reason why the spaces Cb (I × Ω; K) and Cb (I × Ω; K) (k ∈ N) are called parabolic H¨ older spaces. Proposition 1.2.4 The following properties are satisfied. (i) Let Ω be Rd , Rd+ or a bounded domain of class C 2+α (α ∈ (0, 1)) and let I ⊂ R be 1+α/2,2+α an interval. Then, the space Cb (I × Ω; K) can be characterized as the set of α/2,α all f ∈ Cb1,2 (I × Ω; K) such that Dt f and Dxβ f belong to Cb (I × Ω; K) for every multi-index β with length two. Moreover, the norm X 1+α/2,2+α ||f ||∞ + ||Dtj Dxβ f ||C α/2,α (I×Ω;K) , f ∈ Cb (I × Ω; K), b

2j+|β|=2 1+α/2,2+α

is equivalent to usual norm of Cb

(I × Ω; K).

Semigroups of Bounded Operators and Second-Order PDE’s

11

(ii) If Ω is Rd , Rd+ or a bounded domain of class C 2 , then Cb1,2 (I × Ω; K) is continuously (1+β)/2,1+β embedded into Cb (I × Ω; K) for each β ∈ (0, 1). Moreover, the first-order 1/2,0 1,2 (I × Ω; K) and there exist a spatial derivatives of u ∈ Cb (I × Ω; K) belong to Cb positive constant C, independent of u, such that ||Dj u||C 1/2,0 (I×Ω;K) ≤ C||u||C 1,2 (I×Ω;K) . Proof We prove the proposition in the case when Ω is a bounded domain. The other cases are completely similar. (i) Clearly, we have only to show that, if f ∈ Cb1,2 (I × Ω; K) is a function such that α/2,α Dt f and Dxβ f ∈ Cb (I × Ω; K) for every multi-index |β|, with length two, then Dj f ∈ (1+α)/2,1+α Cb (I × Ω; K) and there exists a positive constant C, independent of f , such that   X j β ||Dt Dx f ||C α/2,α (I×Ω;K) ||Dj f ||C (1+α)/2,1+α (I×Ω;K) ≤ C ||f ||∞ + b

b

2j+|β|=2 1+α/2,2+α

for every j = 1, . . . , d. For this purpose, we fix f ∈ Cb (I × Ω; K), t1 , t2 ∈ I, with t1 < t2 and apply estimate (1.1.2), with (α, β, θ) = (2 + α, 1, α) to the function f (t2 , ·) − f (t1 , ·) ∈ C 2+α (Ω) to obtain that 1−α

1+α

||Dj f (t2 , ·) − Dj f (t1 , ·)||∞ ≤C||f (t2 , ·) − f (t1 , ·)||C 2α (Ω;K) ||f (t2 , ·) − f (t1 , ·)||C 22+α (Ω;K) 1−α

1+α

≤C1 ||f ||C 20,2+α (I×Ω;K) ||f (t2 , ·) − f (t1 , ·)||C 2α (Ω;K)

(1.2.1)

b

for every j = 1, . . . , d and some positive constants C and C1 , independent of f . Next, we observe that Z t2 [Dt f (t, x2 ) − Dt f (t, x1 )] dt |f (t2 , x2 ) − f (t1 , x2 ) − f (t2 , x1 ) + f (t1 , x1 )| = t1 Z t2 |Dt f (t, x2 ) − Dt f (t, x1 )| dt ≤ t1

≤(t2 − t1 )|x2 − x1 |α [Dt f ]C 0,α (I×Ω;K) b

for every x1 , x2 ∈ Ω and, similarly, |f (t2 , x) − f (t1 , x)| ≤ |t2 − t1 |||Dt f ||∞ ,

x ∈ Ω,

(1.2.2)

so that ||f (t2 , ·) − f (t1 , ·)||C α (Ω;K) ≤ |t2 − t1 |||Dt f ||C 0,α (I×Ω;K) .

(1.2.3)

b

Replacing (1.2.3) into (1.2.1), we conclude that 1−α

1+α

||Dj f (t2 , ·) − Dj f (t1 , ·)||∞ ≤ C2 ||f ||C 20,2+α (I×Ω;K) ||Dt f ||C 20,α (I×Ω;K) |t2 − t1 |

1+α 2

(1.2.4)

b

b

for each j = 1, . . . , d, the constant C2 being independent of f , t1 and t2 . Moreover, from (1.1.6), with θ = 0, β = 1 and α = 2, and Young inequality, we deduce that 1

1

2 ||f ||C 1 (Ω;K) ≤C3 ||f ||∞ ||f ||C2 2 (Ω;K) ≤ ε||f ||C 2 (Ω;K) + C4,ε ||f ||∞ b X =ε||f ||C 1 (Ω;K) + ε ||Dβ f ||∞ + C4,ε ||f ||∞

|β|=2

12

Function Spaces

for every ε > 0 and some positive constants C3 and C4 , independent of f and ε (C3 is independent also of ε). Taking ε sufficiently small, we conclude that  X  ||Dβ f ||∞ + ||f ||∞ . ||f ||C 1 (Ω;K) ≤ C5 b

|β|=2 (1+α)/2,0

From this estimate and (1.2.4), it follows that the function Dj f belongs to Cb Ω; K) and   X ||Dtj Dxβ f ||C α/2,α (I×Ω;K) ||Dj f ||C (1+α)/2,0 (I×Ω;K) ≤ C6 ||f ||∞ +

(I ×

b

b

2j+|β|=2

for each j = 1, . . . , d, the constant C6 being independent of f . The arguments in Remark 1.1.2, which show that C 1 (Ω; K) ,→ C α (Ω; K), allow to complete the proof of (i). (ii) We begin by proving that C 1,2 (I × Ω; K) ,→ C (1+β)/2,1+β (I × Ω; K). Using (1.2.2), 1+β we get |u(t2 , x) − u(t1 , x)| ≤ ||Dt u||∞ |t2 − t1 | 2 for all x ∈ Ω and t1 , t2 ∈ I with |t2 − t1 | ≤ 1, which, in view of Lemma 1.1.3 allows to conclude that Cb1,2 (I × Ω; K) ,→ (1+β)/2,0 Cb (I × Ω; K). Moreover, using the embedding C 1 (Ω; K) ,→ C β (Ω; K), we conclude 2 that C (Ω; K) ,→ C 1+β (Ω; K). Summing up, we have shown that [u]C (1+β)/2,0 (I×Ω;K) + b [u]C 0,β (I×Ω;K) +[Dj u]C 0,β (I×Ω;K) ≤ C7 ||u||C 1,2 (I×Ω;K) for every j = 1, . . . , d and some positive b b b constant C7 , independent of u. Finally, using (1.1.2) we can estimate 1

1

2 ||Dj u(t, ·) − Dj u(s, ·)||∞ ≤C8 ||u(t, ·) − u(s, ·)||∞ ||u(t, ·) − u(s, ·)||C2 2 (Ω;K) 1

≤C9 ||u||C 1,2 (I×Ω;K) |t − s| 2 ≤ C9 ||u||C 1,2 (I×Ω;K) |t − s| b

1+β 2

(1.2.5)

b

for all t, s ∈ I with |t − s| ≤ 1, x ∈ Rd and j = 1, . . . , d, where the constants C8 and C9 are independent of u, t and s. Using once more Lemma 1.1.3 we conclude that [Dj u]C β/2,0 (I×Ω;K) ≤ C10 ||u||C 1,2 (I×Ω;K) for every j = 1, . . . , d and some positive constant b

b

C10 , independent of u. The embedding C 1,2 (I × Ω; K) ,→ C (1+β)/2,1+β (I × Ω; K) follows. Moreover, from (1.2.5) also the second part of the statement of property (ii) follows.  For further use, we also need the following result. Proposition 1.2.5 Let Ω be Rd or Rd+ or a bounded open set of class C 2+α (α ∈ (0, 1)). Further, let I ⊂ R be an interval. Then, for every γ ∈ (0, (1 + α)/2] there exists a positive 1+α/2,2+α constant C such that, for each ε ∈ (0, 1) and each function f ∈ Cb (I × Ω; K), it holds that ||f ||C γ,1+α (I×Ω;K) ≤ C(ε||f ||C 1+α/2,2+α (I×Ω;K) + ε−Λα,γ ||f ||∞ ), (1.2.6) b

b

where Λα,γ = (1 + α) ∨ [(2γ + 1)(1 + α − 2γ)−1 ], if γ < (1 + α)/2, and Λα,(1+α)/2 = 1 + α. Similarly, 2 e (1.2.7) ||f ||C 1,2 (I×Ω;K) ≤ C(ε||f ||C 1+α/2,2+α (I×Ω;K) + ε− α ||f ||∞ ) b

b

1+α/2,2+α

for every ε ∈ (0, 1), every f ∈ Cb

e (I × Ω; K) and some positive constant C.

Proof Throughout the proof, C denotes a positive constant, independent of ε and the functions that we consider, which may vary from line to line.

Semigroups of Bounded Operators and Second-Order PDE’s

13

Using the interpolation estimate (1.1.6) with (α, θ, β) = (2 + α, 0, 1 + α), together with the Young inequality, we get ||ζ||C 1+α (Ω;K) ≤ C(ε−(1+α) ||ζ||∞ + ε||ζ||C 2+α (Ω;K) ) b

b

for every ζ ∈ Cb2+α (Ω; K) and ε > 0. Moreover, from (1.1.6) (with (α, β, θ) = (1 + α/2, γ, 0)) and the Young inequality we infer that   2γ 1+α/2 ϕ ∈ Cb (I; K), ||ϕ||Cbγ (I;K) ≤ C ε− 2+α−2γ ||ϕ||∞ + ε||ϕ||C 1+α/2 (I;K) , b

for every ε as above. Applying these estimates with ζ = f (t, ·) and ϕ = f (·, x), we deduce that ||f ||C γ,0 (I×Ω;K) +

d X

b

  ||Dj f ||C 0,α (I×Ω;K) ≤ C ε||f ||C 1+α/2,2+α (I×Ω;K) + ε−(1+α) ||f ||∞ (1.2.8) b

b

j=1

for each ε ∈ (0, 1). To estimate the σ-H¨older norm of the functions Dj f (·, x) (j = 1, . . . , d) for every x ∈ Rd , when σ ∈ {α/2, γ}, we apply the interpolation estimate (1.1.6), with (α, β, θ) = ((1 + α)/2, γ, 0) to the function Dj f (·, x) (j = 1, . . . , d, x ∈ Ω) and again Young’s inequality, to deduce that   2σ ||Dj f ||C σ,0 (I×Ω;K) ≤ C ε||Dj f ||C (1+α)/2,0 (I×Ω;K) + ε− 1+α−2σ ||Dj f ||∞ for every ε ∈ (0, 1). This inequality shows that d X

  2σ ||Dj f ||C σ,0 (I×Ω;K) ≤ C ε||f ||C 1+α/2,2+α (I×Ω;K) + ε− 1+α−2σ |||∇x f |||∞ . b

(1.2.9)

b

j=1

Now, from (1.1.6) (with (α, β, θ) = (2 + α, 1, 0)) and Young’s inequality, we can es1 timate |||∇x f |||∞ ≤ δ||f ||C 1+α/2,2+α (I×Ω;K) + δ − 1+α ||f ||∞ for each δ ∈ (0, 1). Choosing b

δ = ε(1+α)/(1+α−2σ) and replacing this estimate in (1.2.9), we get d X

  ||Dj f ||C γ,0 (I×Ω;K) ≤ C ε||f ||C 1+α/2,2+α (I×Ω;K) + ε−Λα,γ ||f ||∞ . b

(1.2.10)

b

j=1

From (1.2.8) and (1.2.10), estimate (1.2.6) follows at once. To prove estimate (1.2.7) we use (1.1.6), with (α, θ, β) = (1 + α, 0, 1) and (α, θ, β) = (2 + α, 0, 2) to estimate 1

||f ||C 1,0 (I×Ω;K) ≤ ε||f ||C 1+α/2,2+α (I×Ω;K) + Cε− α ||f ||∞ , b

b

2

||f ||C 0,2 (I×Ω;K) ≤ ε||f ||C 1+α/2,2+α (I×Ω;K) + Cε− α ||f ||∞ b

b

for every ε > 0. From these two estimate, (1.2.7) follows at once,

 1+α/2,2+α Cb (I ×

The following proposition provides us with a different norm of the space Ω), which is equivalent to the norm introduced in Definition 1.2.1, and will be used in the next chapters of this book.

14

Function Spaces

Proposition 1.2.6 Let Ω be Rd , Rd+ or a bounded open set of class C 2+α for some α ∈ (0, 1). Further, let I ⊂ R be an interval. Then, there exists a positive constant C such that, 1+α/2,2+α for every u ∈ Cb (I × Ω; K), it holds that  ||u||C 1+α/2,2+α (I×Ω;K) ≤ C ||u||∞ + ||Dt u||C α/2,α (I×Ω;K) + b

b

d X

 ||Dij u||C α/2,α (I×Ω;K) .

i,j=1

b

Proof Using estimate (1.1.6), with (α, β, θ) = (2, 1, 0), and Young’s inequality, it can be easily shown that |||∇v|||∞ ≤ ε||v||C 2 (Ω;K) +ε−1 ||v||∞ for every v ∈ Cb2 (Ω) and every ε > 0. b Choosing ε small enough, from this estimate the assertion follows at once. 

1.2.1

Anisotropic spaces of functions defined on the boundary of a set

In this subsection, mimicking the definition of the spaces C α (∂Ω; K), introduced in Subsection 1.1.1. We introduce only those spaces which we really need in this book. As in Subsection 1.1.1, we introduce a covering {Ux : x ∈ ∂Ω} of ∂Ω such that for every open set Ux there exists a function ψx : Ux → B(0, r) of class C α which is invertible with inverse function which is of class C α as well. Since ∂Ω is a compact set, we can extract a subcovering {Uxj : j = 1, . . . , N } of ∂Ω. Associated with this covering of ∂Ω, we consider a partition of the unity {ηj : j = 1, . . . , N } (see Proposition B.1.1). Moreover, we denote by B 0 (0, r) the ball of Rd−1 centered at zero with radius r. Definition 1.2.7 Let Ω be a bounded open set of class C 1+α for some α ∈ (0, 1). By C (1+α)/2,1+α ([0, T ] × ∂Ω; K), we denote the set of all the functions f : [0, T ] × ∂Ω → K such that the function fk : [0, T ]×B 0 (0, r) → K, defined by fk (t, y) = ηk (ψk−1 (y, 0))f (t, ψk−1 (y, 0)) for every t ∈ [0, T ] and y ∈ B 0 (0, r) belongs to C (1+α)/2,1+α ([0, T ] × B 0 (0, r)) for every k = 1, . . . , N . It is normed by setting ||u||C (1+α)/2,1+α ([0,T ]×∂Ω;K) =

N X

||fk ||C (1+α)/2,1+α ([0,T ]×B 0 (0,r))

k=1

for every u ∈ C (1+α)/2,1+α ([0, T ] × ∂Ω; K). Definition 1.2.8 Let Ω be a bounded open set of class C 2+α for some α ∈ (0, 1). By C 1+α/2,2+α ([0, T ] × ∂Ω; K), we denote the set of all functions f : [0, T ] × ∂Ω → K such that the function fk : [0, T ] × B 0 (0, r) → K, defined by fk (t, y) = ηk (ψk−1 (y, 0))f (t, ψk−1 (y, 0)) for every t ∈ [0, T ] and y ∈ B 0 (0, r) belongs to C 1+α/2,1+α ([0, T ] × B 0 (0, r)) for every k = 1, . . . , N . It is normed by setting ||u||C 1+α/2,2+α ([0,T ]×∂Ω;K) =

N X

||fk ||C 1+α/2,2+α ([0,T ]×B 0 (0,r))

k=1

for every u ∈ C 1+α/2,2+α ([0, T ] × ∂Ω; K). Remark 1.2.9 Arguing as in Remark 1.1.7, it can be easily shown that the above definitions are independent of the choice of the covering of ∂Ω, of the diffemorphisms and of the partition of the unity considered. Changing the quantities above results at most in a change of the norms, which are then replaced by equivalent ones.

Semigroups of Bounded Operators and Second-Order PDE’s

1.3

15

Lp - and Sobolev Spaces

Throughout this book we always consider the Lebesgue measure. Definition 1.3.1 Let Ω be a domain of Rd .

ˆ For every p ∈ [1, ∞), we denote by Lp (Rd , K) the space of all the (equivalence classes of ) measurable functions f : Ω → K such that Z |f |p dx < ∞. Ω

It is a Banach space when endowed with the norm Z

p

|f | dx

||f ||Lp (Ω,K) =

 p1 ,

f ∈ Lp (Ω, K).

Ω

ˆ By L∞ (Ω, K), we denote the space of all the (equivalence classes of ) measurable functions f : Ω → K such that essinf f = inf{Λ > 0 : |f (x)| ≤ Λ, for almost every x ∈ Ω}. Ω

It is a Banach space when endowed with the norm ||f ||∞ = essinf Ω f for every f ∈ L∞ (Ω, K).

ˆ We denote by Lploc (Ω, K) (resp. L∞ loc (Ω, K)) the set of all the (equivalence classes of) measurable functions f : Ω → K which belong to Lp (Ω0 , K) (resp. L∞ (Ω0 , K)) for every bounded domain Ω0 whose closure is contained in Ω. When K = R we simply write Lp (Ω), Lploc (Ω), L∞ (Ω), L∞ loc (Ω). We start by the following standard density result, where we give also a sketch of the proof. Theorem 1.3.2 Cc∞ (Ω, K) is dense in Lp (Ω, K) for every p ∈ [1, ∞). Proof We first consider the case Ω = Rd . Fix u ∈ Lp (Rd ; K) and a function ϑ ∈ Cc∞ (Rd ) with ||ϑ||L1 (Rd ) = 1. For every n ∈ N, set ϑn (x) = nd ϑ(nx) for every x ∈ Rd . Then, the sequence (vn ), where Z vn (x) = ϑn (x − y)u(y) dy, x ∈ Rd , Rd

belongs to C ∞ (Rd ; K) ∩ Lp (Rd ; K) and converges to u in Lp (Rd ; K). Finally, setting un (x) = ϑ(n−1 x)vn (x) for every x ∈ Rd and n ∈ N, we get a sequence in Cc∞ (Rd ; K) which converges to u in Lp (Rd ; K). If Ω is an open subset of Rd , it suffices to apply the above construction to the trivial extension u of u to Rd .  In this book we extensively use also the Sobolev spaces of integer order. Definition 1.3.3 Let Ω be a domain of Rd . Moreover, let k ∈ N and p ∈ [1, ∞) be positive numbers.

16

Function Spaces

ˆ By W k,p (Ω, K) we denote the subspace of Lp (Ω, K) those distributional derivatives up to the order k belong to Lp (Ω; K). It is a Banach space when endowed with the norm ||f ||W k,p (Ω,K) =

 X

|||D

α

f |||pLp (Ω,K)

 p1 ,

f ∈ W k,p (Ω, K).

|α|≤k

ˆ By W0k,p (Ω, K) we denote the closure of Cc∞ (Ω, K) (with respect to the norm of W k,p (Ω, K)) into W k,p (Ω, K). k,p ˆ By Wloc (Ω, K) we denote the set of all the (equivalence classes of ) measurable functions f : Ω → K which belong to W k,p (Ω0 , K) for every bounded domain Ω0 whose closure is contained in Ω. k,p We simply write W k,p (Ω) and Wloc (Ω), when K = R.

In general, W0k,p (Ω, K) is a proper closed subspace of W k,p (Ω, K). As the following theorem shows, when Ω = Rd the two spaces actually coincide. Theorem 1.3.4 Let Ω be Rd , or Rd+ or a bounded open set of class C k for some k ∈ N. Then, the set of the restrictions to Ω of Cc∞ (Rd ; K) functions is dense in W k,p (Ω, K) for every p ∈ [1, ∞). Proof We first consider the case Ω = Rd . Fix u ∈ W k,p (Rd ; K) for some p ∈ (1, ∞), k ∈ N and consider the same sequence (un ) ⊂ Cc∞ (Rd ; K) introduced in the proof of Theorem 1.3.2, i.e., un (x) = ϑ(n−1 x)vn (x) for every x ∈ Rd and n ∈ N, where Z vn (x) = ϑn (x − y)u(y) dy, x ∈ Rd . Rd

Since u belongs to W k,p (Rd ; K) and the function ϑn (x − ·) belongs to Cc∞ (Rd ; K) for every x ∈ Rd , it follows easily that Z Z Dβ vn (x) = (Dxβ ϑn )(x − y)u(y) dy = (−1)|β| (Dyβ ϑn (x − ·))(y)u(y) dy Rd Rd Z = ϑn (x − ·)Dβ u(y) dy Rd

for every multi-index β with length at most k. By assumptions, the function Dβ u belongs to Lp (Rd ; K). Therefore, Dβ vn converges to Dβ u in Lp (Rd ; K). Now, it is straightforward to check that un converges to u in W k,p (Rd ; K) as n tends to ∞. If Ω = Rd+ or Ω is a bounded open set of class C k , we fix u ∈ W k,p (Ω; K) and apply the first part of the proof to the function Ek u which belongs to W k,p (Rd ; K), where Ek is the extension operator in Proposition B.4.5.  The following theorem provides a sufficient condition to ensure that the restriction to Rd+ of a function f ∈ W 1,p (Rd ; K) actually belongs to W01,p (Rd+ ; K). Lemma 1.3.5 Let u ∈ W 1,p (Rd ; K) be a function such that u(x0 , xd ) = −u(x0 , −xd ) for almost every x ∈ Rd . Then, u ∈ W01,p (Rd+ ; K). Proof Let ϑ ∈ Cc∞ (Rd ) be an even function (with respect to the last variable) and for every n ∈ N, let us set ϑn (x) = nd ϑ(nx) for every x ∈ Rd . Further, let us denote by un

Semigroups of Bounded Operators and Second-Order PDE’s

17

the convolution between u and ϑn for every n ∈ N. Clearly, un belongs to C ∞ (Rd ; K) ∩ W 1,p (Rd ; K). Moreover, Z un (x0 , 0) = u(y 0 , yd )ϑn (x0 − y 0 , −yd ) dy 0 dyd Rd Z = u(y 0 , −z)ϑn (x0 − y 0 , z) dy 0 dz Rd Z =− u(y 0 , z)ϑn (x0 − y 0 , z) dy 0 dz d ZR =− u(y 0 , z)ϑn (x0 − y 0 , −z) dy 0 dz = −un (x0 , 0) Rd

for every x ∈ R and n ∈ N. Thus, un (x0 , 0) = 0, so that un belongs to W01,p (Rd ; K) (see Theorem B.3.3). Since un converges to u in W 1,p (Rd ; K), we conclude that u ∈ W01,p (Rd ; K).  0

d−1

In the second part of the book we extensively use the Sobolev embedding theorems, which we list here below. Theorem 1.3.6 Let Ω be Rd , or Rd+ or a bounded domain of class C 1 (see Definition B.2.1). Fix p ∈ [1, ∞) ∪ {∞}. Then, the following properties are satisfied. (i) If p1 − kd > 0, then, W k,p (Ω, K) is continuously embedded into Lq (Ω, K) for every q ∈ [p, p∗ ], where p1∗ = p1 − kd . In particular, if Ω is bounded, then the embedding is compact for every p ∈ [1, p∗ ). (ii) If p1 − kd = 0, then W k,p (Ω, K) is continuously embedded into Lq (Ω, K) for every q ∈ [p, ∞). In particular, if Ω is bounded, then the embedding is compact for each q ∈ [1, ∞). (iii) If p1 − kd < 0, then W k,p (Ω, K) is continuously embedded into L∞ (Ω, K). More precisely, k−d/p

if d/p is not an integer, then W k,p (Ω, K) is continuously embedded in Cb d

(Ω, K).

Proof The core of the proof is the case k = 1 and Ω = R . This is the content of Steps 1 to 4. Then, in Steps 5, we prove properties (i)–(iii) when Ω is Rd+ or an open bounded set of class C 1 . Finally, in Steps 6 and 7 we complete the proof, considering the case when k > 1. Throughout the proof, we denote by r0 the index conjugate to r ∈ (1, ∞), i.e., 1/r+1/r0 = 1. Moreover, the constants that we will consider depend on p but are all independent of the functions that we consider. 0 Step 1. Here, we prove that W 1,1 (Rd ; K) ,→ Ld (Rd ; K). The crucial point consists in proving that, if f1 , . . . , fd are d nonnegative functions in Cc (Rd−1 ; K), then the function f , Qd xj ) for every x ∈ Rd , belongs to L1 (Rd ; K) and ||f ||L1 (Rd ;K) ≤ defined by f (x) = j=1 fj (b Qd bj = (x1 , . . . , xj−1 , xj+1 , . . . , xd ) for every x ∈ Rd and j = j=1 ||fj ||Ld−1 (Rd−1 ;K) . Here, x 1, . . . , d. The proof of this estimate, known as Gagliardo-Nirenberg inequality, is obtained by induction on the dimension d. If d = 2, then the Gagliardo-Nirenberg inequality is just a rewriting of the Fubini-Tonelli theorem. Suppose that the inequality is true for some d > 2 and let us prove it with d being replaced by d + 1. For this purpose, we Q consider d+1 d + 1 nonnegative functions f1 , . . . , fd+1 which belong to Ld (Rd ; K). We set f = k=1 fk , integrate f with respect to the variable xd+1 and apply the generalized H¨older inequality to get  d1 Z d Z d Y Y d f (x, xd+1 ) dxd+1 ≤ fd+1 (x) |fj (b xj , xd+1 )| dxd+1 =: fd+1 (x) gj (b xj ). R

j=1

R

j=1

18

Function Spaces

Integrating over Rd and using H¨ older inequality we obtain d Y

Z

Z f (x, xd+1 ) dx dxd+1 = ||fd+1 ||Ld (Rd ;K) Rd+1

d0

0

 10 d

.

(gj (b xj )) dx

(1.3.1)

Rd j=1

From the inductive assumption we deduce that d Y

Z

d0

0

(gj (b xj )) dx ≤

Rd j=1

d Z Y

1  d−1

d

|gj (b xj )| db xj

=

Rd−1

j=1

d Y

0

||fj ||dLd (Rd ;K) .

j=1

Replacing this inequality into (1.3.1), the assertion follows. Now, we fix u ∈ Cc1 (Rd ; K) and observe that Z xj u(x) = Dj u(x1 , . . . , xj−1 , t, xj+1 , . . . , xd ) dt,

x ∈ Rd ,

−∞

for every j = 1, . . . , d. Thus, 0

|u(x)|d ≤

1  d−1 d Y =: fj (b xj ) |Dj u(x1 , . . . , xj−1 , t, xj+1 , . . . , xd )| dt

d Z Y j=1

R

j=1

for every x ∈ Rd . Applying the Gagliardo-Nirenberg inequality to the functions f1 , . . . , fd , we conclude that ||u||Ld0 (Rd ;K) ≤ C ∗ |||∇u|||L1 (Rd ;K) . By density, we extend this inequality to 0 every u ∈ W 1,1 (Rd ; K). The embedding W 1,1 (Rd ; K) ,→ Ld (Rd ; K) follows. Step 2. Here, we prove property (i) when p > 1. For this purpose, we fix u ∈ Cc1 (Rd ; K) and apply Step 1 to the function v = |u|α , where α > 1 is a constant to be properly chosen. We get Z

αd0

|u| Rd

 10 Z d ∗ ≤C dx

|∇v| dx ≤ αC

Z

∗

Rd

≤αC ∗ |||∇u|||Lp (Rd ;K)

|u|α−1 |∇u| dx

Rd

Z

0

|u|(α−1)p dx

 10 p

.

(1.3.2)

Rd

Choosing αd0 = (α − 1)p0 , i.e., α = (d − 1)p/(d − p), from the previous inequality the assertion follows at once for functions in Cc1 (Rd ; K). Again by density, we extend property (i), with k = 1, to every u ∈ W 1,p (Rd ; K). Step 3. Here, we prove property (ii) when k = 1 and d > 1. For this purpose, we use a bootstrap argument, based on estimate (1.3.2). Fix q ∈ (d, ∞) and apply the quoted estimate with α = d to infer that ||u||Ldd0 (Rd ;K) ≤ C1∗ ||u||W 1,d (Rd ;K) . If q < dd0 then we are done, otherwise we apply again estimate (1.3.2) with α = d + 1 and get ||u||L(d+1)d0 (Rd ;K) ≤ C2∗ ||u||W 1,d (Rd ;K) . If (d + 1)d0 ≥ q, then we are done, otherwise we go on taking α = d + 2. Iterating this procedure n-times we get ||u||L(d+n−1)d0 (Rd ;K) ≤ Cn∗ ||u||W 1,d (Rd ;K) . Since (d + n − 1)d0 diverges to ∞ as n tends to ∞, in a finite number of steps we obtain the wished estimate. Step 4. Here, we prove property (iii) with k = 1. We fix u ∈ Cc1 (Rd ; K), x0 ∈ Rd , r > 0 and first prove that Z |∇u(y)| |u(x) − uB(x0 ,r) | ≤ C dy, x ∈ B(x0 , r), (1.3.3) d−1 B(x0 ,r) |x − y| where uB(x0 ,r) is the average of u on the ball B(x0 , r), κd denotes the Lebesgue measure

Semigroups of Bounded Operators and Second-Order PDE’s

19

of the ball B(0, 1) ⊂ Rd and C = 2d (dκd )−1 . For this purpose, we fix x, y ∈ B(x0 , r), with x 6= y and set g(ρ) = u(x + |x − y|−1 (y − x)ρ) for every ρ ∈ [0, |y − x| ]. Then, |x−y|

Z

g 0 (t) dt

u(y) − u(x) =g(|x − y|) − g(0) = 0 |y−x|

Z

|y − x|−1 h∇u(x + (y − x)|y − x|−1 ρ), y − xi dρ

=

(1.3.4)

0

and, consequently, uB(x0 ,r) − u(x) = =

1 rd κd

Z

1 rd κd

Z

(u(y) − u(x)) dy B(x0 ,r)

Z

|y−x|

|y − x|−1 h∇u(x + (y − x)|y − x|−1 ρ), y − xi dρ

dy B(x0 ,r)

0

for x ∈ B(x0 , r). We extend the gradient of u by zero outside B(x0 , r) and denote by ∇u the so obtained function. Since B(x0 , r) ⊂ B(x, 2r), we can estimate 1 rd κd

Z

1 = d r κd

Z

|uB(x0 ,r) − u(x)| ≤

|y−x|

Z

|∇u(x + |y − x|−1 (y − x)ρ)| dρ

dy B(x,2r)

0

Z

|z|

dy B(0,2r) Z 2r d−1

|∇u(x + ρz|z|−1 )| dρ

0

Z Z ∞ 1 d−1 = d s ds dH (σ) ∇u(x + ρσ)| dρ r κd 0 ∂B(0,1) 0 Z |∇u(x + y)| 2d dy = dκd Rd |y|d−1 Z 2d |∇u(y)| = dy dκd B(x0 ,r) |x − y|d−1 and (1.3.3) follows. Now applying H¨older’s inequality, we can estimate  10 p 1 |uB(x0 ,r) − u(x)| ≤C|||∇u|||Lp (Rd ;K) dy 0 (d−1)p B(x0 ,r) |x − y| Z  10 p 1 . ≤C|||∇u|||Lp (Rd ;K) dy 0 (d−1)p B(x0 ,r)−x |y| Z

We claim that Z B(x0 ,r)−x

1 |y|(d−1)p0

Z dy ≤ B(0,r)

1 |y|(d−1)p0

dy = κd

(1.3.5)

p − 1 p−d r p−1 . p−d

To prove the claim, we set E = B(x0 , r)−x and observe that E = (E \B(0, r))∪(E ∩B(0, r)) and B(0, r) = (B(0, r) \ E) ∪ (E ∩ B(0, r)). Therefore, Z Z 1 1 dy − 0 0 dy (d−1)p (d−1)p E |y| B(0,r) |y| Z Z 1 1 = dy − 0 0 dy (d−1)p (d−1)p E\B(0,r) |y| B(0,r)\E |y|

20

Function Spaces ≤

1 |r|(d−1)p0

(|E \ B(0, r)| − |B(0, r) \ E|) = 0.

From (1.3.5) we can thus infer that d

|uB(x0 ,r) − u(x)| ≤ Cp |||∇u|||Lp (Rd ;K) r1− p .

(1.3.6)

We use (1.3.6) to conclude the proof. First, we estimate the sup-norm of u in terms of its W 1,p (Rd ; K)-norm. For this purpose, we fix x ∈ Rd and take x0 = x. Then, we can write d

d

−1

d

|u(x)| ≤ Cp |||∇u|||Lp (Rd ;K) r1− p + |uB(x,r) | ≤ Cp |||∇u|||Lp (Rd ;K) r1− p + κd p r− p ||u||Lp (Rd ;K) , d

−1

d

so that ||u||∞ ≤ Cp |||∇u|||Lp (Rd ;K) r1− p + κd p r− p ||u||Lp (Rd ;K) . Minimizing with respect to r > 0 gives 1− d

d

p p ||u||∞ ≤ Cp0 ||u||Lp (R d ;K) |||∇u|||Lp (Rd ;K) .

Next, to estimate the (1 − d/p)-H¨older seminorm of u, we fix x, y ∈ Rd , set r = 2|y − x| and apply (1.3.6) with y = x0 to get d

d

|u(x) − u(y)| ≤ |u(x) − uB(y,r) | + |uB(y,r) − u(y)| ≤ 22− p Cp |||∇u|||Lp (Rd ;K) |x − y|1− p . Summing up, we have proved that ep ||u||W 1,p (Rd ;K) . ||u||C 1−d/p (Rd ;K) ≤ C

(1.3.7)

b

Now, we fix u ∈ W 1,p (Rd ; K) and consider a sequence (un ) ⊂ Cc∞ (Rd ; K) converging to u 1−d/p in W 1,p (Rd ; K). The above estimate shows that (un ) is a Cauchy sequence in Cb (Rd ; K). 1−d/p d Hence, it converges to a function v ∈ Cb (R ; K). By uniqueness v = u almost everywhere on Rd . Finally, writing (1.3.7) with u being replaced by un and letting n tend to ∞, we 1−d/p conclude that W 1,p (Rd ; K) ,→ Cb (Rd ; K). Note that the if p = d = 1, the previous arguments show that u ∈ L∞ (R; K). Actually, using the representation formula (1.3.4), we can show that Z |y−x| |u(y) − u(x)| ≤ |u0 (ρ)|dρ 0

for every u ∈ W 1,1 (R; K) and almost every x, y ∈ R. Using this inequality, it can be easily checked that L1 (R; K) is continuously embedded in the set of absolutely continuous functions over R. Step 5. To prove properties (i)–(iii) when k = 1 and Ω is Rd+ or a bounded open set of class C 1 , we fix a function u ∈ W 1,p (Ω; K) and consider the extension operator E1 in Proposition B.4.5. Since E1 u ∈ W 1,p (Rd ; K), applying Steps 1–3 to E1 u and then considering its restriction to Ω, the assertion follows. Step 6. In this and in the next step, Ω is Rd or Rd+ or a bounded open set of class C 1 . We begin by proving property (i) and (ii) for k > 1 by induction. These properties are satisfied for k = 1, as we have proved in Steps 1–3. Suppose that they hold true for some k > 1 and let us prove them with k + 1 instead of k. Assume that p1 − k+1 d ≥ 0. Then, p < d so k+1,p that, applying Steps 1 and 2 to u ∈ W (Ω; K) and its derivatives up to the order k, we pd infer that u ∈ W k,p1 (Ω; K), where p1 = d−p . Moreover, there exists a positive constant Ck,p such that ||u||W k,p1 (Ω;K) ≤ Ck,p ||u||W k+1,p (Ω;K) . If p11 − kd = p1 − k+1 d > 0, then applying the

Semigroups of Bounded Operators and Second-Order PDE’s

21

∗

∗ ||u||W k+1,p (Ω;K) , inductive assumption we infer that u ∈ Lp (Ω; K) and ||u||Lp∗ (Ω;K) ≤ Ck,p 1 k ∗ q for a constant Ck,p . Similarly, if p1 − d = 0, then u belongs to L (Ω; K) and ||u||Lq (Ω;K) ≤ Ck,p,q ||u||W k+1,p (Ω;K) , for every q ∈ (p, ∞) and some positive constant Ck,p,q . Step 7. Finally, we prove property (iii). We first assume p > d and u ∈ Cc∞ (Ω; K). Applying Step 4 to u and its derivatives up to the order k − 1, we conclude that ek,p ||u||W k,p (Ω;K) . Hence, every sequence (un ) ⊂ Cc∞ (Ω; K) converging to ||u||C k−d/p (Ω;K) ≤ C b

k−d/p

u in W k,p (Ω; K) is a Cauchy sequence in Cb (Ω; K), so that un converges to a function k−d/p k−d/p v ∈ Cb (Ω; K) in Cb (Ω; K). Clearly, u = v almost everywhere on Ω. Next, we suppose that p < d and denote by j < d the largest integer such that p1 − dj > 0. We fix u ∈ Cc∞ (Ω; K). Since Cc∞ (Ω; K) ,→ W j,p (Ω; K), then from Step 6, applied to u and its derivatives of order e ∗ ||u||W k,p (Ω;K) , at most k − j, it follows that u ∈ W k−j,pj (Ω; K) and ||u||W k−j,pj (Ω;K) ≤ C k,p where

1 pj

=

1 p

− dj . Note that pj > d. Indeed,

1 pj

−

1 d

=

1 p

−

j+1 d

< 0 (it cannot be zero k−d/p

since, by assumptions d/p is not an integer). Therefore, we infer that u ∈ Cb (Ω; C) and b ||u||C k−d/p (Ω;C) ≤ Ck,p ||u||W k,p (Ω;K) . Again by density, we can complete the proof of this step. b  Remark 1.3.7 Actually, it can be proved that W 1,1 (R; K) can be identified with the space of absolutely continuous function over R. We refer the interested reader to e.g., [6, Section 8.2] and [24, Section 7.2]. In this book, we will use the following interpolation estimates. Proposition 1.3.8 For every p ∈ [1, ∞), the following properties are satisfied. (i) Let Ω be Rd or Rd+ . Then, there exists a positive constant Cp such that 1

1

2 |||∇u|||Lp (Ω;K) ≤ Cp ||u||L2 p (Ω;K) ||u||W 2,p (Ω;K)

(1.3.8)

for every u ∈ W 2,p (Ω; K). In particular, if Ω = Rd or Ω = Rd+ , then ||u||W 2,p (Ω;K) can be replaced by ||D2 u||Lp (Ω;K) in the right-hand side of (1.3.8). (ii) Let Ω be Rd or Rd+ or a bounded open subset of Rd of class C 2 . Then, there exists a positive constant Cp0 such that |||∇u|||Lp (Ω;K) ≤ ε||u||W 2,p (Ω;K) +

Cp0 ||u||Lp (Ω;K) ε

(1.3.9)

for every u ∈ W 2,p (Ω; K) and ε > 0. In particular, if Ω = Rd or Ω = Rd+ then we can replace ||u||W 2,p (Ω;K) with |||D2 u|||Lp (Ω;K) in the right-hand side of (1.3.9). Proof We prove the assertions for real-valued functions. For complex-valued functions it suffices to consider their real and imaginary parts. (i) We begin by considering the case when Ω = Rd . We fix u ∈ C 2 (Rd ) ∩ W 2,p (Rd ), j ∈ {1, . . . , d} and observe that Z h u(x + hej ) = u(x) + Dj u(x)h + Djj u(x + rej )(h − r) dr 0 d

for every x ∈ R and h > 0. Therefore, Dj u(x) =

u(x + hej ) − u(x) 1 − h h

Z

h

(h − r)Djj u(x + rej ) dr. 0

(1.3.10)

22

Function Spaces

Next, we observe that p Z Z h 1 dxj D u(x + re )(h − r) dr jj j h 0 R p  0Z Z h Z h p 1 ≤ p (h − r) dr dxj (h − r)|Djj u(x + hej )|p dr h 0 R 0 Z Z hp−2 h (h − r) dr |Djj u(x + rej )|p dx = p−1 2 R Z 0 hp p |Djj u(x)| dxj , = p 2 R so that p Z h Z 1 hp Djj u(x + rej )(h − r) dr dx ≤ p |Djj u(x)|p dx. 2 Rd Rd h 0

Z

Similarly, Z Z p−1 Z u(x + hej ) − u(x) p 2p p p dx ≤ 2 (|u(x + he )| + |u(x)| ) dx = |u(x)|p dx. j p p h h h d d d R R R Therefore, integrating both sides of (1.3.10) over Rd and using the previous estimates, we get ||Dj u||pLp (Rd ) ≤

hp 2p ||u||pLp (Rd ) + p ||Djj u||pLp (Rd ) . p h 2

Minimizing over (0, ∞), with respect to h, we conclude that 1

1

1

1

||Dj u||pLp (Rd ) ≤ 2||u||L2 p (Rd ) ||Djj u||L2 p (Rd ) ≤ ||u||L2 p (Rd ) ||D2 u||L2 p (Rd ) Hence, estimate (1.3.8) follows for functions u ∈ C 2 (Rd ) ∩ W 2,p (Rd ). Since this space is dense in W 2,p (Rd ), estimate (1.3.8) can be straightforwardly extended to every function u ∈ W 2,p (Rd ). The proof in the case Ω = Rd+ is completely similar. (ii) If Ω = Rd or Ω = Rd+ , then estimate (1.3.9) follows from (1.3.8) using Young’s inequality ab ≤ εa2 + (4ε)−1 b2 , which holds for every a, b ≥ 0 and ε > 0, taking a = 1/2 1/2 ||D2 u||Lp (Ω) and b = Cp ||u||Lp (Ω) . If Ω is a bounded open set of class C 2 , then by Proposition B.4.5, there exists a bounded extension operator E2 mapping Lp (Ω) into Lp (Rd ) and W 2,p (Ω) in W 2,p (Rd ). Therefore, for every u ∈ W 2,p (Ω), we can apply estimate (1.3.9) to the function E2 u and write |||∇u|||Lp (Ω) ≤ε0 ||D2 E2 u||Lp (Rd ) +

Kp ||E2 u||Lp (Rd ) ε0

≤ε0 ||E2 ||L(W 2,p (Ω),W 2,p (Rd )) ||u||W 2,p (Ω) +

Kp ||E2 ||L(Lp (Ω),Lp (Rd )) ||u||Lp (Ω) ε0

for every ε0 > 0 and some positive constant Kp , independent of ε and u. Taking ε = ε0 ||E2 ||L(W 2,p (Ω),W 2,p (Rd )) , estimate (1.3.9) follows immediately.  We conclude this first part of the section with the following propositions that will be used in Chapters 10 and 11.

Semigroups of Bounded Operators and Second-Order PDE’s

23

Proposition 1.3.9 Let Ω be Rd or Rd+ or a bounded open set of class C 1 . Then, for every 0 u ∈ W 1,p (Ω; K) (p ∈ [2, ∞)), the function u|u|p−2 belongs to W 1,p (Ω; K), where 1/p+1/p0 = 1. Moreover, if (ϕn ) ⊂ Cc∞ (Rd ; K) is a sequence converging to u in W 1,p (Ω; K), then the 0 sequence (ϕn |ϕn |p−2 ) converges to u|u|p−2 in W 1,p (Ω; K). Here, v denotes the function conjugate to v. Proof We claim that it suffices to consider the case when Ω = Rd . Indeed, in the other cases we easily reduce to this situation replacing the function u with the function E1 u, where E1 : W 1,p (Ω; K) → W 1,p (Rd ; K) is an extension operator (see Proposition B.4.5). Moreover, we can limit ourselves to assuming that p > 2, since the case p = 2 is trivial. So, let us 0 fix u ∈ W 1,p (Rd ; K) and p > 2. Clearly, the function v = u|u|p−2 belongs to Lp (Rd ; K). 0 To prove that it actually belongs to W 1,p (Rd ; K), we fix a sequence (ϕn ) ⊂ Cc∞ (Rd ; C) converging to u in W 1,p (Rd ; K) and pointwise almost everywhere on Rd (see Theorem 1.3.4). We set vn = ϕn |ϕn |p−2 for every n ∈ N. This function belongs to Cc1 (Rd ; K) and ∇vn =

p p−2 2 ∇ϕn |ϕn |p−2 + ϕn ∇ϕn |ϕn |p−4 2 2

for every n ∈ N. Let us prove that ∇ϕn |ϕn |p−2 and ϕn 2 ∇ϕn |ϕn |p−4 converge, respectively, 0 to ∇u|u|p−2 and u2 ∇u|u|p−4 in Lp (Rd ; Kd ). For this purpose, we observe that Z Z 0 0 0 |∇ϕn |ϕn |p−2 − ∇u|u|p−2 |p dx ≤ |∇u − ∇ϕn |p |u|p (p−2) dx Rd Rd Z 0 0 + |∇ϕn |p ||u|p−2 − |ϕn |p−2 |p dx Rd

= : I1,n + I2,n for every n ∈ N. Applying H¨ older’s inequality, it is easy to check that p0 (p−2)

0

I1,n ≤ |||∇u − ∇ϕn |||pLp (Rd ;K) ||u||Lp (Rd ;K) , so that I1,n vanishes as n tends to ∞. As far as I2,n is concerned, we estimate (using again H¨ older’s inequality) I2,n ≤

0 |||∇ϕn |||pLp (Rd ;K)

Z

p−2 p |u| − |ϕn |p−2 | p−2 dx

 p−2 p−1 ,

n ∈ N.

Rd

Clearly, |||∇ϕn |||Lp (Rd ;K) converges to |||∇u|||Lp (Rd ;K) as n tends to ∞. Moreover, the sequence p (wn ), defined by wn = 21/(p−1) (|u|p + |ϕn |p ) − ||u|p−2 − |ϕn |p−2 | p−2 for every n ∈ N. consists of nonnegative functions and it converges almost everywhere in Rd to the function 2p/(p−1) |u|p . Applying Fatou lemma, we infer that Z p p 1 2 p−1 ||u||pLp (Rd ;K) ≤ lim inf [2 p−1 (|u|p + |ϕn |p ) − ||u|p−2 − |ϕn |p−2 | p−2 ] dx n→∞

=2

1 p−1

Rd p ||u||Lp (Rd ;K)

Z − lim sup n→∞

1

+ 2 p−1 lim ||ϕn ||pLp (Rd ;K)

||u|

n→∞

p−2

p

− |ϕn |p−2 | p−2 dx

Rd

so that Z lim sup n→∞

Rd

p

||u|p−2 − |ϕn |p−2 | p−2 dx ≤ 0.

24

Function Spaces

Hence, I2,n converges to 0 as n tends to ∞. Next, we observe that Z 0 |ϕn 2 ∇ϕn |ϕn |p−4 − u2 ∇u|u|p−4 |p dx d ZR Z 0 0 0 0 ≤ |∇ϕn − ∇u|p |u|p (p−2) dx + |∇ϕn |p |ϕn 2 |ϕn |p−4 − u2 |u|p−4 |p dx Rd

Rd

≤|||∇ϕn − +

0 p0 (p−2) ∇u|||pLp (Rd ;K) ||u||Lp (Rd ;K)

0 |||∇ϕn |||pLp (Rd ;K)

Z

2

p−4

|ϕn |ϕn |

2

p−4

− u |u|

|

p p−2

 p−2 p−1 . dx

Rd

Arguing as in the proof of the convergence of I2,n with the obvious changes, we can show that the last-side of the previous chain of inequalities vanishes as n tends to ∞. We have so proved that ∇vn converges to the function 2−1 p∇u|u|p−2 + 2−1 (p − 0 2 2)u ∇u|u|p−2 in W 1,p (Rd ; K) as n tends to ∞. To complete the proof, we observe that the same arguments used above show that vn 0 0 converges to u|u|p−2 in Lp (Rd ; K). Since vn ∈ Cc1 (Rd ; K) ⊂ W 1,p (Rd ; K) for every n ∈ N, 0 the function u|u|p−2 is in W 1,p (Rd ; K). The proof is complete.  Remark 1.3.10 From the proof of Proposition 1.3.9 it follows easily that, if Ω is an open subset of Rd of class C 1 and u ∈ W01,p (Ω; K) for some p ≥ 2, then the function u|u|p−2 0 belongs to W01,p (Ω; K), where 1/p + 1/p0 = 1. Proposition 1.3.11 Let Ω be a domain of Rd (possibly Ω = Rd ). Then the following properties are satisfied. (i) Let Ω0 be a domain compactly contained in Ω. Then, ||u(r,j) ||Lp (Ω0 ;K) ≤ |||∇u|||Lp (Ω;K) ,

0 < |r| < dist(Ω0 , ∂Ω),

(1.3.11)

for every p ∈ [1, ∞) and u ∈ W 1,p (Ω; K), where u(r,j) is the function defined by u(r,j) (x) := r−1 (u(x + rej ) − u(x)) for every |r| ∈ (0, dist(Ω0 , ∂Ω)) and j ∈ {1, . . . , d}. If Ω = Rd , then (1.3.11) holds true for every r ∈ R \ {0} and with Ω0 being replaced with Rd . If Ω = Rd+ , then (1.3.11) holds true, with Ω0 being replaced by Ω, for every r ∈ R \ {0} and j < d and for every r > 0 when j = d. (ii) Let Ω0 be as in (i). If u ∈ Lp (Ω; K) for some p ∈ (1, ∞) is such that 0 < |r| < dist(Ω0 , ∂Ω), j ∈ {1, . . . , d}, (1.3.12) √ then u ∈ W 1,p (Ω; K) and |||∇u|||Lp (Ω;K) ≤ C d. If Ω = Rd , we assume that (1.3.12) is satisfied by every r ∈ R \ {0}. ||u(r,j) ||Lp (Ω0 ) ≤ C|r|,

Proof (i) Fix p ∈ [1, ∞). We begin by assuming that Ω = Rd and u ∈ Cc∞ (Rd ; K). Then, for every j ∈ {1, . . . , d} and r 6= 0, we can write Z 1 r u(r,j) (x) = Dj u(x + sej ) ds, x ∈ Rd . r 0 Therefore, using Jensen’s inequality, we can estimate Z 1 r (r,j) p |u (x)| ≤ |Dj u(x + sej )|p ds, |r| 0

x ∈ Rd ,

Semigroups of Bounded Operators and Second-Order PDE’s

25

and then, integrating with respect to x on Rd and using Fubini’s theorem to invert the order of integration, we get Z Z r Z 1 (r,j) p p dx |Dj u(x + sej )| ds |u (x)| dx ≤ |r| Rd 0 Rd Z Z r 1 p = dx |Dj u(x + sej )| ds |r| Rd 0 Z Z 1 r p ds |Dj u(x + sej )| dx = ||Dj u||pLp (Rd ;K) , = |r| 0 Rd so that (1.3.11) follows in this case. For a general u ∈ W 1,p (Rd ; K), it suffices to use a density argument, approximating u in W 1,p (Rd ; K) with a sequence (ϕn ) ⊂ Cc1 (Rd ; K) (see Theorem 1.3.4). If Ω = Rd+ , then we can argue as above, considering first the case when u ∈ Cc∞ (Rd+ ; K) and then using a density argument. Finally, for a general domain Ω, observe that given a subdomain Ω0 , compactly contained in Ω, r ∈ R with 0 < |r| < dist(Ω0 , ∂Ω), and u ∈ Cc∞ (Rd ; K), repeating the above arguments we can write Z Z Z 1 r (r,j) p p |u (x)| dx ≤ ds |Dj u(x + sej )| dx |r| 0 0 Ω Ω Z0 Z 1 r p ds |Dj u(y)| dx = |r| 0 0 Ω +sej ≤||Dj u||pLp (Ω0 +B(0,|r|);K) and the set Ω0 + B(0, |r|) is compactly contained in Ω, due to the choice of r. Given u ∈ W 1,p (Ω; K), we extend it in the trivial way to Rd and compute the convolution of this function with a standard sequence of mollifiers. We thus get a sequence (ψn ) of smooth functions, which are not compactly supported in Rd , if Ω is unbounded. In this case, we multiply each function ψn by ϑn , where ϑn (x) = ϑ(nx) for every x ∈ Rd , n ∈ N and ϑ ∈ C ∞ (B(0, 1); K). The sequence (ϑn ψn ) belongs to Cc∞ (Rd ; K), converges to u in Lp (Ω; K) and in W 1,p (Ω00 ; K) for every open set Ω00 b Ω. Hence, again a density argument allows us to get the assertion in this case. (ii) We fix p ∈ (1, ∞), u ∈ Lp (Ω; K), which satisfies (1.3.12), ϕ ∈ Cc∞ (Ω; K), j ∈ {1, . . . , d}. Further, we denote by K b Ω the support of ϕ and by r any positive number less than the distance of K from ∂Ω. Then, the set Ω0 = K +B(0, |r|) is compactly supported in Ω and Z Z Z 1 uD ϕ dx = uD ϕ dx = lim u(x)(ϕ(x + he ) − ϕ(x)) dx j j j h→0 h Ω K K Z = lim u(−h,j) (x)ϕ(x) dx h→0 0 Ω

≤ lim sup ||u(−h,j) ||Lp (Ω0 ;K) ||ϕ||Lp0 (Ω;K) ≤ C||ϕ||Lp0 (Ω;K) , h→0

R where 1/p + 1/p0 = 1. It thus follows that the operator ϕ 7→ Ω uDj ϕ dx is bounded from (Cc∞ (Ω; K), || · ||Lp0 (Ω;K) ) into K. By density, it can be extended uniquely with a functional 0 on Lp (Ω; K), whose operator norm does not exceed C. Hence, there exists a unique vj ∈ Lp (Ω; K) such that Z Z uDj ϕ dx = − vj u dx, ϕ ∈ Cc∞ (Ω; K). Ω

Ω

26

Function Spaces

We deduce that the distributional derivative Dj u belongs to Lp (Ω; K) and ||Dj u||Lp (Ω;K) ≤ C. The assertion is proved in this case. 

1.4

Besov Spaces

In this section, we introduce the Besov spaces when Ω is Rd or the boundary of a bounded and sufficiently smooth domain. An exhaustive treatment of Besov spaces is beyond the scope of this book. Here, we present just those basic results that are used in the forthcoming chapters. For a much detailed description of Besov spaces, we refer the interested reader, e.g., to the monographs [24, 33, 37]. Definition 1.4.1 For every p ∈ (1, ∞) and k ∈ N, the Besov space B k−1/p,p (Rd ; K) is the subset of W k−1,p (Rd ; K) of functions u such that [Dβ u]pp =

d Z X i=1

∞

σ −p dσ

Z

|Dβ u(x + σei ) − Dβ u(x)|p dx < ∞

Rd

0

for every multi-index β with length k − 1. Here, W 0,p (Rd ; K) = Lp (Rd ; K). B k−1/p,p (Rd ; K) is a Banach space with the norm X ||u||B k−1/p,p (Rd ;K) = ||u||W k−1,p (Rd ;K) + [Dβ u]p ,

u ∈ B k−1/p,p (Rd ; K).

|β|=k−1

As the following theorem shows, every function u ∈ B k−1/p,p (Rd ; K) can be approximated by a sequence of smooth functions. Theorem 1.4.2 The space Cc∞ (Rd ; K) is dense in B k−1/p,p (Rd ; K) for every p ∈ (1, ∞) and k ∈ N. Proof First of all, we prove that the functions in B k−1/p,p (Rd ; K) with compact support are dense in B k−1/p,p (Rd ; K). For this purpose, we fix u ∈ B k−1/p,p (Rd ; K), introduce a function ϑ ∈ Cc∞ (Rd ), such that 0 ≤ ϑ ≤ 1 on Rd , and set ϑn (x) = ϑ(n−1 x) for every x ∈ Rd and n ∈ N. It is easy to check that the sequence (vn ), defined by vn = ϑn u for every n ∈ N, belongs to W k−1,p (Rd ; K) and converges to u in W k−1,p (Rd ; K). Moreover, for every multi-index β with length k − 1, we can estimate [Dβ vn − Dβ u]pp Z d Z ∞ X |Dβ u(x + σej ) − Dβ u(x)|p dx ≤ dσ |ϑn (x) − 1|p σp Rd j=1 0 +

d X  Z X β j=1 α 0 and estimate |u(x + σej ) − u(x)|p ≤ 2p−1 (|u(x + σej ) − u(Sj y + (xj + 2−1 σ)ej )|p + |u(Sj y + (xj + 2−1 σ)ej ) − u(x)|p ) for every x ∈ Rd , where Sj y = (y1 , . . . , yj−1 , 0, yj , . . . , yd−1 ) for every y ∈ Rd−1 and j = 1, . . . , d − 1. Computing the average of both sides Pof the previous inequality on the ball B(b xj , σ/2) ⊂ Rd−1 with respect to y, where x bj = i6=j xi ei , we get Z 2p−1 p |u(x + σej ) − u(x)| ≤ d−1 |u(x + σej ) − u(Sj y + (xj + 2−1 σ)ej )|p dy σ κd B(bxj ,σ/2)  Z + |u(Sj y + (xj + 2−1 σ)ej ) − u(x)|p dy , B(b xj ,σ/2)

where κd is the Lebesgue measure of the ball B(0, 1) ⊂ Rd . Therefore, Z ∞ Z |u(x + σej ) − u(x)|p dσ dx σp 0 Rd Z Z Z |u(x + σej ) − u(Sj y + (xj + 2−1 σ)ej )|p 2p−1 ∞ dσ dx dy ≤ κd 0 σ p+d−1 Rd B(b xj ,σ/2) Z Z Z 2p−1 ∞ |u(Sj y + (xj + 2−1 σ)ej ) − u(x)|p + dσ dx dy. κd 0 σ p+d−1 Rd B(b xj ,σ/2) The two integral terms in the right-hand side of the previous inequality can be estimated in the same way. Hence, we limit ourselves to considering the first one, which, by a straightforward change of variables, we can write as Z ∞ Z Z |u(x) − u(Sj y + (xj − 2−1 σ)ej )|p I := dσ dx dy. σ p+d−1 0 Rd B(b xj ,σ/2)

30

Function Spaces

Interchanging the order of integration and then performing the change of variables τ = xj − 2−1 σ, we obtain Z ∞ Z |u(x) − u(Sj y + (xj − 2−1 σ)ej )|p dy dσ σ p+d−1 0 B(b xj ,σ/2) Z Z ∞ |u(x) − u(Sj y + (xj − 2−1 σ)ej )|p = dy dσ σ p+d−1 2|y−b xj | Rd−1 Z Z xj −|y−bxj | |u(x) − u(Sj y + τ ej )|p dy =22−p−d dτ. |τ − xj |p+d−1 −∞ Rd−1 Note that |y−b xj | ≤ |xj −τ | for every τ < xj −|y−b xj |, so that |b xj −y|2 +|xj −τ |2 ≤ 2|xj −τ |2 . Thus, performing the change of unknowns zi = yi for i < j, zj = τ , zi = yi−1 for i > j, we can estimate Z Z xj −|y−bxj | |u(x) − u(Sj y + τ ej )|p dτ dy |τ − xj |p+d−1 −∞ Rd−1 Z Z p+d−1 |u(x) − u(Sj y + τ ej )|p 2 ≤2 dy p+d−1 dτ xj − y|2 + |xj − τ |2 ) 2 Rd−1 R (|b Z p+d−1 |u(x) − u(z)|p =2 2 dz. p+d−1 Rd |x − z| Putting everything together, we conclude that I ≤ Cp,d [[u]]p for some positive constant Cp,d , 0 0 independent of u. We have so proved that ||u||B 1−1/p,p (Rd ;K) ≤ Cp,d [[u]]p , where Cp,d is inde1−1/p,p pendent of u. By density, we can extend this inequality to every u ∈ W (Rd ; K). Indeed, the arguments in the proof of Theorem 1.4.2 can be adapted to prove that Cc∞ (Rd ; K) is dense in W k−1/p,p (Rd ; K) for every k ∈ N. This completes the proof.  We now extend the definition of the Besov spaces to the case when Rd is replaced by the boundary of a bounded smooth domain Ω. For this purpose, we fix a bounded open set Ω of class C k for some k ∈ N and consider the covering {Ux : x ∈ ∂Ω} of ∂Ω provided by Definition B.2.1. Since ∂Ω is compact, we can extract a finite subcovering {Uxj : j = 1, . . . , N }. Associated with this subcovering of ∂Ω, we consider the functions ψxj : Uxj → B(0, r) (j = 1, . . . , N ) which are bijective and of class C k together with their inverse functions. Finally, we fix a partition of the unity subordinated to the previous subcovering of ∂Ω, i.e., we consider the family {ηj : j = 1, . . . , N }, where ηj ∈ Cc∞ (Uxj ) for PN every j = 1, . . . , N and j=1 ηj = 1 on ∂Ω. The following definition is given in terms of the (d − 1)- dimensional Hausdorff measure Hd−1 . For a complete description of the Hausdorff measure and its main properties, we refer the reader to [23]. Definition 1.4.5 Let Ω be a bounded open set of class C k for some k ∈ N. For p ∈ (1, ∞), B k−1/p,p (∂Ω; K) is the set of all functions u ∈ Lp (∂Ω, Hd−1 ; K) such that for every j ∈ {1, . . . , N } the function vj = (ηj u) ◦ ψx−1 (·, 0), extended by zero outside B(0, r), belongs to j B k−1/p,p (Rd−1 , K) for every j = 1, . . . , N . It is endowed by the norm ||u||B k−1/p,p (∂Ω;K) =

N X

||vj ||B k−1/p,p (Rd−1 ;K) ,

u ∈ B k−1/p,p (∂Ω; K).

j=1

Notation. Throughout the rest of this section, for every j = 1, . . . , N we denote by vj both the function (uηj ) ◦ ψx−1 (·, 0) and its trivial extension outside B(0, r), where the functions j ηj , ψxj are introduced before Definition 1.4.5.

Semigroups of Bounded Operators and Second-Order PDE’s

31

Remark 1.4.6 The previous definition is independent of the functions ψxj and ηj (j = 1, . . . , N ). Changing the functions ψj = ψxj and/or the functions ηj produces just a change in the norm of B k−1/p,p (∂Ω; K) in the sense that the norm introduced in Definition 1.4.5 is replaced by an equivalent norm. Indeed, suppose that {Ui∗ : i = 1, . . . , M } is a different covering of ∂Ω, ϕi : Ux∗i → B(0, r) are bijective functions of class C k with inverse of class ∗ C k and let {η1∗ , . . . , ηM } be a partition of the unity subordinated to this new covering. Fix ∗ j ∈ {1,P . . . , M } and denote by Ij the set of all index i ∈ {1,P . . . , N } such that P Ui ∩ Uj 6= ∅. ∗ ∗ ∗ Then, i∈Ij ηi = 1 on ∂Ω ∩ Uj and, consequently, uηj = i∈Ij uηj ηi = i∈Ij ui,j . Note −1 ∗ 0 ∗ that ui,j ◦ ϕ−1 j (·, 0) = (ηj ◦ ψi (·, 0)vi ) ◦ Θi,j on the open set Ωi,j = PRd−1 ϕj (∂Ω ∩ Ui ∩ Uj ) = 0 ∗ 0 d−1 B (0, r) ∩ PRd−1 ϕj (Ui ∩ Uj ), where B (0, r) is the ball of R centered at zero with radius (·, 0) is a diffeomorphism of r, PRd−1 x = x0 for every (x0 , xd ) ∈ Rd and Θi,j = ψi ◦ ϕ−1 j k 0 ∗ class C on Ωi,j . Since the supports of ηi and ηj are compact subsets of Uj and Uj∗ , re0 spectively, the support of the function ui,j ◦ ϕ−1 j (·, 0) is compact in Ωi,j . Note that, arguing as in the proof of Theorem 1.4.2, it can be checked that the trivial extension of the function wij := ηj∗ ◦ ψi−1 (·, 0)vi outside the open set Θij (Ω0ij ) belongs to B k−1/p,p (Rd−1 ; K) and ||wi,j ||B k−1/p,p (Rd−1 ;K) ≤ Cp ||vi ||B k−1/p,p (Rd−1 ;K) . It thus follows that the trivial extenk−1/p,p (Rd−1 ; K)- and its sion (say vi,j ) to Rd+ of the function ui,j ◦ ϕ−1 j (·, 0) belongs to B k−1/p,p d−1 B (R ; K)-norm can be bounded from above by a positive constant, independent of vi , times the B k−1/p,p (Rd−1 ; K)-norm of vi . Hence, setting vj∗ := (ηj∗ u) ◦ ϕ−1 xj (·, 0), we can conclude that ||vj∗ ||B k−1/p (Rd−1 ;K) ≤

X

||vi,j ||B k−1/p,p (Rd−1 ;K) ≤ Cp

N X

||vi ||B k−1/p,p (Rd−1 ;K)

i=1

i∈Ij

for some constant Cp , independent of u, so that M X

||vj∗ ||B k−1/p (Rd−1 ;K) ≤ M Cp ||u||B k−1/p,p (∂Ω;K) .

j=1

Similarly, it can be shown that ep ||u||B k−1/p (∂Ω;K) ≤ N C

M X

||vj∗ ||B k−1/p,p (Rd−1 ;K) ,

j=1

ep is independent of u. where the constant C In the case k = 1, we can give an equivalent definition of the space B 1−1/p,p (∂Ω; K) in the spirit of Theorem 1.4.4, which does not require to deal with local charts. Proposition 1.4.7 For every bounded open set Ω of class C 1 and every p ∈ (1, ∞), B 1−1/p,p (∂Ω; K) is the set of all functions u ∈ Lp (∂Ω, Hd−1 ; K) such that Z Z |u(x) − u(y)|p p d−1 [u]p = dH (x) dHd−1 (y) < ∞. p+d−2 ∂Ω ∂Ω |x − y| Moreover, the norm ||u||∗B 1−1/p,p (∂Ω;K) = ||u||Lp (∂Ω;K) + [u]p is equivalent to the norm of B 1−1/p,p (∂Ω; K). Proof Suppose that u ∈ B 1−1/p,p (∂Ω; K) and split u =

PN

i=1

ui , where ui = ηi u. Note

32

Function Spaces

that each function ui belongs to B 1−1/p,p (∂Ω; K). Indeed, using the area formula we can write Z Z Z |ui (σ)|p dHd−1 (σ) = |ui (σ)|p dHd−1 (σ) = |vi (x0 )|p gi (x0 ) dx0 ∂Ω

B 0 (0,r)

∂Ω∩Uxi

where B 0 (0, r) is the ball in Rd−1 centered at zero and with radius r and gi (x0 ) is the square root of the determinant of the matrix A∗i (x0 )Ai (x0 ), Ai (x0 ) being the Jacobian matrix of the function ψx−1 (·, 0) evaluated at x0 ∈ B 0 (0, r). Note that we can determine a positive i (y 0 , 0)| ≥ C −1 |x0 − y 0 | for every (x0 , 0) − ψx−1 constant C such that ||gi ||∞ ≤ C and |ψx−1 j j 0 0 0 x , y ∈ B (0, r). Therefore, Z Z |ui (σ)|p dHd−1 (σ) ≤ ||gi ||∞ |vi (x0 )|p dx0 ≤ C||vi ||pLp (Rd−1 ;K) . B 0 (0,r)

∂Ω

Since u =

PN

i=1

ui , it thus follows that

||u||Lp (∂Ω,Hd−1 ;K) ≤

N X

1

||ui ||Lp (∂Ω,Hd−1 ;K) ≤ C p

i=1

N X

||vi ||Lp (Rd−1 ;K) .

(1.4.4)

i=1

Arguing similarly, we can show that Z Z |ui (σ) − ui (τ )|p d−1 dHd−1 (τ ) dH (σ) p+d−2 ∂Ω ∂Ω |σ − τ | Z Z |ui (σ) − ui (τ )|p d−1 dH (σ) = dHd−1 (τ ) |σ − τ |p+d−1 ∂Ω∩Uxi ∂Ω∩Uxi Z Z |vi (x0 ) − vi (y 0 )|p g (y 0 ) dy 0 = gi (x0 ) dx0 −1 0 −1 0 p+d−2 i B 0 (0,r) B 0 (0,r) |ψxi (x , 0) − ψxi (y , 0)| Z Z |vi (x0 ) − vi (y 0 )|p 0 ≤C 2−p−d ||gi ||2∞ dx0 dy 0 p+d−2 B 0 (0,r) B 0 (0,r) |x − y| ≤C 4−p−d [vi ]pB 1−1/p,p (Rd−1 ;K) . Hence, |u(σ) − u(τ )|p dHd−1 (τ ) p+d−2 ∂Ω ∂Ω |σ − τ | Z N Z X |ui (σ) − ui (τ )|p ≤Cp,d dHd−1 (σ) dHd−1 (τ ) p+d−1 |σ − τ | ∂Ω i=1 ∂Ω Z

dH

0 ≤Cp,d

d−1

N X

Z

(σ)

[vi ]pB 1−1/p,p (Rd−1 ;K)

(1.4.5)

i=1 0 for some positive constants Cp,d and Cp,d , independent of u, as all the other constants which appear in the proof. From (1.4.4) and (1.4.5) it follows that ||u||∗B 1−1/p,p (∂Ω;K) ≤ ep,d ||u||B 1−1/p,p (∂Ω;K) . C

Viceversa, let us fix u ∈ Lp (∂Ω, Hd−1 ; K) such that ||u||∗B 1−1/p,p (∂Ω;K) < ∞. Then, for every i = 1, . . . , N we can estimate Z Z |ui (σ) − ui (τ )|p dHd−1 (σ) dHd−1 (τ ) p+d−2 ∂Ω ∂Ω |σ − τ |

Semigroups of Bounded Operators and Second-Order PDE’s Z Z |u(σ) − u(τ )|p dHd−1 (τ ) ≤ |ηi (σ)|p dHd−1 (σ) p+d−2 |σ − τ | ∂Ω ∂Ω Z Z |ηi (σ) − ηi (τ )|p p d−1 + |u(σ)| dH (σ) dHd−1 (τ ) p+d−2 ∂Ω ∂Ω |σ − τ | Z Z dHd−1 (τ ) . ≤[u]pp + |||∇ηi |||p∞ |u(σ)|p dHd−1 (σ) d−2 ∂Ω ∂Ω∩Uxi |σ − τ |

33

Note that Z ∂Ω∩Uxi

gi (y 0 ) dy 0 0 , 0)|d−2 − ψx−1 (y i Z 1 2−d dy 0 ≤C ||gi ||∞ 0 − y 0 |d−2 |x 0 B (0,r) Z 1 =C 2−d ||gi ||∞ dy 0 0 |d−2 |y 0 0 B (x ,r) Z 1 2−d ≤C ||gi ||∞ dy 0 , 0 |d−2 |y 0 B (0,2r)

dHd−1 (τ ) = |σ − τ |d−2

Z

−1 0 B 0 (0,r) |ψxi (x , 0)

since B 0 (x0 , r) ⊂ B 0 (0, 2r). Thus, Z Z |ui (σ) − ui (τ )|p d−1 ∗ dH (σ) dHd−1 (τ ) ≤ Cp,d (||u||pLp (∂Ω,Hd−1 ;K) + [u]pp ) |σ − τ |p+d−2 ∂Ω∩Uxi ∂Ω∩Uxi and, consequently, Z

0

gi (x )dx

0

Z

|vi (x0 ) − vi (y 0 )|p g (y 0 ) dy 0 −1 0 0 p+d−2 i |ψx−1 i (x , 0) − ψxi (y , 0)|

B 0 (0,r) B 0 (0,r) p ∗ ≤Cp,d (||u||Lp (∂Ω,Hd−1 ;K) + [u]pp ).

(y 0 , 0)| ≤ (x0 , 0) − ψx−1 Since the infimum over B 0 (0, r) of the function gi is positive and |ψx−1 i i 0 0 0 0 0 −1 |||∇ψxi |||∞ |x −y | for every x , y ∈ B (0, r), from the previous inequality and recalling that e ∗ ||u||∗ 1−1/p vi is compactly supported in B 0 (0, r), we deduce that [vi ]p ≤ C . Showing p,d B (∂Ω;K) ∗ b ||u||Lp (∂Ω,Hd−1 ;K) is similar and even simpler. The assertion follows. that ||vi ||Lp (Rd−1 ;K) ≤ C d,p  To conclude this section, we prove the counterpart of Theorem 1.4.2. Theorem 1.4.8 Let Ω be a bounded open set of class C k . For every p ∈ (1, ∞), the space B k−1/p,p (∂Ω; K) ∩ C k (∂Ω; K) is dense in B k−1/p,p (∂Ω; K). Proof Fix u ∈ B k−1/p,p (∂Ω; K) and consider the covering of ∂Ω, the functions ψxj and ηj , introduced before Definition 1.4.5. For every i = 1, . . . , N , set ui = uηi and vi = ui ◦ ψx−1 (·, 0), this latter function trivially extended outside the ball B 0 (0, r) of Rd−1 . Throughout i the proof, by ζ we denote the trivial extension of the function ζ to Rd . By assumptions, vi belongs to B k−1/p (Rd−1 ; K). By Theorem 1.4.2, we can determine a sequence (vn,i ) ⊂ Cck (Rd−1 ; K) converging to vi in B k−1/p (Rd−1 ; K). Moreover, the proof of the quoted theorem shows that we can assume that the functions vn,i are compactly supported in B 0 (0, r), since vi has compact support in B 0 (0, r). Define by un,i the trivial extension to ∂Ω of the function vn,i ◦ ψxi , which is defined in ∂Ω ∩ Uxi . Finally, set PN un = i=1 un,i . We claim that the sequence (un ) belongs to C k (∂Ω; K) and converges to

34

Function Spaces

u in B k−1/p (∂Ω; K). For this purpose, we fix j ∈ {1, . . . , N } and P denote by Ij the set of all indexes i ∈ {1, . . . , N } such that Uxi ∩ Uxj 6= ∅. Then, un ηj = i∈Ij un,i ηj and, consequently, (un,i ηj )◦ψx−1 (·, 0) = (ηj ◦ψx−1 (·, 0))vn,i ◦Φi,j on Ω0 = B 0 (0, r)∩ψxj (Uxi ∩Uxj ), where j j −1 Φi,j = ψxi ◦ ψxj (·, 0) is a diffeomorphism of class C k defined in Ω0 . Note that outside Ω0 the function (un,i ηj ) ◦ ψx−1 vanishes since its support is compactly contained in the open set Ω0 . j Therefore, we can extend this function in the trivial way outside Ω0 obtaining a function which belongs to C k (Rd−1 ; K) and converges in B k−1/p,p (Rd−1 ; K) to (ηj ◦ ψx−1 j (·, 0))vi ◦ Φi,j (·, 0))v ◦ Φ (see Theorem 1.4.2 for further details). Since of the function (ηj ◦ ψx−1 n,i i,j j P −1 −1 −1 (un ηj ) ◦ ψxj (·, 0) = i∈Ij (un,i ηj ) ◦ ψxj (·, 0), we conclude that (un ηj ) ◦ ψxj (·, 0) converges P in B k−1/p (Rd−1 ; K) to the function i∈Ij (ηj ◦ ψx−1 j (·, 0))vi ◦ Φi,j and this latter function is 0 0 0 vj . Indeed, for x ∈ B (0, r), let Ij (x ) denote the set of the indexes i ∈ {1, . . . , N } such that ψx−1 (x0 , 0) ∈ Uxi . Then, j X X ηj (ψx−1 (ηj (ψx−1 (x0 , 0))vi (Φi,j (x0 )) = (x0 , 0))(uηi )(ψx−1 (x0 , 0)) j j j i∈Ij (x0 )

i∈Ij (x0 )

=vj (x0 )

X i∈Ij

since

1.5

−1 0 i∈Ij (x0 ) ηi (ψxi (x , 0))

P

ηi (ψx−1 (x0 , 0)) = vj (x0 ), j

(x0 )

= 1. The proof is complete.



Exercises

1. Prove the Young inequality ab ≤

1 p 1 q a + b for every a, b ≥ 0 and p, q ∈ (1, ∞) such that p q

1/p + 1/q = 1. 2. Prove that BU C(Rd ; K) is the closure of Cb1 (Rd ; K) in Cb (Rd ; K). 3. Arguing as in the first part of the proof of Proposition 1.1.4, show that 1

1

2 2 ||Djj f ||∞ ||Dj f ||∞ ≤ 2||f ||∞

for every f ∈ Cb (Rd ; K) (or f ∈ Cb (Rd+ ; K)) and every j = 1, . . . , d. 4. Let Ω be an open set (even unbounded). Prove that if f ∈ Cbα (Ω; K) for some α ∈ (0, ∞) \ N, then f can be extended with a α-H¨older continuous and bounded function to Ω. 5. Adapting the arguments in Remark 1.2.2, prove that, if a function f : I × Ω → K, I and Ω being as in Definition 1.2.1 is continuous and admits, for some j ∈ {1, . . . , d}, the derivatives Dj f , Dt f and Dj Dt f , which are continuous in I × Ω, then also the derivative Dt Dj f exists in I × Ω and Dj Dt f = Dt Dj f . (k+α)/2,k+α

6. Let f ∈ Cb (I × Ω; K) for some k ∈ N, k > 1, and α ∈ (0, 1). Assume that I is an interval, Ω = Rd , Ω = Rd+ or Ω is a bounded open set of class C k+α . Prove that, for every multi-index β ∈ Nd of length at most k − 2, the function Dxβ f belongs to 1+α/2,1+α Cb (I × Ω; K) and there exists a positive constant C, independent of f , such that ||Dxβ f ||C 1+α/2,2+α (I×Ω;K) ≤ C||f ||C (k+α)/2,k+α (I×Ω;K) . b

b

Part I

Semigroups of Bounded Operators

Chapter 2 Strongly Continuous Semigroups

In this and in the next chapter, we introduce the concept of semigroup of bounded linear operators and study two important classes of semigroups: the strongly continuous and the analytic semigroups. Semigroups of bounded operators naturally arise in the study of parabolic equations: as we will see a suitable semigroup governs the dynamics of parabolic equations. In several chapters we will also show that semigroups are a very powerful tool in the study of linear parabolic equations. The concept of semigroups of bounded operators generalizes what is known since the first courses of Calculus: the solutions of the system Dt u = Au of d ordinary differential equations with constant coefficients are given by u(t) = etA c for every t ∈ R, where c ∈ Rd is an arbitrary vector and ∞ n X t n A , t ∈ R. (2.0.1) etA = n! n=0 The previous formula can be straightforwardly extended to the case when the matrix A is replaced by a bounded operator in a Banach space X. Indeed,

m+p

m+p X tn

X tn n

≤ A ||A||nL(X)

n! n! L(X) n=m n=m P∞ n for every m, p ∈ N and the real valued series n=0 tn! ||A||nL(X) converges locally uniformly P n ∞ in R (to et||A||L(X) ). Hence, the series n=0 tn! An converges in L(X), locally uniformly with k k respect to t ∈ R. Set ak = t A /k! and bk = sk Ak /k! for every k ∈ N ∪ {0}. Then, n X k=0

ak bn−k = An

n X tk sn−k (t + s)n n = A , k!(n − k)! n!

n ∈ N ∪ {0}.

k=0

Hence, as for the Cauchy product of scalar series, one can see that tA sA

e e

∞ i i X ∞ ∞ X n ∞ X X X sj Aj (t + s)n n tA = · = ak bn−k = A = e(t+s)A = esA etA . i! j! n! n=0 n=0 j=0 i=0 k=0

Based on the above remarks, we can now give the following definition. Definition 2.0.1 A family {T (t) : t ≥ 0} (in the sequel simply denoted by {T (t)}) of bounded linear operators on a Banach space X is called a semigroup of bounded operators if it satisfies the semigroup property, i.e., T (0) = I and T (t + s) = T (t)T (s) for every s, t > 0. It follows that, for each A ∈ L(X), {etA : t ≥ 0} is a semigroup of bounded operators on the Banach space X. As a matter of fact, this class of semigroups, usually referred as uniformly continuous semigroups, is too small and these semigroups are not associated to parabolic equations. For this reason, we need to go further in the study of semigroups. 37

38

Strongly Continuous Semigroups

Throughout this chapter, X will denote a complex Banach space and || · || its norm. The chapter is organized as follows. In Section 2.1, we introduce the strongly continuous semigroups and prove some basic properties. In particular, we show that for every strongly continuous semigroup there exist M ≥ 1 and ω ∈ R such that ||T (t)||L(X) ≤ M eωt for every t > 0. Usually, one refers to this property saying that the semigroup {T (t)} belongs to the class G(M, ω). In Section 2.2, we introduce the concept of the infinitesimal generator A of a strongly continuous semigroup and state its main properties: this operator is densely defined and its resolvent set contains a right-halfplane. Further, the resolvent operator R(λ, A) = (λI − A)−1 can be written in terms of the semigroup, being the (pointwise) Laplace transform of {T (t)}. Section 2.3 is devoted to the well celebrated Hille-Yosida, Lumer-Phillips and Trotter-Kato theorems. The first one completely characterizes the operators which are infinitesimal generator of a strongly continuous semigroup, whereas the Lumer-Phillips theorem provides us with a sufficient (and necessary) condition for an operator to be the infinitesimal generator of a strongly continuous semigroup of contractions. Next, in Section 2.4, we study abstract Cauchy problems associated to generators of strongly continuous semigroups. In Section 2.5 we collect some concluding notes and remarks. Finally, Section 2.6 contains some exercises which may help the reader to become more familiar with strongly continuous semigroups.

2.1

Definitions and Basic Properties

Definition 2.1.1 A family {T (t)} of bounded operators on X, which satisfies the semigroup property, is a strongly continuous semigroup (or C0 -semigroup1 ) if the function t 7→ T (t)x is continuous in [0, ∞) with values in X, for each x ∈ X. The following example is crucial and we will use it in the next chapter to show the main differences between analytic and C0 -semigroups. Example 2.1.2 On X = BU C(R) consider the family {T (t)} of linear operators defined by (T (t)f )(x) = f (x + t) for each x ∈ R, t ≥ 0 and f ∈ BU C(R). This is a C0 -semigroup on X. Indeed, T (t)f tends to T (s)f uniformly in R as t → s if and only if supx∈R |f (x+t)−f (x+s)| vanishes as t → s. But this condition, is a rewriting of the definition of uniform continuity. The semigroup property is straightforward to prove. We stress that this semigroup, called the semigroup of the left-translations, cannot be written in the form (2.0.1) for some bounded operator A. Indeed, if this were the case, then ||etA − I||L(X) ≤

∞ n X t ||A||nL(X) = et||A||L(X) − 1, n! n=1

t ≥ 0,

and, consequently, limt→0+ ||etA − I||L(X) = 0. The semigroup of left translations does not satisfy this property since ||T (t) − I||L(X) = 2 for every t > 0. To check this claim, fix t > 0 and consider the bounded and uniformly continuous function ft : R → R, defined by ft (x) = sin(πx/t) for each x ∈ R. As it is immediately seen, ||ft ||∞ = 1 and ft (x+t) = −ft (x) for every x ∈ R. Hence, ||T (t) − I||L(X) ≥ sup |ft (x + t) − ft (x)| = 2 sup |ft (x)| = 2. x∈R

x∈R

1C aro summable of order zero, which means the continuity property 0 or (C, 0) abbreviates Ces` limt→0 T (t)x = x for every x ∈ X.

Semigroups of Bounded Operators and Second-Order PDE’s

39

On the other hand, ||T (t)f ||∞ = ||f ||∞ for every f ∈ BU C(R) and, hence, ||T (t) − I||L(X) ≤ ||T (t)||L(X) + 1 = 2 for every t ≥ 0. Actually, {T (t)} is a group of bounded operators, since T (t) can be defined, using the same rule, also for negative values of t. Now, we establish some basic properties of strongly continuous semigroups. To begin with, we prove that the function t 7→ ||T (t)||L(X) grows at most exponentially at infinity. Proposition 2.1.3 There exist M ≥ 1 and ω ∈ R such that ||T (t)||L(X) ≤ M eωt ,

t ≥ 0.

Proof The core of the proof consists in showing that there exists δ > 0 such that sup ||T (t)||L(X) < ∞.

(2.1.1)

t∈[0,δ]

Once this property is established, the semigroup property allows us to complete the proof. Indeed, if t > δ, then there exist n ∈ N and r ∈ [0, δ) such that t = nδ + r. By applying the semigroup property, we conclude that T (t) = T (r)(T (δ))n . Hence, denoting by M the supremum in (2.1.1) and observing that M ≥ 1, we get ||T (t)||L(X) ≤||T (r)||L(X) ||T (δ)||nL(X) ≤ M n+1 = M exp(n log(M )) ≤ M e

log(M ) t δ

and the assertion follows with ω = δ −1 log(M ). To prove (2.1.1), we argue by contradiction. We suppose that (2.1.1) does not hold true. Then, we can determine a positive sequence (tn ) converging to zero such that ||T (tn )||L(X) diverges to ∞ as n tends to ∞. Since T (tn )x converges to x as n tends to ∞, for every x ∈ X, the uniform boundedness principle leads us to a contradiction.  Corollary 2.1.4 A semigroup {T (t)} of bounded operators on X is strongly continuous if and only if the function t 7→ T (t)x is continuous at t = 0 for each x ∈ X. Proof Clearly, if the semigroup {T (t)} is strongly continuous, then, in particular, the mapping t 7→ T (t)x is continuous at t = 0 for every x ∈ X. Vice versa, if {T (t)} is a semigroup such that the mapping t 7→ T (t)x is continuous at t = 0, then from the proof of Proposition 2.1.3 we deduce that there exist M ≥ 1 and ω ≥ 0 such that ||T (t)||L(X) ≤ M eωt for every x ∈ X and t > 0. Fix t0 > 0 and observe that, if t ≥ t0 , then ||T (t)x − T (t0 )x|| = ||T (t0 )[T (t − t0 )x − x]|| ≤ M eωt0 ||T (t − t0 )x − x|| and the last side of the previous chain of inequalities vanishes as t tends to t+ 0 . Similarly, if t < t0 , then ||T (t)x − T (t0 )x|| = ||T (t)[x − T (t0 − t)x]|| ≤ M eωt ||T (t0 − t)x − x|| and also in this case the last side vanishes as t tends to t− 0 . Hence, the function t 7→ T (t)x is continuous at t0 . The arbitrariness of t0 > 0 shows that {T (t)} is a strongly continuous semigroup.  In the proof of Proposition 2.1.3 the constant ω is nonnegative. Actually, ω can be also negative. This is, for instance, the case of the C0 -semigroup {Ta (t)} on R, defined by Ta (t)x = e−at x for each x ∈ R and t ≥ 0, where a is a positive number. In view of Proposition 2.1.3 the following definition makes sense. Definition 2.1.5 The growth bound ω0 of a C0 -semigroup {T (t)} is defined as the infimum of the set {ω ∈ R : ∃ M = Mω ≥ 1 such that ||T (t)|| ≤ M eωt for every t ≥ 0}. Remark 2.1.6

(i) We stress that ω0 could be also equal to −∞ (see Exercise 2.6.3).

40

Strongly Continuous Semigroups

(ii) In general, ω0 is just an infimum and semigroup (actually, a group) {T (t)} in  1 T (t) = 0

not a minimum. Consider for instance, the R2 , defined by  t , t ≥ 0. 1

Here, ω0 = 0 and the function t 7→ T (t) is not bounded in [0, ∞) with values in L(R2 ). Example 2.1.7 For f ∈ Lp (R) we define (S(t)f )(s) := f (t + s) for s ∈ R and t ≥ 0. Then S(t) is a linear isometry on Lp (R). Moreover, S(t + s) = S(t)S(s) for every s, t ≥ 0. We call {S(t)} the semigroup of left translations on Lp (R). Furthermore, for p ∈ [1, ∞) semigroup {S(t)} is strongly continuous on Lp (R). Indeed, recall that the semigroup of left translations is strongly continuous on the space of bounded and uniformly continuous functions and that the set of continuous functions with compact support is dense in Lp (R), see Theorem 1.3.2. Taking f ∈ Cc (R) and α, β ∈ R such that supp f ⊂ [α, β], we see that Z ||S(t)f − f ||pp = |f (s) − f (s + t)|p ds ≤ (β − α) sup |f (s) − f (s + t)|p , s∈[α,β]

R

which tends to zero as t → 0 by the uniform continuity of f . Since ||S(t)|| ≤ 1, the statement follows by Corollary 2.1.4.

2.2

The Infinitesimal Generator

In this section, we show that to any C0 -semigroup it is possible to associate a linear operator, called the infinitesimal generator of the semigroup, and we study the main properties of this operator. Definition 2.2.1 The infinitesimal generator A of a C0 -semigroup {T (t)} is the operator defined as follows:    T (t)x − x   D(A) = x ∈ X : ∃ lim ∈ X ,  t t→0+    Ax = lim + T (t)x − x , x ∈ D(A). t→0 t Proposition 2.2.2 The generator A of a C0 -semigroup {T (t)} satisfies the following properties. (i) A is a linear operator satisfying AT (t) = T (t)A on D(A) for every t ≥ 0. Moreover, for each x ∈ D(A), the function u = T (·)x belongs to C 1 ([0, ∞); X) ∩ C([0, ∞); D(A)) and solves the Cauchy problem ( u0 (t) = Au(t), t ≥ 0, (2.2.1) u(0) = x. t

Z (ii) For each t > 0 and x ∈ X,

T (s)x ds ∈ D(A) and 0

Z T (t)x − x = A

t

T (s)x ds. 0

(2.2.2)

Semigroups of Bounded Operators and Second-Order PDE’s In particular, if x ∈ D(A), then Z t Z t A T (s)x ds = T (s)Ax ds. 0

41

(2.2.3)

0

(iii) A is closed and D(A) is dense in X. (iv) The operator A completely characterizes the semigroups {T (t)} in the sense that there exist no other semigroups which admit A as infinitesimal generator. Proof (i) The linearity of the operator A is straightforward to check. Note that D(A) 6= ∅ since 0 ∈ D(A). To complete the proof of property (i), we fix x ∈ D(A) and t > 0. Using the semigroup property and the continuity of the operator T (t), we get lim

h→0+

T (h)x − x T (h)x − x T (h)T (t)x − T (t)x = lim+ T (t) = T (t) lim+ = T (t)Ax. h h h h→0 h→0

Hence, T (t)x ∈ D(A) and AT (t)x = T (t)Ax. Since T (h)T (t) = T (t + h), the above computation shows that the function T (·)x is differentiable from the right in [0, ∞) and the right derivative coincides with the function AT (·)x. To prove that the previous function is also differentiable from the left in (0, ∞), we fix t > 0 and observe that T (t + h)x − T (t)x T (−h)x − x − T (t)Ax =T (t + h) − T (t)Ax h −h   T (−h)x − x − Ax + T (t + h)Ax − T (t)Ax. =T (t + h) −h Hence, taking Proposition 2.1.3 into account, we obtain



T (t + h)x − T (t)x

− T (t)Ax

h

T (−h)x − x

≤||T (t + h)||L(X) − Ax

+ ||T (t + h)Ax − T (t)Ax|| −h



T (−h)x − x

+ ||T (t + h)Ax − T (t)Ax|| ≤M eω(t+h) − Ax

−h for some M ≥ 1 and ω ∈ R. Letting h tend to 0− we conclude that the function T (·)x is differentiable from the left at t. Hence, the function T (·)x is differentiable in [0, ∞) and since its derivative is the function T (·)Ax, which is continuous in [0, ∞), it follows that T (·)x ∈ C 1 ([0, ∞); X) ∩ C([0, ∞); D(A)) and solves problem (2.2.1). Z t (ii) Fix x ∈ X, t > 0, set y = T (s)x ds and observe that 0

 Z t  Z t Z t t T (h)y − y 1 1 = T (h) T (s)x ds − T (s)x ds = T (h + s)x ds − T (s)x ds h h h 0 0 0 0  Z t+h   Z t+h  Z t Z h 1 1 = T (s)x ds − T (s)x ds = T (s)x ds − T (s)x ds . h h h 0 t 0 

Z

Since the function s 7→ T (s)x is continuous, taking the limit as h tend to 0+ gives lim

h→0+

T (h)y − y = T (t)x − x. h

42

Strongly Continuous Semigroups

Hence, y ∈ D(A) and (2.2.2) follows. In the particular case when x ∈ D(A), by property (i), we know that T (s)Ax = AT (s)x = Ds T (s)x for every s ≥ 0. Formula (2.2.3) follows. (iii) Let (xn ) ⊂ D(A) be a sequence converging to some x ∈ X and such that Axn converges to some y ∈ X as n tends to ∞. By property (ii), we know that Z 1 h T (h)xn − xn T (s)Axn ds, n ∈ N, h > 0. (2.2.4) = h h 0 Since ||T (s)Axn − T (s)y|| ≤ M eωs ||Axn − y|| ≤ M eωh ||Axn − y|| for every s ∈ [0, h] and some M ≥ 1, ω ≥ 0, T (·)Axn converges to T (·)y, uniformly in [0, h]. Hence, letting n tend to ∞ in both the sides of (2.2.4), we conclude that Z T (h)x − x 1 h T (s)y ds, h > 0. = h h 0 Since the function T (·)y is continuous in [0, ∞), letting h tend to 0+ it follows that x ∈ D(A) and Ax = y. Hence, A is a closed. To prove that D(A) is a dense subspace of X, we fix x ∈ X and, for each n ∈ N, Z 1/n we set xn = n T (s)x ds. By property (ii), xn ∈ D(A) for every n ∈ N. Moreover, 0

limn→∞ xn = x. (iv) Suppose that {S(t)} is another C0 -semigroup having A as infinitesimal generator. Fix x ∈ D(A), t > 0 and consider the function u : [0, t] → X, defined by u(s) := T (t − s)S(s)x for s ∈ [0, t]. As it is easily seen, u is differentiable in [0, t] with identically vanishing derivative. This implies that u(t) = u(0), i.e., S(t)x = T (t)x. Since D(A) is dense in X and t is arbitrarily fixed in (0, ∞), we conclude that T (t) = S(t) for every t > 0.  The following result shows that the resolvent set of the infinitesimal generator A is not empty and, for each λ ∈ ρ(A), the operator R(λ, A) can be written in terms of the semigroup. Proposition 2.2.3 Let {T (t)} be a C0 -semigroup with infinitesimal generator A and let M ≥ 1 and ω ∈ R be such that ||T (t)||L(X) ≤ M eωt for every t ≥ 0. Then, ρ(A) ⊃ {λ ∈ C : Re λ > ω} and Z ∞

e−λt T (t)x dt

R(λ, A)x =

(2.2.5)

0

for every x ∈ X and λ ∈ C with Reλ > ω. Moreover, ||(R(λ, A))n ||L(X) ≤

M , (Re λ − ω)n

n ∈ N, Re λ > ω.

(2.2.6)

Proof Fix λ ∈ C with Re λ > ω. Then, the operator defined by the right-hand side of (2.2.5) is well defined, linear and continuous in X, since ||e−λt T (t)x|| ≤ M e−(Re λ−ω)t ||x|| for t ≥ 0. In particular,

Z ∞

Z ∞

M −λt

e T (t)x dt ≤ M ||x|| e−(Re λ−ω)t dt = ||x||, x ∈ X. (2.2.7)

Re λ − ω 0 0 To prove that this operator, which we denote by Rλ , is the inverse of λI − A, we observe that, since A is a closed operator and the function t 7→ e−Re λt ||T (t)Ax|| is integrable in [0, ∞) for every x ∈ D(A), Rλ x belongs to D(A) for every x ∈ D(A) and, integrating by parts, we get Z ∞ Z n Rλ Ax = e−λt T (t)Ax dt = lim e−λt Dt T (t)x dt 0

n→∞

0

Semigroups of Bounded Operators and Second-Order PDE’s   Z n e−λt T (t)x dt = −x + λRλ x. = lim e−λn T (n)x − x + λ n→∞

43

0

Hence, Rλ (λI − A)x = x for every x ∈ D(A). This shows that the operator λI − A is injective. To prove that it is also surjective, we fix x ∈ X and prove that Rλ x ∈ D(A) and ARλ x = λRλ − x. For this purpose, we fix h > 0 and observe that  Z ∞ Z ∞ T (h)Rλ x − Rλ x 1 e−λt T (t)x dt e−λt T (t + h)x dt − = h h 0 0 Z ∞ λh Z ∞ e 1 = e−λs T (s)x ds − e−λt T (t)x dt h h h 0 Z Z eλh − 1 ∞ −λs eλh h −λs e e = T (s)x ds − T (s)x ds. h h 0 0 Letting h tend to 0+ , we conclude that lim

h→0+

T (h)Rλ x − Rλ x = λRλ x − x, h

so that Rλ x ∈ D(A) and the claim follows. To complete the proof, let us prove (2.2.6) with n > 1, since the case n = 1 follows from (2.2.7). Fix x ∈ X. By Proposition A.4.3, we know that the function λ 7→ R(λ, A)x is holomorphic in ρ(A) and, applying the dominated convergence theorem to the formula (2.2.5), it can be easily checked that Z ∞ dn R(λ, A)x = (−1)n tn e−λt T (t)x dt dλn 0 for n ∈ N and λ ∈ C with Re λ > ω. Let us compute the derivatives of the function R(·, A)x in a different way, using formula (A.4.1). This formula implies that dn R(λ, A) = (−1)n n!(R(λ, A))n+1 , dλn

λ ∈ C, Reλ > ω.

From these last two formulas it follows that 1 (R(λ, A)) x = (n − 1)! n

Z

∞

tn−1 e−λt T (t)x dt.

0

Hence, ||(R(λ, A))n x|| ≤ which completes the proof.

M ||x|| (n − 1)!

Z 0

∞

tn−1 e−Re λt eωt dt =

M ||x||, (Re λ − ω)n 

Example 2.2.4 If {T (t)} is given by (2.0.1), then it is clearly strongly continuous since the series defining the semigroup converges locally uniformly in [0, ∞). Its infinitesimal generator is the operator A. Indeed,

X

∞ n−1 X

T (t) − I

∞ tn−1 n t et||A||L(X) − 1 n



− A = A ≤ ||A|| = − ||A||L(X) L(X)



t n! n! t L(X) L(X) n=2 n=2 for every t ∈ (0, 1]. It follows that limt→0+ ||t−1 (T (t) − I) − A||L(X) = 0. As a byproduct, t−1 (T (t)x − x) converges to Ax as t → 0+ for each x ∈ X.

44

Strongly Continuous Semigroups

Example 2.2.5 Let us go back to the semigroup of left translations in Example 2.1.2 and show that its infinitesimal generator A is the first-order derivative with BU C 1 (R) as domain. Let f ∈ D(A). Then, t−1 (f (· + t) − f ) converges to Af in BU C(R) as t tends to 0+ . In particular, for each x ∈ R, the ratio t−1 (f (x + t) − f (x)) converges to (Af )(x). It thus follows that f is differentiable from the right in R and f 0 = Af ∈ BU C(R). Thanks to Proposition A.2.7, we conclude that f ∈ BU C 1 (R). Conversely, suppose that f ∈ BU C 1 (R). Then, by the fundamental theorem of calculus, we can write Z 1 t 0 f (x + t) − f (x) f (x + s) ds, = x ∈ R, t > 0. t t 0 Fix ε > 0 and let δ > 0 be such that |f 0 (x2 ) − f 0 (x1 )| ≤ ε for every x1 , x2 ∈ R such that |x2 − x1 | ≤ δ. Then, Z 1 t 0 f (x + t) − f (x) 0 ≤ |f (x + s) − f 0 (x)| ds ≤ ε − f (x) t t 0 for every x ∈ Rd if t ∈ (0, δ]. Hence, t−1 (T (t)f − f ) converges to f 0 uniformly in R as t tends to 0+ . This shows that BU C 1 (R) ⊂ D(A).

2.3

The Hille-Yosida, Lumer-Phillips and Trotter-Kato Theorems

By the results of the previous section, we know that to any C0 -semigroup {T (t)} on X, we can associate an operator A : D(A) ⊂ X → X, the infinitesimal generator, which satisfies the following properties: (i) A is closed and densely defined; (ii) ρ(A) contains the right-halfplane {λ ∈ C : Reλ > ω0 } for some ω0 ∈ R; (iii) for each ω > ω0 there exists a positive constant M such that ||(R(λ, A))n ||L(X) ≤ M (Re λ − ω)−n for every n ∈ N and λ ∈ C with Re λ > ω. A natural question arises: is any linear operator A : D(A) ⊂ X → X, which satisfies the above three properties, the infinitesimal generator of a C0 -semigroup on X? The answer is positive and given by the famous Hille-Yosida theorem Theorem 2.3.1 (Hille-Yosida) Let A : D(A) ⊂ X → X be a linear operator on a Banach space X. Then the following properties are equivalent. (a) A generates a C0 -semigroup {T (t)} on X. (b) A satisfies the above properties (i)–(iii). Proof In view of the remarks at the beginning of this section, we just need to show that (b)⇒(a). Without loss of generality we can assume that ω = 0. Indeed, the operator e = A − ωI, satisfies properties (i)–(iii), with ω = 0. If we prove that A e generates a A C0 -semigroup {S(t)}, then the operator A generates the C0 -semigroup {eωt S(t)}.

Semigroups of Bounded Operators and Second-Order PDE’s

45

Since it is rather long, we split the proof into steps. Step 1. Here, we define the Yosida approximation of the operator A, i.e., the operators An : X → X defined by An = nAR(n, A) for n ∈ N. Since AR(n, A) = (A − nI)R(n, A) + nR(n, A) = nR(n, A) − I, each operator An is bounded in X. We claim that limn→∞ An x = Ax for each x ∈ D(A). To prove the claim, it suffices to show that nR(n, A)y tends to y for each y ∈ X. Indeed, An x = nR(n, A)Ax for x ∈ D(A). First, we suppose that y ∈ D(A). Then, nR(n, A)y = R(n, A)(ny − Ay + Ay) = y + R(n, A)Ay,

n ∈ N,

and ||R(n, A)Ay|| ≤ M n−1 ||Ay|| vanishes as n tends to ∞. Hence, nR(n, A)y tends to y as n tends to ∞. If y ∈ X, then we consider a sequence (ym ) ⊂ D(A), converging to y as m tends to ∞. Since ||nR(n, A)||L(X) ≤ M for every n ∈ N, we can estimate ||nR(n, A)y − y|| ≤||nR(n, A)(y − ym )|| + ||nR(n, A)ym − ym || + ||ym − y|| ≤(M + 1)||ym − y|| + ||nR(n, A)ym − ym || for m, n ∈ N. Hence, lim sup ||nR(n, A)y − y|| ≤(M + 1)||y − ym || + lim ||nR(n, A)ym − ym || = (M + 1)||ym − y|| n→∞

n→∞

for every m ∈ N. Letting m tend to ∞, we conclude that lim sup ||nR(n, A)y − y|| = 0. n→∞

Hence, nR(n, A)y converges to y as n tends to ∞. Step 2. For every n ∈ N, we introduce the uniform continuous semigroup {Tn (t)} on X, defined by Tn (t) = etAn =

∞ k X t k=0

k!

Akn ,

t ≥ 0,

and prove that, for each x ∈ X, (Tn (·)x) is a Cauchy sequence in C([0, T ]; X) for every T > 0. To begin with, we observe that, since An = n2 R(n, A) − nI, it follows that Tn (t) = 2 e−nt etn R(n,A) for each t ≥ 0. Hence, recalling that ||(nR(n, A))k ||L(X) ≤ M for k ∈ N, we deduce that

∞ k

∞ X

X t

(tn)k −nt 2 k ||Tn (t)||L(X) =e (n R(n, A)) ≤ e−nt ||(nR(n, A))k ||L(X) k! k! L(X) k=0

k=0

≤M e−nt ent = M

(2.3.1)

for every t ≥ 0 and n ∈ N. Next, we fix T > 0, t ∈ (0, T ], x ∈ D(A), m, n ∈ N and introduce the function um,n : [0, t] → X, defined by um,n (s) = Tm (t − s)Tn (s)x for every s ∈ [0, t]. As it is easily checked um,n (t) = Tn (t)x and um,n (0) = Tm (t)x. Moreover, um,n is differentiable in [0, t] and u0m,n (s) = −Tm (t − s)Am Tn (s)x + Tm (t − s)Tn (s)An x for every s ∈ [0, t]. Note that Am commutes with An since R(n, A) commutes with R(m, A) (see Proposition A.4.3). As a byproduct, Am commutes with the semigroup {Tn (t)} and, consequently, u0m,n (s) = Tm (t − s)Tn (s)(Am x − An x) for every s ∈ [0, t]. Thus, Z ||Tn (t)x − Tm (t)x|| ≤

t

||Tm (t − s)Tn (s)(Am x − An x)|| ds 0

46

Strongly Continuous Semigroups ≤M 2 t||Am x − An x|| ≤ M 2 T ||Am x − An x||.

Since T has been arbitrarily fixed and x ∈ D(A), by Step 1 (Tn (·)x) is a Cauchy sequence in C([0, T ]; X) for every T > 0. If x ∈ X, then there exists a sequence (xk ) ⊂ D(A) which converges to x as k tends to ∞. Then, taking (2.3.1) into account, for each k, m, n ∈ N and T > 0, we can estimate ||Tn (·)x − Tm (·)x||C([0,T ];X) ≤||Tn (·)(x − xk )||C([0,T ];X) + ||Tn (·)xk − Tm (·)xk ||C([0,T ];X) + ||Tm (·)(x − xk )||C([0,T ];X) ≤2M ||x − xk || + M 2 T ||Am xk − An xk ||. Hence, for every fixed ε > 0, we can fix k0 , n0 ∈ N such that 2M ||x − xk0 || ≤ ε/2 and M 2 T ||Am xk0 − An xk0 || ≤ ε/2 for each m, n ≥ n0 , so that ||Tn (·)x − Tm (·)x||C([0,T ];X) ≤ ε for each m, n ≥ n0 . Hence, (Tn (·)x) is a Cauchy sequence in C([0, T ]; X) for every x ∈ X. Step 3. Here, we define a C0 -semigroup {T (t)}. Since each operator Tn (t) is linear and (Tn (·)x) is a Cauchy sequence in C([0, T ]; X) for each T > 0 and x ∈ X, there exists a family of linear operators {T (t)} such that limn→∞ Tn (t)x = T (t)x for every x ∈ X and the convergence is uniform on each interval [0, T ]. So, the function t 7→ T (t)x is continuous on [0, ∞) for each x ∈ X. As it is immediately seen, T (0) = I, since Tn (0) = I for every n ∈ N. Moreover, from the estimate ||Tn (t)x|| ≤ M ||x||, which holds true for every t ≥ 0, x ∈ X and n ∈ N, it follows that T (t) is a bounded linear operator and ||T (t)||L(X) ≤ M for every t ≥ 0. To conclude that {T (t)} is a C0 -semigroup we need to check the semigroup property. This follows from letting n tend to ∞ in the formula Tn (t + s)x = Tn (t)Tn (s)x which holds true for each t, s > 0, x ∈ X and n ∈ N. Step 4. Here, we complete the proof. Since {T (t)} is a C0 -semigroup on X which satisfies the estimate ||T (t)||L(X) ≤ M for every t ≥ 0, by Propositions 2.2.2 and 2.2.3 it admits an infinitesimal generator B whose resolvent set contains the line (0, ∞). To show that B coincides with the operator A, we begin by showing that D(A) ⊂ D(B) and Bx = Ax for every x ∈ D(A). For this purpose, we fix x ∈ D(A), n ∈ N and observe that the function Tn (·)x is differentiable in [0, ∞) with Dt Tn (t)x = Tn (t)An x for each t ≥ 0. We claim that Dt Tn (·)x converges to the function T (·)Ax locally uniformly in [0, ∞). To check the claim, we observe that ||Dt Tn (·)x − T (·)Ax||C([0,T ];X) ≤ M ||An x − Ax|| + ||Tn (·)Ax − T (·)Ax||C([0,T ];X) for every T > 0 and the right-hand side of the previous inequality vanishes as n tends to ∞ by Steps 1 and 3. The claim is proved. Since Tn (·)x converges to T (·)x, locally uniformly in [0, ∞), the function T (·)x is differentiable in [0, ∞) and Dt T (t)x = T (t)Ax for every t ≥ 0. This, in particular, shows that x ∈ D(B) and Bx = Ax. To show that D(B) = D(A), we observe that 1 ∈ ρ(A) ∩ ρ(B) and X = (I − A)(D(A)) = (I − B)(D(A)) ⊂ (I − B)(D(B)) = X. Hence, (I − B)(D(A)) = (I − B)(D(B)). Since the operator I − B is injective, we obtain that D(A) = D(B) and we are done.  Remark 2.3.2 Given a closed operator A, such that π = {λ ∈ C : Re λ > ω} ⊆ ρ(A) for some ω ∈ R, we need to check the infinitely many conditions ||(R(λ, A))n || ≤ M (Re λ − ω)−n for every λ ∈ π to establish whether A generates a C0 -semigroup or not. In the particular case when M = 1, things are easier since once the previous condition is proved with n = 1, it can be easily extended to every n > 1 just observing that ||(R(λ, A))n ||L(X) ≤ ||R(λ, A)||nL(X) for each n ∈ N.

Semigroups of Bounded Operators and Second-Order PDE’s

47

Remark 2.3.3 At the very beginning of this chapter, we have introduced uniform continuous semigroups, which are actually groups since the operator etA is defined for every real value of t and the semigroup property is satisfied for every s, t ∈ R. On the other hand, C0 -semigroups are in general defined only on the line [0, ∞). Indeed, suppose that the C0 -semigroup is actually a group. Then, {T (t)} and {T (−t)} are C0 -semigroups. Clearly, if A is the infinitesimal generator of {T (t)}, then, −A generates the semigroup {T (−t)}. Hence, the Hille-Yosida theorem implies that the resolvent set of A should contain the halfplanes {λ ∈ C : Reλ > ω} and {λ ∈ C : Reλ < −ω} for some ω ≥ 0. Moreover, ||(R(λ, A))n ||L(X) ≤ M (|Reλ| − ω)−n for every λ ∈ C, such that |Reλ| > ω, every n ∈ N and some positive constant M . These conditions are also sufficient for an operator A to be the generator of a C0 -group. A useful criterium to guarantee that a closed operator generates a C0 -semigroup of contractions is the well celebrated Lumer-Phillips theorem. To state it, we introduce the concept of dissipative operator. Definition 2.3.4 An operator A : D(A) ⊂ X → X is called dissipative if ||λx−Ax|| ≥ λ||x|| for each λ > 0 and x ∈ X. Remark 2.3.5 The property of an operator to be dissipative can be stated also in a different way. In fact, a linear operator A : D(A) ⊂ X → X is dissipative if and only if for every x ∈ D(A) there exists x∗ in the duality set of x (i.e., there exists x∗ ∈ X 0 such that x∗ (x) = ||x||2 = ||x∗ ||2X 0 ) such that Re(x∗ (Ax)) ≤ 0. In particular, if X is a Hilbert space, then A is dissipative if and only if Re hAx, xi ≤ 0 for every x ∈ D(A). Theorem 2.3.6 (Lumer-Phillips theorem) Let A : D(A) ⊂ X → X be a dissipative operator with dense domain and such that ρ(A) ∩ (0, ∞) 6= ∅. Then, A generates a C0 semigroup of contractions on X (i.e., ||T (t)||L(X) ≤ 1 for every t ≥ 0). Proof Fix λ0 ∈ ρ(A)∩(0, ∞). Then, the dissipativity of A implies that ||R(λ0 , A)||L(X) ≤ λ−1 0 . The first part of the proof of Proposition A.4.3, shows that the open ball B(λ0 , r) is contained in ρ(A) when r = ||R(λ0 , A)||−1 L(X) . Since r ≥ λ0 , the interval (0, 2λ0 ) is contained in ρ(A). Now, replacing λ0 with 3λ0 /2 we conclude that the interval (0, 5λ0 /2) is contained in ρ(A). Starting from 2λ0 instead of 3λ0 /2, we obtain that (0, 3λ0 ) ⊂ ρ(A). Iterating this procedure, we can show that (0, ∞) ⊂ ρ(A). The assertion follows now from Theorem 2.3.1 and Remark 2.3.2.  The adjoint of a densely defined operator A : D(A) ⊂ X → X is given by ( D(A∗ ) = {x∗ ∈ X 0 : ∃y ∗ ∈ X 0 such that x∗ (Ax) = y ∗ (x), ∀x ∈ D(A)} A∗ x∗ = y ∗ , x∗ ∈ D(A∗ ). Since D(A) = X, one can easily see that A∗ : D(A∗ ) ⊂ X 0 → X 0 is a well defined linear operator on X 0 . Corollary 2.3.7 If A : D(A) ⊂ X → X is densely defined, dissipative and its adjoint operator A∗ is dissipative as well, then the closure A of A generates a C0 -semigroup of contractions. Proof First of all, we prove that A is closable. For this purpose, we fix a sequence (xn ) ⊂ D(A) converging to zero and such that Axn converges to some y ∈ X as n tends to

48

Strongly Continuous Semigroups

∞. Since D(A) is dense in X, we can determine a sequence (yn ) ⊂ D(A) converging to y as n tends to ∞. From the dissipativity of A we infer that ||k 2 xn − kAxn + kym − Aym || =||k(kxn + ym ) − A(kxn + ym )|| ≥ k||kxn + ym || for every k, m, n ∈ N. Letting n tend to ∞ in the first and last side of the previous chain of inequalities gives || − ky + kym − Aym || ≥ k||ym || for every k, m ∈ N, or, equivalently, dividing by k, || − y + ym − k −1 Aym || ≥ ||ym || for the same values of k and m, so that, letting k tend to ∞, we obtain that ||y − ym || ≥ ||ym ||. Finally, letting m tend to ∞, we can easily infer that y = 0. This shows that A is closable. Next, we show that the range of the operator I − A is dense in X. For this purpose, we fix a functional x∗ ∈ X 0 such that x∗ (x − Ax) = 0,

x ∈ D(A).

(2.3.2)

From (2.3.2) we deduce that x∗ ∈ D(A∗ ) and x∗ − A∗ x∗ = 0. Since A∗ is dissipative, in particular it is a one-to-one operator, it follows that x∗ = 0 and we conclude that (I − A)(D(A)) is dense in X. Since A extends the operator A, also the range of the operator I − A is dense in X for every x ∈ D(A). Actually, (I − A)(D(A)) = X. Indeed, fix y ∈ (I − A)D(A) and consider a sequence (xn ) ⊂ D(A) such that yn = xn − Axn converges to y as n tends to ∞. From the dissipativity of A we infer that ||xn − xm || ≤ ||yn − ym || for every m, n ∈ N so that (xn ) is a Cauchy sequence in X. Therefore it converges to some x ∈ X. As a byproduct also Axn = Axn converges in X to x − y. Since A is a closed operator, x belongs to D(A) and Ax = x − y or, equivalently, x − Ax = y. Thus, y ∈ (I − A)D(A). So, (I − A)D(A) = X. Applying Lumer-Phillips theorem, we conclude that A is the generator of a C0 -semigroup of contractions.  We conclude this section by proving the Post-Widder inverse formula that we will use in Chapter 15. For this purpose, we first prove the following Trotter-Kato approximation theorem. Theorem 2.3.8 (Trotter-Kato theorem) Assume that the C0 -semigroups {T (t)} and {Tn (t)} on X belong to the class G(M, ω) for some M ≥ 1 and ω ∈ R. Let A and An be the generators of {T (t)} and {Tn (t)} respectively. Then the following assertions are equivalent: (a) R(λ, An )x converges to R(λ, A)x, as n tends to ∞, for all x ∈ X and all λ ∈ (ω, ∞). (b) Tn (·)x converges to T (·)x in C([0, T ]; X) as n tends to ∞, for all x ∈ X and T > 0. Proof The implication (b) ⇒ (a) follows from (2.2.5) and the dominated convergence theorem. Let us now prove the implication (a) ⇒ (b). Fix x ∈ X, λ > ω and t ∈ [0, T ], for an arbitrary fixed T > 0. Then ||(Tn (t) − T (t))R(λ, A)x|| ≤||Tn (t)(R(λ, A) − R(λ, An ))x|| + ||R(λ, An )(Tn (t) − T (t))x|| + ||(R(λ, A) − R(λ, An ))T (t)x|| ≤M eωt ||(R(λ, A) − R(λ, An ))x|| + ||R(λ, An )(Tn (t) − T (t))x|| + ||(R(λ, A) − R(λ, An ))T (t)x|| = : En,1 (t, x) + En,2 (t, x) + En,3 (t, x). (2.3.3) Clearly, En,1 (·, x) = 0 tends to zero as n tends to ∞ uniformly with respect to t ∈ [0, T ], and, by the strong continuity of {T (t)}, we deduce that En,3 (·, x) tends to zero in C([0, T ]; X) as

Semigroups of Bounded Operators and Second-Order PDE’s

49

n tends to ∞. We now show that En,2 (·, x) tends to zero in C([0, T ]; X) for every x ∈ D(A). An easy computation and Proposition 2.2.2(i) show that d (Tn (t − s)R(λ, An )T (s)R(λ, A)x) = Tn (t − s)[R(λ, A) − R(λ, An )]T (s)y ds for s ∈ [0, T ] and y ∈ X. This implies that Z t Tn (t − s)[R(λ, A) − R(λ, An )]T (s)y ds. R(λ, An )(T (t) − Tn (t))R(λ, A)y = 0

Thus, Z ||R(λ, An )(T (t) − Tn (t))R(λ, A)y|| ≤

T

||Tn (t − s)||L(X) ||[R(λ, A) − R(λ, An )]T (s)y|| ds 0

≤M eω

+

T

Z

T

||[R(λ, A) − R(λ, An )]T (s)y|| ds. 0

Using the dominated convergence theorem (and the uniform boundedness principle, which, due to property (a), implies that supn∈N ||R(λ, A) − R(λ, An )||L(X) < ∞), we deduce that lim ||R(λ, An )(T (·) − Tn (·))R(λ, A)y||C([0,T ];X) = lim ||En,2 (·, R(λ, A)y)||C([0,T ];X) = 0.

n→∞

n→∞

Summing up, from (2.3.3) it thus follows that ||Tn (·)x − T (·)x||C([0,T ];X) vanishes as n tends to ∞ for each x ∈ D(A2 ). So, the assertion (b) follows from the density of D(A2 ) in X, see Exercise 2.6.7.  Corollary 2.3.9 (Post-Widder inverse formula) Let {T (t)} be a strongly continuous semigroup on a Banach space X and let A : D(A) ⊂ X → X be its infinitesimal generator. Then,   n n n T (t)x = lim R ,A x, x ∈ X, (2.3.4) n→∞ t t and the limit is uniform with respect to t in compact intervals. Proof Without loss of generality, we assume that ||T (t)|| ≤ M eωt for every t > 0 and some M ≥ 1, ω > 0. Set  if r = 0,   I −1 −1 r R(r , A) if r ∈ (0, (2ω)−1 ), F (r) :=   0 if r ≥ (2ω)−1 , and observe that ||F (r)n ||L(X) ≤ r−n ||R(r−1 , A)n ||L(X) ≤

M M = n (1 − ωr)n − ω)

rn ( 1r

for each r > 0 and n ∈ N. Thus, setting Ar = r−1 (F (r) − I) for every r > 0 we can estimate t

||etAr ||L(X) =e− r ||etr

−1

F (r)

t

||L(X) ≤ e− r

∞ X ωt tn ||F (r)n ||L(X) ≤ M e 1−ωr ≤ M e2ωt (2.3.5) n n!r n=0

for all t ≥ 0 and r > 0. Moreover, lim Ar x = lim+ r−1 [r−1 R(r−1 , A)x − x] = lim+ r−1 R(r−1 , A)Ax = Ax

r→0+

r→0

r→0

50

Strongly Continuous Semigroups

for all x ∈ D(A), by Step 1 in the proof of Theorem 2.3.1. Hence, for every λ > ω and x ∈ D(A), we obtain lim R(λ, Ar )(λ − A)x = lim+ (λR(λ, Ar )x − R(λ, Ar )Ax)

r→0+

r→0

= lim+ (x + R(λ, Ar )(Ar x − Ax)) = x r→0

and, as a by product, R(λ, Ar )x converges to R(λ, A)x, as r → 0+ , for every x ∈ X and λ > ω (note that (2.3.5) implies that supr>0 ||R(λ, Ar )||L(X) < ∞). By Theorem 2.3.8, we deduce that e·Ar x converges to T (·)x in C([0, T ]; X) for every T > 0 and x ∈ X. In particular, ||etAt/n x−T (t)x||C([0,T ];X) tends to 0 as n tends to ∞ for every T and x as above. Thus, applying the estimate in Exercise 2.6.8, we obtain

 

 n  n



tA

√

e t/n x−F t x = en(F (t/n)−I) x−F t x ≤ M n F t x − x = tM



√n ||At/n x|| n n n for every x ∈ X. Thus, the assertion follows, since the above estimate gives

 n

tA

t t/n

lim e x−F x ∈ D(A), x

= 0, n→∞ n and the limit is uniform with respect to t varying in compact intervals.

2.4



Nonhomogeneous Cauchy Problems

In this section, we consider the nonhomogeneous Cauchy problem ( u0 (t) = Au(t) + f (t), t ∈ [0, T ], u(0) = x,

(2.4.1)

when A is the infinitesimal generator of a C0 -semigroup {T (t)}, f : [0, T ] → X is a continuous function and x ∈ X. Moreover, we assume that ||T (t)|| ≤ M eωt for every t > 0 and some constants M ≥ 1 and ω ≥ 0 (see Proposition 2.1.3). Definition 2.4.1 A function u : [0, T ] × X is called strict solution of the Cauchy problem (2.4.1) if u ∈ C([0, T ]; D(A)) ∩ C 1 ([0, T ]; X), u(0) = x and u0 (t) = Au(t) + f (t) for every t ∈ [0, T ]. Remark 2.4.2 As it is immediately seen, if u is a strict solution of problem (2.4.1), then x ∈ D(A). The following proposition provides us with a very useful representation formula for solutions of the Cauchy problem (2.4.1). Proposition 2.4.3 Let u be a strict solution of the Cauchy problem (2.4.1). Then, u is given by the variation-of-constants formula Z t u(t) = T (t)x + T (t − s)f (s) ds, t ∈ [0, T ], (2.4.2) 0

where {T (t)} is the C0 -semigroup generated by A in X. In particular, the strict solution of the Cauchy problem (2.4.1), if existing, is unique.

Semigroups of Bounded Operators and Second-Order PDE’s

51

Proof Suppose that u is a strict solution of the Cauchy problem (2.4.1), fix t ∈ (0, T ] and introduce the function v : [0, t] → X defined by v(s) = T (t − s)u(s) for every s ∈ [0, t]. Note that this function is differentiable in [0, t] and v 0 (s) = −T (t − s)Au(s) + T (t − s)u0 (s) = T (t − s)(u0 (s) − Au(s)) = T (t − s)f (s) for every s ∈ [0, t]. Moreover, the function v 0 is continuous on [0, t] as it is easily seen. Thus, integrating over [0, t] we deduce that t

Z

v 0 (s) ds =

u(t) − T (t)x = v(t) − v(0) = 0

Z

t

T (t − s)f (s) ds, 0

and the assertion follows.



Definition 2.4.4 The function u defined by (2.4.2) is called mild solution to problem (2.4.1). Remark 2.4.5 The mild solution to problem (2.4.1) is well defined and continuous on [0, T ] for every x ∈ X and f ∈ C([0, T ]; X). Moreover, ||u(t)|| ≤ M eωT ||x|| + M

eωT − 1 ||f ||∞ , ω

t ∈ [0, T ],

(2.4.3)

where ω −1 (eωT − 1) is replaced by T when ω = 0. As the following example shows, in general the conditions x ∈ D(A) and f ∈ C([0, T ]; X) are not sufficient to guarantee the existence of a strict solution u of the Cauchy problem (2.4.1). Example 2.4.6 Let X = BU C(R) and let A : BU C 1 (R) → BU C(R) be the operator defined by Au = u0 for every u ∈ BU C 1 (R). Then, A is the generator of the semigroup of left-translations (see Example 2.2.5). Take x = 0 and f (t) = T (t)ψ for every t ∈ [0, T ], where ψ ∈ BU C(R) is not differentiable over R. Clearly, f ∈ C([0, T ]; X) and T (t−s)f (s) = T (t − s)T (s)ψ = T (t)ψ. Thus, in view of formula (2.4.2) if problem (2.4.1) admits a strict solution u, then u(t) = tT (t)ψ = tψ(t + ·) for every t ∈ [0, T ], which is not differentiable on [0, T ]. The following theorem provides us with two sufficient conditions for problem (2.4.1) to admit a strict solution. Theorem 2.4.7 Suppose that x ∈ D(A). The following properties are satisfied. (i) If f ∈ C([0, T ]; D(A)), then problem (2.4.1) admits a unique strict solution u. Moreover, there exists a positive constant C, independent of x, u and f , such that ||u||C 1 ([0,T ];X) + ||Au||C([0,T ];X) ≤ C(||x||D(A) + ||f ||C([0,T ];D(A)) ).

(2.4.4)

(ii) If f ∈ C 1 ([0, T ]; X), then problem (2.4.1) admits a unique strict solution u. Moreover, there exists a positive constant C, independent of x, u and f , such that ||u||C 1 ([0,T ];X) + ||Au||C([0,T ];X) ≤ C(||x||D(A) + ||f ||C 1 ([0,T ];X) ).

(2.4.5)

52

Strongly Continuous Semigroups

Proof (i) Let us prove that the mild solution to problem (2.4.1) is continuous on [0, T ] with values in D(A). From Remark 2.4.5, we already know that such a function u is continuous on [0, T ] with values on X. So, let us prove that u(t) belongs to D(A) for every t ∈ [0, T ]. Of course, we can limit ourselves to considering the convolution term in (2.4.2), which we denote by v. We fix t ∈ (0, T ], h ∈ (0, 1] and observe that

Z t

  Z t



T (h) − I

T (t − s) T (h)f (s) − f (s) − Af (s) ds

≤

v(t) − T (t − s)Af (s) ds



h h 0 0

Z t

T (h)f (s) − f (s)

≤M eωT − Af (s)

ds. h 0 Since h−1 (T (h)f (s) − f (s)) converges to Af (s) as h tends to 0+ for every s ∈ [0, T ] and

Z

T (h)f (s) − f (s)

1 h

||T (r)Af (s) − Af (s)|| dr ≤ (M eω + 1)||Af ||∞ , − Af (s) ≤

h h 0 we can apply the dominated convergence theorem and infer that v(t) belongs to D(A) and Z t Av(t) = T (t − s)Af (s) ds. 0

By assumptions, the function f is continuous on [0, T ] with values on D(A). Therefore, Remark 2.4.5 shows that the function Av is continuous in [0, T ] with values in X. In a similar way, we can show that v is differentiable on [0, T ] with values in X. For this purpose, we fix t ∈ [0, T ) and h > 0 such that t + h ≤ T . Then,  Z t+h  Z t v(t + h) − v(t) 1 = T (t + h − s)f (s) ds − T (t − s)f (s) ds h h 0 0 Z Z t 1 t+h T (h) − I T (t − s)f (s) ds. (2.4.6) = T (t + h − s)f (s) ds + h t h 0 Since the function (r, s) 7→ T (r)f (s) is continuous on [0, T ] × [0, T ], the first integral in the right-hand side of (2.4.6) converges to f (t) as h tends to 0. Similarly, since the semigroup {T (t)} leaves D(A) invariant, h−1 (T (h) − I)T (t − s)f (s) converges to AT (t − s)f (s) = T (t − s)Af (s) as h tends to 0+ for every s ∈ [0, t]. Moreover, the same argument used above shows that there exists a positive constant K, independent of h and s such that ||h−1 (T (h)− I)T (t − s)f (s)|| ≤ K. Again, we can invoke the dominated convergence theorem to deduce Z t that the second term in the right-hand side of (2.4.6) converges to T (t − s)Af (s) ds 0

as h tends to 0+ . We have so proved that v is differentiable from the right on [0, T ) and v 0 (t) = Av(t) + f (t). Since the right-hand side of this formula is continuous on [0, T ], this is enough to conclude that v 0 exists and is continuous on [0, T ] (see Proposition A.2.7). Finally,

Z t



≤ M T eωT ||Af ||C([0,T ];X) , ||Av(t)|| = T (t − s)Af (s) ds t ∈ [0, T ], (2.4.7)

0

so that ||Av||C([0,T ];X) ≤ M T eωT ||Af ||C([0,T ];X) . As a byproduct, ||v 0 ||C([0,T ];X) ≤ ||Av||C([0,T ];X) + ||f ||C([0,T ];X) ≤ M T eωT ||Af ||C([0,T ];X) + ||f ||C([0,T ];X) . (2.4.8) From (2.4.3), (2.4.7) and (2.4.8), estimate (2.4.4) follows at once.

Semigroups of Bounded Operators and Second-Order PDE’s

53

(ii) The proof is similar. A straightforward change of variables shows that   Z t+h Z t v(t + h) − v(t) 1 T (s)f (t − s) ds T (s)f (t + h − s) ds − = h h 0 0 Z t Z 1 t+h f (t + h − s) − f (t − s) = T (s) T (s)f (t + h − s) ds + ds h t h 0 for every t ∈ [0, T ] and h ∈ R \ {0} such that t + h ∈ [0, T ]. Applying the dominated convergence theorem, we conclude that v is differentiable at t and Z t v 0 (t) = T (t)f (0) + T (t − s)f 0 (s) ds. 0

In particular, v is continuous on [0, T ] with values in X and ||v 0 (t)|| ≤ C1 ||f ||C 1 ([0,T ];X) ,

t ∈ [0, T ]

(2.4.9)

for some positive constant C1 , independent of f . On the other hand,  Z t−h  Z t 1 T (h) − I v(t) = T (t − s)f (s + h) ds − T (t − s)f (s) ds h h −h 0 Z Z t−h 1 0 f (s + h) − f (s) T (t − s)f (s + h) ds + T (t − s) ds = h −h h 0 Z 1 t − T (t − s)f (s) ds h t−h for every t ∈ (0, T ] and h > 0. Letting h tend to 0+ , by dominated convergence it follows that Z t T (h) − I v(t) = T (t)f (0) + T (t − s)f 0 (s) ds − f (t) = v 0 (t) − f (t). lim h h→0+ 0 We have proved that v(t) belongs to D(A) and Av(t) = v 0 (t) − f (t). We conclude that v ∈ C([0, T ]; D(A)) and ||Av(t)|| ≤ ||v 0 ||C([0,T ];X) + ||f ||C([0,T ];X) ≤ C1 ||f 0 ||C([0,T ];X) + (C1 + 1)||f ||C([0,T ];X) (2.4.10) for every t ∈ [0, T ]. In particular, v solves the differential equation v 0 = Av + f on [0, T ]. Proposition 2.2.2(i) shows that u is a strict solution of problem (2.4.1). Finally, estimate (2.4.5) follows from (2.4.3), (2.4.9) and (2.4.10). 

2.5

Notes and Remarks

Operator semigroups has been widely studied during the last decades and there are many monographs dealing with them. We mention here the excellent graduate texts by Engel and Nagel [13, 14]. The first milestone in the theory was the opus of Hille and Phillips [19]. An important later reference are the books by Belleni-Morante [5], Goldstein [18] and Pazy [29].

54

Strongly Continuous Semigroups

2.6

Exercises

1. On X = `p (p ∈ [1, ∞)) consider the operator A defined by A(xn ) = (an xn ) for every (xn ) ∈ D(A) = {(xn ) ∈ `p : (an xn ) ∈ `p }, where (an ) ⊂ C is a given sequence. Show that: (i) A ∈ L(X) if and only if (an ) ∈ `∞ ; (ii) A is a closed operator with dense domain; (iii) A generates a C0 -semigroup {T (t)} if and only if there exists ω ∈ R such that Rean ≤ ω for all n ∈ N. In this case, T (t)(xn ) = (etan xn ) for each (xn ) ∈ X; (iv) if an = −n2 , then {T (t)} is continuous in the operator norm on (0, ∞), but not right continuous at t = 0. 2. Let X = C0 (R) and q ∈ C(R). Consider the operator (Af )(s) := q(s)f (s) with D(A) := {f ∈ X : qf ∈ X} and make analogous statements as in the previous exercise. Prove these statements. 3. On the Banach space X0 = {f ∈ C([0, 1]) : f (1) = 0}, endowed with the sup-norm, consider the family {T (t)} of operators, defined by ( f (x + t), if x + t ≤ 1, (T (t)f )(x) = 0, otherwise, for every x ∈ [0, 1] and t ≥ 0. Prove that {T (t)} is a C0 -semigroup, called the semigroup of left translations on C([0, 1]), and show that its growth bound ω0 is −∞. 4. Determine which of the following operators generate a C0 -semigroup on X = C([0, 1]): (i) A1 f = f 0 for f ∈ D(A1 ) = {f ∈ C 1 ([0, 1]) : f (0) = 0}; (ii) A2 f = f 00 for f ∈ D(A2 ) = C 2 ([0, 1]); (iii) A3 f = f 00 for f ∈ D(A3 ) = {f ∈ C 2 ([0, 1]) : f (0) = f (1) = 0} (iv) A4 f = f 00 for f ∈ D(A4 ) = {f ∈ C 2 ([0, 1]) : f 00 (0) = 0}. 5. On X = C0 (R) consider the one-parameter family {T (t)} defined by Z t  (T (t)f )(x) = exp q(s) ds f (x − t), x ∈ R, t ≥ 0, x−t

where q : R → R is a bounded and continuous function. Prove that {T (t)} is a C0 semigroup and identify its infinitesimal generator. 6. Prove Remark 2.3.3. 7. For the generator A with domain D(A) of a C0 -semigroup on a Banach space X prove that D(An ) is dense in X, where D(An ) is defined inductively by D(An ) := {x ∈ D(An−1 ) : An−1 x ∈ D(A)},

n ∈ N.

n 8. Prove that for every B ∈ L(X) with √ ||B || ≤ M for some M ≥ 1 and every n ∈ N it n(B−I) n holds that ||e x − B x|| ≤ M n||Bx − x|| for every x ∈ X and n ∈ N. P∞ m [Hint: use the formula m=0 nm! (m − n)2 = nen , n ∈ N.]

Chapter 3 Analytic Semigroups

In this chapter, we keep on the study of semigroups of bounded operators introducing analytic semigroups and analyzing their main properties. Section 3.1 is a prelude to analytic semigroups and it is aimed at explaining the definition of analytic semigroups via the Dunford integral, starting from the easiest class of semigroups (which are also analytic): the uniformly continuous semigroups introduced in Chapter 2. Then, in Section 3.2 we introduce the sectorial operators, which are the operators which are associated to analytic semigroups. Via the Dunford integral, we define the analytic semigroup associated with the sectorial operator A and study its main properties. As Proposition 3.2.5 shows, in general analytic semigroups are not strongly continuous semigroups since the operator A is not required to be densely defined in the Banach space X. Proposition 3.2.8 provides us with a very useful (in view of applications) condition to guarantee that a closed operator A is sectorial. Section 3.3 is devoted to introducing and giving equivalent characterization of the interpolation space DA (α, ∞), which plays a relevant role in the study of abstract Cauchy problems associated with sectorial operators. We also define the space DA (1 + α, ∞) as the domain of the part of A in DA (α, ∞), i.e., the set of all x ∈ D(A) such that Ax ∈ DA (α, ∞). Based on all the previous results, in Section 3.4 we study the abstract Cauchy problem ( u0 (t) = Au(t) + f (t), t ∈ [0, T ], (3.0.1) u(0) = x when f ∈ C([0, T ]; X) and x ∈ X. Differently, from the case when A is the generator of a strongly continuous semigroup, here we can relax the assumptions on x and f due to the nicer properties of analytic semigroups. We can also introduce a slightly weaker definition of solution to problem (3.0.1): the so called classical solution, where we require the differential equation in (3.0.1) to be solved only in the interval (0, T ]. As Example 2.4.6 and Theorem 2.4.7 show, if f ∈ / C 1 ([0, T ]; X) ∪ C([0, T ]; D(A)) and/or x ∈ D(A), then we can not expect that problem (3.0.1) admits a solution. Even in the case when A is sectorial, the condition f ∈ C([0, T ]; X) is not sufficient to guarantee the existence of a classical solution to problem (3.0.1) as Example 3.4.5 shows. On the other hand, the conditions x ∈ D(A) and f ∈ C α ([0, T ]; X) or f ∈ C([0, T ]; X) and bounded in [0, T ] with values in DA (α, ∞) for some α ∈ (0, 1) are sufficient to guarantee the existence of a classical solution to problem (3.0.1). Under additional conditions (which turn out to be also necessary), the classical solution to problem (3.0.1) is actually a strict solution and enjoys optimal regularity properties. The assumptions are of two different type depending on what kind of regularity of the solution u one is interested in, i.e., time regularity or spatial regularity. Notation. Throughout the chapter, by X we still denote a complex Banach space with norm || · ||. Moreover, given a curve γ : I → C (I ⊂ R being an interval) and a function f defined at least on the support of f , we set Z Z f (λ) dλ := − f (λ) dλ. −γ

γ

55

56

Analytic Semigroups

3.1

Prelude

In Chapter 2, we have seen that, to every bounded operator on X, we can associate a uniformly continuous semigroup by setting etA =

∞ n X t n A , n! n=0

t ∈ R.

(3.1.1)

This formula cannot be extended to the case when A is not defined in the whole X since the domains of the powers An become smaller and smaller. On the other hand, when ρ(A) is not empty, the resolvent operator is defined and bounded in X. Hence, the idea is to look for a formula for etA which involves the operators R(λ, A). Lemma 3.1.1 For every operator A ∈ L(X) the following property holds: σ(A) ⊆ B(0, ||A||L(X) ). Proof Fix λ ∈ C and observe that λI − A = λ(I − λ−1 A). Hence, from Lemma A.4.4 it follows that, if |λ−1 |||A||L(X) < 1, then λI − A is invertible in X, and we are done.  Now, we can prove the following integral representation formula for uniformly continuous semigroups. Proposition 3.1.2 Let A ∈ L(X) and let γr (r > ||A||L(X) ) be the curve defined by γr (t) = reit for t ∈ [0, 2π]. Then, Z 1 etA = etλ R(λ, A) dλ, t ∈ R. (3.1.2) 2πi γr Proof From (3.1.1) and Lemma A.4.4, we can write R(λ, A) =

∞ X Ak λk+1

k=0

for every |λ| > ||A||L(X) . Hence, eλt R(λ, A) =

∞ n n X ∞ X t λ Ak . n! λk+1 n=0 k=0

Integrating both sides of the previous formula along the curve γr and observing that the series and the integral commute, we get 1 2πi

Z

etλ R(λ, A) dλ =

γr

Z ∞ ∞ 1 X tn X k A λn−k−1 dλ = etA , 2πi n=0 n! γr k=0

since Z γr

λn−k−1 dλ =



2πi, 0,

n = k, otherwise. 

Semigroups of Bounded Operators and Second-Order PDE’s

57

Note that the right-hand side of formula (3.1.2) makes sense for each closed operator A : D(A) ⊂ X → X whose spectrum is bounded, and is independent of r > 0. Moreover, if we set T (0) = I, then the family {T (t)} defines a semigroup on X. Indeed, take s, t > 0 and let r > 0 be such that σ(A) ⊂ B(0, r). Then, Z Z 1 etλ R(λ, A) dλ T (t)T (s) = − 2 esµ R(µ, A) dµ 4π γ2r γr Z Z 1 etλ dλ esµ R(λ, A)R(µ, A) dµ =− 2 4π γ2r γr Z Z esµ 1 etλ dλ (R(λ, A) − R(µ, A)) dµ =− 2 4π γ2r γr µ − λ Z Z Z Z 1 1 esµ etλ =− 2 etλ R(λ, A) dλ dµ − 2 esµ R(µ, A) dµ dλ, 4π γ2r 4π γr γr µ − λ γ2r λ − µ where in the last integral term we have changed the order of integration. Since λ ∈ / B(0, r), −1 sµ the function µ 7→ (µ − λ) e is holomorphic in B(0, r) and, consequently, by the Cauchy R integral theorem, γr (µ − λ)−1 esµ dµ = 0. On the contrary, the function λ 7→ (λ − µ)−1 etλ R has a simple pole at λ = µ ∈ B(0, 2r). Hence, by the residue theorem, γ2r (λ − µ)−1 etλ dλ = (2πi)etµ . It thus follows that Z 1 T (t)T (s) = e(s+t)µ R(µ, A) dµ = T (t + s). 2πi γ2r In general, the spectrum of a closed operator is not bounded. Hence, formula (3.1.2) cannot used to define a semigroup associated to an unbounded operator A. But, as we will show in the next section, we can overcome this difficulty changing the path of integration, provided that the operator A satisfies nice spectral properties.

3.2

Sectorial Operators and Analytic Semigroups

As announced, here we introduce an important class of closed operators, the so-called, sectorial operators, to which we can associate a semigroup through a variant of formula (3.1.2). Definition 3.2.1 A linear operator A : D(A) ⊂ X → X is called sectorial in X if there exist ω ∈ R, θ0 ∈ (π/2, π) and M > 0 such that the resolvent set of A contains the sector Σω,θ0 = {λ ∈ C : λ 6= ω, |arg(λ − ω)| < θ0 } and ||R(λ, A)||L(X) ≤ M |λ − ω|−1 for every λ ∈ Σω,θ0 . Throughout this chapter, we denote by S(ω, θ0 , M ) the set of all the sectorial operators A which satisfy the condition in Definition 3.2.1. Given A ∈ S(ω, θ0 , M ), we can define an operator T (t), for each t > 0, by using formula (3.1.2), where we replace the curve γr by the “union” of three curves −γ1,r,η,ω , γ2,r,η,ω and γ3,r,η,ω , where γ2k+1,r,η,ω : [r, ∞) → C (k = 0, 1) is defined by γ2k+1,r,η,ω (ρ) = ω + k+1 ρe(−1) iη , for each ρ ≥ r, and γ2,r,η,ω : [−η, η] → C is defined by γ2,r,η,ω (θ) = ω + reiθ for each θ ∈ [−η, η]. Here, r > 0 and η ∈ (π/2, θ0 ) are arbitrarily fixed. More precisely, we set

58

Analytic Semigroups

σ(A) ω ω+r

FIGURE 3.1: The support of the union of the three curves γ1,r,η,ω , γ2,r,η,ω and γ3,r,η,ω . Z 1 e R(λ, A) dλ + eλt R(λ, A) dλ 2πi γ2,r,η,ω γ1,r,η,ω Z 1 + eλt R(λ, A) dλ 2πi γ3,r,η,ω Z eωt ∞ ρ cos(η)t iρ sin(η)t = e e R(ω + ρeiη , A) dρ 2πi r Z eωt ∞ ρ cos(η)t −iρ sin(η)t e e R(ω + ρe−iη , A) dρ − 2πi r Z eωt r η (r cos(θ)+ir sin(θ))t + e R(ω + reiθ , A)eiθ dθ. 2πi −η

1 T (t) = − 2πi

Z

λt

(3.2.1)

R To ease the notation, we denote the above integral by γr,η,ω etλ R(λ, A)dλ. This integral is usually referred to as Dunford integral. Note that T (t) is well defined since the definition is independent of r and η. Indeed, by Proposition A.4.3, the function λ 7→ v(λ) = etλ R(λ, A) is holomorphic in the sector Σω,θ0 , with values in L(X). This in particular shows that the integral of v over γ2,r,η,ω is well defined. Similarly, since ||etρ cos(η)±itρ sin(η) R(ω + ρ cos(η) ± iρ sin(η), A)||L(X) ≤ M ρ−1 etρ cos(η)

(3.2.2)

for every ρ ≥ r and cos(η) < 0, it follows immediately that also the integrals of v over the curves γ1,r,η,ω and γ3,r,η,ω are well defined. Now, we fix r0 > 0 and η 0 ∈ (π/2, θ0 ) and denote by D be the region lying between the supports of the curves γj,r,η,ω and γj,r0 ,η0 ,ω (j = 1, 2, 3) and by Dn (n ∈ N) the intersection of D with the closed ball centered at zero with radius n. Denote by γ a curve which parameterizes Dn , obtained from the curves γj,r,η,ω , γj,r0 ,η0 ,ω and the canonical parametrization of the arcs of ∂B(0, n). The Cauchy integral theorem implies that Z etλ R(λ, A) dλ = 0. γ

Semigroups of Bounded Operators and Second-Order PDE’s

59

γr0 ,η0 ,ω

γr,η ,ω Dn

n

ω

FIGURE 3.2: The region Dn . γr,η,ω denotes the union of the curves γj,r,η,ω (j = 1, 2, 3). Same meaning for γr0 ,η0 ,ω By estimate (3.2.2), the integrals on the two arcs contained in ∂B(0, n) vanish as n tends to ∞. From these remarks, we deduce that Z Z etλ R(λ, A) dλ = etλ R(λ, A) dλ γr,η,ω

γr0 ,η0 ,ω

as claimed. We can now study the main properties of the operators T (t), t ≥ 0. Theorem 3.2.2 Let A ∈ S(ω, θ0 , M ) and let T (t) be given by (3.2.1) for t > 0. Then, the following properties hold true. (i) For each x ∈ X, k ∈ N and t > 0, T (t)x belongs to D(Ak ). Further, if x ∈ D(Ak ), then Ak T (t)x = T (t)Ak x for every t ≥ 0. (ii) If we set T (0) = I, then the family {T (t) : t ≥ 0} (in the sequel, simply denoted by {T (t)}) is a semigroup of bounded linear operators. (iii) There exist positive constants Mk (k ∈ N ∪ {0}) such that ||tk (A − ωI)k T (t)||L(X) ≤ Mk eωt ,

t > 0, k ∈ N ∪ {0}.

(3.2.3)

(iv) The function t 7→ T (t) belongs to C ∞ ((0, ∞); L(X)) and Dtk T (t) = Ak T (t) for every t > 0. Moreover, the function t 7→ T (t) admits an analytic extension to the sector Σ0,θ0 −π/2 , given by Z 1 T (z) = eλz R(λ, A) dλ, z ∈ Σ0,θ0 −π/2 , 2πi γr,θ0 ,ω z

where θz0 is arbitrarily fixed in (π/2, θ − arg(z)).

60

Analytic Semigroups

Proof As in the proof of Hille-Yosida theorem (see Theorem 2.3.1), we replace A with the operator A − ωI, which is sectorial in Σ0,θ . (i) The core of the proof is the case k = 1. Fix t > 0 and x ∈ X. Since etλ AR(λ, A) = λt −e I + λeλt R(λ, A), ||etλ AR(λ, A)||L(X) =|| − eλt I + λeλt R(λ, A)||L(X) ≤etRe λ (1 + ||λR(λ, A)||L(X) ) ≤ etRe λ (1 + M ) and cos(η) < 0, the function λ 7→ etλ AR(λ, A)x is integrable along the curves γ1,r,η,0 and γ3,r,η,0 for each x ∈ X. Of course, being a continuous function it is also integrable along the curve γ2,r,η,0 . Therefore, we can invoke Proposition A.2.5, which guarantees that T (t)x ∈ D(A) and Z Z Z x 1 1 etλ AR(λ, A)x dλ = − etλ dλ + λetλ R(λ, A)x dλ AT (t)x = 2πi γr,η,0 2πi γr,η,0 2πi γr,η,0 Z 1 λetλ R(λ, A) dλ, (3.2.4) = 2πi γr,η,0 since Z

etλ dλ = 0.

(3.2.5)

γr,η,0

If x ∈ D(A), then AR(λ, A)x = R(λ, A)Ax and, hence, AT (t)x = T (t)Ax. Iterating this argument, we can show that T (t)x ∈ D(Ak ) for every k ∈ N and Z 1 k A T (t)x = λk eλt R(λ, A)x dλ (3.2.6) 2πi γr,η,0 and, if x ∈ D(Ak ), then Ak T (t)x = T (t)Ak x. (ii) The semigroup property can be proved arguing as in the last part of Section 3.1. Fix s, t > 0. Writing Z Z 1 1 T (t) = eλt R(λ, A) dλ, T (s) = eλs R(λ, A) dλ 2πi γr,η,0 2πi γ2r,η0 ,0 for each r > 0, π/2 < η 0 < η < θ and using the resolvent identity, it follows that Z Z 1 eµs T (t)T (s) = − 2 eλt R(λ, A) dλ dµ 4π γr,η,0 γ2r,η0 ,0 µ − λ Z Z 1 eλt + 2 eµs R(µ, A) dµ dλ. 4π γ2r,η0 ,0 γr,η,ω µ − λ Since

Z

eµs

γ2r,η0 ,0

and

Z γr,η,0

dµ = 2πiesλ for λ ∈ γr,η,0 µ−λ

eλt

dλ = 0 for µ ∈ γ2r,η0 ,0 , µ−λ

we conclude that T (t)T (s) = T (t + s).

(3.2.7)

(3.2.8)

Semigroups of Bounded Operators and Second-Order PDE’s

61

(iii) We fix t > 0 and take the norm of the three integrals which define T (t) (see (3.2.1)). Since we are assuming that ω = 0, from the estimate in Definition 3.2.1 we get   Z Z M η tr cos(θ) M etr cos(η) M ∞ tρ cos(η) −1 e ρ dρ + e dθ ≤ ||T (t)||L(X) ≤ + etr π r 2π −η π tr| cos(η)| for each r > 0 and η ∈ (π/2, θ0 ). Note that, if we take the same r for every t > 0, then we end up with an estimate for ||T (t)||L(X) which is singular as t tends to 0. To overcome this difficulty, it suffices to replace r by 1/t and we get (3.2.3) with M0 = M π −1 (| cos(η)|−1 ecos(η) +e). Estimating the norm of AT (t) is easier. We do not need to take a radius which depends on t. It is enough to use formula (3.2.4) observing that ||λR(λ, A)||L(X) ≤ M . We thus get   Z M r η tr cos(θ) M etr cos(η) e dθ + ||AT (t)||L(X) ≤ π t| cos(η)| 2π −η for each r > 0. Letting r tend to 0+ , by dominated convergence we get ||AT (t)||L(X) ≤

M , πt| cos(η)|

t > 0.

(3.2.9)

To estimate the operator norm of Ak T (t) for k > 1 it suffices to use (3.2.9), property (i) and the semigroup property to write Ak T (t) = (AT (t/k))k and, hence, estimate ||Ak T (t)||L(X) ≤ ||AT (t)||kL(X) . (iv) By applying repeatedly the dominated convergence theorem, it can be easily shown that the function t 7→ T (t) belongs to C ∞ ((0, ∞); L(X)) and Z 1 Dtk T (t) = λk eλt R(λ, A) dλ, t > 0. 2πi γr,η,0 From this formula and (3.2.6) it follows that Dtk T (t) = Ak T (t) for every t > 0 and k ∈ N. To complete the proof, we fix α ∈ (0, θ0 − π/2). Then, the function Z 1 z 7→ T (z) = ezλ R(λ, A) dλ 2πi γr,θ0 −α,0 is well defined and holomorphic in the sector Σ0,θ0 −π/2−α . Indeed, if λ = ρei(θ0 −α) and z = |z|eiφ belongs to Σ0,θ−π/2−α , then Re(zλ) = ρ|z| cos(θ0 − α + φ) and cos(θ0 − α + φ) is negative since θ0 − α + φ ∈ (π/2, 3π/2). Hence, we can differentiate under the integral sign, taking the dominated convergence theorem into account, S and conclude that the map z 7→ T (z) is holomorphic in Σ0,θ0 −π/2−α . Since Σ0,θ0 −π/2 = α∈(0,θ0 −π/2) Σ0,θ0 −π/2−α , the conclusion follows.  Remark 3.2.3 From property (iii) in Theorem 3.2.2 it follows that, for every k ∈ N, there exists a positive constant Ck,ε such that ||tk Ak T (t)||L(X) ≤ Ck,ε e(ω+ε)t for t > 0. Indeed, let us fix k ∈ N and observe that k   X k k−n k k A = (A − ωI + ωI) = ω (A − ωI)n n n=0 and, therefore, ||Ak T (t)||L(X) ≤

k   k   X X k k |ω|k−n ||(A − ωI)n T (t)||L(X) ≤ eωt |ω|k−n Mn t−n n n n=0 n=0

62

Analytic Semigroups

for t > 0. If t ≤ 1, then from the previous estimate we immediately get ||Ak T (t)||L(X) ≤ t−k eωt

k   X k n=0

n

|ω|k−n Mn .

On the other hand, if t > 1, then for every ε > 0 there exists a positive constant Cε such that t−n |ω|k−n ≤ Cε t−k eωt for every n < k. We thus conclude that k

||A T (t)||L(X) ≤ Cε t

−k (ω+ε)t

e

k   X k n=0

n

Mn

and we are done. In view of Theorem 3.2.2, we can now give the following definition. Definition 3.2.4 Let A be a sectorial operator. The family of operators {T (t)}, defined by (3.2.1) for t > 0 and such that T (0) = I, is called analytic semigroup generated by A (in X). A natural question arises: is each analytic semigroup strongly continuous? The answer is in general negative since D(A) may be not dense in X. In any case, for each x ∈ X, T (t)x converges to x as t → 0+ in a “weak” sense as the following proposition shows. Proposition 3.2.5 Let {T (t)} be an analytic semigroup associated with an operator A ∈ S(ω, θ0 , M ). Then, limt→0+ T (t)x = x if and only if x ∈ D(A). As a byproduct, for each λ ∈ ρ(A) and x ∈ X, limt→0+ R(λ, A)T (t)x = R(λ, A)x. Proof By replacing A with A−ωI, we can assume without loss of generality that ω = 0. Using the Cauchy integral theorem, it follows that Z Z  1 1 eλt 1  T (t)x − x = eλt R(λ, A)x − x dλ = R(λ, A)Ax dλ 2πi γr,η,0 λ 2πi γr,η,0 λ for x ∈ D(A) and t > 0. Since ||λ−1 R(λ, A)Ax|| ≤ C||Ax|||λ|−2 , from the dominated convergence theorem and Cauchy’s theorem we conclude that Z 1 1 lim+ T (t)x − x = R(λ, A)Ax dλ = 0 2πi γr,η,0 λ t→0 for x ∈ D(A) and, hence, for x ∈ D(A) by density, since ||T (t)|| ≤ M for every t ∈ [0, 1]. The other implication is obtained from the inclusion T (t)X ⊂ D(A) for every t > 0, see Theorem 3.2.2. The last statement follows from the fact that R(λ, A) commutes with T (t) for every t > 0 and λ ∈ ρ(A).  Remark 3.2.6 By Theorem 3.2.2, T (t) maps X into D(A) for every t > 0. Hence, it leaves D(A) invariant. Moreover, by Proposition 3.2.5, T (t)x converges to x as t → 0+ for every x ∈ D(A). It follows that the restriction of {T (t)} to D(A) is a C0 -semigroup. Note that D(A) is the largest subspace of X, where the restriction of {T (t)} is a C0 -semigroup. Indeed, as already remarked, T (t)x ∈ D(A) for each x ∈ X and t > 0. Hence, if T (t)x converges to x as t → 0+ , then, necessarily, x ∈ D(A).

Semigroups of Bounded Operators and Second-Order PDE’s

63

In the following proposition, we show other interesting properties of analytic semigroups. In particular, we show that the infinitesimal generator of the restriction of {T (t)} to D(A) is the part of A in D(A), i.e., the operator defined in D := {x ∈ D(A) : Ax ∈ D(A)} by A|D(A) x = Ax for each x ∈ D. Proposition 3.2.7 The following properties hold true. Z t T (s)x ds ∈ D(A) and (i) For each x ∈ X and t > 0, 0

Z

t

T (s)x ds = T (t)x − x.

A

(3.2.10)

0

If, in addition, x ∈ D(A), then Z

t

T (t)x − x =

T (s)Ax ds,

t ≥ 0.

(3.2.11)

0

(ii) If λ ∈ C with Re λ > ω, then Z

∞

e−λt T (t) dt.

R(λ, A) =

(3.2.12)

0

(iii) If x ∈ D(A) and Ax ∈ D(A), then (T (t)x − x)/t tends to Ax as t tends to 0+ . Conversely, if z := limt→0+ (T (t)x − x)/t exists, then x ∈ D(A) and Ax = z ∈ D(A). Proof (i) Fix t > 0 and x ∈ X. Since the function T (·)x is differentiable in (0, ∞) and Dt T (·)x = AT (·)x (see Theorem 3.2.2) then Z t Z t T (s)x ds = ((ω + 1)I − A)R(ω + 1, A)T (s)x ds ε ε Z t Z t d (R(ω + 1, A)T (s)x) ds =(ω + 1) R(ω + 1, A)T (s)x ds − ds ε ε Z t =(ω + 1)R(ω + 1, A) T (s)x ds − T (t)R(ω + 1, A)x + T (ε)R(ω + 1, A)x ε

for each ε ∈ (0, t). Recalling that the function T (·)x is bounded in [0, t] and continuous in (0, ∞) and taking Proposition 3.2.5 into account, we can let ε tend to 0 and obtain Z t Z t T (s)x ds = (ω + 1)R(ω + 1, A) T (s)x ds − R(ω + 1, A)(T (t)x − x). (3.2.13) 0

0

Z

t

T (s)x ds ∈ D(A) and applying the operator (ω + 1)I − A to both the sides of

Thus, 0

(3.2.13), formula (3.2.10) follows. If x ∈ D(A), then the function AT (·)x is continuous in (0, t] and bounded in [0, t]. Hence, Z t Z t Z t A T (s)x ds = AT (s)x ds = T (s)Ax ds ε

ε

ε

and, letting ε tend to 0+ , we conclude that Z t Z t T (s)x ds = T (s)Ax ds. lim A ε→0+

ε

0

64

Analytic Semigroups

Formula (3.2.11) follows. (ii) Fix λ ∈ C with Re λ > ω and consider the operator A − λI. This operator is sectorial and the associated semigroup {S(t)} is defined by S(t) = e−λt T (t) for t ≥ 0. From formula (3.2.10), we know that Z (A − λ)

t

S(r)x dr = S(t)x − x,

t > 0, x ∈ X.

0

Since ||T (t)||L(X) ≤ M eωt for t ≥ 0 (see Theorem 3.2.2), the function S(t)x vanishes as t tends to ∞. Hence, Z t lim (A − λ) S(r)x dr = −x t→∞

Z

0

t

Z

for every x ∈ X. Moreover, S(r)x dr converges to 0 Z ∞ Z S(r)x dr belongs to D(A) and (λ − A) of A, 0

∞

S(r)x dr. Hence, by the closedness 0 ∞

e−λt T (t)x dt = x for every x ∈ X.

0

Since we know that λ ∈ ρ(A), formula (3.2.12) follows at once. (iii) Fix x ∈ D(A) such that Ax ∈ D(A). Then, by property (i), we can write T (t)x − x 1 = A t t

Z

t

T (s)x ds = 0

1 t

Z

t

T (s)Ax ds.

(3.2.14)

0

Since Ax ∈ D(A), by Proposition 3.2.5 the function T (·)Ax is continuous in [0, ∞). Hence, T (t)x − x letting t tend to 0+ in (3.2.14), we conclude that lim+ = Ax. t t→0 Vice versa, suppose that the limit z := limt→0+ (T (t)x − x)/t exists. Then, clearly, T (t)x converges to x as t → 0+ , so that x belong to D(A). Since T (t)x ∈ D(A) for every t > 0, also z belongs to D(A). Moreover, using property (i) and recalling that R(ω + 1, A) commutes with T (s), we get Z t T (t)x − x 1 R(ω + 1, A)z = lim R(ω + 1, A) = lim R(ω + 1, A)A T (s)x ds t→0 t→0 t t 0 Z Z 1 t 1 t = − lim T (s)x ds + (ω + 1) R(ω + 1, A)T (s)x ds. (3.2.15) t→0 t 0 t 0 Z 1 t T (s)x ds = x. t→0+ t 0 Similarly, R(ω + 1, A)T (t)x converges to R(ω + 1, A)x as t → 0+ . Hence, from (3.2.15) it follows that x = R(ω + 1, A)((ω + 1)x − z) ∈ D(A) and Ax = z.  Since x ∈ D(A), the function T (·)x is continuous in [0, ∞). Hence lim

We now provide a sufficient condition for a closed operator to be sectorial. This criterium is particularly useful in the applications. Proposition 3.2.8 Let A : D(A) ⊂ X → X be a closed operator such that Π = {λ ∈ C : Re λ ≥ ω} ⊂ ρ(A) for some ω ∈ R, and ||λR(λ, A)||L(X) ≤ M,

λ ∈ Π,

for some constant M ≥ 1. Then, A is a sectorial operator.

(3.2.16)

Semigroups of Bounded Operators and Second-Order PDE’s

65

Proof The same argument as in the proof of Proposition A.4.3 (which uses Lemma A.4.4) shows that the open ball B(ω ± ir, |ω + ir|/M ) is contained in ρ(A) for each r > 0. Since |ω + ir| ≥ r, the union of such balls and the halfplane Π properly contains the sector Σω,θ2M , where θ2M = π − arctan(2M ). Observe that each λ ∈ Σω,θ2M with Re λ ≤ ω can be written in the form λ = ω ± ir − (2M )−1 (θr) for some θ ∈ [0, 1). Moreover, since R(λ, A) = R(ω ± ir, A)(I − (2M )−1 θrR(ω ± ir, A))−1 and ||(I − (2M )−1 θrR(ω ± ir, A))−1 ||L(X) ≤ 2, it follows that √ 4M 2 + 1 2M 2M ≤ ≤ . ||R(λ, A)||L(X) ≤ |ω ± ir| r |λ − ω| On the other hand, if λ ∈ Σω,θ2M with Re λ > ω, then from (3.2.16) it follows easily that ||R(λ, A)||L(X) ≤ M |λ − ω|−1 and this completes the proof.  A useful consequence is the following result. Corollary 3.2.9 If there exists θ ∈ (0, π2 ) such that e±iθ A generate C0 -semigroups on X, then A : D(A) → X is a sectorial operator. Proof Without loss of generality we can assume that e±iθ A generates a bounded C0 semigroup on X. Let e−iθ = a − ib with a, b > 0. Using the Hille-Yosida Theorem 2.3.1 we deduce that ||R(t + is, A)|| = ||e−iθ R(e−iθ (t + is), e−iθ A)|| ≤

f M M ≤ at + bs s

f = b−1 M . Since eiθ A generates a C0 -semigroup on X, we obtain the for all t, s > 0, with M f|s|−1 for all t > 0 and all s ∈ R \ {0}. same estimate for s < 0. Thus, ||R(t + is, A)|| ≤ M −1 c c, |s|−1 M f} ≤ M 0 |λ|−1 Moreover, ||R(t + is, A)|| ≤ M t and, hence, ||R(λ, A)|| ≤ max{t−1 M 0 for all λ with Re λ > 0 and some constant M ≥ 1. So, the assertion follows from Proposition 3.2.8.  Remark 3.2.10 An inspection of the proof of the above result one can see that the assumption that e±iθ A generates a C0 -semigroup is not actually needed. It can be replaced by the two conditions {λ ∈ C : Reλ > 0} ⊂ ρ(e±iθ A),

||R(λe±iθ , e±iθ A)|| ≤

M , Re(λe±iθ )

λ ∈ C, Re λ > 0,

which should be satisfied by some θ ∈ (0, π2 ).

3.3

Interpolation Spaces

Interpolation spaces play a crucial role in the study of analytic semigroups. They can defined in three equivalent ways. The first definition is given in terms of the behaviour of the function t 7→ AT (t)x as t tends to 0+ . By Remark 3.2.3, we can estimate ||AT (t)x|| ≤ C1 t−1 ||x||,

t ∈ (0, 1]

for each x ∈ X. On the other hand, if x ∈ D(A), then AT (t)x = T (t)Ax for every t > 0 (see Theorem 3.2.2(i)). Thus, from (3.2.3) we infer that ||AT (t)x|| ≤ C2 ||x||D(A) ,

t ∈ (0, 1].

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Analytic Semigroups

The constants C1 and C2 are independent of t and x. The interpolation space DA (α, ∞) (α ∈ (0, 1)) is roughly speaking the set of x ∈ X for which the function t 7→ AT (t)x exhibits an intermediate behaviour (as t → 0+ ). More precisely, Definition 3.3.1 For each α ∈ (0, 1), the interpolation space DA (α, ∞) is the set of all x ∈ X such that [x]DA (α,∞) := sup t1−α ||AT (t)x|| < ∞. t∈(0,1]

As it is easily seen, DA (α, ∞) is a Banach space when endowed with the norm ||x||DA (α,∞) = ||x|| + [x]DA (α,∞) . Remark 3.3.2 From Theorem 3.2.2, it follows easily that D(A) is continuously embedded into DA (α, ∞) for every α ∈ (0, 1). Remark 3.3.3 Splitting Ak T (t)x = Ak−1 T ((k − 1)t/k)AT (t/k)x for every t ∈ (0, 1], x ∈ DA (α, ∞) and k ≥ 2, we easily conclude that ||Ak T (t)x|| ≤ ||Ak−1 T ((k − 1)t/k)||L(X) ||AT (t/k)x|| ≤ Ck tα−k [x]DA (α,∞) . e1 ||x|| for every t ∈ (0, 1] and From Theorem 3.2.2, we also know that ||T (t)x − x|| ≤ C e2 t||x||D(A) for x ∈ X. On the other hand, if x ∈ D(A), then (3.2.11) yields ||T (t)x − x|| ≤ C e e every t ∈ (0, 1]. Here, the constants C1 and C2 are independent of x and t. As the following proposition shows, the space DA (α, ∞) can be also characterized as the set of x ∈ X such that the function t 7→ ||T (t)x − x|| exhibits an intermediate behaviour. Proposition 3.3.4 For each α ∈ (0, 1), DA (α, ∞) is the set of all x ∈ X such that [x]α = supt∈(0,1] t−α ||T (t)x − x|| < ∞. Equivalently, DA (α, ∞) is the set of x ∈ X such that the function t 7→ T (t)x is α-H¨ older continuous in [0, T ] for each T > 0. Finally, x 7→ ||x|| + [x]α is a norm which is equivalent to the “usual” norm of DA (α, ∞). Proof Fix x ∈ D(A). Since the function t 7→ AT (t)x is continuous in (0, 1] and the function t ∈ ||AT (t)x|| belongs to L1 ((0, 1)), then from Propositions 3.2.7(ii) and A.2.5(ii) we can write

Z t

Z t

Z t



≤

= ||AT (s)x|| ds AT (s)x ds ||T (t)x − x|| = A T (s)x ds



0

0

t

Z ≤[x]DA (α,∞) 0

0

1 [x]DA (α,∞) t1−α . s−α ds = 1−α

We have so proved that [x]α ≤ (1 − α)−1 [x]DA (α,∞) . Vice versa, let us suppose that [x]α < ∞. We observe that  Z t  Z t 1 1 AT (t)x =AT (t) (x − T (s)x) ds + AT (t) T (s)x ds t 0 t 0  Z t  1 1 (x − T (s)x) ds + T (t)(T (t)x − x). =AT (t) t 0 t Therefore, ||AT (t)x|| ≤||AT (t)||L(X)

1 t

Z 0

t

1 ||x − T (s)x|| ds + ||T (t)||L(X) ||T (t)x − x|| t

Semigroups of Bounded Operators and Second-Order PDE’s Z t 1 sα ds + C2 tα−1 [x]α ≤ C3 tα−1 [x]α ≤C1 2 [x]α t 0

67

for some positive constants C1 , C2 , C3 and every t ∈ (0, 1] (see Theorem 3.2.2 and Remark 3.2.3). From this chain of inequalities it follows that x belongs to DA (α, ∞) and [x]DA (α,∞) ≤ C3 [x]α . So the equivalent characterization of DA (α, ∞) is proved as well as the equivalence between the classical norm of DA (α, ∞) and the norm x 7→ ||x|| + [x]α . To conclude the proof, we observe that, if the function t 7→ T (t)x is H¨older continuous in [0, T ] for every T > 0, then, in particular, [x]α = sup t−α ||T (t)x − x|| ≤ [T (·)]C α ([0,T ];X) , t∈(0,1]

so that x ∈ DA (α, ∞). On the other hand, if x ∈ DA (α, ∞) and s, t ∈ [0, T ], with 0 < t − s ≤ 1, then ||T (t)x − T (s)x|| ≤ ||T (s)||L(X) ||T (t − s)x − x|| ≤ C(t − s)α . If t − s > 1, then we simply estimate ||T (t)x − T (s)x|| ≤ 2 sup ||T (t)||L(X) ||x|| ≤ 2 sup ||T (t)||L(X) ||x|||t − s|α . t∈[0,T ]

t∈[0,T ]

The proof is now complete.



Remark 3.3.5 As in the above proof, it can be seen, using Proposition 3.2.7, that Z t T (t)x − x = AT (s)x ds 0

for all x ∈ DA (α, ∞) and t ≥ 0. The interpolation space DA (α, ∞) can be also characterized in terms of the behaviour of the function λ 7→ AR(λ, A)x as λ tends to ∞, as the following proposition shows. Proposition 3.3.6 For every α ∈ (0, 1), the interpolation space DA (α, ∞) is the set of all x ∈ X such that lim supλ→∞ λα ||AR(λ, A)x|| < ∞. Moreover, for every λ0 > ω + , where ω is the constant in Theorem 3.2.2, the norm of DA (α, ∞) is equivalent to the norm x 7→ ||x|| + supλ≥λ0 λα ||AR(λ, A)x||. Proof We first prove that, if lim supλ→∞ λα ||AR(λ, A)x|| < ∞, then x ∈ DA (α, ∞). Fix λ as in the statement. Then, the previous condition is equivalent to saying that −1 [[x]]α := supλ≥λ0 λα ||AR(λ, A)x|| < C. For every t ∈ (0, λ−1 R(t−1 , A)x − 0 ), we split x = t −1 AR(t , A)x and write AT (t)x =t−1 AT (t)R(t−1 , A)x − AT (t)AR(t−1 , A)x =t−1 T (t)AR(t−1 , A)x − AT (t)AR(t−1 , A)x. Therefore, ||AT (t)x|| ≤t−1 ||T (t)||L(X) ||AR(t−1 , A)x|| + ||AT (t)||L(X) ||AR(t−1 , A)x|| ≤tα−1 (||T (t)||L(X) + ||tAT (t)||L(X) )[[x]]α for every t ∈ (0, λ−1 0 ]. If λ0 ≤ 1, then we conclude that [x]DA (α,∞) ≤

sup

(||T (t)||L(X) + ||tAT (t)||L(X) )[[x]]α .

t∈(0,λ−1 0 )

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If λ0 > 1, then, we can estimate ||AT (t)x|| ≤ supt∈(λ−1 ,1] ||AT (t)||L(X) ||x||. Therefore, x ∈ 0 DA (α, ∞) and ||x||DA (α,∞) ≤ C1 (||x|| + [[x]]α ) for some positive constant C1 , independent of x. Vice versa, let us suppose that x ∈ DA (α, ∞). By Proposition 3.3.4, we know that supt∈(0,1] t−α ||T (t)x − x|| ≤ C2 ||x||DA (α,∞) for some positive constant C2 , independent of x. We fix λ ≥ λ0 ∨ 1, split λα AR(λ, A)x = λα AR(λ, A)T (λ−1 )x + λα AR(λ, A)(x − T (λ−1 x)) and estimate λα ||AR(λ, A)x|| =λα ||AR(λ, A)T (λ−1 )x|| + λα ||AR(λ, A)(x − T (λ−1 x))|| =λα ||R(λ, A)AT (λ−1 )x|| + λα ||AR(λ, A)(x − T (λ−1 x))|| ≤λα ||R(λ, A)||L(X) ||AT (λ−1 )x|| + λα ||AR(λ, A)||L(X) ||T (λ−1 )x − x|| ≤C3 ([x]DA (α,∞) + [x]α ) ≤C4 ||x||DA (α,∞) , (3.3.1) where the constants C3 and C4 are independent of x and λ. If λ0 ≥ 1, then we conclude that [[x]]α ≤ C4 ||x||DA (α,∞) and the assertion follows. If λ0 < 1, then we estimate λα ||AR(λ, A)x|| ≤ max ||AR(λ, A)||L(X) ||x||, λ∈[λ0 ,1]

λ ∈ [λ0 , 1).

(3.3.2)

From (3.3.1) and (3.3.2), we deduce that [[x]]α ≤ C5 ||x||DA (α,∞) for some constant C5 , independent of x. The assertion follows.  The last way to introduce the space DA (α, ∞) is through the so-called K-method used to define the interpolation spaces between two given Banach spaces X and Y , with Y ,→ X. It is from this definition that the name “interpolation space”, which we use to call the space DA (α, ∞), comes. We refer the reader to Section A.5 for the definition of the interpolation spaces and some basic properties. Proposition 3.3.7 For each α ∈ (0, 1), DA (α, ∞) = (X, D(A))α,∞ with equivalence of the corresponding norms. Proof Fix x ∈ DA (α, ∞) t ∈ (0, 1] and split x = (x − T (t)x) + T (t)x. Since T (t)x belongs to D(A), it follows that K(t, x) ≤ ||x − T (t)x|| + t||T (t)x||D(A) .

(3.3.3)

Due to Proposition 3.3.4, there exists a positive constant C1 , independent of t and x such that ||T (t)x − x|| ≤ C1 t−α ||x||DA (α,∞) . Moreover, ||T (t)x||D(A) =||T (t)x|| + ||AT (t)x|| ≤ C2 (||x||X + tα−1 [x]DA (α,∞) ) ≤ C2 tα−1 ||x||DA (α,∞) , the constant C2 being independent of t and x. Replacing these two inequalities in (3.3.3), we conclude that [x](X,D(A))α,∞ ≤ C2 ||x||DA (α,∞) so that DA (α, ∞) is continuously embedded into (X, D(A))α,∞ . Vice versa, fix x ∈ (X, D(A))α,∞ . Then, K(t, x) ≤ tα [x](X,D(A))α,∞ for every t ∈ (0, 1]. Thus, for each t ∈ (0, 1] we can determine zt ∈ X and yt ∈ D(A) such that x = zt + yt and ||zt || + t||yt ||D(A) ≤ 2tα [x](X,D(A))α,∞ . Based on this inequality, we can estimate ||AT (t)x|| =||AT (t)zt || + ||AT (t)yt || ≤ ||AT (t)zt || + ||T (t)Ayt || ≤C3 t−1 (||zt || + t||yt ||D(A) ) ≤ 2C1 tα−1 [x](X,D(A))α,∞ for every t ∈ (0, 1] and some positive constant C3 , independent of x and t. Therefore, x belongs to DA (α, ∞) and [x]DA (α,∞) ≤ C1 [x](X,D(A))α,∞ . The assertion follows. 

Semigroups of Bounded Operators and Second-Order PDE’s

69

Remark 3.3.8 From the definition, it follows that (X, Y )α,∞ remains unchanged if we replace the norm of X and/or Y with equivalent norms. Thus, in view of Proposition 3.3.7 it follows immediately that, if {S(t)} and {T (t)} are two analytic semigroups, associated to the sectorial operators A1 and A2 with the property that D(A1 ) = D(A2 ), with equivalence of the corresponding norms, then DA1 (α, ∞) = DA2 (α, ∞), with equivalence of the corresponding norms, for every α ∈ (0, 1). To complete this section and for further uses, we introduce a new family of interpolation spaces. Definition 3.3.9 For each α ∈ (0, 1), DA (1 + α, ∞) is the set of all x ∈ D(A) such that Ax ∈ DA (α, ∞). DA (1 + α, ∞) is a Banach space when endowed with the norm ||x||DA (1+α,∞) = ||x||X + ||Ax||DA (α,∞) . Remark 3.3.10 Arguing as in Remark 3.3.3, it can be easily proved that, for every t ∈ (0, 1], every k ≥ 2 and every x ∈ DA (1 + α, ∞), it holds that ||Ak T (t)x|| ≤ Ck tα+1−k ||x||DA (1+α,∞) ,

t ∈ (0, 1].

The following proposition provides us with an equivalent characterization of the space DA (1 + α, ∞). Proposition 3.3.11 For each α ∈ (0, 1), DA (1 + α, ∞) is the set of all x ∈ X such that the function t 7→ T (t)x belongs to C 1+α ([0, T ]) for every T > 0. Proof Suppose that x ∈ DA (1 + α, ∞). Then, the function t 7→ v(t) = T (t)x is differentiable in [0, T ] and v 0 = T (·)Ax. Since Ax ∈ DA (α, ∞), from Proposition 3.3.4 it follows that the function v 0 belongs to C α ([0, T ]; X) for each T > 0. We have so proved that v ∈ C 1+α ([0, T ]; X) and ||v||C 1+α ([0,T ];X) ≤ CT ||x||DA (1+α,∞) for some constant CT , independent of x. Vice versa, let us suppose that the function v belongs to C 1+α ([0, T ]; X) for every T > 0. In particular, v is differentiable at t = 0 so that x ∈ D(A). Moreover, v 0 = T (·)Ax belongs to C α ([0, T ]; X). Therefore, Ax ∈ DA (α, ∞) and we are done.  Finally, we show that the part of A in DA (α, ∞) is still sectorial and the associated semigroup is the restriction of the semigroup T (t) to DA (α, ∞). Definition 3.3.12 Let A : D(A) ⊂ X → X be a linear operator and let Y be a subspace of D(A). The part of A in Y is the operator A|Y defined by: ( D(A|Y ) = {x ∈ D(A) : Ax ∈ Y }, A|Y x = Ax, x ∈ D(A|Y ). Theorem 3.3.13 Let A be a sectorial operator in X. Then, for every α ∈ (0, 1), the part of A in DA (α, ∞) has DA (α + 1, ∞) as domain. Moreover, σ(A|DA (α,∞) ) ⊂ σ(A) and R(λ, A|DA (α,∞) ) = R(λ, A)|DA (α,∞) for every λ ∈ ρ(A). In particular, A|DA (α,∞) is sectorial in DA (α, ∞) and the associated semigroup is the restriction to DA (α, ∞) of the semigroup associated with A.

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Analytic Semigroups

Proof By the definition of the space DA (1 + α, ∞), it follows immediately that D(A|DA (α,∞) ) coincides with DA (α + 1, ∞). Next, we fix λ ∈ ρ(A) and y ∈ DA (α, ∞). Then, the equation λx − Ax = y admits x = R(λ, A)y as its unique solution in X. By difference, Ax = λx − y belongs to DA (α, ∞), so that x ∈ DA (1 + α, ∞). Since any solution of the equation λx − Ax = y in DA (α + 1, ∞) is, in particular, a solution in D(A), we infer that λ ∈ ρ(A|DA (α,∞) ) and the equality R(λ, A|DA (α,∞) ) = R(λ, A)|DA (α,∞) holds true. Finally, recalling that each operator T (t) commutes with R(λ, A), we can write t1−α ||AT (t)R(λ, A)x|| = t1−α ||R(λ, A)AT (t)x|| ≤ ||R(λ, A)||L(X) t1−α ||AT (t)x|| and, consequently, [R(λ, A)x]DA (α,∞) ≤ ||R(λ, A)||L(X) [x]DA (α,∞) . It thus follows that ||R(λ, A|DA (α,∞) )||L(DA (α,∞)) ≤ ||R(λ, A)L(X) . Thus, A|DA (α,∞) is sectorial and we can associate a semigroup with it through the Dunford integral (3.2.1), which converges in DA (α, ∞). Since DA (α, ∞) ,→ X, such integral converges also in X and this completes the proof. 

3.4

Nonhomogeneous Cauchy Problems

In this section, which is the counterpart of Section 2.4, we consider the Cauchy problem ( u0 (t) = Au(t) + f (t), t ∈ [0, T ], (3.4.1) u(0) = x, when A is a sectorial operator, f : [0, T ] → X is a continuous function and x ∈ X. Moreover, we denote by {T (t)} the analytic semigroup associated with the operator A in X and assume that ||T (t)|| ≤ M eωt for every t ≥ 0 and some constants M ≥ 1 and ω ∈ R (see Theorem 3.2.2(iii)). Besides the concept of strict solution, introduced in Definition 2.4.1, we introduce the concept of classical solution. Definition 3.4.1 A function u : [0, T ] → X is a classical solution of the Cauchy problem (3.4.1) if u ∈ C([0, T ]; X)∩C((0, T ]; D(A))∩C 1 ((0, T ]; X), u(0) = x and u0 (t) = Au(t)+f (t) for every t ∈ (0, T ]. Remark 3.4.2 We stress that (i) if u is a classical solution of the Cauchy problem (3.4.1), then u(t) belongs to D(A) for every t ∈ (0, T ] and u(t) tends to x in X as t tends to 0+ . Hence, x ∈ D(A); (ii) if u is a strict solution of the Cauchy problem (3.4.1), then u is continuous on [0, T ] with values in D(A). Since u(0) = x, it follows that x ∈ D(A). Moreover, t−1 (u(t) − x) ∈ D(A) for every t ∈ (0, T ]. Therefore, D(A) 3 lim

t→0+

u(t) − x = u0 (0) = Au(0) + f (0) = Ax + f (0). t

The following proposition shows that strict and classical solutions to problem (3.4.1) are given by the variation-of-constants formula.

Semigroups of Bounded Operators and Second-Order PDE’s

71

Proposition 3.4.3 Let f ∈ C([0, T ], X) and let x ∈ D(A) be given. If u is a classical solution (and, in particular, if u is a strict solution) of the Cauchy problem (3.4.1), then Z

t

T (t − s)f (s) ds,

u(t) = T (t)x +

t ∈ [0, T ].

(3.4.2)

0

As a byproduct, the classical solution to problem (3.4.1) is unique, if existing. Proof The proof is similar to that of Proposition 2.4.3. Let u be a classical solution to the Cauchy problem (3.4.1). Then, for every t ∈ (0, T ], the function v : [0, t] → X defined by v(s) = T (t − s)u(s), for every s ∈ [0, T ], belongs to C([0, t]; X) ∩ C 1 ((0, t), X), and v 0 (s) = T (t − s)f (s) for each s ∈ (0, t). Integrating between ε and t − ε, where ε ∈ (0, t/2) is arbitrarily fixed, gives Z t−ε T (t − s)f (s) ds. (3.4.3) T (ε)u(t − ε) − T (t − ε)u(ε) = ε

Since ||T (t − s)f (s)|| ≤ M e(ω∨0)T ||f ||∞ for every t ∈ [0, s], by dominated convergence we infer that Z t−ε Z t lim T (t − s)f (s) ds = T (t − s)f (s) ds. ε→0+

ε

0

Similarly, ||T (ε)u(t − ε) − T (t − ε)u(ε) − u(t) + T (t)x|| ≤||T (ε)[u(t − ε) − u(t)]|| + ||T (ε)u(t) − u(t)|| + ||T (t − ε)[u(ε) − x]|| + ||T (t)x − T (t − ε)x|| +

≤M eω ||u(t − ε) − u(t)|| + ||T (ε)u(t) − u(t)|| + M eω

+

T

||u(ε) − x|| + ||T (t)x − T (t − ε)x||.

Since u(t) ∈ D(A) and the function T (·)x is continuous on (0, ∞) (by Theorem 3.2.2(iv) and Proposition 3.2.5), the last side of the previous chain of inequalities vanishes as ε tends to 0+ . Thus, we can let ε tend to 0+ in both the sides of (3.4.3) and conclude the proof.  Definition 3.4.4 The function u defined by (3.4.2) is called mild solution to the Cauchy problem (3.4.1). As in the case of strongly continuous semigroups, the existence of a classical or strict solution of the Cauchy problem (3.4.1) is reduced to the problem of regularity of the mild solution. It is easy to check that the mild solution to the Cauchy problem (3.4.1) is continuous on (0, T ] if f ∈ C([0, T ]; X) (and it is continuous also at 0 if x ∈ D(A)) and satisfies the estimate eωT − 1 ||u(t)|| ≤ M e(ω∨0)T ||x|| + M ||f ||∞ , t ∈ [0, T ], (3.4.4) ω where ω −1 (eωT − 1) is replaced by T when ω = 0. In general, even in the case when x = 0 the continuity of f is not sufficient to guarantee that the mild solution is actually a classical solution of problem (3.4.1), as the following example shows. Example 3.4.5 P Let `2 denote the space of all the complex-valued sequences x = (xn ) such ∞ that ||(xn )||2`2 := n=1 x2n < ∞. On `2 we consider the operator A : D(A) ⊂ `2 → `2 defined by Ax = (−nxn ) for every sequence x ∈ D(A). It is easy to see that the spectrum of A is

72

Analytic Semigroups

−N. Moreover, if y ∈ `2 and λ ∈ / σ(A), then R(λ, A)y = ((λ + n)−1 yn ). Hence, if Re λ > 0, then we can estimate ||R(λ, A)y||2`2 =

∞ X |yn |2 −2 ≤ |λ| yn2 = |λ|−2 ||y||2`2 . 2 |λ + n| n=1 n=1 ∞ X

Proposition 3.2.8 implies that A is sectorial on `2 . Moreover, D(A) is dense in `2 . Indeed, denote by ek the sequence whose unique component different Pn from zero is the k-th one. As it is immediately seen, given y ∈ `2 the sequence yn = k=1 yk ek belongs to D(A) and converges to y in `2 . By contradiction, we assume that the Cauchy problem ( u0 (t) = Au(t) + f (t), t ∈ [0, T ], u(0) = 0, admits a unique classical (and, hence, strict since D(A) is dense in `2 ) solution for every f ∈ C([0, T ]; `2 ). Then, the operator T : C([0, T ]; `2 ) → C([0, T ]; D(A)), defined by Z t (T f )(t) = T (t − s)f (s) ds, t ∈ [0, T ], 0

is continuous. Indeed, it is continuous from C([0, T ]; `2 ) into itself, so that it is, in particular, closed; the closed graph theorem yields the continuity of T from C([0, T ]; `2 ) into C([0, T ]; D(A)). This implies that there exists a positive constant C such that ||AT f ||C([0,T ];`2 ) ≤ C||f ||C([0,T ];`) ,

f ∈ C([0, T ]; `2 ).

(3.4.5)

Let us violate estimate (3.4.5) by providing a bounded sequence (fn ) ⊂ C([0, T ]; `2 ) such that the sup-norm of AT fn blows up as n tends to ∞. For this purpose, we begin by −tn observing that R(λ, A)en = λ−1 en for every n en for every n ∈ N. Therefore, T (t)en = e t > 0 and n ∈ N. We now define the sequence (fn ) by setting fn (t) =

n X

g(2k (T − t))e2k ,

t ∈ [0, T ], n ∈ N,

k=1

where g ∈ Cb ([0, ∞)) is a nonnegative function which identically vanishes outside [T /2, T ]. Due to the above remarks, and denoting by M the L1 -norm of the function t 7→ e−t g(t), we can estimate  n Z T n X X k (T fn )(T ) = e−2 (T −s) g(2k (T − s)) ds e2k = M e2k . 0

k=1

k=1

Therefore, (AT fn )(T ) = M Ae2k and ||(AT fn )(T )||

`2

X

X  21 n

n

k

, = M Ae2k = M 2 k=1

k=1

which implies that ||(AT fn )(T )||`2 blows up as n tends to ∞. On the other hand, since the supports of the functions g(2k (T − ·)) are pairwise disjoint, it follows that ||fn ||C([0,T ];X) ≤ ||g||∞ for every n ∈ N. Condition (3.4.5) is thus violated and we get to a contradiction.

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73

Remark 3.4.6 Even if the mild solution to problem (3.4.1) might be not a classical solution, it solves that problem in a integral sense. More precisely, Z t Z t u(t) = x + A u(s) ds + f (s) ds, t ∈ [0, T ]. (3.4.6) 0

0

Indeed, since u is continuous on [0, T ] and the function (r, s) 7→ T (r)f (s) is continuous on (0, T ] × [0, T ], changing the order of integration we can write Z t Z t Z t Z s u(s) ds = T (s)x ds + ds T (s − r)f (r) dr 0 0 0 0 Z t Z t Z t = T (s)x ds + dr T (s − r)f (r) ds 0

0

for every t ∈ (0, T ]. Since Z t

Z T (s − r)f (r) dr =

r

r

t−r

T (σ)f (r) dr 0

by Proposition 3.2.7 this term belongs to D(A) and Z t A T (s − r)f (r) dr = T (t − r)f (r) − f (r). r

We Z t can thus invoke Proposition A.2.5 and again Proposition 3.2.7 to conclude that u(s) ds ∈ D(A) and 0

Z

t

t

Z u(s) ds = T (t)x − x +

A 0

Z T (t − r)f (r) dr −

0

t

f (r) dr,

t ∈ [0, T ].

0

From this formula, (3.4.6) follows at once. In the case of strongly continuous semigroups, the integral term in the definition of the mild solution, i.e., the function v : [0, T ] → X, defined by Z t v(t) = T (t − s)f (s) ds, t ∈ [0, T ], 0

is, in general, just a continuous function with values on X (see Example 2.4.6). The picture is different in the case of analytic semigroups as the following proposition shows. For further use, we introduce the constants Mk0 =

sup

||tk Ak T (t)||L(X) ,

k = 0, 1, 2,

0 0). Strongly continuous semigroups do not enjoy this property. Think for instance to the semigroup of left translations on X = BU C(R). Since T (t)f = f (·+t) for every t > 0, T (t) has no smoothing effects on f . For this reason, as we will see in the next chapters, analytic semigroups are naturally associated to solutions to parabolic equations. For more details on sectorial operators and analytic semigroups, we refer the reader, e.g., to [13, 18, 26, 29, 36]. Finally, for more details on real interpolation theory, we refer the reader, e.g., to [37].

3.6

Exercises

1. Prove formulas (3.2.5), (3.2.7), (3.2.8). 2. Let A : D(A) ⊂ X → X be a sectorial operator, let α ∈ C, and set B : D(B) := D(A) → X, Bx = Ax − αx, C : D(C) = D(A) → X, Cx = αAx. Prove that the operator B is sectorial, and that the associated semigroup {S(t)} is defined by S(t) = e−αt T (t) for every t ≥ 0. For which α the operator C is sectorial? 3. Let A : D(A) ⊂ X → X be a sectorial operator and let x ∈ D(A) be an eigenvector of A with eigenvalue λ. (i) Prove that R(µ, A)x = (µ − λ)−1 x for every µ ∈ ρ(A). (ii) Prove that T (t)x = eλt x for every t > 0. (iii) Prove that if A and −A are sectorial operators in X, then A is bounded. 4. Let Xk , k = 1, . . . , n be Banach spaces, and let Ak : D(Ak ) → Xk be sectorial operators. Set X=

n Y k=1

Xk ,

D(A) =

n Y k=1

D(Ak ),

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81

A(x1 , . . . , xn ) = (A1 x1 , . . . , An xn ), and show that A is a sectorial operator in X, endowed
Pn 2 1/2 . with the product norm ||(x1 , . . . , xn )|| = k=1 ||xk || 5. Let X bep a real Banach space. Prove that the function f : X × X → R defined by f (x, y) = ||x||2 + ||y||2 for every x, y ∈ X, does not satisfy, in general, the homogeneity property. 6.

(i) Prove that if A with domain D(A) generates a C0 -group on a Banach space X, then A2 generates an analytic semigroup on X. (ii) Prove that the operator Af = f 0 for f ∈ D(A) = W 1,p (R; C) generates a C0 -group on Lp (R; C), 1 < p < ∞. 00

(iii) Deduce that the operator Bf = f for f ∈ D(B) = W 2,p (R; C) generates an analytic semigroup on Lp (R; C), 1 < p < ∞. 7. (a) Prove that the operator A, defined by Af = f 00 for every f ∈ D(A) = Cb2 (R; C), is sectorial in Cb (R; C) and, hence, generates an analytic semigroups. (b) Prove that the operator A, defined by Af = f 00 for every f ∈ D(A) = {u ∈ Cb2 ([0, 1]; C) : u(0) = u(1) = 0}, generates an analytic semigroup in C([0, 1]; C). Is this semigroup strongly continuous? (c) Prove that the operator A, defined by Af = f 00 for every f ∈ D(A) = {u ∈ Cb2 ([0, 1]; C) : u0 (0) = u0 (1) = 0}, generates an analytic semigroup in C([0, 1]; C). Is this semigroup strongly continuous?

Part II

Parabolic Equations

Chapter 4 Elliptic and Parabolic Maximum Principles

This chapter is dedicated to the weak and strong parabolic maximum principles for secondorder uniformly elliptic operators on bounded domains. Maximum principles are the fundamental tool that we use in this book to prove the uniqueness of solutions to parabolic Cauchy problems as well as elliptic boundary value problems. On an open subset Ω of Rd (which can be also unbounded if not otherwise specified) we define the second-order elliptic operator Aψ(x) =

d X

qij (x)Dij ψ(x) +

i,j=1

d X

bj (x)Dj ψ(x) + c(x)ψ(x)

j=1

=Tr(Q(x)D2 ψ(x)) + hb(x), ∇x ψ(x)i + c(x)ψ(x) on sufficiently smooth functions ψ : Ω → R, where Q(x) = (qij (x)) and b(x) = (b1 (x), . . . , bd (x)) for x ∈ Ω. Throughout the chapter, we assume the following conditions on the coefficients of A. Hypotheses 4.0.1 (i) The coefficients qij = qji , bj (i, j = 1, . . . , d) and c are real valued functions and belong to C(Ω); (ii) there exists µ > 0 such that hQ(x)ξ, ξi ≥ µ|ξ|2 for each x ∈ Ω, ξ ∈ Rd . The chapter is split into two sections. In the first one we will deal with solutions to the inequality Dt u − Au ≤ 0, whereas in the second one we consider functions which satisfy the inequality Au ≤ 0. The main results that we present are the weak and the strong maximum principles which hold true in both the parabolic and elliptic case. Notation. For T > 0, we set ΩT := (0, T ] ×Ω and denote by ΓT = ΩT \ΩT the so-called parabolic boundary of ΩT . It is clear that ΓT = ({0} × Ω) ∪ ((0, T ) × ∂Ω). We also denote by c0 the supremum over Ω of the potential c of the operator A.

4.1

The Parabolic Maximum Principles

In this section, we prove first the weak maximum principle and then the strong maximum principle for the parabolic operator Dt − A and some of their most relevant consequences.

4.1.1

Parabolic weak maximum principle

Theorem 4.1.1 (Weak maximum principle I) Assume that c ≡ 0 in Ω and fix u ∈ C 1,2 (ΩT ) ∩ Cb (ΩT ). The following assertions hold: (i) if Dt u − Au ≤ 0 in ΩT , then sup u = sup u; ΩT

ΓT

85

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Elliptic and Parabolic Maximum Principles

(ii) if Dt u − Au ≥ 0 in ΩT , then inf u = inf u. ΓT

ΩT

In particular, if Dt u − Au = 0 in ΩT , then ||u||Cb (ΩT ) = ||u||Cb (ΓT ) . Proof It is obvious that the assertion (ii) can be deduced from (i) by considering −u instead of u in (i). Let u be as in the statement of property (i) and consider the function ϕ : Rd → R, defined by ϕ(x) = 1 + |x|2 for every x ∈ Rd . Clearly, ϕ(x) blows up as |x| tends to ∞. Moreover, since Aϕ(x) = 2Tr(Q(x)) + 2hb(x), xi for every x ∈ Ω, we can estimate |Aϕ(x)| ≤ 2

d X

||qjj ||∞ +

X d

j=1

||bj ||2∞

 12 |x|,

x ∈ Ω,

(4.1.1)

j=1

and conclude that Aϕ < λϕ on Rd for some positive constant λ. For each n ∈ N, let vn be the function defined by vn (t, x) = e−λt v(t, x) − n−1 ϕ(x) for every (t, x) ∈ ΩT , where v = u − supΓT u. We claim that vn ≤ 0 on ΩT for all n ∈ N. For this purpose we observe that each function vn is as smooth as u is and lim vn (t, x) = −∞,

|x|→∞ x∈Ω

uniformly with respect to t ∈ [0, T ]. As a consequence, for each n ∈ N vn has a maximum point (tn , xn ) ∈ ΩT . If (tn , xn ) ∈ ΓT , then vn is clearly nonpositive on ΩT . Suppose that (tn , xn ) ∈ ΩT . Then, v(tn , xn ) should be nonpositive as well. Indeed, the function vn satisfies the differential inequality Dt vn −Avn +λvn < 0 in ΩT . Moreover, Dt vn (tn , xn ) ≥ 0 (since tn could be equal to T ), Dj vn (tn , xn ) = 0 for each j = 1, . . . , d, and the matrix (Dij vn (tn , xn )) is negative semi-definite. Thus, Dt vn (tn , xn ) − Avn (tn , xn ) = Dt vn (tn , xn ) −

d X

qij (xn )Dij vn (tn , xn ) ≥ 0

i,j=1

and, as a byproduct, we infer that λvn (tn , xn ) ≤ Avn (tn , xn ) − Dt vn (tn , xn ) ≤ 0. From this inequality, we conclude that vn (tn , xn ) ≤ 0, so that vn is nonpositive in ΩT . Letting n tend to ∞, we deduce that v is nonpositive on ΩT , which is the claim. Finally, to prove the last statement, we observe that, if Dt u−Au = 0, then it follows from (i) and (ii) that supΩT u = supΓT u ≤ supΓT |u| and supΩT (−u) = supΓT (−u) ≤ supΓT |u|. Hence, sup |u| ≤ sup |u| and the assertion follows.  ΩT

ΓT

Corollary 4.1.2 Suppose that c ≡ 0 and u ∈ C 1,2 (ΩT ) ∩ Cb (ΩT ) solves the equation Dt u − Au = f for some f ∈ Cb (ΩT ). Then, ||u||Cb (ΩT ) ≤ ||u||Cb (ΓT ) + 2T ||f ||Cb (ΩT ) .

(4.1.2)

Proof Let v : ΩT → R be the function defined by v(t, x) = u(t, x) − t||f ||Cb (ΩT ) for every (t, x) ∈ ΩT . Clearly, v ∈ C 1,2 (ΩT ) ∩ Cb (ΩT ) and Dt v − Av = f − ||f ||Cb (ΩT ) ≤ 0 on ΩT . Therefore, from Theorem 4.1.1(i) it follows that sup v = sup v ≤ sup u + T ||f ||Cb (ΩT ) . ΩT

ΓT

ΓT

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Since u ≤ v + T ||f ||Cb (ΩT ) on ΩT , we conclude that sup u ≤ sup u + 2T ||f ||Cb (ΩT ) . ΩT

ΓT

Applying this estimate to −u, (4.1.2) follows at once.



Without any condition on c, the weak maximum principle cannot hold true as the following example shows. Example 4.1.3 Consider the one dimensional operator A defined by Aψ = ψ 00 + ψ. The function u defined by u(t, x) = t cos(x) for every (t, x) ∈ (0, 1) × (−π/2, π/2) solves the inequality Dt u − Au ≥ 0, vanishes on ({0} × [−π/2, π/2)) ∪ {(0, 1) × {−π/2, π/2} but it does not identically vanish in (0, 1) × (−π/2, π/2). Assuming that c is nonpositive on Ω, the following maximum principle can be proved. Theorem 4.1.4 (Weak maximum principle II) Let c ≤ 0 in Ω and u ∈ C 1,2 (ΩT ) ∩ Cb (ΩT ). The following properties hold true: (i) if Dt u − Au ≤ 0 in ΩT , then sup u ≤ sup u+ ; ΓT

ΩT

(ii) if Dt u − Au ≥ 0 in ΩT , then inf u ≥ inf u− . ΓT

ΩT

In particular, if Dt u − Au = 0 in ΩT , then supΩT |u| = supΓT |u|. Proof We just need to adapt the arguments in the proof of Theorem 4.1.1. For this purpose, we limit ourselves to proving property (i). We consider the same function ϕ as in the quoted theorem and observe that we can estimate |Aϕ(x)| ≤ 2

d X j=1

||qjj ||∞ +

X d

||bj ||2∞

 12 |x| + ||c||∞ ϕ(x),

x ∈ Ω.

j=1

Therefore, also in this case we can determine a positive constant λ such that Aϕ < λϕ on Ω. From now on the proof follows the same lines as that of Theorem 4.1.1. We just notice that since c ≤ 0, Dt v − Av is nonpositive in ΩT , if v = u − supΓT u+ .  One of the first applications of the above weak maximum principle is the uniqueness of the classical solution to the parabolic problem  t ∈ (0, T ], x ∈ Ω,   Dt u(t, x) = Au(t, x) + g(t, x), u(t, x) = h(t, x), t ∈ (0, T ], x ∈ ∂Ω, (4.1.3)   u(0, x) = f (x), x ∈ Ω, where g ∈ Cb (ΩT ), h ∈ C((0, T ] × ∂Ω) and f ∈ C(Ω), and the potential c is bounded from above on Ω. Here by classical solution we mean a function u ∈ C 1,2 (ΩT ) ∩ Cb (ΩT ), which satisfies (4.1.3). Corollary 4.1.5 For every g ∈ Cb (ΩT ), h ∈ Cb ((0, T ] × ∂Ω) and f ∈ Cb (Ω) there exists at most one classical solution to problem (4.1.3). Moreover, there exists a positive constant CT such that ||u||Cb (ΩT ) ≤ CT (||f ||Cb (ΩT ) + ||h||Cb ((0,T ]×∂Ω) + ||g||Cb ((0,T ]×Ω) ).

(4.1.4)

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Elliptic and Parabolic Maximum Principles

Proof We first assume that c0 ≤ 0 and introduce the function v : ΩT → R defined by v(t, x) = u(t, x) − t||g||Cb (ΩT ) for every (t, x) ∈ ΩT . Clearly, v ∈ C 1,2 (ΩT ) ∩ Cb (ΩT ) and Dt v(t, x) − Av(t, x) ≤ c0 t||g||Cb (ΩT ) + g(t, x) − ||g||Cb (ΩT ) ≤ 0 for every (t, x) ∈ ΩT . Therefore, from Theorem 4.1.4(i) it follows that sup v ≤ sup v + ≤ sup |u| + T ||g||Cb (ΩT ) . ΩT

ΓT

ΓT

Since u ≤ v + T ||g||Cb (ΩT ) on ΩT , we conclude that sup u ≤ sup |u| + 2T ||g||Cb (ΩT ) ≤ ||f ||Cb (Ω) + ||h||Cb ((0,T ]×Ω) + 2T ||g||Cb (ΩT ) . ΩT

ΓT

Applying this estimate to −u, estimate (4.1.4) follows at once. If c0 > 0, then we set w(t, x) = e−c0 t u(t, x) for every (t, x) ∈ ΩT and observe that the function w solves the equation Dt w(t, x) = Aw(t, x) − c0 w(t, x) + e−c0 t g(t, x) for every (t, x) ∈ ΩT . Since c − c0 ≤ 0 on Ω, we can apply the first part of the proof to obtain e−c0 t u(t, x) ≤

sup (t,x)∈(0,T ]×Ω

e−c0 t |h(t, x)| + ||f ||Cb (ΩT ) + 2T

sup

e−c0 t |g(t, x)|

(t,x)∈ΩT

for every (t, x) ∈ Ω, so that u(t, x) ≤ ec0 t ||h||Cb ((0,T ]×Ω) + ec0 t ||f ||Cb (Ω) + 2T ec0 t ||g||Cb (ΩT ) ,

(t, x) ∈ ΩT .

Replacing u by −u, estimate (4.1.4) follows also in this case.



Also a comparison principle can be proved from the weak maximum principles. Corollary 4.1.6 (Comparison principle) Assume that f, g : ΩT × R → R are continuous functions of the variables t, x and y, which satisfy the following condition: f (t, x, y) − g(t, x, z) ≤ β(y − z),

(t, x) ∈ ΩT , y, z ∈ R,

for some β ∈ R. If u, v ∈ C 1,2 (ΩT ) ∩ Cb (ΩT ) satisfy the inequalities Dt u ≤ Au + f (t, x, u) and Dt v ≥ Av + g(t, x, v) in ΩT , and u ≤ v in ΓT , then u ≤ v in ΩT . Proof From the assumption, it follows immediately that Dt (u − v) ≤ A(u − v) + f (t, x, u) − g(t, x, v) ≤ A(u − v) + β(u − v). Next, we introduce the function w : ΩT → R, defined by w(t, x) := e−(c0 +β)t (u(t, x) − v(t, x)),

(t, x) ∈ ΩT ,

which belongs to C 1,2 (ΩT ) ∩ Cb (ΩT ) and satisfies the inequality Dt w ≤ Aw − c0 w in ΩT . As usually in this chapter, c0 denotes the supremum of c over Ω. Hence, since c − c0 ≤ 0, it follows from Theorem 4.1.4 and the assumption (u − v)+ = 0 in ΓT that sup w ≤ sup w+ = 0. ΩT

ΓT

Thus, u ≤ v in ΩT . This proves the claim.



In Corollary 4.1.5 we have proved the uniqueness of the classical solution of the CauchyDirichlet problem associated with the operator A. In the forthcoming chapters, we will consider also different sets of boundary conditions, such as Neumann boundary conditions. To prove the uniqueness of the classical solution to such problems, the following result, which is known as parabolic Hopf ’s lemma plays a crucial role.

Semigroups of Bounded Operators and Second-Order PDE’s

89

Proposition 4.1.7 Let Ω be a bounded domain of class C 2 . Further, suppose that u ∈ C 1,2 (ΩT ) ∩ Cb0,1 ((0, T ] × Ω) solves the differential inequality Dt u − Au ≤ 0 in ΩT and attains its maximum value only at some point (t0 , x0 ) ∈ (0, T ] × ∂Ω. Finally, suppose that c ≡ 0 and let η be a direction such that β = hη, ν(x0 )i > 0, where ν(x0 ) is the unit outward normal vector to ∂Ω at x0 . Then, the derivative of u(t0 , ·) at x0 in the direction η is positive. The same conclusion holds if c0 ≤ 0 and u(t0 , x0 ) ≥ 0. Proof Since Ω is a domain of class C 2 , it satisfies the interior sphere condition (see Proposition B.2.11). Thus, there exists r1 > 0 such that the ball B(x0 − r1 ν(x0 ), r1 ) is contained in Ω and x0 is the only point on the boundary of Ω, which belongs also to ∂B(x1 , r1 ), where x1 := x0 − r1 ν(x0 ). We also consider the ball B(x0 , r0 ) with r0 < r1 and denote by D the intersection of the two previous balls. We split the boundary of D into two parts ∂1 D and ∂2 D, where ∂1 D = ∂B(x0 , r0 ) ∩ B(x1 , r1 ) and ∂2 D = ∂B(x1 , r1 ) ∩ B(x0 , r0 ). Since u < u(t0 , x0 ) on [0, T ] × ∂1 D, which is a compact set, we can determine a positive constant γ such that u ≤ u(t0 , x0 ) − γ on [0, T ] × ∂1 D. On the other hand u ≤ u(t0 , x0 ) on [0, T ] × ∂2 D. 2 2 Next, we introduce the function ζ, defined by ζ(x) = e−α|x−x1 | −e−αr1 for every x ∈ Rd , where α > 0 to be properly chosen. As it is immediately seen, o n 2 2 1 Aζ(x) = 4α2 |Q(x) 2 (x−x1 )|2 −2α[Tr(Q(x))+hb(x), x−x1 i]+c(x) e−α|x−x1 | − c(x)e−αr1 for every x ∈ Ω. Thus, since c ≤ 0 on Ω and |x − x1 | ≥ r1 − r0 for every x ∈ D, we can estimate   d d X X 2 ||bj ||∞ − ||c||∞ e−α|x−x1 | ||qjj ||∞ − 2α(r1 − r0 ) Aζ(x) ≥ 4α2 µ(r1 − r0 )2 − 2α j=1

j=1

for every x ∈ D, so that we can make Aζ positive on D choosing α large enough. With this choice of α, the function wε = u + εζ satisfies the inequality Dt wε ≤ Awε on (0, T ] × D. Moreover, wε = u on [0, T ] × ∂2 D, for every ε > 0, and wε = u + εζ ≤ u(t0 , x0 ) − γ + 2 εe−α(r1 −r0 ) < u(t0 , x0 ) on [0, T ] × ∂1 D, provided that ε is chosen sufficiently small (say less than ε0 ). Thus, by Theorem 4.1.1 (if c ≡ 0) or Theorem 4.1.4 (if c0 ≤ 0 and u(t0 , x0 ) ≥ 0), for every ε ∈ (0, ε0 ) the function wε attains its maximum in D at (t0 , x0 ). We now observe that the point x0 +sη belongs to D if s ∈ (s0 , 0), where s0 = −r0 ∧(2r1 β). As a consequence, the function g : (s0 , 0] → R, defined by g(s) = wε (t0 , x0 + sη) for every s ∈ (s0 , 0], is differentiable and has a maximum point at s = 0. Hence, g 0 (0) ≥ 0, i.e., ∂u ∂ζ (t0 , x0 ) + ε (x0 ) ≥ 0. ∂η ∂η ∂ζ (x0 ) is negative. For this purpose, we split ∂η η = βν(x0 ) + η1 , where hη1 , ν(x0 )i = 0. Since ν(x0 ) = r1−1 (x0 − x1 ), we deduce that To conclude the proof, it suffices to show that

∂ζ (x0 ) =h∇ζ(x0 ), ηi = βh∇ζ(x0 ), ν(x0 )i + h∇ζ(x0 ), η1 i ∂η 2

2

= − 2αβhx0 − x1 , ν(x0 )ie−αr1 − 2αhx0 − x1 , η1 ie−αr1 2

2

= − 2αr1 (β + hν(x0 ), η1 i)e−αr1 = −2αβr1 e−αr1 and the claim follows.



90

Elliptic and Parabolic Maximum Principles

Corollary 4.1.8 Let Ω be a bounded domain of class C 2 . Further, suppose that u ∈ C 1,2 (ΩT ) ∩ Cb0,1 ((0, T ] × Ω) ∩ C(ΩT ) solves the differential inequality Dt u − Au ≤ 0 in ΩT . Further, assume that u(0, ·) is nonpositive on Ω and ∂u (t, x) + a(x)u(t, x) ≤ 0, ∂η

(t, x) ∈ (0, T ] × ∂Ω,

where a ∈ C(∂Ω) is nonnegative and η ∈ C(∂Ω, Rd ) satisfies the conditions |η(x)| = 1 and hν(x), η(x)i > 0 for every x ∈ ∂Ω. Then, u is nonpositive on [0, T ] × Ω. Note that we are not assuming that c is nonpositive on Ω. Proof Let w : [0, T ] × Ω → R be the function defined by w(t, x) = e−(c0 +1)t u(t, x) for every (t, x) ∈ [0, T ] × Ω. Clearly, w belongs to C 1,2 (ΩT ) ∩ Cb0,1 ((0, T ] × Ω) ∩ C(ΩT ). Moreover, Dt w ≤ A0 w on (0, T ] × D, where A0 = A − (c0 + 1)I has a negative potential term. Clearly, w cannot admit a positive maximum on ΩT , since at such point A0 w would be null and Dt w would be positive, contradicting the condition Dt v − A0 v ≤ 0. A positive maximum can neither be attained at (0, T ] × ∂Ω. Indeed, in view of Proposition 4.1.7 the derivative of w along the direction η at such point would be positive contradicting the ∂w ≤ 0 on (0, T ] × ∂Ω. Thus, the maximum of w on ΩT (i.e., the maximum condition aw + ∂η of u) is nonpositive and the claim follows.  Proposition 4.1.9 Suppose that Ω is either Rd+ or a domain of class C 2 . Further, suppose that u ∈ C 1,2 (ΩT ) ∩ Cb0,1 (ΩT ) solves the Cauchy problem  Dt u(t, x) = Au(t, x) + g(t, x),    ∂u a(x)u(t, x) + (t, x) = h(t, x),  ∂η   u(0, x) = f (x),

t ∈ (0, T ],

x ∈ Ω,

t ∈ (0, T ],

x ∈ ∂Ω, x ∈ Ω,

where f ∈ Cb (Ω), g ∈ Cb ((0, T ] × Ω), a ≥ 0 on ∂Ω and inf x∈∂Ω hη(x), ν(x)i > 0. Here, ν(x) denotes the unit exterior normal vector to ∂Ω at x. Then, there exists a positive constant C such that ||u||Cb (ΩT ) ≤ C(||f ||Cb (Ω) + ||h||Cb ((0,T ]×∂Ω) + ||g||Cb ((0,T ]×Ω) ).

(4.1.5)

+

We can take C = 2e(c0 +λ)T [1 ∨ (inf ∂Ω hη, νi)−1 ∨ T ], where the constant λ is defined by Pd λ = 2 j=1 ||qjj ||∞ + |||b|||∞ + ||c||∞ . In particular, if Ω is bounded, then we can take λ = 0. Proof We split the proof into some steps. Step 1. Here, we consider the easiest case when Ω is bounded, c ≤ 0 on Ω and a ≥ a0 on ∂Ω for some positive constant a0 . We introduce the function w : ΩT → R, defined by w(t, x) = u(t, x) − t||g||∞ for every (t, x) ∈ ΩT and observe that it satisfies the differential inequality Dt w(t, x) =Dt u(t, x) − ||g||∞ = Au(t, x) + g(t, x) − ||g||∞ =Aw(t, x) + tc||g||∞ + g(t, x) − ||g||∞ ≤ Aw(t, x) for every (t, x) ∈ ΩT since c ≤ 0. Moreover, Bw(t, x) := a(x)w(t, x) +

∂w (t, x) = h(t, x) − ta(x)||g||∞ =: e h(t, x) ∂η

Semigroups of Bounded Operators and Second-Order PDE’s

91

for every (t, x) ∈ (0, T ] × ∂Ω and w(0, ·) = f . Clearly, |w| is continuous on ΩT . Hence it attains its maximum value at some point (t0 , x0 ) ∈ ΩT . Suppose that ||w||∞ = w(t0 , x0 ). Then, from Theorem 4.1.4, we deduce that ||w||∞ = maxΓT w so that t0 = 0 or t0 > 0 and ∂w x0 ∈ ∂Ω. In the second case, using Proposition 4.1.7, it follows that (t0 , x0 ) > 0 and, ∂η since a ≥ a0 > 0 on ∂Ω, we can estimate ∂w ||e h||∞ ≥ e h(t0 , x0 ) = a(x0 )w(t0 , x0 ) + (t0 , x0 ) > a0 w(t0 , x0 ), ∂η −1 e i.e., ||w||∞ ≤ a−1 0 ||h||∞ ≤ a0 ||h||∞ +T ||g||∞ . On the other hand, if t0 = 0, then ||w||∞ ≤ ||f ||∞ . −1 Therefore, ||w||∞ ≤ ||f ||∞ + a−1 0 ||h||∞ + T ||g||∞ , so that (4.1.5) follows with C = 1 ∨ a0 ∨ T . Arguing similarly, we obtain the same estimate when ||w||∞ = −w(t0 , x0 ). Step 2. If a ≥ a0 > 0 on Ω (bounded) but c is not everywhere nonpositive on Ω, we introduce the function v : ΩT → R, defined by v(t, x) = e−c0 t u(t, x) for every (t, x) ∈ ΩT . Function v belongs to C(ΩT ) ∩ C 1,2 (ΩT ) and solves the Cauchy problem  e Dt v(t, x) = Av(t, x) + gc0 (t, x), t ∈ (0, T ], x ∈ Ω,     ∂v a(x)v(t, x) + (t, x) = hc0 (t, x), t ∈ (0, T ], x ∈ ∂Ω,  ∂η    u(0, x) = f (x), x ∈ Ω,

where Ae = A − c0 I, gc0 (t, x) = e−c0 t g(t, x) for every (t, x) ∈ (0, T ] × Ω and hc0 (t, x) = e−c0 t h(t, x) for every (t, x) ∈ (0, T ] × ∂Ω. By the first part of the proof, we infer that ||v||Cb (ΩT ) ≤ (1 ∨ a−1 0 ∨ T )(||f ||Cb (Ω) + ||hc0 ||Cb ((0,T ]×∂Ω) + ||gc0 ||Cb ((0,T ]×Ω) ) c0 T and we obtain (4.1.5), with C = ec0 T (1 ∨ a−1 ||v||∞ , 0 ∨ T ), observing that ||u||∞ ≤ e ||hc0 ||∞ ≤ ||h||∞ and ||gc0 ||∞ ≤ ||g||∞ . Step 3. Now, we remove the condition a ≥ a0 on ∂Ω and we just assume that a is nonnegative. We introduce the function φ : Ω → R defined by φ(x) = 1 − (2 + ϑ(x)d(x))−1 for every x ∈ Ω, where d(x) = d(x, ∂Ω) and ϑ is a smooth function compactly supported in {x ∈ Rd : d(x) ≤ δ} and δ is sufficiently small to guarantee that the function d(·) is of class C 2 in {x ∈ Ω : d(x) ≤ δ} (see Proposition B.2.14). Moreover, ϑ(x) ∈ [0, 1] for every x ∈ Rd and ϑ ≡ 1 in a neighborhood of ∂Ω. As it is immediately seen, φ(x) ≥ 1/2 for every x ∈ Ω. Moreover, φ ∈ C 2 (Ω) and ∇φ(x) = −ν(x)/4 on ∂Ω. Let us set v(t, x) = u(t, x)φ(x) for every t ∈ [0, T ] and x ∈ Ω. Such a function belongs to C([0, T ] × Ω) ∩ Cb0,1 ((0, T ] × Ω) ∩ C 1,2 ((0, T ) × Ω) and a straightforward computation reveals that  b Dt v(t, x) = Av(t, x) + φg(t, x), t ∈ (0, T ), x ∈ Ω,     ∂v 1 [a + 2−1 hν(x), η(x)i]v + = h(t, x), t ∈ (0, T ), x ∈ ∂Ω,  ∂η 2    v(0, x) = f (x)φ(x), x ∈ Ω,

where Ab = Tr(QD2 ) + hbb, ∇i + b c and bb = b − 2 hQ∇φ, ∇φi, φ

b c=c−

1 2 (Aφ − cφ) + 2 hQ∇φ, ∇φi. φ φ

Due to the condition hη(x), ν(x)i > 0 for every x ∈ ∂Ω, the infimum of the function

92

Elliptic and Parabolic Maximum Principles

hη, νi over ∂Ω is positive, so that the infimum of the function a + 2−1 hν, ηi over ∂Ω is positive as well. We can thus apply Step 2 and conclude that ||v||Cb (ΩT ) ≤2e

c+ 0 T

  −1  1 ∨ inf hν, ηi ∨ T (||φf ||∞ + ||h||∞ + ||φg||∞ ) ∂Ω

  −1  + ≤2ec0 T 1 ∨ inf hν, ηi ∨ T (||f ||∞ + ||h||∞ + ||g||∞ ). ∂Ω

Since |u| ≤ 2|v|, estimate (4.1.5) follows with C = 4e

c+ 0 T

   −1 1 ∨ inf Ω hν, ηi ∨T .

Step 4. Finally, we consider the case when Ω = Rd+ . Thanks to the same argument as above, we can assume that c ≤ 0 on Ω and a ≥ a0 > 0 (in this case, we can take φ(x) = 1 − (xd + 2)−1 for every x ∈ Rd+ ), so we confine ourselves to this case. The main difference is when the supremum of u over [0, T ] × Rd+ is not a maximum. If ||u||∞ = sup(0,T )×Rd+ u, then we consider the function ϕ, defined by ϕ(x) = 1 + |x|2 for every x ∈ Rd . As it has been shown in the proof of Theorem 4.1.1 (see (4.1.1)), there exists a positive constant λ such that Aϕ ≤ λϕ on Rd+ . Moreover, Bϕ is bounded from below on Rd+ by a positive constant β0 . Therefore, the function wn , defined by wn (t, x) = e−λt u(t, x) − n−1 ϕ(x) for every (t, x) ∈ [0, T ] × Rd+ , satisfies the problem  Dt wn (t, x) ≤ Awn (t, x) − λwn (t, x) + gλ (t, x),     ∂wn 1 a(x)wn (t, x) + (t, x) ≤ hλ (t, x) − β0 ,  ∂η n    wn (0, x) ≤ f (x),

t ∈ (0, T ],

x ∈ Rd+ ,

t ∈ (0, T ],

x ∈ ∂Rd+ , x ∈ Rd+ ,

where gλ and hλ are defined as the functions gc0 and hc0 , replacing c0 by λ. Since the potential term of the operator A − λ is nonpositive and wn attains its maximum over [0, T ]×Rd+ , due to the fact that lim|x|→∞ wn (t, x) = −∞ uniformly with respect to t ∈ [0, T ], we conclude, as in Step 1, that   β0 −1 c+ T 0 wn (t, x) ≤ 2e (1 ∨ ηd ∨ T ) ||f ||∞ + ||g||∞ + ||h||∞ + n for every (t, x) ∈ (0, T ) × Rd+ and n ∈ N. Letting n tend to ∞ gives +

u(t, x) ≤ 2e(c0 +λ)T (1 ∨ ηd−1 ∨ T )(||f ||∞ + ||g||∞ + ||h||∞ ) for every (t, x) ∈ (0, T ) × Rd+ , and the assertion follows. Note that estimate (4.1.1) shows Pd that we can take λ = 2 j=1 ||qjj ||∞ + |||b|||∞ + ||c||∞ . Finally, if ||u||∞ = sup(0,T )×Rd+ (−u), then we can apply the same argument with u being replaced by −u. 

4.1.2

The strong maximum principle

To prove the strong maximum principle for the parabolic operator Dt − A, we take advantage of the following easy preliminary result. In this subsection we assume that Ω is a bounded domain. Lemma 4.1.10 For each pair of points x0 , x1 ∈ Ω there exists a polygonal, with support contained in Ω, joining x0 and x1 .

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Proof Let x0 and x1 be as in the statement of the lemma. Since Ω is open and connected, we can determine a curve γ : [0, 1] → Ω such that γ(0) = x0 and γ(1) = x1 . Clearly, for each r > 0, less than the distance of γ([0, 1]) from ∂Ω, the family of balls B(γ(t), r) is an open covering of γ([0, 1]), which is compact in Ω. Thus, there exist 0 = t0 < . . . < tn = 1 ∈ [0, 1] such that n [

γ([0, 1]) ⊂

B(γ(tj ), r) b Ω.

j=0

We can construct a polygonal contained in the union of the previous balls easily as follows. For each i = 1, . . . , n − 1 we select a point yi ∈ B(γ(ti ), r) ∩ B(γ(ti+1 ), r) and then consider the polygonal with vertices at x0 , y1 , . . . yn−1 , x1 . This polygonal is clearly contained in the previous union of balls and, hence, in Ω.  Theorem 4.1.11 (Strong maximum principle in the case c ≡ 0) Let u ∈ C 1,2 (ΩT )∩ Cb (ΩT ) be such that Dt u − Au ≤ 0 (resp. Dt u − Au ≥ 0) in ΩT . If there exists a point x0 ∈ Ω such that u(T, x0 ) = supΩT u (resp. u(T, x0 ) = inf ΩT u), then u is constant in ΩT . Proof The proof that we present follows the same lines as in [32]. Since it is rather long and technical, we split the proof into two steps. Clearly, it suffices to consider the case when Dt u − Au ≤ 0 on ΩT , since the other one follows by applying this case to the function −u. To prove that u is constant in ΩT , the main step (Step 1) consists in showing that, if u attains its maximum value at some point (t∗ , x∗ ) ∈ (0, T ) × Ω, then u is constant on the set {t∗ } × Ω. Here, the weak maximum principle will be of great help. Throughout the proof, we denote by M the supremum over ΩT of the function u. Step 1. Here, we prove that if u(t∗ , x∗ ) ∈ (0, T )×Ω and u(t∗ , x∗ ) = M , then u(t∗ , x) = M for each x ∈ Ω. Thanks to Lemma 4.1.10, it suffices to prove that u(t∗ , ·) is constant on every segment, contained in Ω, having x∗ as endpoint. We argue by contradiction assuming that there exists x0 ∈ Ω such that u(t∗ , x0 ) < M and xs = (1 − s)x0 + sx∗ ∈ Ω for every s ∈ [0, 1]. We denote by s∗ the smallest value of s such that u(t∗ , xs∗ ) = M and, by d0 , any positive number less then the minimum between |xs∗ − x0 | and the minimum of the distances of the point (t∗ , xs ) from ∂ΩT , when s ∈ [0, s∗ ]. We finally denote by sˆ the unique value of the parameter s such that |xsˆ − xs∗ | = d0 . We can now introduce the function R : (ˆ s, s∗ ) → R, defined as follows: R(s) is the distance of (t∗ , xs ) from the closest point in (0, T ) × Ω, where the value of u is exactly M . By definition of s∗ , u(t∗ , xs ) is less than M for every s ∈ [0, s∗ ). Moreover, R(s) ≤ |xs − xs∗ | for every s ∈ (ˆ s, s∗ ) so that the range of R is contained in (0, d0 ). As a byproduct, the ball B((t∗ , xs ), R(s)) is compactly contained in (0, T ) × Ω for every s ∈ (ˆ s, s∗ ) and on its boundary there exists a point where u attains its maximum value M . We claim that this point is (t∗ − R(s), xs ) or (t∗ + R(s), xs ). To prove the claim we argue by contradiction, assuming that the maximum is attained at some point, (t, x), with x 6= xs . Up to replacing B((t∗ , xs ), R(s)) with a smaller ball (tangent to B((t∗ , xs ), R(s)) at (t∗ , x)), if necessary, we can assume that this is the only point where u attains its maximum value in B((t∗ , xs ), R(s)). We introduce the ball B((t, x), R1 ) ⊂ (0, T ) × Ω, with R1 < |xs − x|, and the function v1 : Rd+1 → R, defined by 2

v1 (t, x) = e−α|x−xs |

−α(t−t∗ )2

− e−αR

2

(s)

=: vˆ1 (t, x) − e−αR

2

(s)

,

(t, x) ∈ Rd+1 ,

where α is a positive constant to be fixed to guarantee that Dt v1 − Av1 is negative in B((t, x), R1 ). This can be done, observing that Dt v1 − Av1 = 2αψˆ v1 in B((t, x), R1 ), where ψ(t, x) = −2αhQ(x)(x − xs ), x − xs i − t + t∗ + Tr(Q(x)) + 2αhb(x), x − xs i

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for each (t, x) ∈ Rd+1 . Taking into account Hypothesis 4.0.1 and observing that R(s)−R1 ≤ |x − xs | ≤ R(s) + R1 for every (t, x) ∈ B((t, x), R1 ), we can easily check that α can be fixed sufficiently large to guarantee that ψ < 0 in B((t, x), R1 ). Now, for each ε > 0 we introduce the function wε = u + εv1 , which clearly satisfies the inequality Dt wε − Awε < 0 in B(t, x), R1 ). Moreover, on the part of the boundary of B((t, x), R1 ) which is outside B((t∗ , xs ), R(s)), v1 is negative and u is less than M . Therefore, wε is strictly less than M on this part of ∂B((t, x), R1 ). On the other hand, u is less than M on the closure of the part of ∂B((t, x), R1 ) which lies inside B((t∗ , xs ), R(s)). Therefore, we can determine M1 < M such that u ≤ M1 on this set. It is clear that ε can be fixed to guarantee that wε < M on this part of ∂B((t, x), R1 ). Summing up, wε is less than M on the boundary of B((t, x), R1 ) and it attains value M on the center of the ball. This contradicts Theorem 4.1.1. We have so proved that the maximum of u on the closure of the ball B((t∗ , xs ), R(s)) is attained at (t∗ − R(s), xs ) or at (t∗ + R(s), xs ). Hence, now we can estimate p R(s1 ) ≤|(t∗ , xs1 ) − (t∗ + R(s2 ), xs2 )| = (R(s2 ))2 + |xs2 − xs1 |2 ≤R(s2 ) +

(s2 − s1 )2 |x∗ − x0 |2 |xs2 − xs1 |2 = R(s2 ) + 2R(s2 ) 2R(s2 )

(4.1.6)

for each s1 , s2 ∈ (ˆ s, s∗ ). From the first line of the previous chain of inequalities we also deduce that p R(s2 ) ≥ (R(s1 ))2 − |xs2 − xs1 |2 (4.1.7) for the same values of s1 and s2 . We now assume that s2 > s1 and split the interval [s1 , s2 ] into n subintervals each of them with length (s2 − s1 )/n. To ease the notation, we set τj = s1 + n−1 j(s2 − s1 ) for each j = 0, . . . , n. From (4.1.6) and (4.1.7) it follows that R(s2 ) − R(s1 ) =

n−1 X

[R(τj+1 ) − R(τj )] ≥ −(s2 − s1 )2 |x∗ − x0 |2

j=0

n−1 1 1 X 2n2 j=0 R(τj+1 )

≥ − (s2 − s1 )2 |x∗ − x0 |2

1 1 [R(s1 )2 − n(s2 − s1 )2 |x∗ − x0 |2 ]− 2 . 2n

R(s2 ) − R(s1 ) ≤ (s2 − s1 )2 |x∗ − x0 |2

1 1 [R(s1 )2 − n(s2 − s1 )2 |x∗ − x0 |2 ]− 2 , 2n

Similarly,

so that, letting n tend to ∞ in the previous two formulas, we conclude that R(s2 ) = R(s1 ), so that R is constant. This and the inequality R(s) ≤ |x∗ − xs | which holds for each s ∈ (ˆ s, s∗ ), show that R identically vanishes in (ˆ s, s∗ ) which is the contradiction we were looking for. Step 2. Now, we complete the proof. Again, we argue by contradiction and assume that there exists (t0 , y0 ) ∈ (0, T ) × Ω such that u(t0 , y0 ) < M . Then, by Step 1, u(t0 , x) < M for every x ∈ Ω. In particular, u(t0 , x0 ) < M . We consider the set {t ∈ [t0 , T ] : u(t, x0 ) < M } and denote by τ its supremum. By continuity u(τ, x0 ) = M . We then fix R small enough such that B− ((τ, x0 ), R) := B((τ, x0 ), R) ∩ {(t, x) ∈ Rd+1 : t ≤ τ } is contained in ΩT and observe that in B− ((τ, x0 ), R) there exist no points where u attains its maximum value M . Otherwise, again from Step 1, we would conclude that u(t, x0 ) = M for some t < τ , contradicting the definition of τ . Next, we introduce the function v2 : Rd+1 → R defined by 2

v2 (t, x) = e−α(t−τ )−|x−x0 | − 1,

(t, x) ∈ Rd+1 ,

where α is fixed so large to ensure that Dt v2 − Av2 < 0 on B− ((τ, x0 ), R).

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We now consider the compact set K which is the intersection of B− ((τ, x0 ), R) and the set {(t, x) ∈ Rd+1 : α(t − τ ) + |x − x0 |2 ≤ 0}. Note that (τ, x0 ) belongs to the boundary of K, which splits into two parts: one contained in the boundary of B− ((τ, x0 ), R) and the other contained in the boundary of {(t, x) ∈ Rd+1 : α(t − τ ) + |x − x0 |2 ≤ 0}. We call ∂1 K and ∂2 K, respectively, the closure of these two parts of ∂K. Since u < M on ∂1 K and v2 vanishes on ∂2 K, arguing as in Step 1 we can choose ε > 0 small enough such that the function w = u + εv2 is less than M on ∂K. Since Dt w − Aw < 0 inside K, the weak maximum principle in Theorem 4.1.1 reveals that w < M inside K. It thus follows that (τ, x0 ) is the maximum point of u on K. Therefore Dt w(τ, x0 ) ≥ 0, which means that Dt u(τ, x0 ) ≥ −εDt v2 (τ, x0 ) = αε > 0. Since u attains its maximum positive value at (τ, x0 ), it follows that Au(τ, x0 ) ≤ 0 so that Dt u(τ, x0 ) − Au(τ, x0 ) > 0, contradicting the assumption.  Theorem 4.1.12 (Strong maximum principle in the case c 6≡ 0) Let Ω be a bounded domain of Rd . Let u ∈ C 1,2 (ΩT ) ∩ Cb (ΩT ) be such that Dt u − Au ≤ 0 (resp. Dt u − Au ≥ 0) in ΩT and c ≤ 0 (resp. c ≥ 0) on Ω . If there exists a point x0 ∈ Ω such that 0 ≤ u(T, x0 ) = supΩT u (resp. 0 ≥ u(T, x0 ) = inf ΩT u), then u is constant in ΩT . Proof We follow the same lines as in the proof of Theorem 4.1.11. For this reason, we just sketch the proof, pointing out the main differences. Clearly, it is enough to consider the case when c ≤ 0 and u attains its positive maximum at (T, x0 ). Step 1 in the proof of Theorem 4.1.11 can be repeated verbatim. The only difference is that now 2

Dt v1 − Av1 = (2αψ − c)ˆ v1 + ce−αR ≤ (2αψ − c)ˆ v1 and, since c is bounded, we can fix α sufficiently large such that 2αψ − c is negative in B((t, x), R1 ). Moreover, to get to a contradiction we have to use the Theorem 4.1.4(i), which shows that inside B((t, x), R1 ) function wε cannot attain a positive maximum. Similar changes are required to Step 2.  Corollary 4.1.13 Suppose that u ∈ C 1,2 (ΩT ) ∩ Cb (ΩT ) solves the differential equation Dt u = Au in ΩT , is nonnegative on ΓT and it does not identically vanish in any Γs , s ∈ (0, T ]. Then, u is strictly positive in ΩT . Proof Of course, it suffices to consider the case when the potential c of operator A is nonpositive on Ω. Indeed, in the general case, it suffices to replace u with the function v : ΩT → R, defined by v(t, x) = e−c0 t u(t, x) for every (t, x) ∈ ΩT , as in the proof of Corollary 4.1.5. So, let us suppose that c ≤ 0 in Ω. Since u is nonnegative on ΓT , it follows from Theorem 4.1.4 that u is nonnegative on ΩT . Let us prove now that u is strictly positive in ΩT . We argue by contradiction and assume that there exists (t0 , x0 ) ∈ ΩT such that u(t0 , x0 ) = 0. So, by Theorem 4.1.12, u identically vanishes on Ωt0 . But this would imply that u identically vanishes on Γt0 , which contradicts our assumptions.  In the case when A is the Laplacian, we can provide a slightly different proof of the strong maximum principle assuming that u is bounded and continuous in (0, T ] × Ω. Alternative proof of Theorem 4.1.11 We fix u ∈ C 1,2 (ΩT ) ∩ Cb ((0, T ] × Ω) such that Dt u − ∆u ≤ 0 in ΩT and consider the function v = supΩT u − u. The main step of the proof consists in showing that Z r2 ρ(t, x − x0 )v(t, x) dx ≤ v(T, x0 ), if 0 < T − t ≤ , (4.1.8) 2d B(x0 ,r0 )

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for every x0 ∈ Ω and every r0 > 0 such that B(x0 , r0 ) ⊂ Ω and (2d)−1 r02 < T . Here, ρ(t, x) := K(T −t, x)−K(T −t, r0 e1 ) for (t, x) ∈ [0, T )×Rd and K is the Gauss-Weierstrass kernel d

K(t, x) := (4πt)− 2 e−

|x|2 4t

,

t > 0, x ∈ Rd ,

which we study in great details in Chapter 5. For this purpose, we observe that, for each ϕ ∈ C(Ω), a change of variables reveals that Z Z √ |y|2 1 K(T − t, x − x0 )ϕ(x) dx = e− 4 ϕ(x0 + y T − t) dy d (4π) 2 Br0 /√T −t (0) B(x0 ,r0 ) Z √ |y|2 1 = χBr /√T −t (0) (y)e− 4 ϕ(x0 + y T − t) dy. d 0 (4π) 2 Rd So, by applying the dominated convergence theorem, we deduce that Z K(T − t, x)ϕ(x) dx = ϕ(x0 ). lim t→T

(4.1.9)

B(x0 ,r0 )

Since u(t, ·) converges to u(T, ·) when t → T , and limt→T K(T − t, r0 ) = 0, it follows by (4.1.9) that Z ρ(t, x)v(t, x) dx = v(T, 0). lim ζ(t) := lim t→T

t→T

B(x0 ,r0 )

Next, we observe that ρ ∈ C ∞ ([0, T )×Ω), is positive in [0, T )×Ω, and vanishes on ∂Ω×[0, T ). Moreover, it can be checked that (Dt + ∆)K(T − t, x) = 0 in (0, T ) × Rd (see Chapter 5 for further details) and (Dt + ∆)ρ(t, x) = [r02 − 2d(T − t)]

K(T − t, r0 e1 ) ≥0 4(T − t)2

for every t ∈ [T − (2d)−1 r, T ) and x ∈ Rd . Thus, for such values of t, differentiating under the integral sign we can show that Z ζ 0 (t) = (v(t, x)Dt ρ(t, x) + ρ(t, x)Dt v(t, x)) dx B(x0 ,r0 ) Z ≥ (ρ(t, x)∆v(t, x) − v(t, x)∆ρ(t, x)) dx. B(x0 ,r0 )

To prove the last inequality we have used the positivity of v and ρ, and the inequalities Dt ρ ≥ −∆ρ, Dt v ≥ ∆v. Using Green’s formula, we can continue the previous chain of inequalities and prove that   Z ∂v ∂ρ 0 ζ (t) ≥ ρ(t, σ) (t, σ) − v(t, σ) (t, σ) dσ ∂ν ∂ν ∂B(x0 ,r0 ) Z ∂ρ ≥− v(t, σ) (t, σ) dσ ≥ 0, ∂ν ∂B(x0 ,r0 ) since the normal derivative of ρ is nonpositive on ∂B(x0 , r0 ). Therefore, the function ζ is nondecreasing in [T − (2d)−1 r02 , T ). This implies that v(T, x0 ) = lim ζ(t) ≥ ζ(t) for each t ∈ [T − (2d)−1 r02 , T ) and (4.1.8) follows.

t→T

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We can now complete the proof of the theorem. For this purpose, we suppose that there exists a point x0 ∈ Ω such that u(T, x0 ) = supΩT u and prove that u is constant in ΩT . Let us introduce the set ET = {x ∈ Ω : v(T, x) = 0}. It is clear that ET is closed and not empty. We prove that ET is also open. For this purpose fix x ∈ ET and a ball B(x0 , r0 ) ⊂ Ω. Applying (4.1.8), we deduce that Z r2 0≤ ρ(t, y − x0 )v(t, y) dy ≤ v(T, x) = 0 if 0 < T − t ≤ 0 . 2d B(x0 ,r0 ) Hence, ρ(t, y − x0 )v(t, y) = 0, if y ∈ B(x0 , r0 ) and 0 < T − t ≤

r02 . 2d

Since ρ(t, ·) is positive in B(x0 , r0 ), it follows that w(t, y) = 0 for each y ∈ B(x0 , r0 ) and 0 < T −t0 ≤ (2d)−1 r02 . By continuity, we deduce that v(T, y) = 0 in B(x0 , r0 ), which implies that B(x0 , r0 ) ⊂ ET . Since Ω is connected, we conclude that ET = Ω. The above argument also shows that w(t, ·) = 0 for every t ∈ [T − (2d)−1 r02 , T ]. Iterating k-times this procedure, with k such that (2d)−1 kr02 ≥ T , we obtain that v identically vanishes in ΩT .  Remark 4.1.14 If u(t0 , x0 ) = supΩT u (or u(t0 , x0 ) = inf ΩT u) for some (t0 , x0 ) ∈ ΩT , then we can only conclude that u is constant in (0, t0 ] × Ω even in the case when Dt u − ∆u = 0 and u ∈ C(ΩT ). For example, assume that u ∈ C 1,2 ((0, T ] × (0, 1)) ∩ C([0, T ] × [0, 1]) solves the heat equation  Dt u(t, x) = ∆u(t, x), t ∈ (0, T ], x ∈ (0, 1),     u(t, 1) = β(t), t ∈ (0, T ],  u(t, 0) = α(t), t ∈ (0, T ],    u(0, x) = 0, x ∈ (0, 1). We will prove later that a classical solution to the above heat equation exists. Assume that α(t) = β(t) = 0 in (0, t0 ] for some t0 ∈ (0, T ) and α(t), β(t) > 0 in (t0 , T ]. Then u(t, x) = 0 in (0, t0 ] × (0, 1), by Corollary 4.1.5. We claim that u(t, x) > 0 for each x ∈ (0, 1) and t ∈ (t0 , T ]. By the weak maximum principle, we know that u(t, x) ≥ 0 in ΩT . If there existed (t1 , x1 ) ∈ (t0 , T ] × (0, 1) such that u(t1 , x1 ) = 0 (= inf ΩT u), then by the strong maximum principle u would identically vanish in (0, t1 ] × (0, 1) and this contradicts the fact that u(t, 0) = α(t) > 0 in (t0 , T ].

4.2

Elliptic Maximum Principles

In this section, we prove some maximum principles for the elliptic operator A. The proofs are straightforward consequences of the results of the previous section if deal with functions u ∈ C 2 (Ω) ∩ C(Ω). Indeed, if u satisfies the inequality Au ≥ 0 (resp. Au ≤ 0) then, clearly, Dt u − Au ≤ 0 (resp. Dt u − Au ≥ 0). As a matter of fact, in general, the solutions of the elliptic equations that we consider in this book are less regular than C 2 (Ω). So, we need to establish weak and strong maximum principles also for functions which are less smooth. Going through the proof of the parabolic maximum principles it comes out that the main

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tool to prove them is the fact that if c identically vanishes in Ω (resp. c ≤ 0 in Ω) then at any local maximum (resp. interior positive maximum) x0 ∈ Ω of a smooth function v, Av(x0 ) ≤ 0. This result can be extended to a more general class of functions as described by the following proposition. 2,p Proposition 4.2.1 Let Ω be a domain of Rd . Suppose that u ∈ Wloc (Ω), for every p ∈ [1, ∞), and that Au ∈ C(Ω). Then, the following properties are satisfied.

(i) If c ≡ 0 and x0 ∈ Ω is a local maximum (resp. minimum) of u, then Au(x0 ) ≤ 0 (resp. Au(x0 ) ≥ 0). (ii) If c is nonpositive in Ω and x0 ∈ Ω is a local negative minimum (resp. local positive maximum) of u, then Au(x0 ) ≥ 0 (resp. Au(x0 ) ≤ 0). Proof We limit ourselves to proving property (ii) in the case when u has a local positive maximum at x0 . The other cases are similar and even simpler. Let us fix a function ψ ∈ Cc∞ (Rd ) with support in B(x0 , r) and such that ψ(x0 ) = 1, ∇ψ(x0 ) = 0, D2 ψ(x0 ) = 0 and 0 ≤ ψ(x) < 1 for each x ∈ B(x0 , r) \ {x0 }. As it is easily seen, x0 is the unique maximum point of the function v := ψu, and Av ∈ C(Ω). Moreover, since ∇ψ(x0 ) = 0 and D2 ψ(x0 ) = 0, we deduce that Au(x0 ) = Av(x0 ). Now, let ϑ ∈ Cc∞ (Rd ) be a smooth function, supported in B(0, 1), such that 0 ≤ ϑ(x) ≤ 1 for every x ∈ Rd and ||ϑ||L1 (Rd ) = 1. Further, fix n0 ∈ N such that B(x0 , r + n−1 0 ) ⊂ Ω. For each n ≥ n0 and x ∈ Rd , we set ϑn (x) = nd ϑ(nx) and vn = ϑn ? v ∈ Cc∞ (Ω), where “?” denotes the convolution operator. Since v ∈ Cc (Ω), vn converges to v, uniformly in B(x0 , r + n−1 0 ), as n tends to ∞. Moreover, for every n as above, vn attains its maximum at some point xn ∈ B(x0 , r + n−1 ) so that Avn (xn ) ≤ 0. A straightforward computation reveals that Avn (x) = (ϑn ? Av)(x) + (ϑn ? Tr((Q(x) − Q(·))D2 v))(x) + (ϑn ? hb(x) − b(·), ∇vi)(x) + (ϑn ? (c(x) − c(·))v)(x) =: (ϑn ? Av)(x) + ψn (x) for each x ∈ B(x0 , r + n−1 ). Since Av ∈ Cc (Ω), ϑn ? Av tends to Av uniformly in B(x0 , r + n−1 0 ) as n tends to ∞. Moreover, ψn vanishes as n tends to ∞, uniformly with respect to x ∈ B(x0 , r + n−1 0 ). For simplicity, we just consider the term (ϑn ? (c(x) − c)v)(x) (the other ones being completely similar) and observe that, since c is uniformly continuous on every compact subset of Ω, for each ε > 0 we can determine n sufficiently large such that |c(x) − c(x − n−1 y)| ≤ ε for every x ∈ B(x0 , r + n−1 0 ) and y ∈ B(0, 1). Thus, for every −1 x ∈ B(x0 , r + n0 ) and n ∈ N sufficiently large we can estimate Z |(ϑn ? ((c(x) − c)v)(x)| ≤ |ϑ(y)||c(x) − c(x − n−1 y)||v(x − n−1 y)| dy B(0,1) Z ≤ε |v(x − n−1 y)| dy ≤ ε||v||L1 (B(0,r+n−1 +n−1 )) . 0

B(0,1)

This shows that supB(x0 ,r+n−1 ) |(ϑn ? ((c(x) − c)v))(x)| vanishes as n tends to ∞. 0 We have so proved that vn and Avn converge, respectively, to v and Av uniformly in B(x0 , r + n−1 0 ) as n tends to ∞. Since (xn ) is a bounded sequence, up to a subsequence (xn ) converges to some point x ˆ ∈ B(x0 , r) as n tends to ∞. By continuity, v(xn ) tends to v(ˆ x) as n tends to ∞. Hence, v(ˆ x) = lim v(xn ) = lim vn (xn ) = lim ||vn ||∞ = ||v||∞ = v(x0 ). n→∞

n→∞

n→∞

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99

Recalling that x0 is the unique point where v attains its maximum value, it follows that x ˆ = x0 . Since Avn (xn ) ≤ 0 for each n ∈ N and it converges uniformly to Av, we conclude that Au(x0 ) = Av(x0 ) ≤ 0.  Corollary 4.2.2 The following properties are satisfied. (i) If the function u ∈ Cb (Rd ) ∩ C 2 (Rd ) solves the equation λu − Au = f for some f ∈ Cb (Rd ) and λ > c0 , then, ||u||∞ ≤ (λ − c0 )−1 ||f ||∞ . (ii) If the function u ∈ Cb (Rd+ ) ∩ C 2 (Rd+ ) solves the equation λu − Au = f for some f ∈ Cb (Rd+ ) and λ > c0 , then, ||u||∞ ≤ (λ − c0 )−1 (||f ||∞ + C||u||Cb (∂Rd+ ) ), where the constant C ≥ 1 is independent of u and λ. Proof (i) Without loss of generality, we can assume that ||u||∞ = supRd u. Indeed, if this is not the case, it suffices to replace u with the function −u which solves the equation λu − Au = −f . Since u might not admit absolute maximum over Rd , as in the proof of Theorem 4.1.1 we introduce, for every n ∈ N, the function vn = u − n−1 ϕ, where ϕ(x) = 1 + |x|2 for every x ∈ Rd . Then, Aϕ ≤ λ0 ϕ for some λ0 > 0. If λ ≥ λ0 , then each function vn solves the inequality λvn − Avn ≤ f . Moreover, since it diverges to −∞ as x tends to ∞, vn admits a maximum over Rd at some point xn for every n in N. From Proposition 4.2.1 we deduce that Avn (xn ) ≤ c(xn )vn (xn ), so that (λ−c0 )vn (xn ) ≤ λvn (xn )−Avn (xn ) ≤ f (xn ) ≤ ||f ||∞ . Since λ > c0 , it follows that u(x) −

1 1 ϕ(x) ≤ vn (xn ) ≤ ||f ||∞ , n λ − c0

x ∈ Rd , n ∈ N.

Letting n tend to ∞ the assertion follows in this case. If λ ∈ (0, λ0 ) then we write λ0 u − Au = f + (λ0 − λ)u. Therefore, we can estimate   1 λ − c0 ||u||∞ ≤ ||f ||∞ + 1 − ||u||∞ , λ 0 − c0 λ0 − c0 from which the inequality ||u||∞ ≤ (λ − c0 )−1 ||f ||∞ follows at once. (ii) The proof is similar to that of property (i). So, for every n ∈ N we introduce the function vn and denote by xn ∈ Rd+ a point where vn attains its maximum. If xn ∈ ∂Rd+ , then 1 vn (xn ) = u(xn ) − ϕ(xn ) < u(xn ) ≤ ||u||Cb (∂Rd+ ) . (4.2.1) n On the other hand, if xn ∈ Rd+ , and λ ≥ λ0 , then the arguments in the proof of property (i) show that 1 vn (xn ) ≤ ||f ||∞ . (4.2.2) λ − c0 From (4.2.1) and (4.2.2) it follows that u(x) −

1 1 ϕ(x) ≤ ||f ||∞ + ||u||Cb (∂Rd+ ) , n λ − c0

x ∈ Rd+ .

Letting n tend to ∞ the assertion follows if λ0 ≤ c0 . If λ ∈ (0, λ0 ) then we apply the same argument as in the last part of the proof of (i) to complete the proof.  Using the previous proposition, the proofs of Theorems 4.1.1, 4.1.4, 4.1.11 and 4.1.12 can be easily adapted and they lead to the following results.

100

Elliptic and Parabolic Maximum Principles

Theorem 4.2.3 (Weak maximum principle I) Let Ω be a bounded domain and assume 2,p (Ω), for every p ∈ [1, ∞), such that Au ∈ C(Ω). The that c ≡ 0 in Ω. Fix u ∈ C(Ω) ∩ Wloc following assertions hold: (i) if Au ≥ 0 in Ω, then max u = max u; Ω

∂Ω

(ii) if Au ≤ 0 in Ω, then min u = min u. Ω

∂Ω

In particular, if Au = 0 in Ω, then max |u| = max |u|. Ω

∂Ω

Theorem 4.2.4 (Weak maximum principle II) Let Ω be a bounded domain. Let c ≤ 0 2,p (Ω), for every p ∈ [1, ∞), be such that Au ∈ C(Ω). The following in Ω and u ∈ C(Ω) ∩ Wloc properties hold true: (i) if Au ≥ 0 in Ω, then max u ≤ max u+ ; Ω

∂Ω

(ii) if Au ≤ 0 in Ω, then min u ≥ min u− . Ω

∂Ω

In particular, if Au = 0 in Ω, then maxΩ |u| = max∂Ω |u|. Corollary 4.2.5 Let Ω be a bounded domain. Then, there exists a positive constant C such that ||u||∞ ≤ C||λu − Au||∞ + ||u||C(∂Ω) for every u ∈ C 2 (Ω) ∩ C(Ω), such that Au is bounded on Ω, and every λ ≥ c0 . Proof We introduce the function φ : Rd → R defined by φ(x) = cosh(αx1 ) for every x ∈ Rd , where α is a positive constant to be properly determined. Note that (A − λ)φ(x) =α2 q11 (x) cosh(αx1 ) + αb1 (x) sinh(αx1 ) + (c(x) − λ) cosh(αx1 ) ≥(µα2 − ||b1 ||∞ α − ||c||∞ − λ) cosh(αx1 ) for every x ∈ Ω. We fix α > 0 such that µα2 − ||b1 ||∞ α − ||c||∞ − λ > 1. This choice of α implies that (A − λ)φ ≥ 1 on Ω. Now, we consider the function ψ = cosh(αR) − φ, where R is any positive constant such that max{|x1 | : x ∈ Ω} ≤ R. Then, ψ is positive on Ω and (A − λ)ψ = (A − λ) cosh(αR) − (A − λ)φ ≤ (c − λ) cosh(αR) − 1 ≤ −1,

on Ω.

Let us fix u ∈ C(Ω) ∩ C 2 (Ω) and set v = u − ||Au − λu||∞ ψ − max∂Ω |u|. Clearly, v is nonpositive on ∂Ω. Moreover, (A − λ)v =(A − λ)u − ||Au − λu||∞ (A − λ)ψ − (c − λ) max |u| ∂Ω

≥(A − λ)u + ||(A − λ)u||∞ ≥ 0. From Theorem 4.2.4(i), it follows that v ≤ 0 on Ω. It thus follows that u ≤ ||λu − Au||∞ ||ψ||∞ + ||u||C(∂Ω) . Replacing u with −u, we complete the proof.

 Rd+ .

A similar result can be proved when Ω is replaced with In the following proposition, we assume that the potential term of the operator A is nonpositive over Rd+ .

Semigroups of Bounded Operators and Second-Order PDE’s

101

Proposition 4.2.6 Suppose that u ∈ Cb (Rd+ ) ∩ C 2 (Rd+ ) is such that Au ∈ Cb (Rd+ ). Then, λ||u||∞ ≤ ||λu − Au||∞ + C||u||Cb (∂Rd+ ) for every λ > 0 and some positive constant C, independent of u and λ. The following two results are the natural elliptic counterpart of Proposition 4.1.7 and Corollary 4.1.8. The first one is known as the elliptic Hopf lemma. Their proofs follow using the same arguments and taking Proposition 4.2.1 into account. 2,p Proposition 4.2.7 Let Ω be a domain of class C 2 . Further, suppose that u ∈ Wloc (Ω) ∩ C 1 (Ω) for every p ∈ [1, ∞) solves the differential inequality Au ≥ 0 in Ω and attains its maximum value only at some point x0 ∈ ∂Ω. Finally, suppose that c ≡ 0, Au ∈ C(Ω), and let η be a direction such that β = hη, ν(x0 )i > 0. Then, the derivative of u at x0 along the direction η is positive. The same conclusion holds if c0 ≤ 0 and u(x0 ) ≥ 0.

Corollary 4.2.8 Let Ω be a bounded domain of class C 2 . Further, suppose that c0 ≤ 0 and 2,p u ∈ Wloc (Ω) ∩ C 1 (Ω) for every p ∈ [1, ∞) solves the differential inequality Au ≥ 0 in Ω with Au ∈ C(Ω). Further, assume that a(x)u(x) +

∂u (x) ≤ 0, ∂η

x ∈ ∂Ω,

where a ∈ C(∂Ω) is nonnegative and η ∈ C(∂Ω) satisfies |η(x)| = 1 and hν(x), η(x)i > 0 for every x ∈ ∂Ω. Then, u is nonpositive on Ω. Corollary 4.2.9 Let Ω be a bounded set of class C 2 and let λ > c0 . Suppose that u ∈ C 2 (Ω) ∩ C 1 (Ω) solves the boundary value problem    λu − Au = f, in Ω, ∂u  = g,  au + ∂η

on ∂Ω

for some f ∈ C(Ω) and some g ∈ C(∂Ω). Then, the following properties are satisfied: (i) if a ≥ a0 > 0 on ∂Ω, then ||u||∞ ≤

1 1 ||f ||∞ + ||g||∞ ; λ − c0 a0

(ii) if a ≥ 0 and g = 0 on ∂Ω, then ||u||∞ ≤

1 ||f ||∞ . λ − c0

Proof (i) Let x0 ∈ Ω be such that ||u||∞ = |u(x0 )|. Suppose that u(x0 ) > 0. If x0 ∈ Ω, then Au(x0 ) ≤ c(x0 )u(x0 ) ≤ c0 u(x0 ), so that (λ − c0 )u(x0 ) ≤ λu(x0 ) − Au(x0 ) = f (x0 ) ≤ ||f ||∞ i.e., ||u||∞ ≤

1 ||f ||∞ . λ − c0

(4.2.3)

102

Elliptic and Parabolic Maximum Principles

On the other hand, we claim that if x0 ∈ ∂Ω, then w =u−

1 λ−c0 ||f ||∞ .

∂u (x0 ) > 0. Indeed, consider the function ∂η

Then Aw − λw =

λ−c ||f ||∞ − f ≥ ||f ||∞ − f ≥ 0. λ − c0

Since λ > c0 , all assumptions of Proposition 4.2.7 hold and so, ∂u ∂w (x0 ) = (x0 ) > 0. ∂η ∂η

(4.2.4)

Therefore, ||g||∞ ≥ g(x0 ) = a(x0 )u(x0 ) +

∂u (x0 ) ≥ a0 u(x0 ) ∂η

so that

1 ||g||∞ . (4.2.5) a0 Combining (4.2.3) and (4.2.5) the assertion follows immediately. If u(x0 ) < 0, then if suffices to apply the above arguments to the function −u. (ii) The proof is very close to that of property (i). The only difference is that now x0 does not belong to ∂Ω if u does not identically vanish on Ω. Indeed, since |u(x0 )| > 0, then, by (4.2.4), the equation ||u||∞ ≤

a(x0 )u(x0 ) +

∂u (x0 ) = 0 ∂η

leads us to a contradiction both in the case when ||u||∞ = u(x0 ) (since in this case the left-hand side of the previous equation is positive) and in the case when ||u||∞ = −u(x0 ) (since in this case the left-hand side of the previous equation is negative).  A similar result can be proved in the case when Ω is replaced by Rd+ . Proposition 4.2.10 Suppose that u ∈ C 2 (Rd+ )∩C 1 (Rd+ ) solves the boundary value problem  d   λu − Au = f, in R+ , ∂u  = g, on ∂Rd+  au + ∂η for some f ∈ Cb (Rd+ ) and g ∈ C(∂Rd+ ), where a and η are bounded and continuous functions on Rd−1 , |η(x)| = 1 for every x ∈ Rd−1 and supx∈Rd−1 ηd (x) =: κ < 0. Then, the following properties are satisfied: (i) if a ≥ a0 > 0 on ∂Rd+ and λ > c0 , where c0 is maximum over Rd+ of the function c, then 1 1 ||u||∞ ≤ ||f ||∞ + ||g||∞ ; λ − c0 a0 (ii) if a ≥ 0 then there exists a positive constant e c0 , which depends on c0 and the sup-norm of the functions bd and qdd , such that ||u||∞ ≤ for every λ > e c0 .

2 4 ||f ||∞ + ||g||∞ λ−e c0 |κ|

Semigroups of Bounded Operators and Second-Order PDE’s

103

Proof (i) We introduce the function φ : Rd+ → R, defined by φ(x) = σ + x2d for every x ∈ Rd+ and some positive constant σ to be properly chosen. Since λφ − Aφ(x) =(λ − c(x))(σ + x2d ) − 2qdd (x) − 2bd (x)xd ≥(λ − c0 )(σ + x2d ) − 2||qdd ||∞ − 2||bd ||∞ |x| 1 ≥(λ − c0 − ε)x2d − 2||qdd ||∞ − ||bd ||2∞ + (λ − c0 )σ ε for every ε > 0. Choosing ε = λ − c0 and, then, σ sufficiently large such that (λ − c0 )σ − 2||qdd ||∞ −

1 ||bd ||2∞ > 0, (λ − c0 )

we conclude that λφ − Aφ > 0 on Rd+ . Now, we fix R > 0, u as in the statement of the proposition, and introduce the function w : B+ (0, R) → R, defined by w(x) = u − (σ + R2 )−1 ||u||∞ φ(x) for every x ∈ B+ (0, R), where B+ (0, R) = B(0, R) ∩ Rd+ . This function belongs to C 2 (B+ (0, R)) ∩ C 1 (B+ (0, R)). Moreover, λw − Aw ≤ f on B+ (0, R) and w is nonpositive on ∂B(0, R) ∩ Rd+ . Finally, a(x)w(x) +

∂w a(x)σ σ||a||∞ (x) = g(x) − ||u||∞ ≤ g(x) + ||u||∞ 2 ∂η σ+R σ + R2

for every x ∈ B(0, R) ∩ ∂Rd+ . Arguing as in the proof of property (i) in Corollary 4.2.9, we infer that   ||f ||∞ 1 ||a||∞ w(x) ≤ ||u|| σ + ||g||∞ + ∞ λ − c0 a0 σ + R2 or, equivalently, u(x) ≤

  ||f ||∞ ||u||∞ 1 ||a||∞ ||u|| σ + φ(x) + ||g||∞ + ∞ 2 λ − c0 a0 σ+R σ + R2

(4.2.6)

for every x ∈ B+ (0, R). We now fix x ∈ Rd+ and R0 > 0 such that x ∈ B+ (0, R0 ). Then, we can write (4.2.6) for every R ≥ R0 . Letting R tend to ∞ we conclude that u(x) ≤

1 1 ||f ||∞ + ||g||∞ λ − c0 a0

and the assertion follows. (ii) We consider the function ψ : Rd+ → R, defined by ψ(x) = 1 − (xd + 2)−1 for every e = ψf , where x ∈ Rd+ . We set w = ψu and observe that λw − Aw   d 2X 1 2 Ae = A − qid Dd ψDi − qdd Ddd ψ + bd Dd ψ − qdd (Dd ψ)2 . ψ i=1 ψ ψ Moreover, u = 2w on ∂Rd+ and ∂w ∂ψ ∂u 1 1 ∂u 1 1 ∂u (x) =u(x) (x) + ψ(x) (x) = ηd (x)u(x) + (x) = ηd (x)w(x) + (x) ∂η ∂η ∂η 4 2 ∂η 2 2 ∂η so that

∂u ∂w (x) = 2 (x) − ηd (x)w(x) for every x ∈ ∂Rd+ . Therefore, ∂η ∂η   1 ∂w 1 a − ηd w + = g 2 ∂η 2

104

Elliptic and Parabolic Maximum Principles

on ∂Rd+ . Since ηd is bounded from above by the negative constant κ, the coefficient a − 12 ηd is bounded from below by −κ/2 in Rd+ . Therefore, if λ > e c0 , where e c0 denotes the supremum e of the potential term of the operator A, then we can apply property (i) and conclude that ||w||∞ ≤

1 2 1 2 ||ψf ||∞ + ||g||∞ ≤ ||f ||∞ + ||g||∞ . λ−e c0 |κ| λ−e c0 |κ|

The assertion follows from observing that ψ ≥ 1/2 on Rd+ .



Finally, we state a version of the elliptic strong maximum principle. Theorem 4.2.11 (Strong maximum principle) Let Ω be a bounded domain of Rd of class C 2 . Let u ∈ C 2 (Ω) be such that Au ≥ 0 (resp. Au ≤ 0) in Ω. (i) If c ≡ 0 and there exists a point x0 ∈ Ω such that u(x0 ) = supΩ u (resp. u(x0 ) = inf Ω u), then u is constant in Ω. (ii) If c ≤ 0 on Ω and there exists a point x0 ∈ Ω such that 0 < u(x0 ) = supΩ u (resp. 0 > u(x0 ) = inf Ω u), then u is constant in Ω. Proof Assume by contradiction that u is not constant in Ω. Then the sets Ω1 := {x ∈ Ω : u(x) < u(x0 )} and Ω2 := {x ∈ Ω : u(x) = u(x0 )} are nonempty. As in the proof of Theorem 4.1.11 we define, for x1 ∈ Ω1 , xs = (1 − s)x0 + sx1 ∈ Ω. Since Ω1 is open, there exists s∗ ∈ (0, 1) such that xs∗ ∈ / Ω1 and xs ∈ Ω1 for s ∈ (s∗ , 1]. Take xs for s ∈ (s∗ , 1] such that |xs − xs∗ | < d(xs , ∂Ω). So, since xs∗ ∈ Ω2 , we have r := d(xs , Ω2 ) < d(xs , ∂Ω). Consider now x e0 ∈ Ω2 such that |e x0 − xs | = r. Hence all assumptions of Proposition 4.2.7 are satisfied for Ω, x0 replaced by Ω1 , x e0 and therefore ∂u (e x0 ) > 0, where η is an outward unit direction tangent to ∂Ω at x0 . This implies that ∂η ∇u(e x0 ) 6= 0 which leads to a contradiction, since x e0 is an interior maximum of u. 

4.3

Notes

For the parabolic (resp. elliptic) weak maximum principle we refer to [15] (resp. [17]) and for the parabolic (resp. elliptic) strong maximum principle we refer to [32] (resp. [17]). For the heat equation another proof of the weak maximum principle can be found in Exercise 4.4.5. For the history of the maximum principles we refer to [32, page 156]. We also refer the reader to the classical monographs [25] and [34].

4.4

Exercises

1. Let B = (bij )1≤i,j≤d , C = (cij )1≤i,j≤d be two real and symmetric matrices. Assume that B is positive semi-definite and C is negative semi-definite. Prove that Tr(BC) =

d X i,j=1

bij cij ≤ 0.

Semigroups of Bounded Operators and Second-Order PDE’s

105

2. Consider the function u : Rd+1 → R, defined by u(t, x) = 1 − x2 − 2t for every (t, x) ∈ Rd+1 . (a) Verify that u is a solution to the heat equation Dt u(t, x) = ∆u(t, x). (b) Determine the minimum and the maximum of u in the closed rectangle ΩT := [0, T ] × [0, 1] for a fixed T > 0 without using the maximum principle. (c) Determine the minimum and the maximum of u in ΩT by using the weak maximum principle. 3. Let u ∈ C 1,2 ((0, ∞) × (− π2 , π2 )) ∩ C([0, ∞) × [− π2 , π2 ]) satisfy the inequality Dt u(x) − ∆u(x) ≥ cos x in (0, ∞) × (− π2 , π2 ). Moreover assume that: u(t, − π2 ) ≥ 0, u(t, π2 ) ≥ 0 for all t > 0 and u(0, x) ≥ 2 cos x for all x ∈ [− π2 , π2 ]. Show that h π πi . u(t, x) ≥ (1 + e−t ) cos x in [0, ∞) × − , 2 2 4. Let u ∈ C 1,2 ((0, T ] × Rd ) ∩ C([0, T ] × Rd ) be a solution to the heat equation ( Dt u(t, x) = ∆u(t, x), (t, x) ∈ (0, T ] × Rd , u(0, x) = g(x), x ∈ Rd 2

satisfying the condition u(t, x) ≤ M ea|x| for every (t, x) ∈ [0, T ]×Rd and some constants M, a ≥ 0. (a) Assume first that 4aT < 1 which implies that, there exists ε > 0 such that 4a(T + ε) < 1. Fix ν > 0 and consider the function v : [0, T ] × Rd → R defined by   ν |x|2 v(t, x) = u(t, x) − exp , (t, x) ∈ [0, T ] × Rd . 4(T + ε − t) (T + ε − t)d/2 (i) Prove that v solves Dt v − ∆v = 0 in (0, T ] × Rd and v ∈ C([0, T ] × Rd ). (ii) By applying the weak maximum principle to the function v in the cylinder [0, T ] × Br (0), show that max v ≤ sup g for sufficiently large r. [0,T ]×Br (0)

Rd

(iii) By letting ν tend to 0, deduce that sup

u = sup g.

[0,T ]×Rd

(4.4.1)

Rd

(b) Prove (4.4.1) without the assumption 4aT < 1. 5. Assume that u0 ∈ C(Ω) and u ∈ C 1,2 ((0, T ] × Ω) ∩ C([0, T ] × Ω) is a solution to the heat equation with Dirichlet boundary conditions  (t, x) ∈ (0, T ] × Ω,   Dt u(t, x) = ∆u(t, x), u(t, x) = 0, (t, x) ∈ (0, T ] × ∂Ω,   u(0, x) = u0 (x), x ∈ Ω. 2

Consider the function Φ : R → R defined by Φ(s) = 1 − e−s if s ≥ 0 and Φ(s) = 0 otherwise. Set Z t Z H(t) := Φ(s) ds, ϕ(t) := H(u(t, x) − K) dx, K := 0 ∨ sup u0 (x). 0

Ω

x∈Ω

106

Elliptic and Parabolic Maximum Principles (a) Prove that Φ is a C 1 -function, increasing and its derivative is bounded by 1. (b) Prove that ϕ(0) = 0, ϕ ≥ 0 in [0, T ] and ϕ ∈ C 1 ((0, T ]) ∩ C([0, T ]). Compute ϕ0 and deduce that u(t, x) ≤ K for every (t, x) ∈ [0, T ] × Ω. 1+α/2,2+α

6. Assume that u ∈ Cb ([0, T ] × Rd ) is a classical solution to the following Cauchy problem ( Dt u(t, x) = Au(t, x) + g(t, x), t > 0, x ∈ Rd , u(0, x) = f (x), x ∈ Rd , where f ∈ Cb (Rd ) and g ∈ Cb ([0, T ] × Rd ). Prove that ||u||Cb ([0,T ]×Rd ) ≤ C(||g||Cb ([0,T ]×Rd ) + ||f ||∞ ) for some constant C > 0. [Hint: use the weak maximum principle for the function (t, x) 7→ θn (x)u(t, x), where θn is a cut-off function.] 7. Taking advantage of the results of this chapter prove that, for each f ∈ Cb (R2 ) and g ∈ Cb ([0, T ] × R2 ), there exists at most a solution u ∈ Cb ([0, T ] × R2 ) to the Cauchy problem Dt u = ∆u on (0, T ] × R2 , u(0, ·) = f on R2 , such that Dt u, Dx u, Dy u, Dxx u, Dyy u are bounded and continuous on (0, T ] × R2 and Dxy u is continuous on (0, T ] × (R2 \ {(0, 0)}).

Chapter 5 Prelude to Parabolic Equations: The Heat Equation and the Gauss-Weierstrass Semigroup in Cb( d)

R

This chapter is the prelude to the analysis of parabolic equations in the whole space and in bounded domains. We will deal with the nonhomogeneous Cauchy problem ( Dt u(t, x) = ∆u(t, x) + g(t, x), t ∈ (0, T ], x ∈ Rd , (5.0.1) u(0, x) = f (x), x ∈ Rd , where f : Rd → R and g : [0, T ] × Rd → R are bounded and continuous functions. The Laplace operator is the prototype of a uniformly elliptic operator with bounded coefficients in Rd and it has the peculiarity that, for every f ∈ Cb (Rd ), there exists an explicit formula for the (bounded) classical solution of the homogeneous Cauchy problem ( Dt u(t, x) = ∆u(t, x), t > 0, x ∈ Rd , (5.0.2) u(0, x) = f (x), x ∈ Rd , (see the forthcoming Definition 5.1.1). More precisely,  t = 0, x ∈ Rd ,   f (x), Z u(t, x) =   K(t, x − y)f (y) dy, t > 0, x ∈ Rd , Rd

where K is the so-called heat kernel, which is explicit (see formula (5.1.1)). This greatly simplifies the analysis of problem (5.0.1) as we will see. The chapter is organized as follows. In Section 5.1, we study the main properties of the heat kernel K and use such results to prove that the function u defined above is the unique classical solution to the Cauchy problem (5.0.2). Next, in Section 5.2, we begin by showing that a semigroup {T (t)} of bounded operators in Cb (Rd ) can be associated in a natural way to the Cauchy problem (5.0.2) setting Z (T (t)f )(x) = K(t, x − y)f (y) dy, t > 0, x ∈ Rd , Rd

and T (0) = I. We show that this semigroup, the so-called Gauss-Weierstrass semigroup, is analytic. We also prove optimal uniform estimate for the spatial gradient of the function T (t)f , when f belongs to Cb (Rd ) and to some relevant subspaces. In Section 5.3, we prove two equivalent characterizations of the H¨older spaces Cbα (Rd ) for α ∈ (0, 1) \ {1/2}, which are given in terms of the Gauss-Weierstrass semigroup. More precisely, we show that Cbα (Rd ) is the space of all functions f ∈ Cb (Rd ) such that the function T (·)f is α/2-H¨ older continuous in [0, ∞) with values in Cb (Rd ). It will be also characterized as the sets of all f ∈ Cb (Rd ) such that the function t 7→ t1−α/2 ||∆T (t)f ||∞ 107

108

Prelude to Parabolic Equations: The Heat Equation

is bounded in (0, ∞). These properties play a crucial role in the analysis of the Cauchy problem (5.0.1), which is addressed in Section 5.4. We prove optimal Schauder estimates for such a problem, i.e., we show that, for each T > 0, each f ∈ Cb2+α (Rd ) and each g ∈ C α/2,α ([0, T ] × Rd ), there exists a unique classical solution to problem (5.0.1) which, in addition, satisfies the estimate ||u||C 1+α/2,2+α ([0,T ]×Rd ) ≤ c(||f ||C 2+α (Rd ) + ||g||C α/2,α ([0,T ]×Rd ) ) b

b

b

for some positive constant c, independent of f and u. The optimality of this estimate is explained in Remark 5.4.3.

5.1

The Homogeneous Heat Equation in Existence and Uniqueness

R . Classical Solutions: d

We begin this section with the definition of classical solution to problem (5.0.2). Definition 5.1.1 A function u : [0, ∞) × Rd → R is a classical solution to problem (5.0.2) if (i) belongs to C([0, ∞) × Rd ); (ii) it is continuously differentiable in (0, ∞) × Rd , once with respect to the time variable and twice with respect to the spatial variables; (iii) it satisfies the differential equation and the initial condition in (5.0.2). We now introduce the so-called fundamental solution to the differential equation in (5.0.2), i.e., the function K : (0, ∞) × Rd → R defined by d

K(t, x) = (4πt)− 2 e−

|x|2 4t

,

t > 0, x ∈ Rd ,

(5.1.1)

and study its main properties. Lemma 5.1.2 The following properties are satisfied. (i) K ∈ C ∞ ((0, ∞) × Rd ); Z (ii) K(t, x) dx = 1 for all t > 0; Rd

xj K(t, x) for all (t, x) ∈ (0, ∞) × Rd ; 2t   xi xj δij (iv) Dij K(t, x) = − K(t, x) for all (t, x) ∈ (0, ∞) × Rd ; 4t2 2t

(iii) Dj K(t, x) = −

(v) Dt K(t, x) = ∆K(t, x) for all (t, x) ∈ (0, ∞) × Rd . Proof We limit ourselves to proving property (v), since the remaining ones are straightforward to prove.

Semigroups of Bounded Operators and Second-Order PDE’s From property (iv) it follows that  2  d |x| − ∆K(t, x) = K(t, x), 4t2 2t

109

(t, x) ∈ (0, ∞) × Rd .

On the other hand, Dt K(t, x) = −(4πt)

−d 2

2 |x|2 d |x| d − |x|2 e 4t + (4πt)− 2 2 e− 4t = 2t 4t



 d |x|2 − K(t, x) 4t2 2t

for all (t, x) ∈ (0, ∞) × Rd . Property (v) follows at once.



Now, we can prove the existence and uniqueness of a classical solution to the Cauchy problem (5.0.2). Theorem 5.1.3 For each f ∈ Cb (Rd ), the Cauchy problem (5.0.2) admits a unique bounded classical solution u given by the following formula:  t = 0, x ∈ Rd ,   f (x), Z u(t, x) = (5.1.2)   K(t, x − y)f (y) dy, t > 0, x ∈ Rd . Rd

Moreover, ||u(t, ·)||∞ ≤ ||f ||∞ for all t ≥ 0. Proof To begin with, we observe that, if u is given by (5.1.2), then Z Z Z K(t, x − y) dy = ||f ||∞ K(t, y) dy = ||f ||∞ K(t, x − y)f (y) dy ≤ ||f ||∞ |u(t, x)| ≤ Rd

Rd

Rd

for every (t, x) ∈ (0, ∞) × Rd , by property (ii) in Lemma 5.1.2. It thus follows that ||u(t, ·)||∞ ≤ ||f ||∞ for all t ≥ 0 as claimed. Next, we observe that, since K ∈ C ∞ ((0, ∞) × Rd ) (see Lemma 5.1.2(i)), by the dominated convergence theorem we conclude that u ∈ C ∞ ((0, ∞) × Rd ), see Exercise 5.6.1, and Z Dt u(t, x) − ∆u(t, x) = [Dt K(t, x − y) − ∆K(t, x − y)]f (y) dy = 0 Rd

for every (t, x) ∈ (0, ∞) × Rd , thanks to Lemma 5.1.2(v). To prove that u is a classical solution to problem (5.0.2) we need to show that u is continuous on {0} ×R Rd , where it equals the function f . For this purpose, we fix t > 0 and, using the fact that Rd K(t, x) dx = 1 for all t > 0, we split Z Z |u(t, x) − f (x0 )| = K(t, y)f (x − y) dy − K(t, y)f (x0 ) dy d Rd ZR ≤ K(t, y)|f (x − y) − f (x0 )| dy d ZR ≤ K(t, y)|f (x − y) − f (x)| dy + |f (x) − f (x0 )| Rd

√ for x, x0 ∈ Rd . By the change of variable z = y/ t we can rewrite Z Z √ |z|2 d K(t, y)|f (x − y) − f (x)| dy =(4π)− 2 e− 4 |f (x − tz) − f (x)| dz Rd

Rd

110

Prelude to Parabolic Equations: The Heat Equation Z √ |z|2 d =(4π)− 2 e− 4 |f (x − tz) − f (x)| dz B(0,r) Z √ |z|2 d + (4π)− 2 e− 4 |f (x − tz) − f (x)| dz Rd \B(0,r) Z √ |z|2 d ≤(4π)− 2 e− 4 |f (x − tz) − f (x)| dz B(0,r) Z |z|2 −d + 2(4π) 2 ||f ||∞ (5.1.3) e− 4 dz Rd \B(0,r)

for each r > 0. Fix ε > 0. Note that we can find out r0 > 0 such that Z |z|2 ε −d 2 2(4π) ||f ||∞ e− 4 dz ≤ . 2 d R \B(0,r0 ) As far as the first term in the last side of (5.1.3) is concerned, we notice that since d f is continuous in Rd , it is uniformly continuous in each √ compact set K of R . Hence, if x ∈ B(x0 , 1), z ∈ B(0, r0 ) and t < 1, then both x − tz and x belong to K = B(0, M ), where M = |x0 | + 1 + r0 . Let δ > 0 be such that |f (z2 ) − f (z1 )|√≤ ε/2 if z1 , z2 ∈ K satisfy the condition |z2 − z1 | ≤ δ. If t ≤ t0 := δ 2 r0−2 , then |f (x − tz) − f (x)| ≤ ε/2 for all x ∈ B(x0 , 1), so that Z Z √ |z|2 |z|2 d d ε (4π)− 2 e− 4 |f (x − tz) − f (x)| dz ≤ (4π)− 2 e− 4 dz 2 B(0,r0 ) B(0,r0 ) Z |z|2 ε ε −d ≤ (4π) 2 e− 4 dz = 2 2 d R and we conclude that |u(t, x)−f (x0 )| ≤ ε+|f (x)−f (x0 )| for every (t, x) ∈ (0, t0 ]×B(x0 , 1). It thus follows that lim sup |u(t, x) − f (x0 )| ≤ ε (t,x)→(0,x0 )

and the arbitrariness of ε > 0 implies that u(t, x) tends to f (x0 ) as (t, x) → (0, x0 ). To complete the proof, we observe that the uniqueness of the classical solution to the Cauchy problem (5.0.2) is implied by the maximum principle in Exercise 4.4.4.  Remark 5.1.4 Up to now we have just verified that formula (5.1.2) defines the (unique) classical solution to the Cauchy problem (5.0.2). A natural question is how formula (5.1.2) can be derived. The (formal) answer is based on the use of the Fourier transform. For each f ∈ L1 (Rd ), the Fourier transform of f is the function F(f ) defined by Z ∞ −d 2 F(f )(ξ) := (2π) e−ihξ,xi f (x) dx, ξ ∈ Rd . −∞

We recall that F(Dj f )(ξ) = iξj F(f )(ξ),

ξ ∈ Rd , j = 1, . . . , d,

for every smooth enough function f . If we take the Fourier transform of both the sides of the equation Dt u = ∆u with respect to x and interchange the actions of F and the time derivative, then we deduce that the function u b, defined by u b(t, ξ) = (F(u(t, ·)))(ξ) for every t ≥ 0 and ξ ∈ Rd , solves the Cauchy problem ( Dt u b(t, ξ) = −|ξ|2 u b(t, ξ), (t, ξ) ∈ (0, ∞) × Rd , u b(0, ξ) = fb(ξ), ξ ∈ Rd .

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111

This is a Cauchy problem for an ordinary differential equation, since ξ plays the role of a parameter, and it is easy to see that 2 u b(t, ξ) = e−t|ξ| fb(ξ),

t ≥ 0, ξ ∈ Rd .

(5.1.4)

To get back to u, we take the inverse Fourier transform of the right-hand side of (5.1.4), recalling that the inverse Fourier transform of the product of two functions is the convolution of the inverse Fourier transforms of the two factors, and the inverse Fourier transform of 2 2 the function ξ 7→ e−t|ξ| is the function x 7→ (4πt)−d/2 e−|x| /(4t) for t > 0.

5.2

The Gauss-Weierstrass Semigroup

The results in Theorem 5.1.3 can be rephrased in the semigroup language. More precisely, if for each f ∈ Cb (Rd ) and t ≥ 0 we set T (t)f = u(t, ·), where u is the unique classical solution to problem (5.0.2), then {T (t)} defines a semigroup of bounded linear operators in Cb (Rd ), usually called the Gauss-Weierstrass semigroup. Indeed, the uniqueness of the classical solution to problem (5.0.2) for each f ∈ Cb (Rd ) shows that each operator T (t) is linear on Cb (Rd ). It also implies the semigroup rule, i.e., T (t + s)f = T (t)T (s)f,

f ∈ Cb (Rd ), t, s ≥ 0.

In other terms the value at time t of the classical solution of problem (5.0.2) with initial condition T (s)f at time zero coincides with the value at time t + s of the classical solution to problem (5.0.2) with f as initial condition at time zero. Moreover, for t > 0 and x ∈ Rd , Z (T (t)f )(x) = K(t, x − y)f (y) dy, t > 0, x ∈ Rd . (5.2.1) Rd

The Gauss-Weierstrass semigroup is not strongly continuous. It turns out that T (t)f converges to f in Cb (Rd ) as t → 0+ if and only if f ∈ BU C(Rd ). Indeed, adapting the arguments in the last part of the proof of Theorem 5.1.3, we can easily show that, if f is uniformly continuous in Rd , then T (t)f converges uniformly in Rd to f as t tends to 0+ . Conversely, the forthcoming Proposition 5.2.5 shows that T (t)f ∈ Cb1 (Rd ) for each t > 0. Hence, if T (t)f converges to f in Cb (Rd ) as t → 0+ , then f belongs to the closure of Cb1 (Rd ) in Cb (Rd ), which is BU C(Rd ). On the other hand, the Gauss-Weierstrass semigroup can be straightforwardly extended to Cb (Rd ; C) through formula (5.2.1) and, as the following theorem shows, the so extended semigroup {T (t)} is analytic. Theorem 5.2.1 The Gauss-Weierstrass semigroup is analytic in Cb (Rd ; C). To prove the theorem, we need the following result. Lemma 5.2.2 Let I ⊂ R be an interval and let ψ : I × Rd → C be a continuous function such that ||ψ(t, ·)||∞ ≤ g(t) for each t ∈ I and some function g ∈ L1 (I). Then, the function Ψ : Rd → R, defined by Z Ψ(x) =

ψ(t, x) dt,

x ∈ Rd ,

I

is bounded and continuous and Z (T (t)ψ(s, ·))(x) ds,

(T (t)Ψ)(x) = I

t > 0, x ∈ Rd .

(5.2.2)

112

Prelude to Parabolic Equations: The Heat Equation

Proof The dominated convergence theorem shows that the function Ψ is well defined and belongs to Cb (Rd ; C), and formula (5.2.2) follows from Fubini theorem. Indeed, the 2 function (s, y) 7→ e−|x−y| /(4t) ψ(s, y) belongs to L1 (I × Rd ) for each x ∈ Rd . Hence, Z  Z |x−y|2 −d − 4t (T (t)Ψ)(x) =(4πt) 2 e ψ(s, y) ds dy d I ZR |x−y|2 d =(4πt)− 2 e− 4t ψ(s, y) ds dy d ZI×R Z |x−y|2 d =(4πt)− 2 ds e− 4t ψ(s, y) dy I Rd Z = (T (t)ψ(s, ·))(x) ds I d

for t > 0 and x ∈ R , and we are done



Proof of Theorem 5.2.1 We denote by Π the right-halfplane, i.e., the set of all λ ∈ C with positive real part. For each λ ∈ Π, we introduce the operator Rλ defined by Z ∞ (Rλ f )(x) = e−λt (T (t)f )(x) dt, x ∈ Rd , f ∈ Cb (Rd ; C). 0

Observe that, by the dominated convergence theorem, Rλ f is a bounded and continuous function in Rd for every f ∈ Cb (Rd ; C) and λ ∈ Π. Moreover, Rλ is a bounded operator for every λ ∈ Π. It is easily seen that {Rλ : λ ∈ Π} is a resolvent family: indeed, taking Lemma 5.2.2 into account, we can show that, for every λ, µ ∈ Π, it holds that   Z ∞ Z ∞ −λt −µs (Rλ Rµ f )(x) = e T (t) e (T (s)f )(·) ds (x) dt 0 Z0 ∞ Z ∞ = dt e−λt−µs (T (t + s)f )(x) ds 0

Z

0 ∞

e−µσ (T (σ)f )(x) dσ

= 0

Z =

∞

e−µσ (T (σ)f )(x)

0

=

e

Z

σ

e(µ−λ)t dt

0 (µ−λ)σ

−1 dσ µ−λ

1 [(Rλ f )(x) − (Rµ f )(x)] µ−λ

for all x ∈ Rd and f ∈ Cb (Rd ; C). Let us prove that Rλ is injective for each λ ∈ Π. For this purpose, we fix λ0 ∈ Π and f ∈ Cb (Rd ) such that Rλ0 f ≡ 0. The resolvent identity proved above shows that Rλ f ≡ 0 for each λ ∈ Π. Since, for every x ∈ Rd , the function λ 7→ (Rλ f )(x) is the Laplace transform of the bounded and continuous function t 7→ (T (t)f )(x), by the uniqueness of the Laplace transform, we conclude that (T (t)f )(x) = 0 for all t ≥ 0. Taking t = 0 we conclude that f (x) = 0. The arbitrariness of x ∈ Rd implies that f ≡ 0. Hence, Rλ is injective. By Proposition A.4.6, it follows that there exists a closed operator A : D(A) ⊂ Cb (Rd ; C) → Cb (Rd ; C), whose resolvent set contains Π, and such that R(λ, A) = Rλ for each λ ∈ Π. We claim that A is a sectorial operator. To check the claim, we will show that there exists a positive constant C such that ||R(λ, A)||L(Cb (Rd ,C)) ≤ C|λ|−1 ,

λ ∈ C, Re λ ≥ 1.

(5.2.3)

Semigroups of Bounded Operators and Second-Order PDE’s

113

Indeed, from (5.2.3) and Proposition 3.2.8, the sectoriality of A follows at once. We begin by observing that, for each f ∈ Cb (Rd ; C) and x ∈ Rd , the function t 7→ (T (t)f )(x) can be extended to the right-halfplane with a holomorphic function. Indeed, 2 for x ∈ Rd , the function K(·, x) : Π → C, defined by K(z, x) = (4πz)−d/2 e−|x| /(4z) is holomorphic in Π. Moreover, K(z, ·) ∈ L1 (Rd , C) for every z ∈ Π and 

Z |K(z, x)| dx = Rd

Rez |z|

− d2 ,

z ∈ Π.

Hence, the function Z x 7→ (T (z)f )(x) =

K(z, x − y)f (y) dz

(5.2.4)

Rd

is well defined, bounded and continuous in Rd , for every z ∈ Π, as it can be easily seen again applying the dominated convergence theorem. Moreover,  ||T (z)f ||Cb (Rd ,C) ≤

Rez |z|

− d2 ||f ||Cb (Rd ;C) ,

z ∈ Π.

In particular, if for each ϑ0 ∈ (0, π/2), we denote by Σϑ0 the sector of all λ ∈ C \ {0} such that |arg(λ)| ≤ ϑ0 , then, from the previous estimate it follows immediately that d

||T (z)f ||Cb (Rd ,C) ≤ [1 + (tan(ϑ0 ))2 ] 4 ||f ||Cb (Rd ;C) ,

z ∈ Σϑ0 .

Now, we can prove (5.2.3). If λ = a + iy with a ≥ 1 and y ≥ 0, then by Cauchy integral theorem (see Theorem A.3.4) we have Z ∞ Z −λt (R(λ, A)f )(x) = e (T (t)f )(x) dt = e−λz (T (z)f )(x) dz, x ∈ Rd , (5.2.5) 0

γ

where γ(s) = s − is for every s ≥ 0. Therefore Z ∞ d d 4 e−(a+y)s ds ≤ 2 4 |λ|−1 ||f ||∞ . ||R(λ, A)f ||∞ ≤ 2 ||f ||∞ 0

If y ≤ 0 then one gets the same estimate replacing the curve γ with the curve γ e, defined by γ e(s) = s + is for every s ≥ 0. Estimate (5.2.3) follows. Denote by {S(t)} the analytic semigroup generated by A in Cb (Rd , C). By Proposition 3.2.7, if λ ∈ C has sufficiently large real part, then Z ∞ Z ∞ e−λt (S(t)f )(x) dt = (R(λ, A)f )(x) = (Rλ f )(x) = e−λt (T (t)f )(x) dt 0

0

for x ∈ Rd , i.e., Z

∞

e−λt [(S(t)f )(x) − (T (t)f )(x)] dt = 0,

x ∈ Rd .

0

Again, the uniqueness of the Laplace transform implies that S(t)f ≡ T (t)f in (0, ∞). We have so proved that the Gauss-Weierstrass semigroup is analytic in Cb (Rd ; C).  Remark 5.2.3 Without much effort, we can show that the sectorial operator A associated with the Gauss-Weierstrass semigroup is an extension of the operator (∆, Cb2 (Rd ; C)).

114

Prelude to Parabolic Equations: The Heat Equation

Indeed, fix f ∈ Cb2 (Rd ; C). Recalling that each operator T (t) commutes with the Laplace operator and integrating by parts, we get Z ∞ e−t (T (t)(f − ∆f ))(x) dt (R(1, A)(f − ∆f ))(x) = 0 Z ∞ = e−t ((T (t)f )(x) − (∆T (t)f )(x)) dt 0 Z ∞ Z ∞ −t = e (T (t)f )(x) dt − e−t (Dt T (t)f )(x) dt 0

0

=(R(1, A)f )(x) + f (x) − (R(1, A)f )(x) = f (x) for x ∈ Rd . Hence, R(1, A)(f − ∆f ) = f . Applying the operator I − A to both the sides of the previous equality we obtain f − ∆f = f − Af , i.e., Af = ∆f . Therefore, A is an extension of the operator (∆, Cb2 (Rd ; C)). If N = 1, then A actually coincides with the second-order derivative with Cb2 (R; C) as domain, see Exercise 5.6.3 . On the other hand, if N > 2, then A is a proper extension 2,p of (∆, Cb2 (Rd ; C)). In fact, D(A) = {u ∈ Cb (Rd ; C) ∩ Wloc (Rd ; C), for all p ∈ [1, ∞), and d ∆u ∈ Cb (R ; C)} and Au = ∆u for each u ∈ D(A). The proof of the above characterization of (A, D(A)) will be provided in Chapter 14.

5.2.1

Estimates of the spatial derivatives of T (t)f

To begin with, let us consider the following lemma. Lemma 5.2.4 Fix t > 0, a ∈ Rd , k ∈ N. Then,  Z  0, hx, aik K(t, x) dx =  (2m)! tm |a|2m , Rd m!

k odd, (5.2.6)

k = 2m, m ∈ N.

Proof Fix λ > 0, t > 0 and observe that Z

eλhx,ai K(t, x) dx =

Rd

Z

1 (4πt)

d 2

Rd

eλhx,ai e−

|x|2 4t

dx =

d Z Y

1 (4πt)

d 2

i=1

x2 i

eλxi ai − 4t dxi . (5.2.7)

R

Note that Z Z Z  √ 2 x √ x2 2 2 2 2 2 √ 2 2 i − √i −λai t eλ ai t dxi = eλ ai t e−s 2 t ds = 4πteλ ai t eλxi ai − 4t dx = e 2 t R

R

R

for i = 1, . . . , d, where we have performed the change of variable s = this formula in (5.2.7), we obtain Z 2 2 eλhx,ai K(t, x) dx = etλ |a| .

x√i −λai 2 t

√ t. Replacing

Rd

Differentiating k-times with respect to λ both sides of the previous formula and then computing the so obtained derivative at λ = 0 gives  k  Z ∂ tλ2 |a|2 e = hx, aik K(t, x) dx. ∂λk d λ=0 R

Semigroups of Bounded Operators and Second-Order PDE’s

115

Now, an easy computation yields 

k



∂ tλ2 |a|2 e ∂λk λ=0

  0, =  (2m)! tm |a|2m , m!

k odd, k = 2m, m ∈ N.

The assertion follows.



Proposition 5.2.5 The following estimates hold true. 1 (i) |||∇x T (t)f |||∞ ≤ √ ||f ||∞ for t > 0 and f ∈ Cb (Rd ); 2t (ii) |||∇x T (t)f |||∞ ≤ |||∇f |||∞ for t > 0 and f ∈ Cb1 (Rd ); p 1 + δij (iii) ||Dij T (t)f ||∞ ≤ ||f ||∞ for t > 0, f ∈ Cb (Rd ) and i, j = 1, . . . , d; 2t 1 (iv) ||Dij T (t)f ||∞ ≤ √ |||∇f |||∞ for t > 0, f ∈ Cb1 (Rd ) and i, j = 1, . . . , d; 2t (v) ||Dij T (t)f ||∞ ≤ ||Dij f ||∞ for t > 0, f ∈ Cb2 (Rd ) and i, j = 1, . . . , d; 1 1 (vi) ||Dijk T (t)f ||∞ ≤ √ [1 + δij + δjk + δik + 2δij δjk ] 2 ||f ||∞ for f ∈ Cb (Rd ) and i, j, k = 8t3 1, . . . , d; p 1 + δij δjk (vii) ||Dijk T (t)f ||∞ ≤ |||∇f |||∞ for t > 0, f ∈ Cb1 (Rd ) and i, j, k = 1, . . . , d; 2t

1 (viii) ||Dijk T (t)f ||∞ ≤ √ ||D2 f ||∞ for t > 0, f ∈ Cb2 (Rd ) and i, j, k = 1, . . . , d. 2t (ix) ||Dijk T (t)f ||∞ ≤ ||Dijk f ||∞ for t > 0, f ∈ Cb3 (Rd ) and i, j, k = 1, . . . , d; Proof (i) Fix f ∈ Cb (Rd ). By differentiating under the integral sign and taking Lemma 5.1.2 into account, we get   Z Z (∇x T (t)f )(x) = ∇x K(t, · − y)f (y) dy (x) = ∇x K(t, x − y)f (y) dy d Rd Z R x−y =− K(t, x − y)f (y) dy 2t d R for all t > 0 and x ∈ Rd . It thus follows that Z 1 h(∇x T (t)f )(x), ai = − hx − y, aiK(t, x − y)f (y) dy, 2t Rd

t > 0, a, x ∈ Rd ,

and, consequently, using Cauchy-Schwarz inequality and (5.2.6), we conclude that Z 2 p p 1 |h(∇x T (t)f )(x), ai| = 2 [hx−y, ai K(t, x−y)] K(t, x−y)f (y) dy 4t d Z R Z 1 2 ≤ 2 hx−y, ai K(t, x−y) dy K(t, x−y)(f (y))2 dy 4t Rd d R Z 1 2 2 K(t, x − y)(f (y)) dy ≤ |a| 2t Rd 2

116

Prelude to Parabolic Equations: The Heat Equation ≤

1 2 2 |a| ||f ||∞ 2t

for all t > 0, a, x ∈ Rd . Taking a = (∇x T (t)f )(x), yields 1 |(∇x T (t)f )(x)|2 ||f ||2∞ . 2t

|(∇x T (t)f )(x)|4 ≤

So, property (i) follows. (ii) As it is immediately seen, if f ∈ Cb1 (Rd ), then (t, x) ∈ [0, ∞) × Rd , i = 1, . . . , d.

(Di T (t)f )(x) = (T (t)Di f )(x),

(5.2.8)

Hence, observing that (T (t)ψ)2 ≤ T (t)ψ 2 on Rd , for every t > 0 and ψ ∈ Cb (Rd ), we can estimate X X  12  21 d d 1 2 2 ≤ = [(T (t)|∇f |2 )(x)] 2 ≤ |||∇f |||∞ |(∇x T (t)f )(x)| = T (t)|Dj f | |T (t)Dj f | j=1

j=1

for every t > 0 and x ∈ Rd , which immediately yields the claim. (iii) Fix f ∈ Cb (Rd ). Applying Lemma 5.1.2(iv) and using again H¨older’s inequality, one obtains Z  2  yi yj δij 2 |(Dij T (t)f )(x)| = − K(t, y)f (x − y) dy 2 4t 2t Rd 2 Z Z  yi yj δij ≤ K(t, y)(f (x − y))2 dy − K(t, y) dy 4t2 2t Rd Rd 2 Z  δij yi yj 2 − K(t, y) dy ≤||f ||∞ 4t2 2t d  R Z Z δij δij 1 2 2 2 y y K(t, y)dy + 2 − 3 yi yj K(t, y) dy . =||f ||∞ 16t4 Rd i j 4t 4t Rd Now, if i = j, then by (5.2.6) with k = 2, 4 and a being the i-th element of the Euclidean basis of Rd , we get Z Z 2 (a) yi K(t, y) dy = 2t, (b) yi4 K(t, y) dy = 12t2 . (5.2.9) Rd

Rd

On the other hand, if i 6= j, then, using the one-dimensional version of formula (5.2.6), we get Z yi2 yj2 K(t, y) dy Rd    Y   Z Z Z 2 yh y2 y2 1 1 1 2 − 4ti 2 − 4ti − 4t √ √ √ dyi dyj dyh = yi e yj e e 4πt R 4πt R 4πt R h6=i,j =4t2 .

(5.2.10)

Hence, 1 16t4

Z

δij δij yi2 yj2 K(t, y) dy + 2 − 3 4t 4t Rd

1 , 4t2 yi yj K(t, y) dy =  1 Rd   , 2t2

Z

   

i 6= j, i = j.

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117

Property (iii) follows. (iv) As it is immediately seen, if f ∈ Cb1 (Rd ), then (Di T (t)f )(x) = (T (t)Di f )(x) for every (t, x) ∈ [0, ∞) × Rd , i = 1, . . . , d and, consequently, thanks to property (i), we can estimate 1 1 ||Dij T (t)||∞ =||Di T (t)Dj f ||∞ ≤ |||∇x T (t)Dj f |||∞ ≤ √ ||Dj f ||∞ ≤ √ |||∇f |||∞ 2t 2t and we are done. (v) From (5.2.8) we easily infer that (Dij T (t)f )(x) = (T (t)Dij f )(x) for every t > 0, x ∈ Rd and from this estimate the claim follows at once. (vi) Fix f ∈ Cb (Rd ). From (5.1.1) we get   δik xj δij xk xi xj xk δjk xi + + − K(t, x) Dijk K(t, x) = 4t2 4t2 4t2 8t3 for t > 0, x ∈ Rd and i, j, k = 1, . . . , d. Hence, 2 δjk yi δik yj δij yk yi yj yk |(Dijk T (t)f )(x)| + + − K(t, y) dy 4t2 4t2 4t2 8t3 d R Z Z δjk (1 + 6δik ) δik =||f ||2∞ yi2 K(t, y) dy + yj2 K(t, y) dy 4 16t4 16t d d R R Z Z δij 1 2 2 2 2 + y K(t, y) dy + y y y K(t, y) dy 16t4 Rd k 64t6 Rd i j k  Z Z δjk + δik δij 2 2 2 2 − y y K(t, y) dy − y y K(t, y) dy . i j 16t5 16t5 Rd i k Rd 2

≤||f ||2∞

Z



Arguing as in the proof of (iii) we get  3   24t , Z  120t3 , yi2 yj2 yk2 K(t, y) dy =  d R   3 8t ,

k 6= i = j or i 6= j = k or j 6= i = k, i = j = k, otherwise.

From this estimate, (5.2.9)(a) and (5.2.10), property (vi) follows. The proofs of properties (vii)–(ix) are left to the reader as an exercise, see Exercise 5.6.4.  Remark 5.2.6 From Proposition 5.2.5(iii), it follows that ∆T (t) is a bounded operator in Cb (Rd ) for every t > 0 and d ||∆T (t)||L(Cb (Rd )) ≤ √ , t 2

t > 0.

(5.2.11)

Now, we want to prove estimates similar to those in Proposition 5.2.5, when Cb (Rd ), and Cb2 (Rd ) are replaced by some subspaces of H¨older continuous functions. Using Propositions 1.1.4 and 5.2.5, we can prove the following result. Cb1 (Rd )

Theorem 5.2.7 For every 0 ≤ α ≤ θ ≤ 3 and ω > 0 there exists a positive constant Cα,θ = Cα,θ (ω) such that ||T (t)f ||Cbθ (Rd ) ≤ Cα,θ t−

θ−α 2

eωt ||f ||Cbα (Rd ) ,

t > 0, f ∈ Cbα (Rd ).

(5.2.12)

118

Prelude to Parabolic Equations: The Heat Equation

Proof We fix ω > 0. The constants that appear in the proof may depend on ω, even if it is not explicitly stressed, but are independent of f and t. In view of Proposition 5.2.5, we can confine ourselves to the case when at least one between α and θ is not an integer. The crucial point is the proof of (5.2.12) with α ∈ (0, 1) and θ = 1. For this purpose, we observe that Z |y−x|2 1 t > 0, x ∈ Rd , i = 1, . . . , d. (yi − xi )e− 4t f (y) dy, (Di T (t)f )(x) = d 2t(4πt) 2 Rd As it is immediately seen, Z (yi −xi )2 (yi − xi )e− 4t dyi = 0,

i = 1, . . . , d,

R

whence Z

(yi − xi )e−

|y−x|2 4t

Z dy =

Rd

(yi − xi )e−

(yi −xi )2 4t

dyi

R

YZ

e−

(yj −xj )2 4t

dyj = 0.

R

j6=i

It thus follows that Z |y−x|2 − 4t (y − x )e (f (y) − f (x)) dy i i d 2t(4πt) 2 Rd Z |y−x|2 1 ≤ |yi − xi |e− 4t |f (y) − f (x)| dy d 2t(4πt) 2 Rd Z |y−x|2 1 α ≤ |y − x|1+α e− 4t dy d [f ]Cb (Rd ) 2t(4πt) 2 Rd Z |z|2 α−1 1 2 = [f ]Cbα (Rd ) |z|1+α e− 4 dz d t 2(4π) 2 Rd 1

|(Di T (t)f )(x)| =

for all x ∈ Rd and t > 0. We conclude that α−1 2

|||∇x T (t)f |||∞ ≤ cα t

||f ||Cbα (Rd ) ,

t > 0.

Since ||T (t)f ||∞ ≤ ||f ||∞ for t > 0, estimate (5.2.12) follows at once in this case. Using the semigroup rule and Proposition 5.2.5(i) we can prove (5.2.12) with θ = 2 and θ = 3. Indeed, we can estimate ||Dij T (t)f ||∞ =||Dij T (t/2)T (t/2)f ||∞ =||Di T (t/2)Dj T (t/2)f ||∞ ≤||Di T (t/2)||L(Cb (Rd )) ||Dj T (t/2)f ||∞ α

≤c0α t 2 −1 [f ]Cbα (Rd ) for all t > 0. Thus, we conclude that ||T (t)f ||Cb2 (Rd ) =||T (t)f ||∞ +

d X

||Dj T (t)f ||∞ +

j=1

≤||f ||∞ + cα t α

α−1 2

d X

||Dij T (t)f ||∞

i,j=1 α

||f ||Cbα (Rd ) + c0α t 2 −1 [f ]Cbα (Rd )

≤e cα t 2 −1 ||f ||Cbα (Rd )

(5.2.13)

for each t ∈ (0, 1]. If t > 1, then we split T (t)f = T (t − 1)T (1)f , use the boundedness of

Semigroups of Bounded Operators and Second-Order PDE’s

119

the Gauss-Weierstrass semigroup together with properties (ii) and (v) in Proposition 5.2.5 and (5.2.13) to estimate eα ||f ||C α (Rd ) . ||T (t)f ||Cb2 (Rd ) ≤ ||T (t − 1)||L(Cb2 (Rd )) ||T (1)f ||Cb2 (Rd ) ≤ C b Now, observing that t1−α/2 e−ωt is bounded in [0, ∞), we can easily complete the proof of estimate (5.2.12) in this case. To prove (5.2.12) with θ = 3, we use Proposition 5.2.5(iii) to get ||Dijk T (t)f ||∞ =||Dij T (t/2)Dk T (t/2)f ||∞ ≤||Dij T (t/2)||L(Cb (Rd )) ||Dk T (t/2)f ||∞ ≤c00α t

α−3 2

[f ]Cbα (Rd )

for all t > 0, i, j, k = 1, . . . , d, and proceed as above. Now, suppose that f ∈ Cbα (Rd ) for some α ∈ [1, 2). Since Di T (t)f = T (t)Di f for all t > 0, i = 1, . . . , d, and Di f ∈ Cbα−1 (Rd ), we can estimate α

||Dij T (t)f ||∞ = ||Dj T (t)Di f ||∞ ≤ cα−1 t 2 −1 [Di f ]C α−1 (Rd )

(5.2.14)

b

for all t > 0 and i, j = 1, . . . , d. Since ||T (t)f ||Cb1 (Rd ) =||T (t)f ||∞ +

d X

||Dj T (t)f ||∞ = ||T (t)f ||∞ +

d X

||T (t)Dj f ||∞ ≤ ||f ||Cb1 (Rd )

j=1

j=1

for all t > 0, from (5.2.14), estimate (5.2.12) with θ = 2 follows. Similarly, ||Dijk T (t)f ||∞ = ||Djk T (t)Di f ||∞ ≤ c0α−1 t−

3−α 2

[f ]C α−1 (Rd ) b

for all t > 0 and i, j, k = 1, . . . , d, and (5.2.12) with θ = 3 follows. The proof of (5.2.12) for α ∈ [2, 3) and θ = 3 is completely similar and, hence, left to the reader. To conclude the proof, we should check (5.2.12) for θ ∈ / N. For this purpose, it suffices to apply Proposition 1.1.4. To fix the ideas, we prove it for θ ∈ (2, 3) and α ∈ (0, 1), but all the other cases can be analyzed just in the same way. By the above results, we know that ||T (t)f ||Cb2 (Rd ) ≤ Cα,2 eωt t

α−2 2

||T (t)f ||Cb3 (Rd ) ≤ Cα,3 eωt t

α−3 2

||f ||Cbα (Rd ) ,

t > 0,

||f ||Cbα (Rd ) ,

t > 0.

and

Moreover, from Proposition 1.1.4 it follows that ||T (t)f ||Cbθ (Rd ) ≤Cθ ||T (t)f ||θ−2 ||T (t)f ||3−θ C 3 (Rd ) C 2 (Rd ) b

for each f ∈

Cbα (Rd ),

b

t > 0 and some positive constant Cθ . Hence,

||T (t)f ||Cbθ (Rd ) ≤Cθ ||T (t)f ||θ−2 ||T (t)f ||3−θ Cb3 (Rd ) Cb2 (Rd ) θ−2  α−2 ωt α−3 ≤Cθ Cα,3, e t 2 ||f ||Cbα (Rd ) [Cα,2,T eωt t 2 ||f ||Cbα (Rd ) ]3−θ θ−2 3−θ ωt =Cθ Cα,3 Cα,2 e t

for all t > 0. This completes the proof.

α−θ 2

||f ||Cbα (Rd ) 

120

Prelude to Parabolic Equations: The Heat Equation

Remark 5.2.8 (i) The arguments in the proof of Theorem 5.2.7 shows that, if α = θ, then estimate (5.2.12) holds true also with ω = 0. (ii) In Theorem 5.2.7 we have assumed that α, θ ≤ 3. This is enough for our purposes (see the forthcoming Theorem 5.4.2). Anyway, this constraint can be removed and formula (5.2.12) can be proved for every α, θ ≥ 0 with α ≤ θ, using the same arguments as in the proofs of Proposition 5.2.5 and Theorem 5.2.7.

5.3

Two Equivalent Characterizations of H¨ older Spaces

Let us introduce the space D∆ (θ, ∞). Definition 5.3.1 For each θ ∈ (0, 1), we denote by D∆ (θ, ∞) the set of all functions f ∈ Cb (Rd ) such that [f ]D∆ (θ,∞) := sup t1−θ ||∆T (t)f ||∞ < ∞. t∈(0,∞)

Remark 5.3.2 For each θ ∈ (0, 1), D∆ (θ, ∞) is a Banach space when endowed with the norm ||f ||D∆ (θ,∞) = ||f ||∞ + [f ]D∆ (θ,∞) , f ∈ D∆ (θ, ∞). To begin with, we prove the following (abstract) equivalent characterization of the space D∆ (θ, ∞). Proposition 5.3.3 For each θ ∈ (0, 1), D∆ (θ, ∞) is the space of all functions f ∈ Cb (Rd ) such that [[f ]]θ = sup t−θ ||T (t)f − f ||∞ < ∞. t>0

Moreover, the norm of D∆ (θ, ∞) is equivalent to the norm || · ||∞ + [[ · ]]. Remark 5.3.4 Since ||T (t)f ||∞ ≤ ||f ||∞ for all t ≥ 0, it is immediate to see that [[f ]]θ < ∞ if and only if [[[f ]]]θ = sup t−θ ||T (t)f − f ||∞ < ∞. t∈(0,1)

Moreover, the norms || · ||∞ + [[ · ]] and || · ||∞ + [[[ · ]]] are equivalent. Proof of Proposition 5.3.3 Suppose that f ∈ D∆ (θ, ∞). Then, in view of the fundamental theorem of calculus we can write Z t Z t (T (t)f )(x) − (T (ε)f )(x) = (Dt T (s)f )(x) ds = (∆T (s)f )(x) ds ε

ε

d

for 0 < ε < t and x ∈ R . Since, by assumptions ||∆T (t)f ||∞ ≤ tθ−1 [f ]D∆ (θ,∞) for t > 0, from the previous formula we infer that Z t |(T (t)f )(x) − (T (ε)f )(x)| ≤ [f ]D∆ (θ,∞) sθ−1 ds = [f ]D∆ (θ,∞) θ−1 (tθ − εθ ). ε

Semigroups of Bounded Operators and Second-Order PDE’s

121

Letting ε tend to 0+ shows that [[f ]]θ ≤ θ−1 [f ]D∆ (θ,∞) .

(5.3.1)

Vice versa, let us suppose that [[f ]]θ < ∞ and let us prove that f belongs to D∆ (θ, ∞). For this purpose, we split Z t Z t 1 1 (f (x) − (T (s)f )(x)) ds + (T (s)f )(x) ds (5.3.2) f (x) = t−ε ε t−ε ε for 0 < ε < t and x ∈ Rd . Applying ∆T (t) to both the sides of the previous formula, by dominated convergence we obtain     Z t Z t 1 1 (f − T (s)f ) ds (x) + T (s)f ds (x). (∆T (t)f )(x) = ∆T (t) ∆T (t) t−ε t−ε ε ε (5.3.3) Taking (5.2.11) into account we can estimate

Z t Z t

1

≤ √ d ∆T (t) (f (·) − (T (s)f )(·)) ds [[f ]] sθ ds θ

t − ε

t 2(t − ε) ε ε ∞ tθ d [[f ]]θ . ≤√ 2(1 + θ) t − ε

(5.3.4)

As far as the second integral term in (5.3.2) is concerned, we observe that Proposition Rt 5.2.5 and the dominated convergence theorem show that the function x 7→ ε (T (s)f )(x) ds belongs to Cb2 (Rd ). Since T (t) commutes with ∆ on Cb2 (Rd ) and ||T (t)||L(Cb (Rd )) = 1, we conclude that



Z t Z t

1

1

(T (s)f (·)) ds = (∆T (s)f )(·) ds

t − ε ∆T (t)

T (t) t − ε ε ε ∞

∞ Z t

1

T (t) (Ds T (s)f )(·) ds =

t−ε ε ∞ 1 ||T (t)(T (t)f − T (ε)f )||∞ = t−ε 1 ≤ ||T (t)f − T (ε)f ||∞ t−ε 1 ≤ (||(T (t)f − f ||∞ + ||T (ε)f − f ||∞ ) t−ε 1 ≤ [[f ]]θ (tθ + εθ ). (5.3.5) t−ε From (5.3.3)–(5.3.5), we conclude that   d tθ 1 ||∆T (t)f ||∞ ≤ [[f ]]θ √ + (tθ + εθ ) 2(1 + θ) t − ε t − ε for every t > 0 and ε ∈ (0, t). Letting ε tend to 0+ , from the previous estimate we infer that   d ||∆T (t)f ||∞ ≤ tθ−1 [[f ]]θ √ +1 2(1 + θ)

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Prelude to Parabolic Equations: The Heat Equation

so that

 [f ]D∆ (θ,∞) ≤

√

 d + 1 [[f ]]θ . 2(1 + θ)

(5.3.6)

We have so proved that f ∈ D∆ (θ, ∞). To conclude the proof it suffices to observe that (5.3.1) and (5.3.6) show that the norms || · ||∞ + [ · ]D∆ (θ,∞) and || · ||∞ + [[ · ]]θ are equivalent.  Remark 5.3.5 In view of Proposition 5.3.3, Remark 5.3.4 and the semigroup property, it follows that D∆ (θ, ∞) can be characterized as the set of all f ∈ Cb (Rd ) such that the function t 7→ T (t)f belongs to Cbθ ([0, ∞); Cb (Rd )), where, as usually, b means bounded. Indeed, if 0 ≤ s < t, and f ∈ D∆ (θ, ∞), then ||T (t)f − T (s)f ||∞ =||T (s)[T (t − s)f − f ]||∞ ≤ ||T (t − s)f − f ||∞ ≤ [[f ]]θ (t − s)θ . We can now give an explicit characterization of the space D∆ (θ, ∞). Theorem 5.3.6 For each θ ∈ (0, 1) \ {1/2} it holds that D∆ (θ, ∞) = Cb2θ (Rd ), with equivalence of the respective norms. Moreover,   |f (x) + f (y) − 2f ((x + y)/2)| 0 and x, y ∈ Rd we can estimate |f (x) − f (y)| ≤|(T (t)f )(x) − f (x)| + |(T (t)f )(x) − (T (t)f )(y)| + |(T (t)f )(y) − f (y)| ≤2[f ]D∆ (θ,∞) tθ + |||∇x T (t)f |||∞ |x − y|.

(5.3.7)

We would like to take t = |x − y|2 in the previous formula. Unfortunately, the estimate in Proposition 5.2.5(i) is not enough sharp for our purposes. Indeed, using that estimate we would get |f (x) − f (y)| ≤ 2[f ]D∆ (θ,∞) |x − y|2θ + 2−1/2 . To overcome this difficulty, we differentiate the formula Z n Z n (T (n)f )(x) − (T (t)f )(x) = (Dt T (s)f )(x) ds = (∆T (s)f )(x) ds t

t

to get Z

n

(Di T (n)f )(x) − (Di T (t)f )(x) =

(Di ∆T (s)f )(x) ds

(5.3.8)

t

for every t ∈ (0, n), x ∈ Rd and i = 1, . . . , d. Since f ∈ D∆ (θ, ∞), we can estimate ||∆T (t)f ||∞ ≤ tθ−1 [f ]D∆ (θ,∞) for all t > 0. Hence, taking Proposition 5.2.5(i) into account, using the semigroup property and the fact that T (t) and ∆ commute on Cb2 (Rd ), we get ||Di ∆T (s)f ||∞ =||Di T (s/2)∆T (s/2)f ||∞ ≤||Di T (s/2)||L(Cb (Rd )) ||∆T (s/2)f ||∞ 3

≤Csθ− 2 [f ]D∆ (θ,∞) ,

(5.3.9)

Semigroups of Bounded Operators and Second-Order PDE’s so that we may let n tend to ∞ in (5.3.8) to get Z ∞ (Di ∆T (s)f )(x) ds, (Di T (t)f )(x) = −

123

t > 0, x ∈ Rd ,

t

and

Z

∞

||Di T (t)f ||∞ ≤ C[f ]D∆ (θ,∞)

3

1

sθ− 2 ds = Cθ tθ− 2 [f ]D∆ (θ,∞)

(5.3.10)

t

for each i = 1, . . . , d. This estimate is what we need to prove that f is 2θ-H¨older continuous in Rd . Indeed, replacing (5.3.10) in (5.3.7) and taking t = |x − y|2 , we obtain eθ ||f ||D (θ,∞) |x − y|2θ , |f (x) − f (y)| ≤ C ∆ eθ ||f ||D (θ,∞) . We have so proved that D∆ (θ, ∞) ,→ so that f ∈ Cb2θ (Rd ) and [f ]Cb2θ (Rd ) ≤ C ∆ 2θ d Cb (R ). This completes the proof in the case θ < 1/2. Let us now suppose that θ ∈ (1/2, 1). As a first step, we prove that f ∈ Cb1 (Rd ) for every f ∈ D∆ (θ, ∞). For this purpose, we fix 0 < s < t and observe that Z t (Di T (t)f )(x) − (Di T (s)f )(x) = (Di ∆T (r)f )(x) dr s

for each x ∈ Rd and i = 1, . . . , d. Using estimate (5.3.9) we deduce that |(Di T (t)f )(x) − (Di T (s)f )(x)| ≤

1 1 2C [f ]D∆ (θ,∞) (tθ− 2 − sθ− 2 ) 2θ − 1

(5.3.11)

for each x ∈ Rd . This estimate shows that (∇x T (1/n)f ) is a Cauchy sequence in (Cb (Rd ))d . On the other hand, applying (5.3.7) with t = |x − y| and Proposition 5.2.5(i), we deduce that D∆ (θ, ∞) ⊂ BU C(Rd ). So, T (1/n)f converges to f uniformly in Rd as n tends to ∞. As a byproduct, we conclude that f is continuously differentiable in Rd . Moreover, from (5.3.11) it follows that eθ [f ]D (θ,∞) tθ− 21 , ||Di T (t)f − Di f ||∞ ≤ C ∆

t > 0, i = 1, . . . , d.

(5.3.12)

Hence, taking Proposition 5.2.5(i) into account, we deduce that eθ [f ]D (θ,∞) + √1 ||f ||∞ ||Di f ||∞ ≤ ||Di T (1)f − Di f ||∞ + ||Di T (1)f ||∞ ≤ C ∆ 2

(5.3.13)

for every i = 1, . . . , d, so that f ∈ Cb1 (Rd ). Fix i ∈ {1, . . . , d}. Since f is differentiable in Rd , taking (5.3.12) into account we can estimate |Di f (x) − Di f (y)| ≤|(Di T (t)f )(x) − Di f (x)| + |(Di T (t)f )(x) − (Di T (t)f )(y)| + |(Di T (t)f )(y) − Di f (y)| 1

eθ [f ]D (θ,∞) tθ− 2 + |||∇x Di T (t)f |||∞ |x − y|. ≤2C ∆

(5.3.14)

With the same arguments used to prove (5.3.9), we can show that e θ−2 [f ]D (θ,∞) , ||Dij ∆T (s)f ||∞ ≤ Cs ∆

s > 0.

Therefore, since Z (Dij T (t)f )(x) = −

∞

(Dij ∆T (s)f )(x) ds, t

t > 0, x ∈ Rd ,

(5.3.15)

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Prelude to Parabolic Equations: The Heat Equation

bθ tθ−1 ||f ||D (θ,∞) . Replacing this estimate using (5.3.15) we obtain that |||∇x Di T (t)f |||∞ ≤ C ∆ in (5.3.14), we get bθ tθ−1 ||f ||D (θ,∞) |x − y| eθ [f ]D (θ,∞) tθ− 12 + C |Di f (x) − Di f (y)| ≤2C ∆ ∆ for x, y ∈ Rd and t > 0. Taking t = |x − y|2 we infer that bθ )||f ||D (θ,∞) . [Di f ]C 2θ−1 (Rd ) ≤ (2Cθ + C ∆

(5.3.16)

b

From (5.3.13) and (5.3.16) we conclude that Di f ∈ Cb2θ−1 (Rd ) and ||Di f ||C 2θ−1 (Rd ) ≤ Kθ ||f ||D∆ (θ,∞) ,

i = 1, . . . , d.

b

The inclusion D∆ (θ, ∞) ,→ Cb2θ (Rd ) follows at once. To complete the proof, let us consider the case θ = 1/2. We fix f ∈ Cb (Rd ) such that [[[f ]]] := sup x6=y

|f (x) + f (y) − 2f ((x + y)/2)| 0 and x ∈ Rd , where we have used the fact that the function K(t, ·) is even. Since ||K(t, ·)||L1 (Rd ) = 1, we easily conclude that Z 1 |(T (t)f )(x) − f (x)| = K(t, y)[f (x − y) + f (x + y) − 2f (x)] dy 2 Rd Z 1 ≤ K(t, y)|f (x − y) + f (x + y) − 2f (x)| dy 2 Rd Z ≤[[[f ]]] K(t, y)|y| dy d √ R ≤C∗ t[[[f ]]] (5.3.17) for each t > 0, x ∈ Rd . In view of Proposition 5.3.3, f belongs to D∆ (1/2, ∞) and e∗ [[[f ]]]. [[f ]]D∆ (1/2,∞) ≤ C Vice versa, let us assume that f ∈ D∆ (1/2, ∞) and fix x, y ∈ Rd , with x 6= y. Then, arguing as in (5.3.7) we get |f (x) + f (y) − 2f ((x + y)/2)| ≤|(T (t)f )(x) − f (x)| + |(T (t)f )(y) − f (y)| + 2|(T (t)f )((x + y)/2) − f ((x + y)/2)| + |(T (t)f )(x) − 2(T (t)f )((x + y)/2) + (T (t)f )(y)| √ ≤4 t[[f ]] 21 + |||∇T (t)f |||∞ |x − y|. (5.3.18) Taking t = |x − y| and using (5.3.10) (which holds true also when θ = 1/2), we conclude that |f (x) + f (y) − 2f ((x + y)/2)| ≤ C∗∗ ||f ||D∆ (1/2,∞) , completing the proof. 

Semigroups of Bounded Operators and Second-Order PDE’s

125

Remark 5.3.7 (i) For every θ ∈ (0, 2), the Zygmund space of order θ is the set Cbθ (Rd ) of all the functions f : Rd → R such that [f ]Cbθ (Rd ) := sup x6=y

|f (x) − 2f ((x + y)/2) + f (y)| < ∞. |x − y|θ

It is a Banach space when endowed with the norm ||f ||Cbθ (Rd ) = ||f ||∞ + [f ]Cbθ (Rd ) for every f ∈ Cbθ (Rd ). Theorem 5.3.6 shows that D∆ (1/2, ∞) = Cb1 (Rd ). For θ 6= 1, Cbθ (Rd ) coincides with the H¨older space Cbθ (Rd ) with equivalence of the corresponding norms, so that we can rephrase Theorem 5.3.6 by saying that D∆ (θ, ∞) = Cb2θ (Rd ) for each θ ∈ (0, 1), with equivalence of the corresponding norms. Indeed, if θ < 1 it is straightforward to check that Cbθ (Rd ) ,→ Cbθ (Rd ). On the other hand, if θ ∈ (1, 2), then we can estimate |f (y) − 2f ((x + y)/2) + f (x)| =|[f (x) − f ((x + y)/2)] + [f (y) − f ((x + y)/2)]| Z Z 1 1 1 = ∇f (x + t(y − x)/2), y − xi dt − ∇f (y + t(x − y)/2), y − xi dt 2 0 0 Z 1 1 ≤ |y − x| |∇f (x + t(y − x)/2) − ∇f (y + t(x − y)/2)| dt 2 0 Z 1 t 1 − 1 dt ≤ |||∇f |||C θ−1 (Rd ) |y − x|θ b 2 2 0 for each f ∈ Cbθ (Rd ) and x, y ∈ Rd , and the inclusion Cbθ (Rd ) ,→ Cbθ (Rd ) follows also in this case. Vice versa, the same arguments as in (5.3.18) show that ||T (t)f − f ||∞ ≤ Ctθ [f ]Cbθ (Rd ) for every t > 0 and f ∈ Cbθ (Rd ). Thus, in view of Proposition 5.3.3, f belongs to D∆ (θ, ∞) and ||f ||D∆ (θ,∞) ≤ C||f ||Cbθ (Rd ) , with C being independent of f . Theorem 5.3.6 yields the inclusion Cbθ (Rd ) ,→ Cbθ (Rd ). (ii) One might think that D∆ (1/2, ∞) = Cb1 (Rd ) or D∆ (1/2, ∞) = Lipb (Rd ) (the space of bounded Lipschitz continuous functions f : Rd → R). But this is not the case. One can easily show that Lipb (Rd ) and, hence, Cb1 (Rd ) are continuously embedded into Cb1 (Rd ), but these ones are proper subspaces of Cb1 (Rd ). Indeed, the function f : Rd → R, defined by f (x) = |x| log(|x|)ϑ(x) for x 6= 0 and f (0) = 0, where ϑ : Rd → R is a cut-off function compactly supported in B(0, 1), belongs to Cb1 (Rd ) but it belongs neither to Cb1 (Rd ) nor to Lipb (Rd ). For further results on the Zygmund spaces, we refer the reader to e.g., [40].

5.4

Optimal Schauder Estimates

In this section we prove the fundamental optimal Schauder estimates for solutions to the Cauchy problem ( Dt u(t, x) = ∆u(t, x) + g(t, x), t ∈ (0, T ], x ∈ Rd , (5.4.1) u(0, x) = f (x), x ∈ Rd ,

126

Prelude to Parabolic Equations: The Heat Equation

proving also that, under suitable assumptions on f and g, it admits a unique bounded classical solution. Definition 5.4.1 A function u : [0, T ] × Rd → R is called “a bounded classical solution” to problem (5.4.1) if (i) u ∈ Cb ([0, T ] × Rd ) ∩ C 1,2 ((0, T ] × Rd ); (ii) u(0, ·) = f in Rd ; (iii) u satisfies the differential equation in (5.4.1). We can now state and prove the main result of this section. Theorem 5.4.2 For each f ∈ Cb2+α (Rd ) and α ∈ (0, 1), the following properties are satisfied. (i) If g ∈ Cb0,α ([0, T ] × Rd ), then the Cauchy problem (5.4.1) admits a unique bounded classical solution u. Further, u belongs to Cb0,2+α ([0, T ] × Rd ), the spatial derivatives α/2,0 belong to Cb ([0, T ] × Rd ) and there exists a positive constant C∗ , independent of u, f and g, such that ||Dt u||Cb ([0,T ]×Rd ) + ||u||C 0,2+α ([0,T ]×Rd ) b

+

d X

||Dj u||C α/2,0 ([0,T ]×Rd ) + b

j=1

d X

||Dij u||C α/2,0 ([0,T ]×Rd )

i,j=1

≤C∗ (||f ||C 2+α (Rd ) + ||g||C 0,α ([0,T ]×Rd ) ). b

(5.4.2)

b

α/2,α

(ii) If g ∈ Cb ([0, T ] × Rd ), then the Cauchy problem (5.4.1) admits a unique bounded 1+α/2,2+α classical solution u. Further, u belongs to Cb ([0, T ] × Rd ) and there exists a positive constant C, independent of u, f and g, such that ||u||C 1+α/2,2+α ([0,T ]×Rd ) ≤ C(||f ||C 2+α (Rd ) + ||g||C α/2,α ([0,T ]×Rd ) ). b

b

(5.4.3)

b

Remark 5.4.3 The above estimates (5.4.2) and (5.4.3) are usually referred to as optimal Schauder estimate. The optimality comes from the following facts: (i) the first-order time derivative and all the second-order derivatives of the solution u have the same degree of smoothness of the datum g; (ii) for each t > 0, the function u(t, ·) preserves the same regularity it has at t = 0; (iii) the assumptions f ∈ Cb2+α (Rd ) and g ∈ Cb0,α ([0, T ] × Rd ) are necessary for u to belong to Cb0,2+α ([0, T ] × Rd ) and Dt u to belong to Cb0,α ([0, T ] × Rd ); α/2,α

(iv) the assumptions f ∈ Cb2+α (Rd ) and g ∈ Cb 1+α/2,2+α belong to Cb ([0, T ] × Rd ).

([0, T ] × Rd ) are necessary for u to

Proof of Theorem 5.4.2 To prove the uniqueness of the classical solution to problem (5.4.1), let us consider two bounded classical solutions u and v to (5.4.1). Then u − v is a bounded classical solution to ( Dt w(t, x) = ∆w(t, x), t ∈ [0, T ], x ∈ Rd , w(0, x) = 0, x ∈ Rd .

Semigroups of Bounded Operators and Second-Order PDE’s

127

So, by Theorem 5.1.3, one obtains u = v. As far as the existence part is concerned, we will show that the function u : [0, T ]×Rd → R, defined by the variation of constants formula Z t (T (t − s)g(s, ·))(x) ds =: u0 (t, x) + u1 (t, x) (5.4.4) u(t, x) = (T (t)f )(x) + 0

for (t, x) ∈ [0, T ] × Rd , solves problem (5.4.1) and satisfies estimate (5.4.2) or (5.4.3), depending on the assumptions on g. Clearly, u(0, ·) = f . Throughout the proof, we denote by c a positive constant, independent of u, f and g, which may vary from line to line. (i) By Theorems 5.1.3 and 5.2.7, u0 is the classical solution to problem (5.4.1) with g ≡ 0 and sup ||u0 (t, ·)||C 2+α (Rd ) ≤ c||f ||C 2+α (Rd ) . (5.4.5) b

t∈[0,T ]

b

Now, we consider the function u1 in (5.4.4), show that it belongs to Cb0,2+α ([0, T ] × Rd ), is continuously differentiable with respect to t on [0, T ] × Rd , solves the Cauchy problem ( Dt u1 (t, x) = ∆u1 (t, x) + g(t, x), t ∈ [0, T ], x ∈ Rd , u1 (0, x) = 0, x ∈ Rd and satisfies the estimate ||u1 ||C 0,2+α ([0,T ]×Rd ) ≤ c||g||C 0,α ([0,T ]×Rd ) .

(5.4.6)

b

b

Since the proof of these properties is rather long, we split it into three steps. Finally, in Step 4, we complete the proof of (i), showing that the spatial derivatives of u are α/2H¨ older continuous with respect to time, uniformly with respect to x ∈ Rd , and showing that estimate (5.4.2) holds true. Step 1. Here, we prove that u1 is continuous in [0, T ] × Rd . To begin with, we observe that, for each t > 0 and x ∈ Rd , the function s 7→ (T (t − s)g(s, ·))(x) is continuous in [0, t]. Hence it can be integrated over [0, t]. This can be easily checked using the representation formula (5.2.1) and the dominated convergence theorem. To prove that the function u1 is continuous in [0, T ] × Rd , by a straightforward change of variable we rewrite it in the form Z t u1 (t, x) = (T (s)g(t − s, ·))(x) ds, (t, x) ∈ [0, T ] × Rd . 0

Fix (t0 , x0 ) ∈ [0, T ] × Rd . To ease the notation, we write (t, x) → (t+ 0 , x0 ) (resp. (t, x) → (t− 0 , x0 )) to denote that (t, x) tends to (t0 , x0 ) with t > t0 (resp. t < t0 ). As a first step, we prove that u1 (t, x) converges to u1 (t0 , x0 ) as (t, x) → (t+ 0 , x0 ). For this purpose, for t > t0 and x ∈ Rd , we split Z t u1 (t, x) − u1 (t0 , x0 ) = (T (s)g(t − s, ·))(x) ds t0

Z + =:

t0

[(T (s)g(t − s, ·))(x) − (T (s)g(t0 − s, ·))(x0 )] ds

0 I1+ (t, x)

+ I2+ (t, x).

Since ||T (s)g(t − s, ·)||∞ ≤ ||g(t − s, ·)||∞ ≤ ||g||Cb ([0,T ]×Rd ) for each 0 ≤ s ≤ t, we immediately

128

Prelude to Parabolic Equations: The Heat Equation

+ conclude that I1+ (t, x) vanishes as (t, x) tends to (t+ 0 , x0 ). As far as I2 (t, x) is concerned, using (5.2.1) we can write

(T (s)g(t − s, ·))(x) − (T (s)g(t0 − s, ·))(x0 ) Z
|x0 −y|2 |x−y|2 d =(4πs)− 2 e− 4s g(t − s, y) − e− 4s g(t0 − s, y) dy. Rd

This formula and the dominated convergence theorem imply that (T (s)g(t − s, ·))(x) = (T (s)g(t0 − s, ·))(x0 )

lim

(t,x)→(t+ 0 ,x0 )

|x0 −y|2

|x−y|2

for every s ∈ [0, t0 ]. Indeed, e− 4s g(t − s, y) − e− 4s g(t0 − s, y) vanishes as (t, x) → d (t+ 0 , x0 ) for every s ∈ [0, t0 ] and y ∈ R . Moreover, if x ∈ B(x0 , 1), then |x − y| ≥ |y| − |x| ≥ |y| − |x0 | − 1. Therefore, if |y| ≥ 2(|x0 | + 1), then we can estimate |e−

|x−y|2 4s

g(t − s, y) − e−

|x0 −y|2 4s

|y|2

g(t0 − s, y)| ≤ 2e− 16s ||g||Cb ([0,T ]×Rd )

for every s ∈ [0, t0 ]. On the other hand, if |y| ≤ 2(|x0 | + 1), then we get |e−

|x−y|2 4s

g(t − s, y) − e−

|x0 −y|2 4s

g(t0 − s, y)| ≤ 2||g||Cb ([0,T ]×Rd ) .

Hence, |e− ≤2(e

|x−y|2 4s

|y|2 − 16s

g(t − s, y) − e−

|x0 −y|2 4s

g(t0 − s, y)|

χRd \B(0,2(|x0 |+1)) + χB(0,2(|x0 |+1)) )||g||Cb ([0,T ]×Rd )

for every y ∈ Rd and the function in the right-hand side of the previous inequality belongs to L1 (Rd ). Since |(T (s)g(t − s, ·))(x) − (T (s)g(t0 − s, ·))(x0 )| ≤ 2||g||Cb ([0,T ]×Rd ) for every 0 ≤ s ≤ t0 ≤ t and x, x0 ∈ Rd , we can apply the dominated convergence theorem once more and deduce that I2+ (t, x) tends to 0 as (t, x) tends to (t+ 0 , x0 ). Hence, u1 (t, x) converges to u1 (t0 , x0 ) as (t, x) → (t+ , x ). If t tends to t from the left, then the proof is similar. We 0 0 0 split Z t0 u1 (t, x) − u1 (t0 , x0 ) = − (T (s)g(t0 − s, ·))(x) ds t Z t0 + [(T (s)g(t − s, ·))(x) − (T (s)g(t0 − s, ·))(x0 )]χ[0,t] (s) ds 0

= : I1− (t, x) + I2− (t, x). Again I1− (t, x) tends to 0 and [(T (s)g(t − s, ·))(x) − (T (s)g(t0 − s, ·))(x0 )]χ[0,t] (s) vanishes as (t, x) → (t− 0 , x0 ), for every s ∈ [0, t0 ]. The dominated convergence theorem shows that also I2− (t, x) vanishes as (t, x) → (t− 0 , x0 ), and the continuity of u1 at (t0 , x0 ) is proved. Moreover, since each operator T (t) is a contraction, we deduce that ||u1 ||Cb ([0,T ]×Rd ) ≤ T ||g||Cb ([0,T ]×Rd ) . Step 2. Here, we prove that the function u1 is twice continuously differentiable with respect to the spatial variables, u1 (t, ·) ∈ Cb2+α (Rd ) for each t ∈ [0, T ] and estimate (5.4.6) is satisfied. By Theorem 5.2.7, for each 0 ≤ s < t ≤ T the function T (t − s)g(s, ·) is differenα−1 tiable in Rd and |||∇x T (t − s)g(s, ·)|||∞ ≤ c(t − s) 2 ||g||C 0,α ([0,T ]×Rd ) . Hence, the dominated b

convergence theorem shows that u1 (t, ·) is differentiable in Rd for each t ∈ [0, T ] and Z t (∇x T (t − s)g(s, ·))(x) ds, (t, x) ∈ [0, T ] × Rd . ∇x u1 (t, x) = 0

Semigroups of Bounded Operators and Second-Order PDE’s

129

α

Similarly, since ||Dx2 T (t − s)g(s, ·)||∞ ≤ c(t − s) 2 −1 ||g||C 0,α ([0,T ]×Rd ) , we deduce that u1 (t, ·) b

is twice differentiable in Rd and Z t 2 Dx u1 (t, x) = (Dx2 T (t − s)g(s, ·))(x) ds,

(t, x) ∈ [0, T ] × Rd .

0

Thus, |Dxj u1 (t, x)| ≤

Z 0

=

t

(Dxj T (t − s)g(s, ·))(x) ds ≤ ||g||C 0,α ([0,T ]×Rd )

Z

b

t

(t − s)

α−j 2

ds

0

α−j+2 2 T 2 ||g||C 0,α ([0,T ]×Rd ) b α−j+2

for all (t, x) ∈ [0, T ] × Rd and j = 1, 2. Moreover, for every θ ∈ (0, α), t ∈ [0, T ] and x1 , x2 ∈ Rd we can estimate Z t |Dx2 u1 (t, x2 ) − Dx2 u1 (t, x1 )| ≤ |(Dx2 T (t − s)g(s, ·))(x2 ) − (Dx2 T (t − s)g(s, ·))(x1 )| ds 0 Z t α−θ ≤C||g||C 0,α ([0,T ]×Rd ) |x2 − x1 |θ (t − s)−1+ 2 ds b

0

α−θ 2C ≤ T 2 |x2 − x1 |θ ||g||C 0,α ([0,T ]×Rd ) , b α−θ

which shows that, for each t ∈ [0, T ], u1 (t, ·) belongs to Cb2+θ (Rd ) and [Dx2 u1 (t, ·)]Cbθ (Rd ) ≤ 2C(α − θ)−1 T (α−θ)/2 ||g||C 0,α ([0,T ]×Rd ) . b

Note that, as θ tends to α from the left, the right-hand side of the previous inequality blows up. This prevents us from concluding that u1 (t, ·) ∈ Cb2+α (Rd ) for each t ∈ [0, T ]. To overcome this difficulty, we perform a different strategy based on Theorem 5.3.6: to prove that Dij u1 (t, ·) ∈ Cbα (Rd ) for every t ∈ [0, T ] and i, j = 1, . . . , d, we will show that α

sup τ 1− 2 ||∆T (τ )Dij u1 (t, ·)||∞ < ∞,

t ∈ [0, T ].

τ >0

For this purpose, using Lemma 5.2.2 we can easily show that Z t Z t (T (τ )Dij u1 (t, ·))(x) = (T (τ )Dij T (t − s)g(s, ·))(x) ds = (Dij T (t + τ − s)g(s, ·))(x) ds 0

0

(5.4.7) for (t, x) ∈ [0, T ] × Rd and τ > 0. Thanks to (5.4.7) we can write Z (∆T (τ )Dij u1 (t, ·))(x) =

t

(∆Dij T (t + τ − s)g(s, ·))(x) ds 0

for (t, x) ∈ [0, T ] × Rd and τ > 0. Hence, by Proposition 5.2.5 and Theorem 5.2.7, we get

   

t+τ −s t+τ −s

Dij T g(s, ·) ||∆Dij T (t + τ − s)g(s, ·)||∞ = ∆T

2 2 ∞

   



t+τ −s

Dij T t + τ − s g(s, ·) ≤ ∆T



2 2 L(Cb (Rd )) ∞

130

Prelude to Parabolic Equations: The Heat Equation α

≤c(t + τ − s) 2 −2 ||g||C 0,α ([0,T ]×Rd ) , b

so that τ

1− α 2

||∆T (τ )Dij u1 (t, ·)||∞ ≤ cτ

1− α 2

Z ||g||C 0,α ([0,T ]×Rd ) b

t

α

(t + τ − s) 2 −2 ds

(5.4.8)

0

for all t ∈ [0, T ], τ > 0 and i, j = 1, . . . , d. Performing the change of variable t − s = τ σ, we can estimate Z τt Z ∞ Z t α α α α α α 2 (1 + σ) 2 −2 dσ ≤ τ 2 −1 (1 + σ) 2 −2 dσ = τ 2 −1 . (t + τ − s) 2 −2 ds =τ 2 −1 2 − α 0 0 0 (5.4.9) From (5.4.8) and (5.4.9) we deduce that α

τ 1− 2 ||∆T (τ )Dij u1 (t, ·)||∞ ≤ c||g||C 0,α ([0,T ]×Rd ) ,

t ∈ [0, T ], τ > 0,

b

i.e., each second-order derivative of u1 (t, ·) belongs to D∆ (α/2, ∞) = Cbα (Rd ) and ||Dij u1 (t, ·)||Cbα (Rd ) ≤ c||g||C 0,α ([0,T ]×Rd ) ,

t ∈ [0, T ], i, j = 1, . . . , d.

b

Summing up, we have proved that u1 is bounded in [0, T ] with values in Cb2+α (Rd ) and satisfies (5.4.6). Step 3. Here, we prove that u1 is continuously differentiable with respect to t in [0, T ] × Rd . For notational convenience, we denote by Dt+ (resp. Dt− ) the right (resp. left) time derivative. For each ε ∈ (0, 1) we introduce the function u1,ε : [0, T ] × Rd → R defined by Z u1,ε (t, ·) =

εt

(T (t − s)g(s, ·))(x) ds,

(t, x) ∈ [0, T ] × Rd .

0

Note that u1,ε converges to u1 as ε → 1− , uniformly in [0, T ] × Rd . Indeed, Z t (T (t − s)g(s, ·))(x) ds ≤ ||g||Cb ([0,T ]×Rd ) (1 − ε)T |u1 (t, x) − u1,ε (t, x)| = εt

for all (t, x) ∈ [0, T ] × Rd and all ε ∈ (0, 1). Let us prove that u1,ε is differentiable in [0, T ] × Rd with respect to t. Fix t ∈ [0, T ) and h ∈ (0, T − t). Then, u1,ε (t + h, ·) − u1,ε (t, x) 1 = h h

Z

ε(t+h)

(T (t + h − s)g(s, ·))(x) ds εt εt

(T (t + h − s)g(s, ·))(x) − (T (t − s)g(s, ·))(x) ds h 0 = : J1+ (h) + J2+ (h). Z

+

We claim that J1+ (h) converges to ε(T ((1 − ε)t)g(εt, ·))(x) as h → 0+ . Indeed, J1+ (h) − ε(T ((1 − ε)t)g(εt, ·))(x) Z 1 ε(t+h) = (T (t + h − s)g(s, ·))(x) ds − ε(T ((1 − ε)t)g(εt, ·))(x) h εt

Semigroups of Bounded Operators and Second-Order PDE’s Z 1 ε(t+h) [(T (t + h − s)g(s, ·))(x) − (T ((1 − ε)t)g(s, ·))(x)] ds = h εt Z ε(t+h) 1 + (T ((1 − ε)t)(g(s, ·) − g(εt, ·)))(x) ds. h εt

131

It is clear that the second integral term in the last side of the previous chain of equalities converges to 0 as h → 0+ . As far as the first integral term is concerned, we observe that an easy computation (based on the representation formula for the Gauss-Weierstrass semigroup) reveals that the function (r, s) 7→ (T (r)g(s, ·))(x) is continuous on [0, T ] × [0, T ]. Therefore, it is therein uniformly continuous. Hence, for each ρ > 0 there exists δ > 0 such that |(T (r2 )g(s, ·))(x) − (T (r1 )g(s, ·))(x)| ≤ ρ if |r2 − r1 | ≤ δ and s ∈ [0, T ]. Consequently, if |h| ≤ δ, then |(T (t + h − s)g(s, ·))(x) − (T ((1 − ε)t)g(s, ·))(x)| ≤ ρ for s ∈ [εt, ε(t + h)] and1 Z 1 ε(t+h) [(T (t + h − s)g(s, ·))(x) − (T ((1 − ε)t)g(s, ·))(x)] ds ≤ ερ. h εt Therefore, 1 lim h→0 h

Z

ε(t+h)

[(T (t + h − s)g(s, ·))(x) − (T ((1 − ε)t)g(s, ·))(x)] ds = 0. εt

As far as J2+ (h) is concerned (which is significant only in the case t > 0), we observe that the function under the integral sign converges to (Dt T (t−s)g(s, ·))(x) = (∆T (t−s)g(s, ·))(x) as h → 0+ . Moreover, using the mean value theorem, we estimate (T (t + h − s)g(s, ·))(x) − (T (t − s)g(s, ·))(x) =|(∆T (t + ξ − s)g(s, ·))(x)| h c ≤ ||g|| d t − s Cb ([0,T ]×R ) for all s ∈ [0, εt], where ξ is a suitable point on the line joining 0 to h. Hence, we can apply the dominated convergence theorem and conclude that Z εt lim+ J2+ (h) = (∆T (t − s)g(s, ·))(x) ds. h→0

0

We have so proved that u1,ε is differentiable from the right in [0, T ) × Rd , with respect to t, and Z tε + Dt u1,ε (t, x) =ε(T ((1 − ε)t)g(εt, ·))(x) + (∆T (t − s)g(s, ·))(x) ds 0

=ε(T ((1 − ε)t)g(εt, ·))(x) + ∆u1,ε (t, x). We can argue similarly to prove that u1,ε is differentiable from the left in (0, T ] × Rd , with respect to t and Z tε − Dt u1,ε (t, x) =ε(T ((1 − ε)t)g(εt, ·))(x) + (∆T (t − s)g(s, ·))(x) ds 0

=ε(T ((1 − ε)t)g(εt, ·))(x) + ∆u1,ε (t, x). 1 Note

that |t + h − s − (1 − ε)t| = |h − s + εt| ≤ |h| if s ∈ [εt, ε(t + h)].

132

Prelude to Parabolic Equations: The Heat Equation

Summing up, u1,ε is differentiable with respect to t in [0, T ] × Rd and tε

Z Dt u1,ε (t, x) =ε(T ((1 − ε)t)g(εt, ·))(x) +

(∆T (t − s)g(s, ·))(x) ds 0

=ε(T ((1 − ε)t)g(εt, ·))(x) + ∆u1,ε (t, x) for all (t, x) ∈ [0, T ] × Rd . Now, we observe that Z t lim− Dt u1,ε (t, x) = g(t, x) + (∆T (t − s)g(s, ·))(x) ds = g(t, x) + ∆u1 (t, x) ε→1

0

for each (t, x) ∈ [0, T ] × Rd . Since u1,ε converges to u1 uniformly in [0, T ] × Rd , it follows that u1 is differentiable with respect to t in [0, T ] × Rd and therein Dt u1 = ∆u1 + g. The proof is now complete. Step 4. To conclude the proof, we need to show that Di u and Dij u (i, j = 1, . . . , d) belong α/2,0 to Cb ([0, T ] × Rd ). For this purpose, we begin by observing that, for each t1 , t2 ∈ [0, T ], with t1 ≤ t2 and x ∈ Rd , we can write Z t2 Z t2 u(t2 , x) − u(t1 , x) = Dt u(s, x) ds = (∆u(s, x) + g(s, x)) ds. t1

t1

Using estimates (5.4.5) and (5.4.6) we conclude that ||u(t2 , ·) − u(t1 , ·)||∞ ≤ c(||f ||C 2+α (Rd ) + ||g||C 0,α ([0,T ]×Rd ) )|t2 − t1 |. b

b

Similarly, |u(t2 , x) − u(t1 , x) − u(t2 , y) + u(t1 , y)| Z t2 (∆u(s, x) + g(s, x) − ∆u(s, y) − g(s, y)) ds = t1 Z t2 ≤ (|∆u(s, x) − ∆u(s, y)| + |g(s, x) − g(s, y)|) ds t1

≤c sup ([∆u(t, ·)]Cbα (Rd ) + [g(t, ·)]Cbα (Rd ) )|x − y|α |t2 − t1 | t∈[0,T ]

≤c(||f ||C 2+α (Rd ) + ||g||C 0,α ([0,T ]×Rd ) )|x − y|α |t2 − t1 | b

b

for all x, y ∈ Rd . Summing up, we have proved that ||u(t2 , ·) − u(t1 , ·)||Cbα (Rd ) ≤ c(||f ||C 2+α (Rd ) + ||g||C 0,α ([0,T ]×Rd ) )|t2 − t1 | b

b

for 0 ≤ t1 ≤ t2 ≤ T . Using the interpolation estimate (1.1.2) together with (5.4.5) and (5.4.6) we deduce that 1− α

α

2 ||u(t2 , ·) − u(t1 , ·)||Cb2 (Rd ) ≤c||u(t2 , ·) − u(t1 , ·)||C2 α (Rd ) ||u(t2 , ·) − u(t1 , ·)||C 2+α (Rd ) b

≤c(||f ||C 2+α (Rd ) + ||g||C 0,α ([0,T ]×Rd ) )|t2 − t1 | b

b

α 2

(5.4.10)

b

for all t1 , t2 ∈ [0, T ]. Hence, for every x ∈ Rd the functions Di u(·, x) and Dij u(·, x) (i, j = 1, . . . , d) are α/2-H¨ older continuous uniformly with respect to x ∈ Rd . We thus conclude α/2,2+α that u ∈ Cb ([0, T ]×Rd ) and, from (5.4.5), (5.4.6) and (5.4.10), estimate (5.4.2) follows at once.

Semigroups of Bounded Operators and Second-Order PDE’s

133

α/2,α

(ii) Since Cb ([0, T ]×Rd ) is continuously embedded into Cb0,α ([0, T ]×Rd ), by property (i), we already know that the function u given by formula (5.4.4) solves the Cauchy problem (5.4.1), belongs to C 1,2 ([0, T ] × Rd ) ∩ C 0,2+α ([0, T ] × Rd ) and satisfies estimate (5.4.2). To conclude the proof, it suffices to observe that, since Dt u = ∆u + g and both the functions α/2,α α/2,α ∆u and g belong to Cb ([0, T ] × Rd ), the function Dt u belongs to Cb ([0, T ] × Rd ). Moreover, from (5.4.2) we deduce that ||Dt u||C α/2,α ([0,T ]×Rd ) ≤ c(||f ||C 2+α (Rd ) + ||g||C α/2,α ([0,T ]×Rd ) ) b

b

b

and estimate (5.4.3) follows, if we take Proposition 1.2.4(i) into account.



Remark 5.4.4 The proof of Theorem 5.4.2 shows that the constants C∗ in (5.4.2) and C in (5.4.3) depend on T and the functions T 7→ C∗ (T ), T 7→ C(T ) are locally bounded. The following result, which we will use in Chapter 6 can be proved adapting the arguments in the proof of Theorem 5.4.2. Theorem 5.4.5 Fix α, θ ∈ (0, 1) and β ∈ [0, 2 + α). Then, for each f ∈ Cbβ (Rd ) and each g ∈ C((0, T ] × Rd ) such that supt∈(0,T ] tθ ||g(t, ·)||Cbα (Rd ) < ∞, the Cauchy problem (5.4.1) admits a unique classical solution u ∈ Cb ([0, T ] × Rd ) ∩ C 1,2 ((0, T ] × Rd ) such that   2+α−β sup tθ∨ 2 ||u(t, ·)||C 2+α (Rd ) ≤ c ||f ||C β (Rd ) + sup tθ ||g(t, ·)||Cbα (Rd ) (5.4.11) b

t∈(0,T ]

b

t∈(0,T ]

for some positive constant c, independent of f , g and u. Moreover, u is bounded on [0, T ] with values in Cbγ (Rd ), where γ = β ∧ (2 + α − 2θ), and   sup ||u(t, ·)||Cbγ (Rd ) ≤ c∗ ||f ||C β (Rd ) + sup tθ ||g(t, ·)||Cbα (Rd ) (5.4.12) b

t∈[0,T ]

t∈(0,T ]

for some positive constant c∗ , independent of f , g and u. In particular, if γ is not an integer, then u belongs to Cb0,γ ([0, T ] × Rd ). Proof The uniqueness part of the statement follows arguing as in the proof of Theorem 5.4.2. As far as the existence part is concerned, we prove that the function u : [0, T ]×Rd → R, defined by Z t u(t, x) = (T (t)f )(x) + (T (t − s)g(s, ·))(x) ds =: u0 (t, x) + u1 (t, x) 0

for (t, x) ∈ [0, T ] × Rd , solves problem (5.4.1) and satisfies estimate (5.4.11). By Theorems 5.1.3 and 5.2.7, the function u0 is a classical solutions to the homogeneous Cauchy problem ( Dt u(t, x) = ∆u(t, x), t > 0, x ∈ Rd , u(0, x) = f (x), x ∈ Rd . Moreover, u0 belongs to Cb2+α (Rd ) and ||u0 (t, ·)||C 2+α (Rd ) ≤ C1 t− b

2+α−β 2

||f ||C β (Rd ) ,

t ∈ (0, T ]

(5.4.13)

b

for some positive constant C = C(T ). To avoid cumbersome notation in the rest of the proof, we still denote by C a positive constant, which may vary from line to line and from

134

Prelude to Parabolic Equations: The Heat Equation

estimate to estimate. Such a constant may depend on T but is independent of the functions that we consider and it is also independent of s, t, τ and ε. Next, we consider the function u1 and observe that α

||Dij T (t − s)g(s, ·)||∞ ≤ Cs−θ (t − s) 2 −1 sup rθ ||g(r, ·)||Cbα (Rd ) r∈(0,T ]

α

for every s ∈ (0, t) and every t ∈ (0, T ]. Since the function s 7→ s−θ (t − s) 2 −1 belongs to L1 ((0, t)), the dominated convergence theorem shows that u1 is twice continuously differentiable on (0, T ] × Rd with respect to the spatial variables and t

Z

θ

||u1 (t, ·)||Cb2 (Rd ) ≤C sup r ||g(r, ·)||Cbα (Rd ) r∈(0,T ]

α

s−θ (t − s) 2 −1 ds

0

α

=Ct 2 −θ sup rθ ||g(r, ·)||Cbα (Rd ) r∈(0,T ]

=Ct

α 2 −θ

θ

1

Z

α

r−θ (1 − r) 2 −1 dr

0

sup r ||g(r, ·)||

(5.4.14)

Cbα (Rd )

r∈(0,T ]

for every t ∈ (0, T ]. To prove that the function u1 (t, ·) belongs to Cb2+α (Rd ) for every t ∈ (0, T ], it suffices α to show that the function τ 7→ τ 1− 2 ||∆T (τ )Dij u1 (t, ·)||∞ is bounded on (0, ∞) for every i, j = 1, . . . , d and t ∈ (0, T ]. Since Z

t

(∆Dij T (t + τ − s)g(s, ·))(x) ds

∆T (τ )Dij u1 (t, x) = 0

for every τ > 0 and every (t, x) ∈ (0, T ] × Rd , and α

||∆Dij T (t + τ − s)g(s, ·)||∞ ≤ Cs−θ (t + τ − s) 2 −2 sup rθ ||g(r, ·)||Cbα (Rd ) , r∈(0,T ]

we deduce that τ

1− α 2

||∆T (τ )Dij u1 (t, ·)||∞ ≤Cτ

1− α 2

sup ||g(r, ·)||C 2+α (Rd ) b

r∈(0,T ]

Note that Z t Z α s−θ (t + τ − s) 2 −2 ds = 0

t 2

α

s−θ (t + τ − s) 2 −2 ds +

Z

α

s−θ (t + τ − s) 2 −2 ds.

0

t

α

s−θ (t + τ − s) 2 −2 ds t 2

0

≤

t

Z

t 2

+τ

 α2 −2 Z

t 2

s−θ ds + 2θ t−θ

Z

t

α

(t + τ − s) 2 −2 ds t 2

0

Z 2τt  α2 −2  t 1−θ α α 1 t = +τ + 2θ t−θ τ 2 −1 (1 + r) 2 −2 dr 1−θ 2 2 0 Z ∞ α α 2θ α −1 −θ θ −θ −1 −2 ≤ τ 2 t +2 t τ 2 (1 + r) 2 dr, 1−θ 0 from which it follows that τ

1− α 2

Z 0

t

α

s−θ (t + τ − s) 2 −2 ds ≤ Ct−θ

Semigroups of Bounded Operators and Second-Order PDE’s

135

for every t ∈ (0, T ], every τ > 0. From this estimate we conclude that α

sup τ 1− 2 ||∆T (τ )Dij u1 (t, ·)||∞ ≤ C sup rθ ||g(r, ·)||Cbα (Rd ) , τ >0

r∈(0,T ]

so that Dij u1 (t, ·) belongs to Cbα (Rd ) and ||Dij u1 (t, ·)||Cbα (Rd ) ≤ C sup sθ ||g(s, ·)||Cbα (Rd ) ,

t ∈ (0, T ], i, j = 1, . . . , d.

(5.4.15)

s∈(0,T ]

From (5.4.14) and (5.4.15) it follows that u1 (t, ·) ∈ Cb2+α (Rd ) for every t ∈ (0, T ] and ||u1 (t, ·)||C 2+α (Rd ) ≤ Ct−θ sup rθ ||g(r, ·)||Cbα (Rd ) . b

(5.4.16)

r∈(0,T ]

Combining (5.4.13) and (5.4.16), estimate (5.4.11) follows at once. Since the function u1 is continuous on [0, T ] × Rd , the same arguments as in Step 4 of the proof of Theorem 5.4.2 show that the first- and second-order spatial derivatives of the function u1 are continuous on (0, T ] × Rd . Let us now prove that the function u1 is continuously differentiable with respect to time on (0, T ]×Rd . As in the proof of Theorem 5.4.2, for each ε ∈ (0, 1) we introduce the function u1,ε : [0, T ] × Rd → R defined by Z

εt

u1,ε (t, ·) =

(t, x) ∈ [0, T ] × Rd ,

(T (t − s)g(s, ·))(x) ds, 0

which converges to u1 as ε → 1− , uniformly in [0, T ] × Rd , since ||u1 − u1,ε ||∞ ≤

T 1−θ sup rθ ||g(r, ·)||Cbα (Rd ) (1 − ε1−θ ), 1 − θ r∈(0,T ]

ε ∈ (0, 1].

To prove that u1,ε is differentiable in (0, T ] × Rd with respect to t, we can repeat the same arguments as in the proof of Theorem 5.4.2. The only difference is in the estimate of the term ∆h (t, s, x) := h−1 [(T (t + h − s)g(s, ·))(x) − (T (t − s)g(s, ·))(x)], which, due to the new assumption on g, is α

∆h (t, s, x) ≤ Cs−θ (t − s) 2 −1 sup rθ ||g(r, ·)||Cb (Rd ) r∈(0,T ]

for all s ∈ (0, εt]. Since this function is integrable on (0, εt), we can follow verbatim the arguments in the proof of the quoted theorem to conclude that the function u1,ε is differentiable with respect to the time variable on (0, T ] × Rd and Z Dt u1,ε (t, x) = ε(T ((1 − ε)t)g(εt, ·))(x) +

εt

(∆T (t − s)g(s, ·))(x) ds 0

for all the pairs (t, x) ∈ (0, T ] × Rd . The continuity of the function (r, s) 7→ (T (r)g(s, ·))(x) on [0, T ] × (0, T ] together with the estimate (5.4.14) allow us to infer that Dt u1,ε (t, x) converges to g(t, x) + ∆u1 (t, x), for every (t, x) ∈ [0, T ] × Rd , as ε tends to 1= . Since u1,ε converges to u1 uniformly in [0, T ] × Rd , we conclude that u1 is differentiable with respect to t in (0, T ] × Rd and therein Dt u1 = ∆u1 + g. To complete the proof, let us show that the function u belongs to Cb0,γ ([0, T ] × Rd ), where γ = β ∧ (2 + α − 2θ). The function u0 is easier to analyze. Indeed, due to Theorem

136

Prelude to Parabolic Equations: The Heat Equation

5.2.7, such a function is bounded on [0, T ] with values in Cbβ (Rd ). As far as the function u1 is concerned, arguing as in the proof of (5.4.14) we can show that Z t s−θ (t − s)θ−1 ds ||u1 (t, ·)||C 2+α−2θ (Rd ) ≤C sup sθ ||g(s, ·)||Cbα (Rd ) b

s∈(0,T ]

0

=C sup sθ ||g(s, ·)||Cbα (Rd ) s∈(0,T ]

1

Z

s−θ (1 − s)θ−1 ds

(5.4.17)

0

for every t ∈ (0, T ]. Summing up, we have proved that u is bounded on [0, T ] with values in Cbγ (Rd ). Moreover, from (5.2.12) and (5.4.17), estimate (5.4.12) follows at once. Finally, we observe that, if γ is not an integer and greater than one, then applying estimate (1.1.6), with Ω = B(0, R), R > 0 being arbitrarily fixed, we can estimate ||u(t2 , ·) − u(t1 , ·)||C [γ] (B(0,R)) ≤C||u(t2 , ·) − u(t1 , ·)||

γ−[γ] γ

C(B(0,R))

≤C||u(t2 , ·) − u(t1 , ·)||

[γ]

||u(t2 , ·) − u(t1 , ·)||Cγγ (B(0,R))

γ−[γ] γ

C(B(0,R))

for every t1 , t2 ∈ [0, T ]. Since the function u is continuous on [0, T ] × Rd , we immediately conclude that the spatial derivatives up to the order [2+α−2θ] are continuous on [0, T ]×Rd , so that u ∈ Cb0,γ ([0, T ] × Rd ). The proof is now complete. 

5.5

Notes

The interested reader may find in [7] a very exhaustive and nice exposition of the heat equation in the one dimensional case. For the multidimensional case we refer the reader to [4].

5.6

Exercises

1. Prove that the function Z K(t, x − y)f (y) dy,

u(t, x) =

t > 0, x ∈ Rd ,

Rd

belongs to C ∞ ((0, ∞) × Rd ) for each f ∈ Cb (Rd ). 2. Prove that the function z 7→ T (z)f , defined in (5.2.4) is holomorphic in the sector Σϑ for every ϑ ∈ (0, π/2). 3. Prove that the sectorial operator associated with the one-dimensional Gauss-Weierstrass semigroup is the second-order derivative with Cb2 (R) as domain. 4. Prove properties (vii)–(ix) in Proposition 5.2.5. 5. Use (5.2.3) and Proposition 3.2.8 to prove that the operator A + b(·)∇ with domain D(A) generates an analytic semigroup in Cb (Rd ; C), where b ∈ Cb (Rd , Rd ) and A is the generator of the Gauss-Weierstrass semigroup {T (t)} in Cb (Rd ; C).

Chapter 6 Parabolic Equations in

R

d

In this chapter, we consider the nonhomogeneous Cauchy problem ( Dt u(t, x) = Au(t, x) + g(t, x), t ∈ (0, T ], x ∈ Rd , u(0, x) = f (x), x ∈ Rd ,

(6.0.1)

where A is the operator defined on smooth functions ψ : Rd → R by Aψ(x) =

d X

qij (x)Dij ψ(x) +

i,j=1

d X

bj (x)Dj ψ(x) + c(x)ψ(x)

j=1

=Tr(Q(x)D2 ψ(x)) + hb(x), ∇x ψ(x)i + c(x)ψ(x), with Q(x) = (qij (x))1≤i,j≤d for x ∈ Rd . Throughout the chapter, we assume the following conditions on the coefficients of the operator A. Hypotheses 6.0.1 (i) The coefficients qij = qji , bj (i, j = 1, . . . , d) and c are bounded and α-H¨ older continuous in Rd for some α ∈ (0, 1); (ii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for all x, ξ ∈ Rd . Remark 6.0.2 The condition (ii) in Hypothesis 6.0.1 is usually rephrased saying that the operator A is uniformly elliptic in Rd . Our aim consists in extending the results in Chapter 5 to the more general Cauchy problem (6.0.1). Two are the main tools which we use to prove the counterpart of Theorem 5.4.2 for the solution to problem (6.0.1): (i) some a priori estimates for solutions to the Cauchy problem (6.0.1); (ii) the continuity method. The continuity method is presented in Section 6.1, whereas Section 6.2 is devoted to the a priori estimates which are of the form ||u||C 1+α/2,2+α ((0,T )×Rd ) ≤ K(||u(0, ·)||C 2+α (Rd ) + ||Dt u − Au||C α/2,α ((0,T )×Rd ) ) b

b

(6.0.2)

b

for each function u ∈ C 1+α/2,2+α ([0, T ] × Rd ) and some positive constant K, independent of u. As a consequence of (6.0.2), we prove in Subsections 6.2.2 and 6.4.1 useful interior Schauder estimates for solutions of parabolic problems in domains. Subsection 6.2.1 contains the optimal regularity results for the Cauchy problem (6.0.1). α/2,α More precisely, if f ∈ Cb2+α (Rd ) and g ∈ Cb ((0, T ) × Rd ), then such a problem admits a 1+α/2,2+α 1,2 d unique solution u ∈ Cb ([0, T ] × R ), which in addition belongs to Cb ([0, T ] × Rd ) and, clearly, satisfies estimate (6.0.2). In Section 6.3, we weaken the assumptions on f and g and show that problem (6.0.1) 137

138

Parabolic Equations in Rd

still admits a solution u ∈ Cb ([0, T ] × Rd ) ∩ C 1,2 ((0, T ] × Rd ). Such assumptions still require that g(t, ·) belongs to Cbα (Rd ) for each t ∈ (0, T ], but the H¨older norm of g(t, ·) is allowed to blow up as t tends to 0+ with a certain rate. Also the assumption on f can be weakened. In Example 6.3.1 we show that, in dimension d ≥ 2 problem (6.0.1) admits, in general, no classical solution if g is merely continuous. Things are different if g ≡ 0. In such a case, in Section 6.4 we prove that the Cauchy problem (6.0.1) admits, for each function f ∈ Cb (Rd ), a unique solution u ∈ C([0, ∞) × 1+α/2,2+α Rd ) ∩ Cloc ((0, ∞) × Rd ), which is bounded in every strip [0, T ] × Rd . This allows us to associate in a natural way a semigroup of bounded operators {T (t)} in Cb (Rd ) with the operator A, as it has been done in Chapter 5 for the Laplacian operator. This semigroup is not strongly continuous in Cb (Rd ) but as we will show in Chapter 14, it is analytic. On the other hand, {T (t)} preserves BU C(Rd ) and its restriction to this space defines a strongly continuous semigroup. Some other remarkable properties of the semigroup are studied in Theorems 6.4.2 and 6.4.3. Finally, in Section 6.5, we prove that the more the data f and g are smooth, the more the solution u to the Cauchy problem (6.0.1) is itself smooth.

6.1

The Continuity Method

Here, we state and prove the continuity method. Theorem 6.1.1 (Continuity method) Let X, Y be two Banach spaces and let L0 , L1 be two bounded linear operators mapping X into Y . For each σ ∈ [0, 1], set Lσ = (1 − σ)L0 + σL1 . Suppose that the operator L0 is invertible and there exists a positive constant C such that ||Lσ x||Y ≥ C||x||X , x ∈ X, σ ∈ [0, 1]. (6.1.1) −1 . Then, each operator Lσ is invertible and ||L−1 σ ||L(Y,X) ≤ C

Proof To begin with, we observe that estimate (6.1.1) implies that each operator Lσ is injective. To prove that Lσ is also surjective for all σ ∈ [0, 1], we prove that, if Lσ0 is invertible for some σ0 ∈ [0, 1), then there exists δ > 0, independent of σ0 , such that Lσ is invertible for all σ ∈ [σ0 , σ0 + δ ∧ (1 − σ0 )]. This is enough for our purposes. Indeed, since L0 is invertible, starting from σ0 = 0 and moving to the right by steps of length δ we reach σ = 1 in a finite number of steps. This shows that all the operators Lσ are invertible. So, let us suppose that Lσ0 is invertible and fix σ ∈ (0, 1]. Observe that −1 Lσ = Lσ0 + Lσ − Lσ0 = Lσ0 (I + L−1 σ0 (Lσ − Lσ0 )) = Lσ0 (I + (σ − σ0 )Lσ0 (L1 − L0 )).

Since Lσ0 is invertible, the operator Lσ is invertible if and only if the bounded operator I + (σ − σ0 )L−1 σ0 (L1 − L0 ), mapping X into itself, is invertible. Applying Lemma A.4.4 and (6.1.1) with σ = σ0 , we conclude that, if |σ − σ0 |||L1 − L0 ||L(X,Y ) < C, then the operator −1 I + (σ − σ0 )L−1 C||L1 − L0 ||−1 σ0 (L1 − L0 ) is invertible. Hence, if δ = 2 L(X,Y ) , then, for each σ ∈ [σ0 , σ0 + δ ∧ (1 − σ0 )] the operator Lσ is invertible. Since δ is independent of σ0 , we are done. 

Semigroups of Bounded Operators and Second-Order PDE’s

6.2

139

A priori Estimates

This section is devoted to prove the following result. Theorem 6.2.1 Let the coefficients of the operator A satisfy Hypotheses 6.0.1. Then, there exists a positive constant K, which depends only on d, T and, in a continuous way, on µ (the ellipticity constant of the operator A) and the C α -norm of the coefficients qij , bj (i, j = 1, . . . , d) and c, such that ||u||C 1+α/2,2+α ((0,T )×Rd ) ≤ K(||u(0, ·)||C 2+α (Rd ) + ||Dt u − Au||C α/2,α ((0,T )×Rd ) ) b

b

(6.2.1)

b

for each u ∈ C 1+α/2,2+α ([0, T ] × Rd ) In view of Theorem 5.4.2(ii), we know that estimate (6.2.1) is satisfied when A is the Laplacian. By a straightforward change of variables, we can show that (6.2.1) holds true when A = Tr(QD2 ) and Q is a positive definite and constant matrix. Lemma 6.2.2 Suppose that A = Tr(QD2 ) for some constant symmetric and positive definite matrix Q. Then, estimate (6.2.1) holds true. Proof We split the proof into two steps. Step 1. Here, we suppose that Q = diag(λ1 , . . . , λd ) and denote, by λmax the maximum eigenvalue of Q. Note that the best possible choice of µ is the minimum eigenvalue of Q. 1+α/2,2+α Given u ∈ Cb × Rd ), we introduce the function v : [0, T ] × Rd → R defined √ ([0, T ] √ by v(t, x) = u(t, λ1 x, . . . , λd xd ) for every (t, x) ∈ [0, T ] × Rd . Clearly, v belongs to 1+α/2,2+α Cb ([0, T ] × Rd ). Hence, applying estimate (5.4.3), we get ||v||C 1+α/2,2+α ((0,T )×Rd ) ≤ K1 (||v(0, ·)||C 2+α (Rd ) + ||Dt v − ∆v||C α/2,α ((0,T )×Rd ) ), b

b

(6.2.2)

b

where the constant K1 depends only on d, T and α. Now, we observe that p p p p Dt v(t, x) − ∆v(t, x) = Dt u(t, λ1 x1 , . . . , λd xd ) − Tr(QD2 u(t, λ1 x1 , . . . , λd xd )). Therefore, ||Dt v − ∆v||C α/2,α ((0,T )×Rd ) ≤||Dt u − Au||C α/2,0 ([0,T ]×Rd ) + λα/2 max [Dt u − Au]C 0,α ([0,T ]×Rd ) b

b

b

≤(1 ∨ λα/2 max )||Dt u − Au||C α/2,α ((0,T )×Rd ) . b

Similarly, ||v||C 1+α/2,2+α ((0,T )×Rd ) ≥||u||Cb ([0,T ]×Rd ) + b

d √ X µ ||Dj u||Cb ([0,T ]×Rd ) + ||Dt u||C α/2,0 ([0,T ]×Rd ) b

j=1

+µ

d X

α

2 ||Dij u||C α/2,0 ([0,T ]×Rd ) + µ 2 [Dt u]C 0,α ([0,T ]×Rd )

α

+ µ1+ 2

b

b

i,j=1 d X

2 [Dij u]C 0,α ([0,T ]×Rd ) b

i,j=1 α

≥(µ1+ 2 ∧ 1)||u||C 1+α/2,2+α ((0,T )×Rd ) b

Parabolic Equations in Rd

140 and

1+ α

||v(0, ·)||C 2+α (Rd ) ≤ (1 ∨ λmax2 )||u(0, ·)||C 2+α (Rd ) . b

b

Replacing these three estimates in (6.2.2), we get α

||u||C 1+α/2,2+α ((0,T )×Rd ) ≤K1 (µ−1− 2 ∨ 1)(1 ∨ λ1+α/2 max ) b

× (||u(0, ·)||C 2+α (Rd ) + ||Dt u − Au||C α/2,α ((0,T )×Rd ) ) b

b

1+α/2

α

and (6.2.1) follows with K = (µ−1− 2 ∨ 1)(1 ∨ λmax )K1 . Step 2. Now, we consider the general case and fix a matrix P such that P P ∗ = P ∗ P = I, the (d × d)-identity matrix, and P QP ∗ = Dλ := diag(λ1 , . . . , λd ). Then, the function v, 1+α/2,2+α defined by v(t, x) = u(t, P ∗ x) for (t, x) ∈ [0, T ] × Rd , belongs to Cb ((0, T ) × Rd ). By Step 1, we know that ||v||C 1+α/2,2+α ((0,T )×Rd ) ≤ K(||v(0, ·)||C 2+α (Rd ) + ||Dt v − Tr(Dλ Dx2 v)||C α/2,α ((0,T )×Rd ) ). b

b

b

(6.2.3) Since ∇x v(t, x) = P ∇x u(t, P ∗ x), Dx2 v(t, x) = P Dx2 u(t, P ∗ x)P ∗ for (t, x) ∈ [0, T ] × Rd and P is an isometry with operator norm equal to 1, it is immediate to check that ||Dj u||∞ ≤ |||∇x u|||∞ = |||∇x v|||∞ ≤

d X

||Dj v||∞ ,

j=1

and ||Dij u||C α/2,α ((0,T )×Rd ) ≤ b

d X

||Dhk u||C α/2,α ((0,T )×Rd ) b

h,k=1

=

d X

||(P ∗ Dx2 v(t, P ·)P )hk ||C α/2,α ((0,T )×Rd ) b

h,k=1

≤

d X

||Dhk v||C α/2,α ((0,T )×Rd ) b

h,j=1

for every i, j = 1, . . . , d. Hence, ||u||C 1+α/2,2+α ((0,T )×Rd ) ≤||v||∞ + b

d X j=1

d X

||Dj v||∞ +

i,j=1

||Dij v||C α/2,α ((0,T )×Rd ) b

+ ||Dt v||C α/2,α ((0,T )×Rd ) b

≤||v||C 1+α/2,2+α ((0,T )×Rd ) . b

Moreover, Dt v(t, x) − Tr(Dλ Dx2 v(t, x)) =Dt u(t, P ∗ x) − Tr(Dλ P Dx2 u(t, P ∗ x)P ∗ ) =Dt u(t, P ∗ x) − Tr(P ∗ Dλ P Dx2 u(t, P ∗ x)) =Dt u(t, P ∗ x) − Tr(QDx2 u(t, P ∗ x)).

(6.2.4)

Semigroups of Bounded Operators and Second-Order PDE’s

141

Hence, ||Dt v − Tr(Dλ Dx2 v)||C α/2,α ((0,T )×Rd ) = ||Dt u − Tr(QDx2 u)||C α/2,α ((0,T )×Rd ) .

(6.2.5)

b

b

Replacing (6.2.4) and (6.2.5) in (6.2.3) the assertion follows also in this case.



For the proof of Theorem 6.2.1 we also need the following lemma. Lemma 6.2.3 For every ε > 0, α ∈ (0, 1) and T > 0 there exists a positive constant Cα,ε,T such that ||u||C 1,2 ([0,T ]×Rd ) ≤ ε||u||C 1+α/2,2+α ((0,T )×Rd ) + Cα,ε,T (||Dt u − Au||Cb ([0,T ]×Rd ) + ||u(0, ·)||∞ ) b

b

(6.2.6) 1+α/2,2+α

for all u ∈ Cb T.

((0, T ) × Rd ) and the constant Cα,ε,T is a locally bounded function of 1+α/2,2+α

Proof We fix u ∈ Cb ((0, T ) × Rd ) and begin by observing that, since u is a classical solution to the Cauchy problem ( Dt u(t, x) = Au(t, x) + g(t, x), t > 0, x ∈ Rd , u(0, x) = f (x), x ∈ Rd , with f = u(0, ·) and g = Dt u − Au, Exercise 4.4.6 implies that ||u||Cb ([0,T ]×Rd ) ≤ C3 (||Dt u − Au||Cb ([0,T ]×Rd ) + ||u(0, ·)||∞ )

(6.2.7)

for some positive constant C3 , independent of u, which is a locally bounded function of T . Now, applying the interpolation estimate (1.1.2), with β = 2 and θ = 0, to the function u(t, ·) and Young’s inequality, we get α

2

2+α ||u||C 0,2 ([0,T ]×Rd ) ≤C||u||C2+α d ||u|| 0,2+α b ([0,T ]×R ) C ([0,T ]×Rd ) b

b

ε bε,1 ||u||C ([0,T ]×Rd ) ≤ ||u||C 0,2+α ([0,T ]×Rd ) + C b b 2 bε,1 , independent of u. for all ε > 0 and some positive constant C Similarly, estimate (1.1.2), with Rd replaced by [0, T ], and again Young’s inequality yield α

2

2+α ||u||C 1,0 ([0,T ]×Rd ) ≤C||u||C2+α d ||u|| 1+α/2,0 b ([0,T ]×R )

Cb

b

([0,T ]×Rd )

ε bε,2 ||u||C ([0,T ]×Rd ) ≤ ||u||C 1+α/2,0 ([0,T ]×Rd ) + C b b 2 bε,2 , independent of u. Hence, for all ε > 0 and some positive constant C ||u||C 1,2 ([0,T ]×Rd ) ≤||u||C 0,2 ([0,T ]×Rd ) + ||u||C 1,0 ([0,T ]×Rd ) b b b ε ≤ (||u||C 0,2+α ([0,T ]×Rd ) + ||u||C 1+α/2,0 ([0,T ]×Rd ) ) b b 2 b b + (Cε,1 + Cε,2 )||u||Cb ([0,T ]×Rd ) bε,1 + C bε,2 )||u||C ([0,T ]×Rd ) . ≤ε||u||C 1+α/2,2+α ((0,T )×Rd ) + (C b b

Replacing (6.2.7) in this estimate, (6.2.6) follows at once.



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142

Now, we are ready to prove Theorem 6.2.1 in its full generality. Proof of Theorem 6.2.1 Throughout the proof, we denote by C a positive constant, which may vary from line to line and depends at most only on T , and in a continuous way on µ and the C α -norm of the coefficients of the operator A. For each x0 ∈ Rd and r > 0, we introduce a cut-off function ϑx0 ,r ∈ Cc∞ (Rd ), such that χB(x0 ,r/2) ≤ ϑx0 ,r ≤ χB(x0 ,r) and r|||∇ϑx0 ,r |||∞ + r2 ||D2 ϑx0 ,r ||∞ + r3 ||D3 ϑx0 ,r ||∞ ≤ M1 (6.2.8) for some positive constant M1 , independent of x0 and r. As a byproduct, there exists a positive constant M2 , independent of x0 and r, such that rα ||ϑx0 ,r ||Cbα (Rd ) + r1+α ||ϑx0 ,r ||C 1+α (Rd ) + r2+α ||ϑx0 ,r ||C 2+α (Rd ) ≤ M2 . b

(6.2.9)

b

1+α/2,2+α

We fix u ∈ Cb ([0, T ] × Rd ), x0 ∈ Rd , r ∈ (0, 1) and apply Lemma 6.2.2 to the function ϑx0 ,r u and the operator Ax0 = Tr(Q(x0 )D2 ), obtaining ||ϑx0 ,r u||C 1+α/2,2+α ((0,T )×Rd ) b

e ≤K(||ϑ x0 ,r Dt u − Ax0 (ϑx0 ,r u)||C α/2,α ((0,T )×Rd ) + ||ϑx0 ,r u(0, ·)||C 2+α (Rd ) ).

(6.2.10)

b

b

Let us estimate the right-hand side of (6.2.10). To begin with, we observe that ϑx0 ,r Dt u − Ax0 (ϑx0 ,r u) = ϑx0 ,r (Dt u − Ax0 u) − uAx0 ϑx0 ,r − 2hQ(x0 )∇ϑx0 ,r , ∇x ui. Hence, taking (6.2.9) into account, we can estimate ||ϑx0 ,r Dt u − Ax0 (ϑx0 ,r u)||C α/2,α ((0,T )×Rd ) b

≤||ϑx0 ,r ||Cbα (Rd ) ||Dt u − Ax0 u||C([0,T ]×B(x0 ,r)) + ||ϑx0 ,r ||∞ ||Dt u − Ax0 u||C α/2,α ((0,T )×B(x0 ,r)) + ||Aϑx0 ,r ||Cbα (Rd ) ||u||C α/2,α ((0,T )×B(x0 ,r)) b

+ 2||Q(x0 )||||∇ϑx0 ,r ||Cbα (Rd ;Rd ) ||∇x u||C α/2,α ((0,T )×B(x0 ,r);Rd ) ≤||Dt u − Ax0 u||C α/2,α ((0,T )×B(x0 ,r)) + M2 r−α ||Dt u − Ax0 u||Cb ([0,T ]×Rd ) + Cr−2−α ||u||C α/2,α ((0,T )×Rd ) + Cr−1−α ||∇x u||C α/2,α ((0,T )×Rd ;Rd ) b

b

≤||Dt u − Ax0 u||C α/2,α ((0,T )×B(x0 ,r)) + Cr−α ||u||C 1,2 ([0,T ]×Rd ) + Cr−2−α ||u||C α/2,1+α ((0,T )×Rd ) b

b

≤||Dt u − Ax0 u||C α/2,α ((0,T )×B(x0 ,r)) + Cr−2−α ||u||C 1,2 ([0,T ]×Rd ) . b

(6.2.11) Analogously, since r ∈ (0, 1), it follows that ||ϑx0 ,r u(0, ·)||C 2+α (Rd ) ≤ Cr−2−α ||u(0, ·)||C 2+α (Rd ) . b b From this estimate, (6.2.10), (6.2.11) and observing that ||u||C 1+α/2,2+α ((0,T )×B(x0 ,r)) ≤ ||ϑx0 ,r u||C 1+α/2,2+α ((0,T )×Rd ) , b

we deduce that ||u||C 1+α/2,2+α ((0,T )×B(x0 ,r))
≤C ||Dt u − Ax0 u||C α/2,α ((0,T )×B(x0 ,r)) + r−2−α ||u||C 1,2 ([0,T ]×Rd ) + r−2−α ||u(0, ·)||C 2+α (Rd ) . b b (6.2.12)

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Next, we replace the operator Ax0 with the operator A in the right-hand side of (6.2.12). For this purpose, we observe that ||Dt u − Ax0 u||C α/2,α ((0,T )×B(x0 ,r)) ≤||Dt u − Au||C α/2,α ((0,T )×B(x0 ,r)) + ||Au − Ax0 u||C α/2,α ((0,T )×B(x0 ,r)) ≤||Dt u − Au||C α/2,α ((0,T )×B(x0 ,r)) + ||Tr((Q − Q(x0 ))Dx2 u)||C α/2,α ((0,T )×B(x0 ,r)) +

d X

||bj ||Cbα (Rd ) ||Dj u||C α/2,α ((0,T )×Rd ) + ||c||Cbα (Rd ) ||u||C α/2,α ((0,T )×Rd ) .

(6.2.13)

b

j=1

Since the diffusion coefficients of the operator A are α-H¨older continuous in Rd , we can estimate ||Tr((Q−Q(x0 ))Dx2 u(t, ·))||C(B(x0 ,r)) ≤ Crα ||Dx2 u||C([0,T ]×B(x0 ,r)) , [Tr((Q−Q(x0 ))Dx2 u(t, ·))]C α(B(x0 ,r)) ≤

d X

||qij −qij (x0 )||C α(B(x0 ,r)) ||Dij u||C([0,T ]×B(x0 ,r))

i,j=1

+

d X

||qij −qij (x0 )||C(B(x0 ,r)) [Dij u]C 0,α ([0,T ]×B(x0 ,r))

i,j=1

≤C

d X

||qij ||Cbα (Rd ) ||Dx2 u||C([0,T ]×B(x0 ,r))

i,j=1 d X

+ Crα

||qij ||Cbα (Rd ) ||Dij u||C 0,α ([0,T ]×B(x0 ,r))

i,j=1

for all t ∈ [0, T ] and [Tr((Q(x) − Q(x0 ))Dx2 u(·, x))]C α/2 ((0,T )) ≤ rα

d X

||qij ||Cbα (Rd ) [Dij u]C α/2,0 ((0,T )×B(x0 ,r))

i,j=1

for all x ∈ B(x0 , r). Hence, we conclude that ||Tr((Q−Q(x0 ))Dx2 u)||C α/2,α ((0,T )×B(x0 ,r)) ≤C||Dx2 u||Cb ([0,T ]×Rd ) + Crα

d X i,j=1

||Dij u||C α/2,α ((0,T )×Rd ) , b

which replaced in (6.2.13), gives ||Dt u − Ax0 u||C α/2,α ((0,T )×B(x0 ,r)) b

≤||Dt u − Au||C α/2,α ((0,T )×Rd ) + C||u||C 0,2 ([0,T ]×Rd ) b

+ C||u||C α/2,1+α ((0,T )×Rd ) + Crα b

d X i,j=1

||Dij u||C α/2,α ((0,T )×Rd ) .

(6.2.14)

b

Recalling once more that r ∈ (0, 1), from (6.2.12) and (6.2.14), we obtain ||u||C 1+α/2,2+α ((0,T )×B(x0 ,r)) ≤C ||Dt u − Au||C α/2,α ((0,T )×Rd ) + rα ||u||C 1+α/2,2+α ((0,T )×Rd ) b

b

Parabolic Equations in Rd

144


+ r−2−α ||u||C 1,2 ([0,T ]×Rd ) + r−2−α ||u(0, ·)||C 2+α (Rd ) . b b (6.2.15) Since the constant C in the previous estimate is independent of x0 ∈ Rd , we immediately deduce that ||u||C 1+α/2,2 ([0,T ]×Rd ) ≤ C ||Dt u − Au||C α/2,α ((0,T )×Rd ) + rα ||u||C 1+α/2,2+α ((0,T )×Rd ) b b b
+ r−2−α ||u||C 1,2 ([0,T ]×Rd ) + r−2−α ||u(0, ·)||C 2+α (Rd ) . (6.2.16) b

b

d

Moreover, for every (t, x), (t, y) ∈ [0, T ] × R , with |x − y| ≤ estimate

r 2,

and i, j = 1, . . . , d, we can

|Dij u(t, x) − Dij u(t, y)| + |Dt u(t, x) − Dt u(t, y)|
≤ [Dij u(t, ·)]C α (B(x,r/2)) + [Dt u(t, ·)]C α (B(x,r/2)) |x − y|α ≤||u||C 1+α/2,2+α ((0,T )×B(x,r)) |x − y|α . By (6.2.15) and Lemma 1.1.3, we conclude that sup [Dij (t, ·)]Cbα (Rd ) + sup [Dt (t, ·)]Cbα (Rd )

t∈[0,T ]

t∈[0,T ]

≤C ||Dt u − Au||C α/2,α ((0,T )×Rd ) + rα ||u||C 1+α/2,2+α ((0,T )×Rd ) b b
+ r−2−α ||u||C 1,2 ([0,T ]×Rd ) + r−2−α ||u(0, ·)||C 2+α (Rd )

(6.2.17)

b

b

and, as byproduct, by (6.2.16) and (6.2.17), ||u||C 1+α/2,2+α ((0,T )×Rd ) ≤C ||Dt u − Au||C α/2,α ((0,T )×Rd ) + rα ||u||C 1+α/2,2+α ((0,T )×Rd ) b b b
+ r−2−α ||u||C 1,2 ([0,T ]×Rd ) + r−2−α ||u(0, ·)||C 2+α (Rd ) . b

b

Cr0α

α

Now, we fix r0 ∈ (0, 1) such that ≤ 1/2 to move the term Cr ||u||C 1+α/2,2+α ((0,T )×Rd ) b to the left-hand side of the previous estimate and thus conclude that ||u||C 1+α/2,2+α ((0,T )×Rd ) ≤ 2Cr0−2−α ||Dt u − Au||C α/2,α ((0,T )×Rd ) b

b
+ ||u||C 1,2 ([0,T ]×Rd ) + ||u(0, ·)||C 2+α (Rd ) .

(6.2.18)

b

b

Using Lemma 6.2.3 we estimate the C 1,2 -norm of u and from (6.2.18) we get ||u||C 1+α/2,2+α ((0,T )×Rd ) b

≤2Cr0−2−α [||Dt u − Au||C α/2,α ((0,T )×Rd ) + ε||u||C 1+α/2,2+α ((0,T )×Rd ) b

b

0 + Cα,ε (||Dt u − Au||Cb ([0,T ]×Rd ) + ||u(0, ·)||∞ ) + ||u(0, ·)||C 2+α (Rd ) ] b

for all ε > 0. Choosing ε0 > 0 sufficiently small, we conclude that ||u||C 1+α/2,2+α ((0,T )×Rd ) ≤ C ||Dt u − Au||C α/2,α ((0,T )×Rd ) + ||u(0, ·)||C 2+α (Rd ) b




b

b

and the proof is complete.



Remark 6.2.4 We stress that the constant K in (6.2.1) can be split as   α K = (µ−1− 2 ∨ 1)C d, T, max ||qij ||Cbα (Rd ) , max ||bj ||Cbα (Rd ) , ||c||Cbα (Rd ) , i,j=1,...,d

j=1,...,d

Semigroups of Bounded Operators and Second-Order PDE’s

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where C(d, ·, ·, ·, ·) is a locally bounded function. Indeed, the ellipticity constant µ appears e of the estimate (6.2.10) and the proof of Lemma 6.2.2 shows that only in the constant K this constant is given by   d X e T, α) 1 ∨ (1 ∨ µ−1−α/2 )C(d, ||qij ||α ∞ . i,j=1

On the other hand the constants C in estimates (6.2.11) and (6.2.14) are locally bounded functions of the Cbα (Rd )-norm of the coefficients of the operator A. Similarly, the constants in (6.2.10), (6.2.12), (6.2.15)–(6.2.18) are locally bounded functions of T . Adapting the proof of Theorem 6.2.1, the following result can be proved. Theorem 6.2.5 For each T > 0 there exists a positive constant C such that ||u||C 0,2+α ([0,T ]×Rd ) ≤ C(||u(0, ·)||C 2+α (Rd ) + ||Dt u − Au||C 0,α ([0,T ]×Rd ) ) b

b

(6.2.19)

b

for all u ∈ Cb1,2+α ([0, T ] × Rd ).

6.2.1

Solving problem (6.0.1)

Now, we are in a position to prove the main result of this chapter. Theorem 6.2.6 Let Hypotheses 6.0.1 be satisfied. Then, for every f ∈ Cb2+α (Rd ) and g ∈ α/2,α Cb ((0, T ) × Rd ) there exists a unique classical solution u to problem (6.0.1). In addition, 1+α/2,2+α u belongs to Cb ((0, T ) × Rd ) and there exists a positive constant C, independent of f and g, such that ||u||C 1+α/2,2+α ((0,T )×Rd ) ≤ C(||f ||C 2+α (Rd ) + ||g||C α/2,α ((0,T )×Rd ) ). b

b

(6.2.20)

b

Proof The uniqueness of the classical solution to problem (6.0.1) follows from the maximum principle in Chapter 4 (see Exercise 4.4.6). The existence of a solution to problem (6.0.1) will be proved, as already claimed, using the continuity method. For this purpose, for every σ ∈ [0, 1], we introduce the operator Aσ defined on smooth functions ψ : Rd → R by Aσ ψ =

d X

qij (·, σ)Dij ψ +

i,j=1

d X

bj (·, σ)Dj ψ + c(·, σ)ψ,

j=1

where qij (·, σ) = (1 − σ)δij + σqij , bj (·, σ) = σbj and c(·, σ) = σc for every i, j = 1, . . . , d. As it is immediately seen, A0 = ∆ and A1 = A. Each operator Aσ is elliptic since d X i,j=1

qij (x, σ)ξi ξj = (1 − σ)|ξ|2 + σ

d X

qij (x)ξi ξj ≥ (1 − σ)|ξ|2 + σµ|ξ|2 ≥ min{1, µ}|ξ|2

i,j=1

(6.2.21) for every x, ξ ∈ Rd and σ ∈ [0, 1]. Moreover, ||qij (·, σ)||Cbα (Rd ) ≤ ||qij ||Cbα (Rd ) ∨ 1, ||bj (·, σ)||Cbα (Rd ) ≤ ||bj ||Cbα (Rd ) ,

(6.2.22) ||c(·, σ)||Cbα (Rd ) ≤ ||c||Cbα (Rd )

(6.2.23)

Parabolic Equations in Rd

146

for each i, j = 1, . . . , d and σ ∈ [0, 1]. 1+α/2,2+α α/2,α Consider the operators L0 , L1 : Cb ((0, T ) × Rd ) → Cb ((0, T ) × Rd ) × 2+α d Cb (R ) defined by L0 u = (Dt u − ∆u, u(0, ·)) and L1 u = (Dt u − Au, u(0, ·)) for ev1+α/2,2+α ery u ∈ Cb ((0, T ) × Rd ). Note that the operator Lσ = (1 − σ)L0 + σL1 is associated with the operator Aσ in the sense that Lσ u = (Dt u − Aσ u, u(0, ·)) for each 1+α/2,2+α u ∈ Cb ((0, T ) × Rd ). By estimate (6.2.1) we known that ||u||C 1+α/2,2+α ((0,T )×Rd ) ≤C(||Dt u − Aσ u||C α/2,α ((0,T )×Rd ) + ||u(0, ·)||C 2+α (Rd ) ) b

b

b

=C||Lσ u||C α/2,α ((0,T )×Rd )×C 2+α (Rd ) , b

b

where the constant C is independent of σ in view of Remark 6.2.4 and the estimates (6.2.21)(6.2.23). To conclude the proof, we observe that Theorem 5.4.2(ii) implies that the operator L0 1+α/2,2+α α/2,α is invertible from Cb ((0, T ) × Rd ) into Cb ((0, T ) × Rd ) × Cb2+α (Rd ). Hence, we can apply Theorem 6.1.1 which shows that the operator L1 is invertible, i.e., for each α/2,α f ∈ Cb2+α (Rd ) and g ∈ Cb ((0, T ) × Rd ), the Cauchy problem (6.0.1) admits a unique 1+α/2,2+α solution u ∈ Cb ((0, T ) × Rd ), which satisfies estimate (6.2.20). 

6.2.2

Interior Schauder estimates for solutions to parabolic equations in domains: Part I

As a byproduct of Theorem 6.2.1, we can prove some interesting interior Schauder estimates satisfied by the solutions to the differential equation Dt u = Au + g in (0, T ] × Ω, Ω ⊂ Rd being an open connected set. Such estimates are of local type, i.e., they allow to estimate the parabolic H¨ older norm of u and its derivatives in every compact subset of (0, T ] × Ω (or even of [0, T ] × Ω) in terms of the sup-norm of the solution itself in a larger compact set. We assume the following conditions on the coefficients of the operator A. α Hypotheses 6.2.7 The coefficients qij = qji , bj and c (i, j = 1, . . . , d) belong to Cloc (Ω) and there exists a positive continuous function µ : Ω → R such that d X

qij (x)ξi ξj ≥ µ(x)|ξ|2 ,

ξ ∈ Rd , x ∈ Ω.

(6.2.24)

i,j=1

Remark 6.2.8 Note that condition (6.2.24) implies that the matrix Q(x) is positive definite d X in Ω and, for every compact set K ⊂ Ω, qij (x)ξi ξj ≥ µK |ξ|2 for every ξ ∈ Rd and x ∈ K, i,j=1

where µK = inf x∈K µ(x). Note also that, under the previous assumptions, it may happen that inf x∈Ω µ(x) = 0. Now, we state precisely the first result of this subsection. Theorem 6.2.9 Let u ∈ C 1,2 ((0, T ) × Ω) be a classical solution to the differential equation α/2,α 1+α/2,2+α Dt u = Au + g, corresponding to g ∈ Cloc ((0, T ) × Ω). Then, u ∈ Cloc ((0, T ) × Ω) and for every pair of bounded closed intervals I1 = [τ, T0 ] and I2 = [σ, T0 ], such that 0 < σ < τ < T0 < T , and every pair of compact sets K1 ⊂ K2 ⊂ Ω, such that d(∂K1 , ∂K2 ) > 0, there exists a positive constant C which depends on I1 , I2 , K1 , K2 and, in a continuous

Semigroups of Bounded Operators and Second-Order PDE’s

147

way on the α-H¨ older norm of the coefficients of the operator A over a neighborhood1 of K2 , such that ||u||C 1+α/2,2+α (I1 ×K1 ) ≤ C(||u||C(I2 ×K2 ) + ||g||C α/2,α (I2 ×K2 ) ). (6.2.25) Proof The main step of the proof consists in showing that, for each x0 ∈ Ω, r > 0, such that 2r < dist(x0 , ∂Ω), 0 < σ < τ < T0 < T , estimate (6.2.25) is satisfied with K1 = B(x0 , r) and K2 = B(x0 , 2r). This is the content of Step 1. Indeed, once this estimate is proved, a covering argument will allow us, in Step 2, to obtain estimate (6.2.25) in its full generality. Step 1. Fix σ, τ , T0 and r > 0 as above and let us prove (6.2.25) with I1 , I2 , K1 and K2 as above. Throughout the proof, we denote by C a positive constant, independent of u, g and n, which may vary from line to line. We introduce the real sequences (rn ) and (tn ), defined by rn = (2 − 2−n )r and tn = σ + 2−n (τ − σ), for every n ∈ N ∪ {0}. Note that the previous sequences are, respectively, increasing and decreasing. Next, we consider two functions ϕ, ϑ ∈ C ∞ (R) which satisfy the conditions χ[2,∞) ≤ ϕ ≤ χ[1,∞) and χ(−∞,1] ≤ ϑ ≤ χ(−∞,2] . For every n ∈ N ∪ {0}, we set   t − tn+1 , t ∈ R, ϕn (t) = ϕ 1 + tn − tn+1   |x − x0 | − rn ϑn (x) = ϑ 1 + , x ∈ Rd . rn+1 − rn As it is easily seen, ϕn (t) = 1 if t ≥ tn and it vanishes if t ≤ tn+1 . Similarly, ϑn (x) = 1 if / B(x0 , rn+1 ). Moreover, we introduce the operator x ∈ B(x0 , rn ) and ϑn (x) = 0 if x ∈ Ae =

d X

qeij Dij +

i,j=1

d X

ebj Dj + e c,

j=1

where qeij = %qij + (1 − %)δij , ebj = %bj (i, j = 1, . . . , d), e c = %c, and % ∈ Cc∞ (Rd ) is compactly supported in Ω with 0 ≤ % ≤ 1 and % ≡ 1 in B(x0 , 2r). Clearly, the coefficients of e extended in the trivial way to the whole Rd and still denoted in the same the operator A, way, belong to Cbα (Rd ) and coincide with the coefficients of the operator A in B(x0 , 2r). Moreover, d X i,j=1

qeij (x)ξi ξj =%(x)

d X

qij (x)ξi ξj + (1 − %)|ξ|2

i,j=1

≥[%(x)µ(x) + 1 − %(x)]|ξ|2 ≥ µ0 |ξ|2 for all x, ξ ∈ Rd , where µ0 = 1 ∧ inf x∈supp% µ(x) > 0. From the above arguments (and in particular the fact that ϑn (x) = 0 for x ∈ / B(x0 , 2r)) it follows that each function vn = uϕn ϑn belongs to Cb1,2 ([0, T ] × Rd ), vanishes at t = 0 and satisfies the equation e n + gn in (0, ∞) × Rd , where Dt vn = Av gn = ϕn ϑn g − ϕn u(Aϑn − cϑn ) − 2ϕn hQ∇ϑn , ∇x ui + ϕ0n ϑn u. Since the coefficients of the operator Ae satisfy Hypotheses 6.0.1, it follows from Theorem 6.2.6 that the Cauchy problem ( e Dt u(t, x) = Au(t, x) + gn (t, x), t ∈ (0, T0 ], x ∈ Rd , u(0, x) = 0, x ∈ Rd , 1 The

neighborhood U b Ω can be arbitrarily fixed.

Parabolic Equations in Rd

148

1+α/2,2+α

admits a unique solution which belongs to Cb 1+α/2,2+α belongs to Cb ((0, T0 ) × Rd ) and

((0, T0 ) × Rd ). Hence, the function vn

||vn ||C 1+α/2,2+α ((0,T0 )×Rd ) ≤ C||gn ||C α/2,α ((0,T0 ]×Rd ) , b

b

n ∈ N,

where the constant C is independent of n. In particular, taking n = 0, we deduce that u belongs to C 1+α/2,2+α ((τ1 , T0 ) × B(x0 , r)). The arbitrariness of τ and r imply that u ∈ C 1+α/2,2+α (J × K) for each closed interval J b (0, T0 ] and each compact set K ⊂ Rd . α/2,α Estimating the Cb ((0, T0 ) × Rd )-norm of the function gn , we get ||vn ||C 1+α/2,2+α ((0,T0 )×Rd )

b
≤C||ϑn ||Cb3 (Rd ) ||ϕn ||Cb2 (R) ||u||C α/2,1+α ((tn+1 ,T0 )×B(x0 ,rn+1 )) + ||g||C α/2,α ((tn+1 ,T0 )×B(x0 ,rn+1 ))   d X 5n ≤2 C ||vn+1 ||C α/2,α ((0,T0 )×Rd ) + ||Dj vn+1 ||C α/2,α ((0,T0 )×Rd ) +||g||C α/2,α ((σ,T0 )×B(x0 ,2r)) , b

j=1

b

where we have used the inequalities ||ϑn ||Cb3 (Rd ) ≤ 23n C and ||ϕn ||Cb2 (R) ≤ 22n C, which hold true for all n ∈ N ∪ {0}, and observed that vn+1 ≡ u in [tn+1 , T0 ] × B(x0 , rn+1 ). This estimate and (1.2.6) yield ||vn ||C 1+α/2,2+α ((0,T0 )×Rd ) b

≤25n C(ε||vn+1 ||C 1+α/2,2+α ((0,T0 )×Rd ) + ε−(1+α) ||vn+1 ||∞ + ||g||C α/2,α ((σ,T0 )×B(x0 ,2r)) ) b

for each ε ∈ (0, 1). We fix η ∈ (0, 2−5(2+α) ) and choose ε = εn = 2−5n C −1 η; from the previous estimate we obtain ζn ≤ ηζn+1 + 25n(2+α) Cη ||u||C([σ,T0 ]×B(x0 ,2r)) + 25n C||g||C α/2,α ((σ,T0 )×B(x0 ,2r)) ,

(6.2.26)

where ζn = ||vn ||C 1+α/2,2+α ((0,T0 )×Rd ) . Multiplying both sides of (6.2.26) by η n and summing b from 1 to m ∈ N, we get ζ0 − η m+1 ζm+1 ≤Cη ||u||C([σ,T0 ]×B(x0 ,2r))

m X

(25(2+α) η)n

n=0

+ C||g||C α/2,α ((σ,T0 )×B(x0 ,2r))

m X

(32η)n

n=0

≤C(||u||C([σ,T0 ]×B(x0 ,2r)) + ||g||C α/2,α ((σ,T0 )×B(x0 ,2r)) ),

(6.2.27)

since both the two series converge, due to the choice of η. To conclude, we observe that η m+1 ζm+1 tends to 0 as m tends to ∞. Indeed, we have shown that u ∈ C 1+α/2,2+α (J × K) for every J b (0, ∞) and K b Rd . In particular, u belongs to C 1+α/2,2+α ((σ, T0 )×B(x0 , 2r)). Hence, ζn =||vn ||C 1+α/2,2+α ((0,T0 )×Rd ) b

≤C||u||C 1+α/2,2+α ((σ,T0 )×B(x0 ,2r)) ||ϕn ||C 2 ([0,T0 ]) ||ϑn ||Cb3 (Rd ) b

≤25n C||u||C 1+α/2,2+α ((σ,T0 )×B(x0 ,2r)) . b

The choice of η implies that η m+1 ζm+1 vanishes as m → ∞. Letting m tend to ∞ in (6.2.27)

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we conclude that ζ0 ≤ C(||u||C([σ,T0 ]×B(x0 ,2r)) + ||g||C α/2,α ((σ,T0 )×B(x0 ,2r)) ) and this implies that ||u||C 1+α/2,2+α ((τ,T0 )×B(x0 ,r)) ≤ C(||u||C([σ,T0 ]×B(x0 ,2r)) + ||g||C α/2,α ((σ,T0 )×B(x0 ,2r)) ), i.e., estimate (6.2.25) with the above choices of Ij and Kj (j = 1, 2) follows. Step 2. Without loss of generality, we can assume that K1 is the closure of a connected bounded open set Ω1 . Let us set δ = dist(∂K1 , ∂K2 ). As it is immediately, [ K1 ⊂ B(x0 , δ/2) ⊂ K2 . Since K1 is a compact set, we can find out m ∈ N and x0 ∈K1

points x1 , . . . , xm ∈ K1 such that K1 ⊂

m [

B(xj , δ/2). We claim that

j=1

||u||C 1+α/2,2+α (I1 ×K1 ) ≤

m X

||u||C 1+α/2,2+α (I1 ×B(xk ,δ/2)) .

(6.2.28)

k=1

This estimate yields easily (6.2.25). It is clear that ||u||C 1,2 (I1 ×K1 ) ≤

max ||u||C 1,2 (I1 ×B(xk ,δ/2)) ,

(6.2.29)

k=1,...,m

||Dt u||C α/2,0 (I1 ×K1 ) ≤

m X

||Dt u||C α/2,0 (I1 ×B(xk ,δ/2)) ,

(6.2.30)

k=1

||Dij u||C α/2,0 (I1 ×K1 ) ≤

m X

||Dij u||C α/2,0 (I1 ×B(xk ,δ/2)) ,

i, j = 1, . . . , d.

(6.2.31)

k=1

Let us show that ||Dt u||C 0,α (I1 ×K1 ) ≤

m X

||Dt u||C 0,α (I1 ×B(xk ,δ/2)) ,

(6.2.32)

k=1

||Dij u||C 0,α (I1 ×K1 ) ≤

m X

||Dij u||C 0,α (I1 ×B(xk ,δ/2)) ,

i, j = 1, . . . , d.

(6.2.33)

k=1

We limit ourselves to proving (6.2.32) since (6.2.33) can be proved in the same way. We fix t ∈ I1 and x, y ∈ Ω1 and estimate |Dt u(t, x) − Dt u(t, y)|. Since Ω1 is a domain of Rd , it is connected by arcs. Hence, there exists a curve γ : [0, 1] → Ω1 such that γ(0) = x and γ(1) = y. Fix a ball B(z1 , δ/2) of the covering of K1 which contains x and denote by t1 ∈ (0, 1] the supremum of all t ∈ [0, T ] such that γ(t) ∈ B(z1 , δ/2). If t1 = T , then both x and y belong to B(z1 , δ/2) and we can estimate |Dt u(t, x) − Dt u(t, y)| ≤ [Dt (t, ·)]C α (B(z1 ,δ/2)) |x − y|α . If t1 < T , then we denote by B(z2 , δ/2) one of the balls of the covering of K1 which contains γ(t1 ). If y ∈ B(z2 , δ/2), then we can estimate |Dt u(t, x) − Dt u(t, y)| ≤|Dt u(t, x) − Dt u(t, γ(t1 ))| + |Dt u(t, γ(t1 )) − Dt u(t, y)| ≤[Dt u(t, ·)]C α (B(z1 ,δ/2)) |x − γ(t1 )|α + [Dt u(t, ·)]C α (B(z2 ,δ/2)) |γ(t1 ) − y|α ≤([Dt u(t, ·)]C α (B(z1 ,δ/2)) + [Dt u(t, ·)]C α (B(z2 ,δ/2)) )|x − y|α . Otherwise, the supremum of the set {t ∈ [t1 , T ] : γ(t1 ) ∈ B(z2 , δ/2)} is less than T . We determine a ball B(z3 , δ/2) with contains γ(t2 ) and proceed as above. In at most m steps the procedure ends and (6.2.32) follows.

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Now, from (6.2.29)–(6.2.33) and Proposition 1.2.4(i), estimate (6.2.28) follows immediately.  The same arguments as in the proof of Theorem 6.2.9 can be used to prove the following result. For the reader’s convenience, we point out the differences. Theorem 6.2.10 Let u ∈ C 1,2 ([0, T ] × Ω) be a classical solution to the differential equation α/2,α 2+α Dt u = Au + g, corresponding to g ∈ Cloc ([0, T ] × Ω) and such that u(0, ·) ∈ Cloc (Ω). 1+α/2,2+α Then, u ∈ Cloc ([0, T ] × Ω) and for every pair of compact sets K1 ⊂ K2 ⊂ Ω such that d(∂K1 , ∂K : 2) > 0, there exists a positive constant C = C(T, µ, K1 , K2 ) such that ||u||C 1+α/2,2+α ([0,T ]×K1 ) ≤ C(||u(0, ·)||C 2+α (K2 ) + ||u||C([0,T ]×K2 ) + ||g||C α/2,α ([0,T ]×K2 ) ). (6.2.34) As a function of T , C is locally bounded in (0, ∞). Proof The only slight differences with respect to the proof of Theorem 6.2.9 are in the proof of (6.2.34) with K1 = B(x0 , r) and K2 = B(x0 , 2r). Hence, here we consider only this case. Throughout the proof, we denote by C a positive constant, independent of u, g and n, which may vary from line to line, and consider the same sequence (ϑn ) and the same operator Ae as in the proof of Theorem 6.2.9. For every n ∈ N the function vn = uϑn belongs to Cb1,2 ([0, T ] × Rd ) and satisfies the e n + gn in (0, ∞) × Rd , where equation Dt vn = Av e n − cϑn ) − 2hQ∇ϑ e n , ∇x ui gn = ϑn g − u(Aϑ

(6.2.35)

e = (e and Q qij ). Since the coefficients of the operator Ae satisfy Hypotheses 6.0.1, the Cauchy problem ( e Dt w(t, x) = Aw(t, x) + gn (t, x), t ∈ [0, T ], x ∈ Rd , w(0, x) = ϑn (x)u(0, x), x ∈ Rd , 1+α/2,2+α

admits a unique solution which belongs to Cb 1+α/2,2+α belongs to Cb ((0, T ) × Rd ) and

((0, T ) × Rd ). Hence, the function vn

||vn ||C 1+α/2,2+α ((0,T )×Rd ) ≤ C(||ϑn u(0, ·)||C 2+α (Rd ) + ||gn ||C α/2,α ((0,T )×Rd ) ) b

b

b

for every n ∈ N ∪ {0}. In particular, taking n = 0, we deduce that u belongs to C 1+α/2,2+α ((0, T ) × B(x0 , r)). The arbitrariness of r imply that function u belongs to C 1+α/2,2+α ([0, T ] × K) for each compact set K ⊂ Rd . α/2,α Estimating the Cb ((0, T ) × Rd )-norm of the function gn and observing that ||ϑn u(0, ·)||C 2+α (Rd ) ≤ 23n C||ϑn ||Cb3 (Rd ) ||u(0, ·)||C 2+α (B(x0 ,2r)) ,

n ∈ N,

b

we get ||vn ||C 1+α/2,2+α ((0,T )×Rd ) ≤C||ϑn ||Cb3 (Rd ) ||u(0, ·)||C 2+α (B(x0 ,2r)) + ||u||C α/2,1+α ((0,T )×B(x0 ,rn+1 )) b
+ ||g||C α/2,α ((0,T )×B(x0 ,2r)) ≤23n C ||vn+1 ||C α/2,α ((0,T )×Rd ) + b

d X j=1

||Dj vn+1 ||C α/2,α ((0,T )×Rd ) b

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+ ||u(0, ·)||C 2+α (B(x0 ,2r)) + ||g||C α/2,α ((0,T )×B(x0 ,2r)) . This estimate and (1.2.6) yield ||vn ||C 1+α/2,2+α ((0,T )×Rd ) ≤23n C(ε||vn+1 ||C 1+α/2,2+α ((0,T )×Rd ) + ε−(1+α) ||vn+1 ||∞ b

b

+ ||u(0, ·)||C 2+α (B(x0 ,2r)) + ||g||C α/2,α ((0,T )×B(x0 ,2r)) ) for each ε ∈ (0, 1). Fixing η ∈ (0, 2−3(2+α) ), choosing ε = εn = 2−3n C −1 η and arguing as in the proof of Theorem 6.2.9 we can easily show that ||u||C 1+α/2,2+α ((0,T )×B(x0 ,r)) ≤c(||u(0, ·)||C 2+α (B(x0 ,2r)) + ||u||C([0,T ]×B(x0 ,2r)) + ||g||C α/2,α ((0,T )×B(x0 ,2r)) ). By the same covering argument used in the proof of Theorem 6.2.9, we can complete the proof. Moreover, taking Remark 6.2.4 into account, it is easy to check the last assertion.  As a consequence of Theorems 6.2.9 and 6.2.10 we can prove the following result. `+α Proposition 6.2.11 Suppose that the coefficients of the operator A belong to Cloc (Ω) for some ` ∈ N and (6.2.24) is satisfied. Then, the following properties are satisfied. α/2,`+α

1+α/2,2+α

(i) Let g ∈ Cloc ([0, T ] × Ω) and u ∈ Cloc ([0, T ] × Ω) solve the equation Dt u = 1+α/2,`+2+α `+2+α Au + g on [0, T ] × Ω, with u(0, ·) ∈ Cloc (Ω). Then, u ∈ Cloc ([0, T ] × Ω), the function Dt u is continuously differentiable up to the `-th-order in [0, T ] × Ω with respect to the spatial variables, the function Dxβ u is continuously differentiable with respect to time in [0, T ] × Ω for every multi-index β, with length at most `, and, for every pair of compact sets K1 ⊂ K2 ⊂ Ω such that d(∂K1 , ∂K2 ) > 0, there exists a positive constant C depending on T, µ, K1 , K2 and the C α+l -norm of the coefficients of A such that ||u||C 1+α/2,`+2+α ([0,T ]×K1 ) ≤C(||u(0, ·)||C `+2+α (K2 ) + ||u||C([0,T ]×K2 ) + ||g||C α/2,`+α ([0,T ]×K2 ) ). α/2,`+α

(6.2.36)

1+α/2,2+α

(ii) Let g ∈ Cloc ((0, T ) × Ω) and u ∈ Cloc ((0, T ) × Ω) solve the equation 1+α/2,`+2+α Dt u = Au + g on (0, T ) × Ω. Then, u ∈ Cloc ((0, T ) × Ω), the function Dt u is continuously differentiable up to the `-th-order in (0, T ) × Ω with respect to the spatial variables, the function Dxβ u is continuously differentiable with respect to time in (0, T ) × Ω for every multi-index β, with length at most `, and, for every 0 < σ < τ < T0 < T and every compact sets K1 ⊂ K2 ⊂ Ω such that d(∂K1 , ∂K2 ) > 0, there exists a positive constant C depending on T, µ, I1 , I2 , K1 , K2 and the C `+α norm of the coefficients of A such that ||u||C 1+α/2,`+2+α ([τ,T0 ]×K1 ) ≤ C(||u||C([σ,T0 ]×K2 ) + ||g||C α/2,`+α ([σ,T0 ]×K2 ) ).

(6.2.37)

Proof (i) We argue by induction on `. As in the proof of Proposition 6.2.9 it suffices to prove the assertion with K1 = B(x0 , r) and K2 = B(x0 , 2r), where x0 ∈ Ω and 2r < dist(x0 , ∂Ω). Indeed, the arguments in Step 2 of the proof of the quoted theorem allow to complete the proof. We fix r < rb < re < 2r and we first assume that ` = 1. For each h ∈ R \ {0}, such that |h| < (b r − r1 ) ∧ (e r − rb) ∧ (2r − re) ∧ (dist(x0 , ∂Ω) − 2r) and k ∈ {1, . . . , d}, we introduce

Parabolic Equations in Rd

152

the operator τh,k , defined by τh,k ψ = h−1 (ψ(· + hek ) − ψ) for every function ψ. Finally, we denote by vh,k : [0, T ] × B(x0 , r) → R the function defined by vh,k (t, ·) = τh,k (u(t, ·)) for every t ∈ [0, T ]. Clearly, vh,k satisfies the differential equation Dt vh,k = Ah,k vh,k +

d X

(τh,k qij )Dij u +

i,j=1

d X

τh,k bj Dj u + τh,k g + (τh,k c)u

j=1

in [0, T ] × B(x0 , r), where Ah,k vh,k = Tr(Q(· + hek )Dx2 vh,k ) + hb(· + hek ), ∇x vh,k i + c(· + hek )vh,k . Let us observe that 1 τh,k qij (x) = h

h

Z

Dk qij (x + σek ) dσ, 0

x ∈ B(x0 , rb),

so that ||τh,k qij ||C(B(x0 ,br)) ≤ ||Dk qij ||C(B(x0 ,2r)) and Z h 1 |τh,k qij (x) − τh,k qij (y)| ≤ |Dk qij (x + σej ) − Dk qij (y + σej )| dσ h 0 ≤[Dk qij ]C α (B(x0 ,er)) |x − y|α for every x, y ∈ B(x0 , rb) and i, j ∈ {1, . . . , d}. It thus follows that τhk qij belongs to C α (B(x0 , rb)) and ||τh,k qij ||C α (B(x0 ,br)) ≤ ||Dk qij ||C α (B(x0 ,2r)) . Similarly, it can be shown that ||τh,k bj ||C α (B(x0 ,br)) ≤ ||Dk bj ||C α (B(x0 ,2r)) for every j = 1, . . . , d, ||τh,k g||C α/2,α ((0,T )×B(x0 ,br)) ≤ ||Dk g||C α/2,α ((0,T )×B(x0 ,2r)) and ||τh,k c||C α (B(x0 ,br)) ≤ ||Dk c||C α (B(x0 ,2r)) . Applying estimate (6.2.34) with K2 = B(x0 , rb), we thus conclude that  ||vh,k ||C 1+α/2,2+α ((0,T )×B(x0 ,r)) ≤C ||τh,k u(0, ·)||C 2+α (B(x0 ,br)) +||τh,k u||C([0,T ]×B(x0 ,br))  + C1 ||u||C 0,2+α ([0,T ]×B(x0 ,br)) + ||g||C α/2,1+α ((0,T )×B(x0 ,2r))  ≤C ||u(0, ·)||C 3+α (B(x0 ,2r)) +C||u||C 0,2+α ([0,T ]×B(x0 ,br))  + ||u||C 0,1 ([0,T ]×B(x0 ,er)) +||g||C α/2,α ((0,T )×B(x0 ,2r)) , where C1 =

d X i,j=1

||Dk qij ||C α (B(x0 ,2r)) +

d X

||Dk bi ||C α (B(x0 ,2r)) + ||Dk c||C α (B(x0 ,2r)) .

i=1

Using once more estimate (6.2.34), with K1 = B(x0 , re), we infer that ||vh,k ||C 1+α/2,2+α ((0,T )×B(x0 ,r)) ≤ C(||u(0, ·)||C 3+α (B(x0 ,2r)) + ||g||C α/2,α ((0,T )×B(x0 ,2r)) ). Letting h tend to 0 and applying a compactness argument, based on Arzel`a-Ascoli theorem, we conclude that Dk u belongs to C 1+α/2,2+α ((0, T ) × B(x0 , r)) and Dt Dk u = ADj u + Pd i,j=1 Dk qij Dij Dk u + hDk b, ∇x ui + (Dk c)u on [0, T ] × B(x0 , r) for each j = 1, . . . , d. By the arbitrariness of k, the assertion follows in this particular case, observing that the equality Dt u = Au + g can be continuously differentiate in [0, T ] × Ω with respect to the spatial variables, so that Dt Dk u = Dk Dt u in [0, T ] × Ω for every k = 1, . . . , d.

Semigroups of Bounded Operators and Second-Order PDE’s

153

Now, we suppose that the assertion holds for some ` > 1 and prove that it is true also with ` being replaced by ` + 1. For this purpose, we fix a multi-index β with length ` and observe that the function v = Dxβ u solves the differential equation Dt v = Av + gβ , where d d X X β  X X β  γ β−γ + D qij Dij Dx u + Dγ bj Dj Dxβ−γ u γ γ i,j=1 0 0 implies that u ∈ C ∞ (R2 \ {0}). On the other hand, if 2−2k−4 ≤ x2 + y 2 ≤ 2−2k for some k ∈ N, then ϑ(2n x, 2n y) = 0 when n ≥ k + 3. Hence, Dxy u(x, y) =Dxy

k+2 X

n−1 xyϑ(2n x, 2n y)

n=1

=

k+2 X

n−1 ϑ(2n x, 2n y) +

n=1

+

k+2 X

k+2 X

4n n−1 xyDxy ϑ(2n x, 2n y)

n=1

2n n−1 [xDx ϑ(2n x, 2n y) + yDy ϑ(2n x, 2n y)].

n=1

Note that, if n ≤ k, then |2n (x, y)| ≤ 1 so that ϑ(2n x, 2n y) = 1. Therefore, Dxy u(x, y) =

k X 1 1 1 + ϑ(2k+1 x, 2k+1 y) + ϑ(2k+2 x, 2k+2 y) n k + 1 k + 2 n=1

4k+1 4k+2 xyDxy ϑ(2k+1 x, 2k+1 y) + xyDxy ϑ(2k+2 x, 2k+2 y) k+1 k+2 2k+1 + [xDx ϑ(2k+1 x, 2k+1 y) + yDy ϑ(2k+1 x, 2k+1 y)] k+1 +

Semigroups of Bounded Operators and Second-Order PDE’s +

159

2k+2 [xDx ϑ(2k+2 x, 2k+2 y) + yDy ϑ(2k+2 x, 2k+2 y)]. k+2

Hence, k X 1 − 2 sup |zwDxy ϑ(z, w)| Dxy u(x, y) ≥ n (z,w)∈R2 n=1  −2 sup |zDx ϑ(z, w)| + sup (z,w)∈R2

 |wDy ϑ(z, w)| .

(z,w)∈R2

The above estimate shows that Dxy u(x, y) diverges to ∞ as (x, y) tends to (0, 0). Nevertheless, the assumptions on f and g may be weakened as the following theorem shows. Theorem 6.3.2 Let Hypotheses 6.0.1 be satisfied. Suppose that f belongs to Cb2+α−2θ (Rd ) and g ∈ C((0, T ] × Rd ) is such that supt∈(0,T ] tθ ||g(t, ·)||Cbα (Rd ) < ∞ for some θ ∈ (0, 1). Then, the Cauchy problem (6.0.1) admits a unique classical solution u, which is bounded in [0, T ] × Rd . In addition, u(t, ·) ∈ Cb2+α (Rd ) for every t ∈ (0, T ] and sup tθ (||u(t, ·)||C 2+α (Rd ) + ||Dt u(t, ·)||Cbα (Rd ) ) + ||u||C 0,2+α−2θ ([0,T ]×Rd ) b

t∈(0,T ]

b



≤C0 ||f ||C 2+α−2θ (Rd ) + sup tθ ||g(t, ·)||Cbα (Rd ) b



t∈(0,T ]

for some positive constant C0 , independent of u, f and g. In particular, if f and g belong to Cb2+α (Rd ) and C([0, T ] × Rd ), respectively, and supt∈[0,T ] ||g(t, ·)||Cbα (Rd ) < ∞, then, u belongs to Cb1,2 ([0, T ] × Rd ), u(t, ·) ∈ Cb2+α (Rd ) for every t ∈ [0, T ] and   sup ||u(t, ·)||C 2+α (Rd ) + sup ||Dt u(t, ·)||Cbα (Rd ) ≤ C1 ||f ||C 2+α (Rd ) + sup ||g(t, ·)||Cbα (Rd ) t∈[0,T ]

b

b

t∈[0,T ]

t∈(0,T ]

(6.3.1) for some positive constant C1 , independent of u, f and g. Proof We just sketch the proof, which is similar to that of Theorem 6.2.6. Throughout the proof, by C, we denote a positive constant, which is independent of u, f and g, and may vary from line to line. To begin with, we observe that, by Theorem 5.4.5 (with β = 2 + α − 2θ), for every f ∈ Cb2+α−2θ (Rd ) and g ∈ C((0, T ] × Rd ) such that sup tθ ||g(t, ·)||Cbα (Rd ) < ∞,

t∈(0,T ]

the Cauchy problem (

Dt u(t, x) = ∆u(t, x) + g(t, x), u(0, x) = f (x),

t ∈ (0, T ],

x ∈ Rd , x ∈ Rd ,

(6.3.2)

admits a unique solution u ∈ Cb ([0, T ] × Rd ) ∩ C 1,2 ((0, T ] × Rd ) such that   sup tθ (||u(t, ·)||C 2+α (Rd ) +||Dt u(t, ·)||Cbα (Rd ) ) ≤ C ||f ||C 2+α−2θ (Rd ) + sup tθ ||g(t, ·)||Cbα (Rd ) . t∈(0,T ]

b

b

t∈(0,T ]

Parabolic Equations in Rd

160

Using the same arguments as in the proof of Lemma 6.2.2, it can be easily checked that the Cauchy problem (6.3.2) with the Laplacian being replaced with the operator Ax0 = Tr(Q(x0 )Dx2 ), admits a (unique) solution u ∈ Cb ([0, T ] × Rd ) ∩ C 1,2 ((0, T ] × Rd ) for every x0 ∈ Rd . In particular, if f ∈ Cb2+α−2θ (Rd ), then u is bounded in [0, T ] with values in Cb2+α−2θ (Rd ) and
sup tθ ||u(t, ·)||C 2+α (Rd ) + ||Dt u(t, ·)||Cbα (Rd ) + ||u||C 0,2+α−2θ ([0,T ]×Rd ) b

t∈(0,T ]

b



≤C ||f ||C 2+α−2θ (Rd ) + sup tθ ||g(t, ·)||Cbα (Rd ) b



(6.3.3)

t∈(0,T ]

with the constant C being independent of x0 ∈ Rd . Take u ∈ Cb0,2+α−2θ ([0, T ] × Rd ) ∩ C 1,2 ((0, T ] × Rd ) such that
sup tθ ||u(t, ·)||C 2+α (Rd ) + tθ ||Dt u(t, ·)||Cbα (Rd ) < ∞ b

t∈(0,T ]

for some θ ∈ (0, 1). Since u solves the Cauchy problem (6.3.2) with ∆ being replaced by the operator Ax0 , f = u(0, ·) ∈ Cb (Rd ) and g = Dt u − Ax0 u, from (6.3.3) we conclude that sup tθ (||u(t, ·)||C 2+α (Rd ) + ||Dt u(t, ·)||Cbα (Rd ) ) + ||u||C 0,2+α−2θ ([0,T ]×Rd ) b

t∈(0,T ]

b



 ≤C ||u(0, ·)||C 2+α−2θ (Rd ) + sup tθ ||Dt u(t, ·) − Ax0 u(t, ·)||Cbα (Rd ) . b

(6.3.4)

t∈(0,T ]

Fix u as above and apply (6.3.4) to the function v = ϑx0 ,r u, where the function ϑx0 ,r ∈ Cc∞ (Rd ) satisfies (6.2.8) and r ∈ (0, 1), obtaining sup tθ (||u(t, ·)||C 2+α (B(x0 ,r/2)) + ||Dt u(t, ·)||C α (B(x0 ,r/2)) ) t∈(0,T ]

+ ||u||C 0,2+α−2θ ([0,T ]×B(x0 ,r/2)) b   ≤C ||ϑx0 ,r u(0, ·)||C 2+α−2θ (Rd ) + sup tθ ||ϑx0 ,r Dt u(t, ·) − Ax0 (ϑx0 ,r u(t, ·))||Cbα (Rd ) . (6.3.5) b

t∈(0,T ]

We note that (see (6.2.14)) tθ ||ϑx0 ,r Dt u(t, ·) − Ax0 (ϑx0 ,r u(t, ·))||Cbα (Rd ) ≤tθ ||Dt u(t, ·) − Ax0 u(t, ·)||C α (B(x0 ,r)) + M r−α tθ ||Dt u(t, ·) − Ax0 u(t, ·)||∞ + Cr−2−α tθ ||u(t, ·)||Cbα (Rd ) + Cr−1−α tθ

d X

||Dj u(t, ·)||Cbα (Rd )

j=1

≤(1 + M r−α )tθ ||Dt u(t, ·) − Ax0 u(t, ·)||C α (B(x0 ,r)) + Ctθ r−2−α ||u(t, ·)||Cb2 (Rd ) + Ctθ rα ||u(t, ·)||C 2+α (Rd ) , b

where the constant C is independent also of t and r. Next, applying (1.1.2), with β = 2 and θ = 0, we can estimate Ctθ ||u(t, ·)||Cb2 (Rd ) ≤Ctθ (εC −1 ||u(t, ·)||C 2+α (Rd ) + Cε ||u||∞ ) b

≤εtθ ||u(t, ·)||C 2+α (Rd ) + Cε0 T θ ||u(t, ·)||∞ b

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for each ε > 0 and some positive constants Cε and Cε0 . Replacing these estimates in (6.3.5), we conclude that sup tθ (||u(t, ·)||C 2+α (B(x0 ,r/2)) + ||Dt u(t, ·)||C α (B(x0 ,r/2)) ) + ||u||C 0,2+α−2θ ([0,T ]×B(x0 ,r/2))

t∈(0,T ]

 ≤C r−2−α ||u(0, ·)||C 2+α−2θ (Rd ) + (1 + M r−α ) sup tθ ||Dt u(t, ·) − Ax0 u(t, ·)||Cbα (Rd ) b

+ (εr

−2−α

t∈(0,T ]

+ Cr ) sup t ||u(t, ·)||C 2+α (Rd ) + Cε0 T θ ||u(t, ·)||∞ α

θ



b

t∈(0,T ]

for each ε > 0 and some positive constant C, which is also independent of r. Now, arguing as in the proof of Theorem 6.2.1, we can show that sup tθ (||u(t, ·)||C 2+α (Rd ) + ||Dt u(t, ·)||Cbα (Rd ) ) + ||u||C 0,2+α−2θ ([0,T ]×Rd ) b

t∈(0,T ]

b

 ≤C ||u(0, ·)||C 2+α−2θ (Rd ) + Cε0 T θ ||u(t, ·)||∞ + sup tθ ||Dt u(t, ·) − Au(t, ·)||Cbα (Rd ) , 

b

t∈(0,T ]

where the constant C does not blow up as T → 0+ . Hence, if T ≤ T0 , where T0 is any positive constant such that CCε0 T0θ ≤ 1/2, then, sup tθ (||u(t, ·)||C 2+α (Rd ) + ||Dt u(t, ·)||Cbα (Rd ) ) + ||u||C 0,2+α−2θ ([0,T ]×Rd ) b

t∈(0,T ]

b

  ≤2C ||u(0, ·)||C 2+α−2θ (Rd ) + sup tθ ||Dt u(t, ·) − Au(t, ·)||Cbα (Rd ) . b

(6.3.6)

t∈(0,T ]

Now, suppose that T > T0 . Then, (6.3.6) holds true with T being replaced by T0 , i.e., sup tθ (||u(t, ·)||C 2+α (Rd ) + ||Dt u(t, ·)||Cbα (Rd ) ) + ||u||C 0,2+α−2θ ([0,T0 ]×Rd ) b

t∈(0,T0 ]

b



 ≤2C ||u(0, ·)||C 2+α−2θ (Rd ) + sup tθ ||Dt u(t, ·) − Au(t, ·)||Cbα (Rd ) . b

(6.3.7)

t∈(0,T0 ]

Moreover, since u belongs to Cb1,2+α ([T0 , T ] × Rd ), we can apply Theorem 6.2.5, which, together with (6.3.7) yields ||u||C 0,2+α ([T0 ,T ]×Rd ) b

≤C(||u(T0 , ·)||C 2+α (Rd ) + ||Dt u − Au||C 0,α ([T0 ,T ]×Rd ) ) b h b  ≤CT0−θ 2C3 ||u(0, ·)||C 2+α−2θ (Rd ) + sup tθ ||Dt u(t, ·) − Au(t, ·)||Cbα (Rd ) b

θ

t∈(0,T0 ]

+ sup t ||Dt u(t, ·) − Au(t, ·)||Cbα (Rd )

i

t∈[T0 ,T ]

h i ≤CT0−θ 2C3 ||u(0, ·)||C 2+α−2θ (Rd ) + (2C3 + 1) sup tθ ||Dt u(t, ·) − Au(t, ·)||Cbα (Rd ) . b

t∈(0,T ]

Hence, sup tθ ||u(t, ·)||C 2+α (Rd ) b

t∈(T0 ,T ]

h

i ≤CT θ T0−θ 2C3 ||u(0, ·)||C 2+α−2θ (Rd ) + (2C3 + 1) sup tθ ||Dt u(t, ·) − Au(t, ·)||Cbα (Rd ) . b

t∈(0,T ]

(6.3.8)

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From (6.3.7) and (6.3.8), we get (6.3.6) also in this situation. To complete the proof, we introduce the sets n X = u ∈ C 1,2 ((0, T ] × Rd ) ∩ Cb0,2+α−2θ ([0, T ] × Rd ) : sup tθ (||u(t, ·)||C 2+α (Rd ) + ||Dt u(t, ·)||Cbα (Rd ) ) < ∞

o

b

t∈(0,T ]

o n Y =Cb2+α−2θ (Rd ) × g ∈ C((0, T ] × Rd ) : sup tθ ||g(t, ·)||Cbα (Rd ) < ∞ , t∈(0,T ]

endowed with the norms ||u||X = ||u||C 0,2+α−2θ ([0,T ]×Rd ) + sup tθ (||u(t, ·)||C 2+α (Rd ) + ||Dt u(t, ·)||Cbα (Rd ) ), b

b

t∈(0,T ]

||(f, g)||Y = ||f ||C 2+α−2θ (Rd ) + sup tθ ||g(t, ·)||Cbα (Rd ) , b

t∈(0,T ]

and, as in the proof of Theorem 6.2.6, Lσ u = (Dt u − Aσ u, u(0, ·)) for each function u ∈ 1+α/2,2+α Cb ((0, T ) × Rd ) and σ ∈ (0, 1), where Aσ = σA + (1 − σ)∆. Since (X, || · ||X ) and (Y, || · ||Y ) are Banach spaces, using the above results and the continuity method, we can complete the proof in this case. If θ = 0, then the arguments are the same. The starting point is Theorem 5.4.2, which shows that the Cauchy problem (6.3.2) admits a unique solution u ∈ C 1,2 ([0, T ] × Rd ) such that   sup (||u(t, ·)||C 2+α (Rd ) + ||Dt u(t, ·)||Cbα (Rd ) ) ≤ C ||f ||C 2+α−2θ (Rd ) + sup ||g(t, ·)||Cbα (Rd ) . t∈(0,T ]

b

b

t∈(0,T ]

Then, the same technique used above can be used to show that sup (||u(t, ·)||C 2+α (Rd ) + ||Dt u(t, ·)||Cbα (Rd ) ) + ||u||C 0,2+α ([0,T ]×Rd ) b

t∈(0,T ]

b



 ≤C ||u(0, ·)||C 2+α (Rd ) + sup ||Dt u(t, ·) − Au(t, ·)||Cbα (Rd ) . b

t∈(0,T ]

Thanks to this estimate, we can apply the continuity method with n X = u ∈ C 1,2 ([0, T ] × Rd ) ∩ Cb0,2+α ([0, T ] × Rd ) : o sup (||u(t, ·)||C 2+α (Rd ) + ||Dt u(t, ·)||Cbα (Rd ) ) < ∞ b

t∈[0,T ]

o n Y =Cb2+α (Rd ) × g ∈ C([0, T ] × Rd ) : sup ||g(t, ·)||Cbα (Rd ) < ∞ , t∈[0,T ]

endowed with the norms ||u||X = ||u||C 0,2+α ([0,T ]×Rd ) + sup (||u(t, ·)||C 2+α (Rd ) + ||Dt u(t, ·)||Cbα (Rd ) ), b

t∈[0,T ]

b

||(f, g)||Y = ||f ||C 2+α−2θ (Rd ) + sup ||g(t, ·)||Cbα (Rd ) , b

t∈[0,T ]

and the same operator Lσ as above to conclude the proof.



Remark 6.3.3 The last part of the statement of Theorem 6.3.2 is a spatial optimal Schauder estimate: The lack of smoothness of g with respect to the time variable reflects in a lack of smoothness of Dt u with respect to the time variable.

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To conclude this section, we consider the following result which will be used in the next section. Theorem 6.3.4 Let u ∈ C 1,2+α ([0, T ] × Rd ) be a classical solution to the differential equation Dt u = Au, Then, for every T > 0, there exists a positive constant CT such that ||u(t, ·)||C 2+α (Rd ) ≤ CT t−

2+α 2

b

||u(0, ·)||∞ ,

t ∈ (0, T ].

(6.3.9)

Proof We fix s0 ∈ (0, T ] and consider the sequence (tn ) as in the proof of Theorem 6.2.9, with σ = s0 /2 and τ = s0 . We also introduce the same sequence (ϕn ) of functions defined in the proof of the quoted theorem. Observe that ||ϕ0n ||∞ ≤ 2n Cs−1 0 for every n ∈ N, C being independent of n. As it is easily seen, for every n ∈ N the function vn = ϕn u belongs to the space C 1+α/2,2+α ([0, T ]×Rd ), vanishes in [0, s0 /2]×Rd and solves the equation Dt vn = Avn +ϕ0n u in [0, T ] × Rd . By (6.2.19), estimate (1.1.2) and Young’s inequality it follows that ||vn ||C 0,2+α ([0,T ]×Rd ) ≤C||ϕ0n ||∞ ||u||C 0,α ([tn+1 ,T ]×Rd ) b

≤2n Cs−1 0 ||u||C 0,α ([tn+1 ,T ]×Rd ) b

≤2n Cs−1 0 ||vn+1 ||C 0,α ([0,T ]×Rd ) b

α

−2 ||vn+1 ||Cb ([0,T ]×Rd ) ) ≤2n Cs−1 0 (ε||vn+1 ||C 0,2+α ([0,T ]×Rd ) + ε b

for each ε > 0 and n ∈ N, since vn+1 = u on [tn+1 , T ] × Rd . Now, we fix η ∈ (0, 2−1−α/2 ) and choose ε = εn = 2−n C −1 s0 η; from the previous estimate we obtain − 2+α 2

||vn ||C 0,2+α ([0,T ]×Rd ) ≤ η||vn+1 ||C 0,2+α ([0,T ]×Rd ) + 2n(1+ 2 ) C 1+ 2 η − 2 s0 α

b

α

α

b

||u||Cb ([0,T ]×Rd ) .

Multiplying both sides of this estimate by η n and summing from 1 to m ∈ N, we get ||v1 ||C 0,2+α ([0,T ]×Rd ) − η m+1 ||vm+1 ||C 0,2+α ([0,T ]×Rd ) b

≤C

1+ α 2

η

b

−α 2

− 2+α s0 2

m X

1+ α 2

(η2

)k ||u||Cb ([0,T ]×Rd )

k=1 − 2+α ≤C1 s0 2 ||u||Cb ([0,T ]×Rd )

(6.3.10)

for some constant C1 , independent of s0 , due to the choice of η. Letting m tend to ∞ − 2+α 2

in (6.3.10) we conclude that ||v1 ||C 0,2+α ([0,T ]×Rd ) ≤ C1 s0 b

||u||Cb ([0,T ]×Rd ) and, hence,

− 2+α C1 s0 2 ||u||Cb ([0,T ]×Rd ) .

||u||C 0,2+α ([s0 ,T ]×Rd ) ≤ Estimate (6.3.9) follows, from the maxib mum principle in Chapter 4 (see Exercise 4.4.6), which shows that ||u||Cb ([0,T ]×Rd ) ≤ ec0 T ||u(0, ·)||∞ , where c0 denotes the supremum over Rd of the function c. 

6.4

The Semigroup Associated with the Operator A

As a byproduct of Theorems 6.3.2 and 6.3.4, in this section we prove that a semigroup {T (t)} of bounded linear operators in Cb (Rd ) can be associated with the operator A, which satisfies Hypotheses 6.0.1.

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164

Proposition 6.4.1 For each f ∈ Cb (Rd ), the Cauchy problem ( Dt u(t, x) = Au(t, x), t > 0, x ∈ Rd , u(0, x) = f (x), x ∈ Rd ,

(6.4.1)

admits a unique solution u ∈ C 1,2 ((0, ∞) × Rd ) ∩ C([0, ∞) × Rd ), which is bounded in 1+α/2,2+α [0, T ] × Rd for every T > 0. Moreover, u ∈ Cloc ((0, ∞) × Rd ) and ||u(t, ·)||∞ ≤ ec0 t ||f ||∞ ,

t > 0, f ∈ Cb (Rd ),

(6.4.2)

where c0 = supx∈Rd c(x). Proof The uniqueness part and estimate (6.4.2) follow from the maximum principle in Chapter 4 (see Exercise 4.4.6). Hence, we just need to prove the existence part. Since it is rather long, we split the proof into two steps. Step 1. Here, we prove that, for each f ∈ BU C(Rd ), problem (6.4.1) admits a solution u as in the statement of the proposition. We fix f ∈ BU C(Rd ) and let (fn ) ⊂ Cb2+α (Rd ) be a sequence which converges to f uniformly in Rd . (For instance, we can take fn = S(1/n)f , where {S(1/n)} denotes the Gauss-Weierstrass semigroup in Cb (Rd ).) By Theorem 6.2.6, for every n ∈ N the Cauchy problem (6.4.1), with f being replaced by fn , admits a unique classical solution un : [0, ∞)× 1+α/2,2+α Rd → Rd which belongs to Cb ((0, T )×Rd ) for every T > 0. By the quoted maximum principle, it follows that ||un (t, ·)−um (t, ·)||∞ ≤ ec0 t ||fn −fm ||∞ for every t > 0 and m, n ∈ N. Hence, un converges uniformly in [0, T ]×Rd , for every T > 0, to a function u : [0, ∞)×Rd → R, which is bounded and uniformly continuous in [0, T ] × Rd for every T > 0 and, of course, satisfies the condition u(0, ·) = f . From the interior Schauder estimates in Theorem 6.2.9, we deduce that ||un − um ||C 1+α/2,2+α ([ε,T ]×K) ≤ C||un − um ||Cb ([0,T ]×Rd ) ,

m, n ∈ N,

for every 0 < ε < T , every compact set K ⊂ Rd and some constant C, independent of m, n ∈ N, which blows up as ε tend to 0. Hence, un converges to u in C 1+α/2,2+α ([ε, T ] × K) 1+α/2,2+α for every ε, T and K as above. Thus, u belongs to Cloc ((0, ∞) × Rd ). Finally, since d Dt un = Aun in (0, ∞) × R , letting n tend to ∞, we conclude that u solves the differential equation in (6.4.1). Step 2. Here, we complete the proof. For notational convenience, for each g ∈ BU C(Rd ) we denote by ug the solution to problem (6.4.1), with f being replaced by g, determined in Step 1. We fix f ∈ Cb (Rd ), introduce a bounded sequence (fn ) ⊂ BU C(Rd ), converging to f locally uniformly in Rd and set un = ufn for each n ∈ N. Arguing as in Step 1, we conclude that the sequence (un ) is bounded in C 1+α/2,2+α ((1/k, k) × B(0, k)) for each k ∈ N. Applying the Arzel` a-Ascoli theorem to un , we can show that, for each k ∈ N, there exist a function uk ∈ C 1+α/2,2+α ((1/k, k) × B(0, k)) and a subsequence (unkh ) converging to uk in C 1,2 ([1/k, k] × B(0, k)) as h tends to ∞. Without loss of generality, we can assume k that (nk+1 h ) is a subsequence of (nh ). It thus follows that uk ≡ uk+1 in [1/k, k] × B(0, k) for each k ∈ N. Let u : (0, ∞) × Rd → R be the function defined as follows: for each (t, x) ∈ (0, ∞) × Rd , u(t, x) = uk (t, x), where k is any integer such that (t, x) ∈ [1/k, k]×Rd . By the above results u is well defined, it belongs to C 1,2 ((0, ∞) × Rd ) and, if we consider the diagonal sequence (nhh ), then we immediately realize that unhh converges to u in C 1,2 ([a, b] × B(0, M )) for each 0 < a < b and M > 0. In particular, Dt u = Au in (0, ∞) × Rd .

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To prove that u is the solution to problem (6.4.1) we are looking for, we need to show that u can be extended by continuity at t = 0 with u(0, ·) = f . For this purpose, we use an argument from [16]. Given M > 0, we consider a function ϑM ∈ Cc∞ (Rd ) such that χB(0,M ) ≤ ϑM ≤ χB(0,2M ) . By the uniqueness of the solution to problem (6.4.1), we can split h ∈ N. (6.4.3) unhh = uϑM fnh + u(1−ϑM )fnh =: vh + wh , h

h

We claim that |wh (t, ·)| ≤ sup ||fnhh ||∞ (ec0 t − uϑM (t, ·)),

t ≥ 0, h ∈ Rd .

h∈N

Indeed, fix h ∈ N and introduce the function ζh = e−c0 t (wh (t, ·) − sup ||fnkk ||∞ (ec0 t − uϑM (t, ·))),

t > 0.

k∈N

It satisfies the differential inequality Dt ζh ≤ (A − c0 )ζh in (0, ∞) × Rd . Moreover, ζh (0, ·) = (1 − ϑM )fnhh − sup ||fnkk ||∞ (1 − ϑM ) ≤ 0. k∈N

Since the potential term of the operator A − c0 is nonpositive, by the maximum principle, ζh is nonpositive in [0, ∞) × Rd , i.e., wh (t, ·) ≤ sup ||fnkk ||∞ (ec0 t − uϑM (t, ·)),

t > 0,

k∈N

in Rd . The same argument, applied to the function ξh , defined by ξh (t, x) = −e−c0 t (wh (t, x) + sup ||fnkk ||∞ (ec0 t − uϑM (t, x))),

(t, x) ∈ (0, ∞) × Rd ,

k∈N

reveals that ξh ≤ 0, i.e., wh (t, ·) ≥ − sup ||fnkk ||∞ (ec0 t − uϑM (t, ·)),

t > 0,

k∈N

in Rd , and the claim follows. From the claim and (6.4.3) we conclude that |unhh (t, ·) − f | ≤ |vh − f | + sup ||fnhh ||∞ (ec0 t − uϑM (t, ·)),

t > 0.

(6.4.4)

h∈N

Since fnhh converges to f , locally uniformly as h tends to ∞, the function ϑM fnhh converges to ϑM f ∈ BU C(Rd ) uniformly in Rd . Hence, letting h tend to ∞ in both the sides of (6.4.4), we obtain that |u(t, x) − f (x)| ≤ |uϑM f (t, x) − f (x)| + sup ||fnhh ||∞ [ec0 t − uϑM (t, x)]

(6.4.5)

h∈N

for every t > 0 and x ∈ Rd . Fix x0 ∈ B(0, M ). By Step 1, uϑM (t, x) converges to ϑM (x0 ) = 1 in and uϑM f (t, x) converges to (ϑM f )(x0 ) = f (x0 ), as (t, x) tends to (0, x0 ). It thus follows that u(t, x) tends to f (x0 ) as (t, x) tends to (0, x0 ) for every x0 ∈ B(0, M ). By the arbitrariness of M , u can be extended by continuity to {0} × Rd , by setting u(0, x) = f (x) for x ∈ Rd . Function u is the solution to problem (6.4.1) that we were looking for.  As a byproduct of Proposition 6.4.1, we can associate a semigroup of bounded operators {T (t)} in Cb (Rd ), by setting T (t)f = u(t, ·) for each t ≥ 0, where u is the classical solution to problem (6.4.1). Note that the semigroup property is guaranteed by the maximum principle in Chapter 4 (see Exercise 4.4.6). In the following theorems, we state some remarkable properties of such a semigroup.

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166

Theorem 6.4.2 Let (fn ) ⊂ Cb (Rd ) be a bounded sequence converging to some function f ∈ Cb (Rd ), pointwise in Rd . Then, T (·)fn converges to T (·)f in C 1,2 ([a, b] × B(0, M )) for every 0 < a < b and M > 0. If the sequence (fn ) converges to f locally uniformly on Rd , then T (·)fn converges to T (·)f in C 1,2 ([0, b] × B(0, M )) for every b > 0 and M > 0. Proof We fix f ∈ Cb (Rd ), a bounded sequence (fn ) ⊂ Cb (Rd ) and split the proof into two steps. Step 1. Here, we assume that fn converges to f pointwise on Rd . We fix a subsequence of (T (·)fn ) and observe that the arguments in the proof of Proposition 6.4.1 can be used to show that we can extract another subsequence (T (·)fnk ) which converges in C 1,2 ([a, b] × B(0, M )), for every 0 < a < b and M > 0, to some function 1+α/2,2+α u ∈ Cloc ((0, ∞) × Rd ), as k tends to ∞. To identify u with T (·)f , we observe that the operator g 7→ (T (t)g)(x) belongs to the dual of C0 (Rd ) for every t > 0 and x ∈ Rd . Hence, there exists a positive Borel measure on Rd , which we denote by p(t, x, dy), such that Z (T (t)g)(x) = g(y) p(t, x, dy), g ∈ C0 (Rd ), t > 0, x ∈ Rd . Rd

Since each function in Cb (Rd ) is the local uniform limit of a bounded sequence of functions in C0 (Rd ), the previous equality can be extended to each g ∈ Cb (Rd ). Now, by dominated convergence we easily infer that (T (t)fnk )(x) converges to (T (t)f )(x) for all t > 0 and x ∈ Rd , as k tends to ∞. Hence, u ≡ T (·)f . We have so proved that from each subsequence of (T (·)fn ) we can extract a subsequence which converges to T (·)f in C 1,2 ([a, b] × K) for every 0 < a < b and every compact subset K of Rd . This is enough to infer that all the sequence (T (·)fn ) converges locally uniformly on (0, ∞) × Rd . Step 2. Now, we assume that fn converges to f locally uniformly on Rd . We set C = supn∈N ||fn ||∞ and fix M > 0. Then, by estimate (6.4.5) we get ||T (t)fn − fn ||C(B(0,M )) ≤||T (t)(ϑM fn ) − ϑM fn ||C(B(0,M )) + Cec0 t ||1 − T (t)ϑM ||C(B(0,M )) ≤||T (t)(ϑM fn − ϑM f )||∞ + ||(T (t)(ϑM f ) − ϑM f ||∞ + ||fn − f ||C(B(0,M )) + Cec0 t ||1 − T (t)ϑM ||C(B(0,M )) ≤2||fn − f ||C(B(0,2M )) + ||(T (t)(ϑM f ) − ϑM f ||∞ + Cec0 t ||1 − T (t)ϑM ||C(B(0,M )) for all t > 0. Fix ε > 0 and let n0 ∈ N be such that 2||fn − f ||C(B(0,2M )) ≤ ε/4 for every n ≥ n0 . With this choice of n0 it follows that ||T (t)fn − fn ||C(B(0,M )) ≤

ε + ||T (t)(ϑM f ) − ϑM f ||C(B(0,M )) + Cec0 t ||1 − T (t)ϑM ||C(B(0,M )) 4

for every n ≥ n0 . Since ||T (t)(ϑM f ) − ϑM f ||C(B(0,M )) and ||ec0 t − T (t)ϑM ||C(B(0,M )) vanish as t tends to 0+ , we can determine t0 > 0 such that ||T (t)fn − fn ||C(B(0,M )) ≤ ε/2 for all t ∈ [0, t0 ] and n ≥ n0 . Now, we are almost done. Indeed, we can estimate ||T (t)fn − T (t)f ||C(B(0,M )) ≤||T (t)fn − fn ||C(B(0,M )) + ||fn − f ||C(B(0,M )) + ||T (t)f − f ||C(B(0,M ))

(6.4.6)

for t > 0 and n ∈ N. Up to replacing t0 with a smaller value and n0 with a larger integer, if needed, we can assume that the sum of the first two terms in the right-hand side of

Semigroups of Bounded Operators and Second-Order PDE’s

167

(6.4.6) does not exceed ε/4 for t ∈ [0, t0 ] and n ≥ n0 . Hence, from (6.4.6) it follows that ||T (t)fn − T (t)f ||C(B(0,M )) ≤ ε for all t ∈ [0, t0 ] and n ≥ n0 . Now, we fix b > 0 and, without loss of generality, we assume that b > t0 . Then, we can estimate ||T (·)fn − T (·)f ||C([0,b]×B(0,M )) =||T (·)fn − T (·)f ||C([0,t0 ]×B(0,M )) ∨ ||T (·)fn − T (·)f ||C([t0 ,b]×B(0,M )) ≤ε ∨ ||T (·)fn − T (·)f ||C([t0 ,b]×B(0,M ))

(6.4.7)

for n ≥ n0 . Since, ||T (·)fn − T (·)f ||C([t0 ,b]×B(0,M )) vanishes as n tends to ∞, from (6.4.7), it follows that lim sup ||T (·)fn − T (·)f ||C([0,b]×B(0,M )) ≤ ε, n→∞

from which the claim follows.



Theorem 6.4.3 The following properties are satisfied. (i) The semigroup {T (t)} preserves positivity and ||T (t)||L(Cb (Rd )) ≤ ec0 t for each t > 0. In particular, if f ≥ 0 does not identically vanish on Rd , then T (t)f is strictly positive on Rd for every t > 0. (ii) For each 0 ≤ β ≤ γ ≤ 2 + α, each operator T (t) is bounded from Cbβ (Rd ) into Cbγ (Rd ) and there exist two positive constants ωβ,γ and Cβ,γ such that ||T (t)f ||Cbγ (Rd ) ≤ Cβ,γ t−

γ−β 2

eωβ,γ t ||f ||C β (Rd ) ,

t > 0.

(6.4.8)

b

(iii) The restriction of {T (t)} to BU C(Rd ) is a strongly continuous semigroup. Proof (i) The proof of these properties follow from the strong maximum principle in Chapter 4 and the proof of Proposition 6.4.1. (ii) Since the proof is rather long, we split it into 3 steps. Step 1. Here, we prove estimate (6.4.8) for every γ ∈ (2, 2 + α] and β < γ and for γ = β ∈ [0, 2 + α]. Theorem 6.2.6 shows that each operator T (t) is bounded from Cb2+α (Rd ) in itself and ||T (t)f ||L(C 2+α (Rd )) ≤ C1 ||f ||C 2+α (Rd ) , b

t ∈ [0, 1],

b

for some positive constant C1 , independent of f , which is larger than one, since T (0)f = f Using the semigroup property, we extend this estimate to each t > 0. Indeed, if t > 1, then t = n + σ, where n ∈ N and σ ∈ [0, 1). So, T (t) = T (n)T (σ) = (T (1))n T (σ) and ||T (t)||L(C 2+α (Rd )) ≤||T (1)||nL(C 2+α (Rd )) ||T (σ)||L(C 2+α (Rd )) b

b

b

≤C1n+1 = exp((n + 1) log(C1 )) ≤ C1 eω2+α,2+α t , where ω2+α,2+α = log C1 . Similarly, taking g ≡ 0 and choosing θ properly in Theorem 6.3.2 we can show that T (t) is bounded from Cbγ (Rd ) in itself for each γ ∈ (0, 2 + α), and it satisfies (6.4.8) (with β = γ). This is clear if γ ∈ (α, 2 + α). If γ ∈ (0, α], then we choose 0 for every γ0 ∈ (0, γ). With this choice of θ the above assertion follows also θ = 1 − γ−γ 2 in the case γ ∈ (0, α]. On the other hand, Theorem 6.3.2 also yields estimate (6.4.8) for β ≤ 2 < γ ≤ 2 + α, such that β > γ − 2.

Parabolic Equations in Rd

168

To prove it with β ≤ γ − 2, we use again the semigroup property and the results so far obtained. More precisely, we have proved that β

||T (t)||L(C β (Rd ),C 2+β/2 (Rd )) ≤ Cβ,2+β/2 t−1+ 4 eωβ,2+β/2 t ,

t > 0,

b

b

γ 2

and, by Theorem 6.3.2 with α = γ − 2 and θ = ||T (t)||L(C 2+β/2 (Rd ),C γ (Rd )) ≤ C2+β/2,γ t b

− 1 − β4 , we have

−2γ+4+β 4

eω2+β/2,γ t ,

t > 0.

b

Hence, ||T (t)||L(C β (Rd ),C γ (Rd )) ≤||T (t/2)||L(C β (Rd ),C 2+β/2 (Rd )) ||T (t/2)||L(C 2+β/2 (Rd ),C γ (Rd )) b

b

≤2

t

− γ−β 2

b

b

b

γ−β 2

Cβ,2+β/2 C2+β/2,γ e

b

1 2 (ωβ,2+β/2 +ω2+β/2,γ )t

and we are done. Step 2. Here, we prove estimate (6.4.8) for 0 < β < γ ≤ 2. Applying the estimate (see Chapter 1) 4+β−2γ 4−β Cbβ (Rd )

||g||Cbγ (Rd ) ≤ C2 ||g||

2(γ−β) 4−β 2+β/2 Cb (Rd )

||g||

2+β/2

g ∈ Cb

,

(Rd ),

with g = T (t)f , we deduce that ||T (t)f ||Cbγ (Rd ) ≤C2 ||T (t)f ||

4+β−2γ 4−β

Cbβ (Rd )

2(γ−β) 4−β 2+β/2 Cb (Rd )

||T (t)f ||

≤C2 (Cβ,β eωβ,β t ||f ||C β (Rd ) )

4+β−2γ 4−β

≤C2 Cβ,β

2(γ−β) 4−β

Cβ,2+β/2 t

2(γ−β) 4−β

b

b

4+β−2γ 4−β

β

(Cβ,2+β/2 t−1+ 4 eωβ,2+β/2 t ||f ||C β (Rd ) )

ωβ,γ t − γ−β 2

e

||f ||C β (Rd ) b

+ ωβ,2+β/2 2(γ−β) for all t > 0, where ωβ,γ = ωβ,β 4+β−2γ 4−β 4−β . Step 3. Finally, we prove estimate (6.4.8) for β = 0 and γ ∈ (0, 2]. First we suppose that f ∈ BU C(Rd ) and fix a sequence (fn ) ⊂ Cb2+α (Rd ), for some α ∈ (0, 1), converging to f uniformly on Rd . By Theorem 6.3.4, it follows that ||T (t)fn − T (t)fm ||C 2+α (Rd ) ≤ Ct− b

2+α 2

||fn − fm ||∞ ,

t ∈ (0, 1],

(6.4.9)

for every m, n ∈ N. Since T (t)fn converges to T (t)f locally uniformly as n tends to ∞, from (6.4.9) we conclude that T (t)f ∈ Cb2+α (Rd ) for every t ∈ (0, 1] and ||T (t)f ||C 2+α (Rd ) ≤ b

2+α

Ct− 2 ||f ||∞ for every t ∈ (0, 1]. Using the semigroup property and arguing as above, we easily show that T (t)f ∈ Cb2+α (Rd ) for every t > 0 and estimate (6.4.8) follows with β = 0. Observing that 2+α−γ

γ

||g||Cbγ (Rd ) ≤ ||g||∞2+α ||g||C2+α 2+α (Rd ) b

Cb2+α (Rd )

for every g ∈ and γ ∈ (0, 2 + α), (see Chapter 1) and arguing as above, estimate (6.4.8) follows with β = 0 and γ ∈ (0, 2 + α). Now, suppose that f ∈ Cb (Rd ). Let (fn ) ⊂ BU C(Rd ) be a sequence converging to f locally uniformly in Rd and such that ||fn ||∞ ≤ ||f ||∞ for all n ∈ N. By Theorem 6.4.2, T (t)fn converges to T (t)f in C 2 (B(0, M )) for every M > 0 and t > 0. Since γ

γ

||T (t)fn ||Cbγ (Rd ) ≤ C0,γ t− 2 eω0,γ t ||fn ||∞ ≤ C0,γ t− 2 eω0,γ t ||f ||∞ ,

t > 0,

Semigroups of Bounded Operators and Second-Order PDE’s

169

for all n ∈ N and γ ∈ (0, 2 + α], letting n tend to ∞, we conclude that γ

||T (t)f ||Cbγ (Rd ) ≤ C0,γ t− 2 eω0,γ t ||f ||∞ for all t > 0, completing the proof of property (ii). (iii) By estimate (6.4.8), with β = 0 and γ = 1, it follows that each operator T (t) maps Cb (Rd ) into Cb1 (Rd ) ,→ BU C(Rd ). To conclude that the restriction of {T (t)} to BU C(Rd ) is a strongly continuous semigroup, we should prove that T (t)f converges to f uniformly on Rd as t tends to 0+ for every f ∈ BU C(Rd ). For this purpose, we fix such a function f and denote by (fn ) ⊂ Cb2+α (Rd ) a sequence converging to f , uniformly in Rd . Note that |(T (t)f )(x) − f (x)| ≤ sup ||T (r)fn − T (r)f ||∞ + |(T (t)fn )(x) − fn (x)| + ||fn − f ||∞ r∈[0,1]

(6.4.10) for all t ∈ [0, 1], x ∈ Rd , n ∈ N, and, by Step 1 in the proof of Proposition 6.4.1, T (·)fn converges to T (·)f uniformly in [0, T ] × Rd for every T > 0. Moreover, using again (6.4.8) and recalling that Dt T (·)fn = AT (·)fn for every n ∈ N, we get Z t |(T (t)fn )(x) − fn (x)| ≤ |(AT (s)fn )(x)| ds ≤ C||fn ||C 2+α (Rd ) t b

0

for every t ∈ [0, 1], x ∈ Rd , n ∈ N and some positive constant C, independent of t ∈ [0, 1]. The previous inequality, replaced in (6.4.10), gives ||T (t)f − f ||∞ ≤ sup ||T (r)fn − T (r)f ||∞ + C||fn ||C 2+α (Rd ) t + ||fn − f ||∞ b

r∈[0,1]

for all t ∈ [0, 1] and n ∈ N. Thus, lim sup ||T (t)f − f ||∞ ≤ sup ||T (r)fn − T (r)f ||∞ + ||fn − f ||∞ , t→0+

n ∈ N,

r∈[0,1]

and letting n tend to ∞, gives lim supt→0+ ||T (t)f − f ||∞ = 0, i.e., T (t)f converges to f uniformly in Rd as t → 0+ , showing that the restriction of {T (t)} to BU C(Rd ) is strongly continuous.  Remark 6.4.4 Note that we can take as ωβ,γ in estimate (6.4.8) any constant greater than c0 , i.e., for every ε > 0, there exists a positive constant Cβ,γ,ε such that ||T (t)f ||Cbγ (Rd ) ≤ Cβ,γ,ε t−

γ−β 2

e(c0 +ε)t ||f ||C β (Rd ) ,

t > 0.

(6.4.11)

b

Indeed, fix t > 1 and split T (t)f = T (1)T (t − 1)f for every f ∈ Cbβ (Rd ). Then, ||T (t)f ||Cbγ (Rd ) ≤||T (1)||L(Cb (Rd ),Cbγ (Rd )) ||T (t − 1)f ||∞ ≤||T (1)||L(Cb (Rd ),Cbγ (Rd )) ec0 (t−1) ||f ||∞ ≤||T (1)||L(Cb (Rd ),Cbγ (Rd )) ec0 (t−1) ||f ||C β (Rd ) . b

e ε), Since supt≥1 tγ/2 e−εt < ∞ for every ε > 0, we can determine a positive constant C(γ, which depends also on c0 , such that e ε)t− γ2 e(c0 +ε)t ||f || β d , ||T (t)f ||Cbγ (Rd ) ≤ C(γ, C (R ) b

t ≥ 1.

Parabolic Equations in Rd

170 Finally, we observe that

||T (t)f ||Cbγ (Rd ) ≤ Cβ,γ eωβ,γ t−

γ−β 2

−

||f ||C β (Rd ) ≤ Cβ,γ e2(ωβ,γ −c0 ) t− b

γ−β 2

||f ||C β (Rd ) b

for every t ∈ (0, 2]. Combining these two estimates, we easily obtain (6.4.11). Clearly, estimate (6.4.11) is much more useful than estimate (6.4.8) when c0 is negative, since it shows that also the Cbγ (Rd )-norm of T (t)f tends to zero as t tends to ∞. Remark 6.4.5 We remark that the semigroup {T (t)} is not strongly continuous in Cb (Rd ). Indeed, as it has been remarked in the proof of property (iii) in Theorem 6.4.3, each operator T (t) maps Cb (Rd ) into Cb1 (Rd ) ,→ BU C(Rd ), which is a closed subset of Cb (Rd ). Hence, T (t)f converges to f uniformly on Rd if and only if f ∈ BU C(Rd ). If f ∈ Cb (Rd ) is not uniformly continuous, then T (t)f converges to f as t tends to 0+ , only locally uniformly on Rd , as the proof of Proposition 6.4.1 shows. Actually, as we will show in Chapter 14, {T (t)} can be extended with an analytic semigroup in Cb (Rd ; C) whose associated sectorial operator A is defined as follows: 2,p D(A) = {u ∈ Cb (Rd ; C) ∩ Wloc (Rd ; C), for each p < ∞ and Au ∈ Cb (Rd ; C)}

and Au = Au for every u ∈ D(A).

6.4.1

Interior Schauder estimates for solutions to parabolic equations in domains: Part II

To conclude this section we prove the following a priori estimate up to t = 0. Theorem 6.4.6 Let Ω ⊂ Rd be an open set and let u ∈ Cb ([0, T ] × Ω) ∩ C 1,2 ((0, T ] × Ω) α/2,α solve the equation Dt u = Au + g in (0, T ) × Ω for some g ∈ Cloc ([0, T ] × Ω). Further, assume that the function t 7→ t||u(t, ·)||Cb2 (Ω) is bounded in (0, T ). Then, for every pair of compact sets K1 ⊂ K2 ⊂ Ω such that d(∂K1 , ∂K2 ) > 0, there exists a positive constant C = C(µ, T, K1 , K2 ) such that, for every t ∈ (0, T ), √ t||Dx2 u(t, ·)||C(K1 ) + t |||∇x u(t, ·)|||C(K1 ) ≤ C(||u||Cb ([0,T ]×K2 ) + ||g||C α/2,α ([0,T ]×K2 ) ). (6.4.12) Proof Throughout the proof, we denote by C a positive constant independent of n, u and t, which can vary from line to line. The main part of the proof consists of proving estimate (6.4.12) when K1 = B(x0 , r) and K2 = B(x0 , 2r). We also consider the same sequence of cut-off functions (ϑn ) as in the proof of Theorem 6.2.9. Let us set un := ϑn u and observe that each function un vanishes on [0, T ] × ∂B(0, rn+1 ) e n + gn in (0, T ) × Rd , where the operator Ae is the same as in the proof of and Dt un = Au Theorem 6.2.10 and the function gn is given by (6.2.35). In view of the variation-of-constants-formula it thus follows that Z t un (t, x) = (T (t)(ϑn u(0, ·)))(x) + (T (t − s)gn (s, ·))(x) ds 0 d

for every t ∈ [0, T ] and x ∈ R , where {T (t)} is the semigroup associated in Cb (Rd ) with e From (6.4.8) we deduce that the operator A. α

t1− 2 ||T (t)ψ||Cb2 (Rd ) ≤ C||ψ||Cbα (Rd )

Semigroups of Bounded Operators and Second-Order PDE’s

171

for every ψ ∈ Cbα (Rd ) and t ∈ (0, T ). Since gn (s, ·) ∈ Cbα (Rd ) for every s ∈ (0, T ), we can estimate Z t α (t − s)−1+ 2 ||gn (s, ·)||Cbα (Rd ) ds (6.4.13) t||un (t, ·)||Cb2 (Rd ) ≤ C||u||∞ + C 0

for every t ∈ (0, T ). Note that ||gn (s, ·)||Cbα (Rd ) ≤ C||ϑn ||C 2+α (Rd ) (||u(s, ·)||C 1+α (Rd ) + ||g(s, ·)||Cbα (Rd ) ) b

b

for every s ∈ (0, T ). Moreover, for every σ ∈ (s, T ), ε > 0 and n ∈ N it holds that 1

1

1

1

2 2 ||u(s, ·)||C 1 (B(x0 ,rn+1 )) ≤ Cs− 2 ||u||∞ ≤ s− 2 (Cε−1 ||u||∞ + εζn+1 ), ζn+1

||∇x u(s, ·)||C α (B(x0 ,rn+1 )) ≤ Cs−

α+1 2

1−α

1+α

2 ≤ s− ||u||∞2 ζn+1

α+1 2

1+α

(Cε− 1−α ||u||∞ + εζn+1 ),

where ζn := sups∈(0,T ) s||u(s, ·)||C 2 (B(x0 ,rn )) . Since ||ϑn ||C 2+α (Rd ) ≤ C8n for every n ∈ N, b from these last three estimates we conclude that ||gn (s, ·)||Cbα (Rd ) ≤8n s−

α+1 2

1+α

(Cε− 1−α ||u||∞ + εζn+1 ) + C8n ||g(s, ·)||C α (B(x0 ,rn+1 ))

for every s ∈ (0, T ) and ε > 0. Replacing (6.4.14) into (6.4.13) yields   1+α ζn ≤ C||u||∞ + 8n C ε− 1−α ||u||∞ + εζn+1 + ||g||C α/2,α ([0,T ]×B(x0 ,2r))

(6.4.14)

(6.4.15)

for every n ∈ N and ε ∈ (0, 1). Let us fix η ∈ (0, 64−1/(1−α) ) and ε = 8−n C −1 η. Multiplying both sides of (6.4.15) by n η and summing from 1 to m ∈ N, and arguing as in the last part of the proof of Theorem 6.2.9, we get the assertion. 

6.5

Higher-Order Regularity Results

In this section, we are going to prove that, the more the data of the Cauchy problem (6.0.1) are smoother, the more the solution to such a Cauchy problem is itself smooth. We first consider the following result which considers the case when f is smoother up to t = 0. For this purpose, we need to refine Hypotheses 6.0.1. Hypotheses 6.5.1 (i) The coefficients qij = qji , bj (i, j = 1, . . . , d) and c belong to Cb`+α (Rd ) for some α ∈ (0, 1) and ` ∈ N; (ii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for all x, ξ ∈ Rd . Theorem 6.5.2 Under Hypotheses 6.5.1, suppose that f ∈ Cb`+α (Rd ). Then, the following properties are satisfied. (`+α)/2,`+α

(i) If g ∈ Cb ((0, T ) × Rd ), then the classical solution of the Cauchy problem (`+2+α)/2,`+2+α (6.0.1) belongs to Cb ((0, T ) × Rd ) and there exists a positive constant C` , which depends also on µ, α and the C α -norm of the coefficients of A, such that ||u||C (`+2+α)/2,`+2+α ((0,T ])×Rd ) ≤ C` (||f ||C `+2+α (Rd ) + ||g||C (`+α)/2,`+α ((0,T )×Rd ) ). (6.5.1) b

b

b

Parabolic Equations in Rd

172

(ii) If g ∈ Cb0,`+α ([0, T ] × Rd ), then the classical solution of the Cauchy problem (6.0.1) belongs to Cb0,`+2+α ([0, T ] × Rd ) and there exists a positive constant C`0 , which depends also on µ, α and the norm of the coefficients of A, such that ||u||C 0,`+2+α ([0,T ]×Rd ) ≤ C`0 (||f ||C 2+`+α (Rd ) + ||g||C 0,`+α ([0,T ]×Rd ) ). b

b

(6.5.2)

b

Proof Throughout the proof, we denote by C a positive constant, which is independent of f , g and u, and may vary from line to line. (i) The proof is similar to that of Proposition 6.2.12. For the reader’s convenience, we sketch the main points. Step 1. The crucial step is the case ` = 1, which we consider here. In view of Theorem 6.2.6, we already know that the Cauchy problem (6.0.1) admits a solution 1+α/2,2+α u ∈ Cb ([0, T ] × Rd ) and it satisfies estimate (6.2.20). To prove that u ∈ (3+α)/2,3+α Cb ([0, T ] × Rd ), we begin by proving that each spatial derivative Dk u belongs 1+α/2,2+α to Cb ([0, T ] × Rd ). For this purpose, for each h ∈ R \ {0} we introduce the operator τh : Cb (Rd ) → Cb (Rd ), defined by τh ψ = h−1 [ψ(· + hek ) − ψ] for each ψ ∈ Cb (Rd ). Clearly, this operator is bounded in Cb (Rd ). Further, let vh be the function defined by vh (t, ·) = τh u(t, ·) for every t ∈ [0, T ]. Function vh belongs to C 1+α/2,2+α ((0, T ) × Rd ) and solves the Cauchy problem ( Dt vh (t, x) = Avh (t, x) + ψ(t, x, h), t ∈ [0, T ], x ∈ Rd , vh (0, x) = (τh f )(x), x ∈ Rd , where ψ(t, ·, h) =τh (g(t, ·)) +

d X

τh (qij )Dij u(t, · + hej )

i,j=1

+

d X

τh (bj )Dj u(t, · + hej ) + τh (c)u(t, · + hej ).

j=1

Since ||τh (qij )||Cbα (Rd ) ≤ ||Dk qij ||Cbα (Rd ) , ||τh (c)||Cbα (Rd ) ≤ ||Dk c||Cbα (Rd ) , ||τh (g)||C α/2,α ((0,T )×Rd ) ≤ ||Dk g||Cbα ([0,T ]×Rd )

||τh (bj )||Cbα (Rd ) ≤ ||Dk bj ||Cbα (Rd ) , ||τh (f )||C 2+α (Rd ) ≤ ||f ||C 3+α (Rd ) , b

b

b

for i, j = 1, . . . , d, we deduce that ψ(·, ·, h) belongs to C α/2,α ([0, T ]×Rd ) and, using (6.2.20), we conclude that ||ψ(·, ·, h)||C α/2,α ((0,T )×Rd ) ≤ C(||g||C (1+α)/2,1+α ((0,T )×Rd ) + ||f ||C 2+α (Rd ) ), b

b

b

where C is independent of h. We can thus apply Theorem 6.2.6, which shows that vh ∈ 1+α/2,2+α Cb ((0, T ) × Rd ) and ||vh ||C 1+α/2,2+α ((0,T )×Rd ) ≤ C(||f ||C 3+α (Rd ) + ||g||C (1+α)/2,1+α ((0,T )×Rd ) ), b

b

b

C being independent of h. A compactness argument allows us to let h tend to 0 and conclude 1+α/2,2+α that Dk u ∈ Cb ((0, T ) × Rd ) and ||Dk u||C 1+α/2,2+α ((0,T )×Rd ) ≤ C(||f ||C 3+α (Rd ) + ||g||C (1+α)/2,1+α ((0,T )×Rd ) ). b

b

b

Semigroups of Bounded Operators and Second-Order PDE’s

173

In particular, the second-order derivatives of u belongs to C (1+α)/2,1+α ((0, T ) × Rd ) and, since Dt u = Au + g, it follows that Dt u belongs to C (1+α)/2,1+α ((0, T ) × Rd ) and ||Dt u||C (1+α)/2,1+α ((0,T )×Rd ) ≤ C(||f ||C 3+α (Rd ) + ||g||C (1+α)/2,1+α ((0,T )×Rd ) ). b

b

b

(3+α)/2,3+α

From the above results, we deduce that u ∈ Cloc ([0, T ] × Ω) and (6.5.1) follows. Step 2. Here, we assume that the assertion is true when ` ∈ {1, . . . , `0 } for some `0 ∈ N (` +1+α)/2,`0 +1+α and prove it with ` = `0 + 1. We begin by observing that, since Cb 0 ((0, T ) × (` +α)/2,` +α ` +3+α ` +2+α 0 0 d d d d 0 0 R ) ,→ Cb ((0, T ) × R ) and Cb (R ) ,→ Cb (R ), it follows that u ∈ (` +2+α)/2,`0 +2+α Cb 0 ((0, T ) × Rd ). Moreover, for every k ∈ {1, . . . , d} the functions Dk g and (` +α)/2,`0 +α Dk f belong, respectively, to Cb`0 +2 (Rd ) and Cb 0 ((0, T ) × Rd ). Since by Step 1, the function v = Dk u solves the Cauchy problem ( Dt v(t, x) = Av(t, x) + ψ(t, x), t ∈ [0, T ], x ∈ Rd , (6.5.3) v(0, x) = Dk f (x), x ∈ Rd , where ψ = Dk g +

d X

(Dk qij )Dij u +

d X (Dk bj )Dj u + (Dk c)u j=1

i,j=1 (`0 +α)/2,`0 +α

and the function ψ belongs to Cb

((0, T ) × Rd ) and satisfies the estimate

||ψ||C (`0 +α)/2,`0 +α ((0,T )×Rd ) b  d X ≤C ||Dk g||C (`0 +α)/2,`0 +α ((0,T )×Rd ) + ||Dij u||C (`0 +α)/2,`0 +α ((0,T )×Rd ) b

+

d X

b

i,j=1

 ||Dj u||C (`0 +α)/2,`0 +α ((0,T )×Rd ) + ||u||C (`0 +α)/2,`0 +α ((0,T )×Rd ) b

j=1

b

≤C(||f ||C `0 +3+α (Rd ) + ||g||C (`0 +1+α)/2,`0 +1+α ((0,T )×Rd ) ) b

b

(`0 +2+α)/2,`0 +2+α

we conclude that Dk u ∈ Cb

((0, T ) × Rd ) and

||Dk u||C (`0 +2+α)/2,`0 +2+α ((0,T )×Rd ) ≤ C(||f ||C `0 +3+α (Rd ) + ||g||C (`0 +1+α)/2,`0 +1+α ((0,T )×Rd ) ). b

b

b

(` +3+α)/2,` +3+α

0 If `0 is even, then this is enough to infer that u ∈ Cb 0 ((0, T ) × Rd ) and it satisfies estimate (6.5.1). On the other hand, if `0 is odd then [(`0 + 3 + α)/2] = (`0 + 3)/2 = [(`0 + 2 + α)/2] + 1 and so we have to prove that u admits the time derivative of order (`0 + 3)/2, which α/2,α belongs to Cb ((0, T ) × Rd ). For this purpose, it suffices to recall that Dt u = Au + g d on [0, T ] × R . By assumptions g admits the time derivative of order (`0 + 1)/2 which α/2,α belongs to Cb ((0, T ) × Rd ). Moreover, since each first-order derivative of u belongs to (`0 +2+α)/2,`0 +2+α (` +1)/2 (` +1)/2 Cb ((0, T ) × Rd ), then the derivatives Dt 0 Di u and Dt 0 Dij u exist α/2,2α and belong to Cb ((0, T ) × Rd ). As a byproduct, Au + g admits the time derivative of (` +3)/2 α/2,α order (`0 + 1)/2 on [0, T ] × Rd so that Dt 0 u belongs to Cb ((0, T ) × Rd ) and

(`0 +3)/2

||Dt

u||C α/2,α ((0,T )×Rd ) ≤ C(||f ||C `0 +3+α (Rd ) + ||g||C (`0 +1+α)/2,`0 +1+α ((0,T )×Rd ) ). b

b

b

Parabolic Equations in Rd

174 (` +3+α)/2,` +3+α

0 Thus, u ∈ Cb 0 ((0, T ) × Rd ) and estimate (6.5.1) follows. (ii) The same arguments as in Step 1 above, show that, if f ∈ Cb0,3+α ([0, T ] × Rd ), then the function ψ(·, ·, h) belongs to Cb0,α ([0, T ] × Rd ) and

||ψ(·, ·, h)||C 0,α ([0,T ]×Rd ) ≤ C(||f ||C 3+α (Rd ) + ||g||C 0,1+α ([0,T ]×Rd ) ). b

b

From Theorem 6.3.2, we conclude that Dk u ∈ C 1,2 ([0, T ] × Rd ) ∩ Cb0,2+α ([0, T ] × Rd ) and estimate 6.5.2 holds true. Suppose that the assertion is true for some ` > 1. Then, the function u belongs to Cb0,`+2+α ([0, T ] × Rd ). Moreover, Dk g ∈ Cb0,`+α ([0, T ] × Rd ). Therefore, the function ψ belongs to Cb0,`+α ([0, T ] × Rd ) and, consequently, the function v = Dk u, which solves the Cauchy problem (6.5.3), belongs to Cb0,`0 +2+α ([0, T ] × Rd ). Due to the arbitrariness of k, we conclude that u ∈ Cb0,`0 +3+α ([0, T ] × Rd ) and estimate (6.5.2) holds true. 

6.6

Notes

The results concerning existence and regularity of solutions to parabolic problems in the whole space Rd with respect to the H¨older norms are well known. For the classical approach we refer to the monographs [1], [21], [22, Chapter IV]. The abstract evolution approach that we have considered here can be also found in [26, Section 5.1.1].

6.7

Exercises

1. Prove that, for every pair of disjoint compact sets K1 and K2 with K1 ⊂ K2 ⊂ Rd , there exists a function ϕ ∈ Cc∞ (Rd ) such that ϕ ≡ 1 in K1 and ϕ ≡ 0 outside K2 . 2. Given x0 ∈ Rd and r > 0, construct a function ϑx0 ,r satisfying (6.2.8) and prove estimate (6.2.9). (k+1+α)/2,k+1+α

3. Prove that for every k ∈ N and α ∈ (0, 1), the space Cloc ([0, T ] × Ω) is (k+α)/2,k+α α/2,k+1+α contained in Cloc ([0, T ] × Ω) and in Cloc ([0, T ] × Ω). Prove also that if (k+1+α)/2,k+1+α (k+1−h+α)/2,k+1−h+α u ∈ Cloc ([0, T ] × Ω), then Dth u ∈ Cloc ([0, T ] × Ω) for every h = 1, . . . , k. 4. Prove Proposition 6.2.12(ii).

Chapter 7 Parabolic Equations in Boundary Conditions

R

d +

with Dirichlet

In this chapter, we deal with the Cauchy-Dirichlet problem  t ∈ (0, T ],   Dt u(t, x) = Au(t, x) + g(t, x), 0 0 u(t, x , 0) = ψ(t, x ), t ∈ (0, T ],   u(0, x) = f (x),

x ∈ Rd+ , x0 ∈ Rd−1 , x ∈ Rd+ ,

(7.0.1)

where we find it convenient to split x = (x0 , xd ) for every x ∈ Rd . Here, A is the elliptic operator defined on smooth functions ψ : Rd+ → R by Aψ(x) =

d X

qij (x)Dij ψ(x) +

d X

bj (x)Dj ψ(x) + c(x)ψ(x)

j=1

i,j=1

=Tr(Q(x)D2 ψ(x)) + hb(x), ∇x ψ(x)i + c(x)ψ(x) for every x ∈ Rd+ , with, as usually, Q(x) = (qij (x))1≤i,j≤d . Throughout the chapter, we assume the following conditions on the coefficients of the operator A. Hypotheses 7.0.1 (i) The coefficients qij = qji , bj (i, j = 1, . . . , d) and c are bounded and α-H¨ older continuous in Rd+ for some α ∈ (0, 1); (ii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for every x ∈ Rd+ and ξ ∈ Rd . The main result of this chapter is the following counterpart of Theorem 6.2.6. Theorem 7.0.2 Let Hypotheses 7.0.1 be satisfied and fix T > 0. Then, for every f ∈ α/2,α ((0, T ) × Rd+ ) and ψ ∈ C 1+α/2,2+α ((0, T ) × Rd−1 ) such that f (·, 0) = Cb2+α (Rd+ ), g ∈ Cb d−1 ψ(0, ·) on R , Dt ψ(0, ·) = Af (·, 0) + g(0, ·, 0) on Rd−1 , there exists a unique bounded 1+α/2,2+α classical solution1 u to problem (7.0.1). In addition, u belongs to Cb ((0, T ) × Rd+ ) and there exists a positive constant C, independent of f , g and u, such that ||u||C 1+α/2,2+α ((0,T )×Rd ) +

b

≤C(||f ||C 2+α (Rd ) + ||g||C α/2,α ((0,T )×Rd ) + ||ψ||C 1+α/2,2+α ((0,T )×Rd−1 ) ). b

+

b

+

(7.0.2)

b

The proof of Theorem 7.0.2 is rather technical. The main core consists in proving that, for 1+α/2,2+α each function φ ∈ C 1+α/2,2+α (Rd−1 (Rd+,T ) T ) there exists a unique function v ∈ C d−1 such that Dt v = Av, v(·, 0) = φ on RT , together with the estimate ||v||C 1+α/2,2+α (Rd+,T ) ≤ C||φ||C 1+α/2,2+α (Rd−1 ) , b

T

1 By bounded classical solution, we mean a function u ∈ C ([0, T ]×Rd )∩C((0, T ]×Rd )∩C 1,2 ((0, T ]×Ω) b + + which solves problem (7.0.1).

175

Parabolic Equations in Rd+ with Dirichlet Boundary Conditions

176

the constant C being independent of v and φ, when the supremum c0 of the potential c of the operator A is negative. This result is proved first in the easiest case A = ∆−I, where an explicit formula for the candidate solution v to the above problem is available, in terms of (a spatial derivative of) the heat kernel. Direct computations allow to estimate the sup-norm of v and the C α/2,α (Rd+,T )-norm of the derivatives Dt v and Dij v (when i, j = 1, . . . , d). Much more efforts are needed to estimate the H¨older seminorm of the derivative Did v (i = 1, . . . , d − 1). For this purpose, we use Lemma 7.1.2, which allows to estimate the C α/2,α (Rd+,T )-norm of the function Did v in terms of the parabolic α-H¨older seminorm of time derivative of v and the spatial derivatives Dij v (i, j = 1, . . . , d − 1) and Ddd v. Freezing the coefficients and using the continuity method we address the more general operator A. In Section 7.4, we first state the natural counterpart of Theorem 6.3.2 and then prove further regularity results for the solution to the Cauchy problem (7.0.1), assuming that the coefficients of the operator A, as well as the data f , g and ψ are smoother. The proofs of such results, that we present, make us of tools from semigroup theory, for this reason they are postponed to Chapter 14. Finally, Section 7.5 is devoted to introducing the semigroup associated with the operator A, with homogeneous Dirichlet boundary conditions, in Cb (Rd+ ).

7.1

Technical Results

To ease the notation, in this section, we set [ζ]α = sup[ζ(t, ·)]Cbα (D) + sup [ζ(·, x)]C α/2 (I) t∈I

α/2,α

for every ζ ∈ Cb

b

x∈D

(I × D), I × D being either the whole Rm+1 or the set Rm +,T (m ≥ 1).

Theorem 7.1.1 There exists a positive constant C such that ||u||C 1+α/2,2+α (Rd+1 ) ≤ C(||Dt u − Au||C α/2,α (Rd+1 ) + ||u||∞ ), b

1+α/2,2+α

for every u ∈ Cb

(7.1.1)

b

(Rd+1 ).

Proof Since it is rather long, we split the proof into some steps. Throughout the proof, C denotes a positive constant, independent of u, which may vary from line to line. Step 1. Here, we show that [Dt u]α +

d X

[Dij u]α ≤ C

sup

sup r−2−α

(t,x)∈Rd+1 r>0

i,j=1

inf

p∈Pol1,2

||u − p||C(Cr (t,x)) =: C[[u]]α

1+α/2,2+α

for every u ∈ Cb (Rd+1 ), where Cr (t, x) denotes the cylinder (t − r2 , t) × B(x, r) and Pol1,2 the set of all the polynomials in (t, x) which are of at most first-order in t and at Pd most second-order in x. To ease the notation a bit more, we set [Dx2 w]α = i,j=1 [Dij w]α . 1+α/2,2+α

We fix u ∈ Cb (Rd+1 ) and begin by estimating [Dij u]α for arbitrarily fixed i, j ∈ {1, . . . , d}. For every h > 0 and (t, x) ∈ Rd+1 we set (Rh u)(t, x) =

1 [u(t, x + h(ei + ej )) − u(t, x + hej ) − u(t, x + hei ) + u(t, x)] h2

Semigroups of Bounded Operators and Second-Order PDE’s

177

and denote by ζt,x (h) the term in square brackets. The function ζt,x belongs to Cb2+α ([0, ∞)) 0 for every (t, x) ∈ Rd+1 . Moreover, ζt,x (0) = ζt,x (0) = 0 and 00 |ζt,x (h) − 2Dij u(t, x)| ≤|Dii u(t, x + h(ei + ej )) − Dii u(t, x + hei )|

+ |Djj u(t, x + h(ei + ej )) − Djj u(t, x + hej )| + 2|Dij u(t, x + h(ei + ej )) − Dij u(t, x)| ≤4hα [Dx2 u]α . Hence, expanding ζt,x by Taylor’s formula centered at zero, we get   1 00 0 (0)h + ζt,x (h∗ )h2 − Dij u(t, x) |Rh u(t, x) − Dij u(t, x)| = h−2 ζt,x (0) + ζt,x 2 1 00 = |ζt,x (h∗ ) − 2Dij u(t, x)| ≤ 2hα [Dx2 u]α (7.1.2) 2 for each h > 0 and (t, x) ∈ Rd+1 and some h∗ ∈ (0, h). Using (7.1.2) we can estimate [Dx2 u]α . Fix (t, x1 ) and (t, x2 ) in Rd+1 and set r = |x2 −x1 |. As it is immediately seen, the points (t, xk ), (t, xk + h(ei + ej )), (t, xk + hei ) and (t, xk + hej ) belong to C2r (t, xk ), for k = 1, 2, if |h| ≤ r. Noting that if p ∈ Pol1,2 , then Rh p is constant and coincides with the coefficients of the monomial xi xj , we can estimate |Dij u(t, x2 ) − Dij u(t, x1 )| ≤

2 X

|Dij u(t, xk ) − Rh u(t, xk )| + |[Rh (u − p)](t, x2 ) − [Rh (u − p)](t, x1 )|

k=1 α

≤4h [Dx2 u]α + 4h−2 ||u − p||C(C2r (t,x1 )) + 4h−2 ||u − p||C(C2r (t,x2 )) for each p ∈ Pol1,2 . Minimizing with respect to p, gives |Dij u(t, x2 ) − Dij u(t, x1 )| ≤ 4hα [Dx2 u]α + 8h−2 (2r)2+α [[u]]α for every h ∈ (0, r). We now choose h = γr, where 4γ α = 1/4 to obtain   1 2 [Dx u]α + C[[u]]α |x2 − x1 |α . (7.1.3) |Dij u(t, x2 ) − Dij u(t, x1 )| ≤ 4 √ Similarly, if we take (t1 , x), (t2 , x) in Rd+1 with t1 < t2 , set r = t2 − t1 and argue as above, then we conclude that   α 1 2 |Dij u(t2 , x) − Dij u(t1 , x)| ≤ [Dx u]α + C[[u]]α |t2 − t1 | 2 . (7.1.4) 4 Estimates (7.1.3) and (7.1.4) yield [Dij u]α ≤ 2−1 [Dx2 u]α +C[[u]]α and summing with respect to i, j we finally get [Dx2 u]α ≤ C[[u]]α . To estimate [Dt u]α we replace Rh u with the differential quotient Sh u = h−2 [u(·−h2 , ·)− u]. Since ||Sh u − Dt u||∞ ≤ hα ||Dt u||C α/2,0 (Rd+1 ) for every h > 0 and Sh p coincides with the b opposite of the coefficient of the monomial t, we can repeat the same arguments used to estimate Dij u and conclude that [Dt u]α ≤ C[[u]]α . Step 2. Here, we prove that [[u]]α ≤ C[Dt u − ∆u]α for every compactly supported 1+α/2,2+α function u ∈ Cb (Rd+1 ). We fix u as above, (t0 , x0 ) ∈ Rd+1 , r > 0, set g := Dt u−∆u and consider the polynomial pu defined by pu (t, x) =u(t0 , x0 ) + Dt u(t0 , x0 )(t − t0 ) + h∇x u(t0 , x0 ), x − x0 i

Parabolic Equations in Rd+ with Dirichlet Boundary Conditions

178

1 + hDx2 u(t0 , x0 )(x − x0 ), (x − x0 )i 2 for every (t, x) ∈ Rd+1 . Clearly, Dt pu − ∆pu = Dt u(t0 , x0 ) − ∆u(t0 , x0 ) = g(t0 , x0 ) on Rd+1 . Next, we fix a function ψ ∈ Cc∞ (Rd+1 ) with ψ ≡ 1 on CrM (t0 , x0 ), where rM = (M + 1)r, with M > 1 to be properly chosen later on. The function u − ψpu is smooth, compactly supported on Rd+1 and Dt (u − ψpu ) − ∆(u − ψpu ) = g − (Dt (ψpu ) − ∆(ψpu )) =: g − g1 . The function u − ψpu is the convolution in Rd+1 of the functions g − g1 and G, where G(t, x) = d

|x|2

(4πt)− 2 e− 4t χ(0,∞) (t) for every (t, x) ∈ Rd+1 , see Exercise 7.7.1. In particular, u − pu = G ? (g − g1 ) on CrM (t0 , x0 ). We split G ? (g − g1 ) into the sum of the integral over CrM (t0 , x0 ) and over its complement on Rd+1 . The so obtained functions are denoted by z1 and z2 α respectively. Since g1 = g(t0 , x0 ) on CrM (t0 , x0 ), we can estimate |z1 | ≤ rM [g]α G?χCrM (t0 ,x0 ) −2 −1 0 0 on CrM (t0 , x0 ). Performing the change of variables (s , y ) = (rM (s − t0 ), rM (y − x0 )) gives Z d+2 2 0 (G ? χCrM (t0 ,x0 ) )(t, x) =rM G(t − t0 − rM s , x − x0 − rM y 0 )χC1 (0,0) (s0 , y 0 ) dy 0 ds0 Rd+1

2 =rM (G

−2 −1 2 ? χC1 (0,0) )(rM (t − t0 ), rM (x − x0 )) ≤ rM

2+α [g]α . for every (t, x) ∈ Rd+1 . Hence, ||z1 ||C(Cr (t0 ,x0 )) ≤ rM M As far as z2 is concerned, we observe that it belongs to C ∞ (CrM (t0 , x0 )) and Dt z2 − ∆z2 identically vanishes on CrM (t0 , x0 ). Indeed, as it is easily seen, Dt z2 − ∆z2 = [(Dt K − ∆K)χ(0,∞) ] ? (g − g1 ) on Rd+1 \ CrM (t0 , x0 ) and K is the heat kernel, which is a classical solution of the homogeneous heat equation on (0, ∞) × Rd (see Lemma 5.1.2(v)). Moreover, using Taylor’s formula we can write

z2 (t, x) =z2 (t0 , x) + Dt z2 (t∗ , x)(t − t0 ) 1 =z2 (t0 , x0 ) + h∇z2 (t0 , x0 ), x − x0 i + hDx2 z2 (t0 , x∗ )(x − x0 ), x − x0 i 2 + Dt z2 (t∗ , x)(t − t0 ) for every (t, x) ∈ CrM (t0 , x0 ) and some suitable point (t∗ , x∗ ) ∈ CrM (t0 , x0 ), depending on (t, x). Hence, if we denote by pz2 the second-order polynomial defined as pu with u being replaced by z2 , then we can estimate |z2 (t, x) − pz2 (t, x)| ≤|Dt z2 (t∗ , x) − Dt z2 (t0 , x0 )||t − t0 | d 1 X (Dij z2 (t0 , x∗ ) − Dij z2 (t0 , x0 ))(x − x0 )i (x − x0 )j + 2 i,j=1 ≤r2 |Dt z2 (t∗ , x) − Dt z2 (t0 , x)| + r2 |Dt z2 (t0 , x) − Dt z2 (t0 , x0 )| + r3

d X

||Dijh z2 ||C(Cr (t0 ,x0 ))

i,j,h=1

≤r4 ||Dt2 z2 ||C(Cr (t0 ,x0 )) + r3

d X j=1

||Djt z2 ||C(Cr (t0 ,x0 )) + r3

d X

||Dijh z2 ||C(Cr (t0 ,x0 ))

(7.1.5)

i,j,h=1

for every (t, x) ∈ Cr (t0 , x0 ). Since (t1 , x1 ) + CM r (0, 0) ⊂ CrM (t0 , x0 ) for every (t1 , x1 ) ∈ Cr (t0 , x0 ) we can apply Exercise 7.7.3 to the function z2 (· + t1 , · + x1 ) and conclude that M 2 r2 |Dij z2 (t1 , x1 )| + M 3 r3 |Dijh z2 (t1 , x1 )| + M 4 r4 |Dijhk z2 (t1 , x1 )| ≤ Λ||z2 ||C(Cr

M

(t0 ,x0 ))

Semigroups of Bounded Operators and Second-Order PDE’s

179

for every i, j, h, k = 1, . . . , d and some positive constant Λ, independent of z2 and (t1 , x1 ). Observing that d X

and Dt2 z2 (t1 , x1 ) =

Djt z2 (t1 , x1 ) = Dj ∆z2 (t1 , x1 )

Diijj z2 (t1 , x1 ),

i,j=1

from the previous estimate, we conclude that M 3 r3 |Djt z2 (t1 , x1 )| ≤ dΛ||z2 ||C(Cr

M

M 4 r4 |Dt2 z2 (t1 , x1 )| ≤ d2 Λ||z2 ||C(Cr

(t0 ,x0 ))

M

,

(t0 ,x0 ))

.

Thus, we can continue estimate (7.1.5) obtaining that |z2 (t, x) − pz2 (t, x)| ≤ C(M −3 + M −4 )||z2 ||C(Cr Let us estimate ||z2 ||C(Cr

M

(t0 ,x0 ))

||u − pu ||C(Cr

M

M

(t0 ,x0 ))

= ||u − pu − z1 ||C(Cr

(t0 ,x0 ))

M

(t, x) ∈ Cr (t0 , x0 ).

, (t0 ,x0 ))

. Since

2+α ≤ CrM ([Dt u]α + [Dx2 u]α )

(which follows arguing as in the first part of (7.1.5)), taking the above estimate of z1 and Step 1 into account we get ||z2 ||C(Cr

M

(t0 ,x0 ))

≤C(M + 1)2+α r2+α ([g]α + [Dt u]α + [Dx2 u]α ) ≤CM 2+α r2+α ([g]α + [[u]]α ).

Summing up, we have proved that ||z2 − pz2 ||C(Cr (t0 ,x0 )) ≤ CM α−1 r2+α ([g]α + [[u]]α ). Now, we are done. Indeed, [[u]]α ≤

sup

sup r−2−α ||u − pu − pz2 ||C(Cr (t0 ,x0 ))

(t0 ,x0 )∈Rd+1 r>0

≤

sup

sup r−2−α ||z1 ||C(Cr (t0 ,x0 )) +

(t0 ,x0 )∈Rd+1 r>0

sup

sup r−2−α ||z2 − pz2 ||C(Cr (t0 ,x0 ))

(t0 ,x0 )∈Rd+1 r>0

≤C(M 2+α [g]α + M α−1 [[u]]α ). Taking M large enough, we conclude that [[u]]α ≤ C[g]α . Step 3. By Steps 1 and 2, we know that [Dt u]α + [Dx2 u]α ≤ C[Dt u − ∆u]α for each u ∈ 1+α/2,2+α Cb (Rd+1 ) with compact support. We now remove the condition on the support of u. 1+α/2,2+α For this purpose, we fix u ∈ Cb (Rd+1 ) and consider a sequence (ϑn ) ⊂ Cc∞ (Rd+1 ) such that χ[−n,n]×B(0,n) ≤ ϑn ≤ 1 on Rd+1 and [Dt ϑn ]α + [Dx2 ϑn ]α vanishes as n tend to ∞. The function un = uϑn satisfies the estimate [Dt un ]α + [Dx2 un ]α ≤ C[Dt un − ∆un ]α for every n ∈ N. Since un = u on [−n, n] × B(0, n), a straightforward computation shows that sup [Dt u(·, x)]C α/2 ([−n,n]) + x∈B(0,n)

+

sup [Dt u(t, ·)]C α (B(0,n)) +

t∈[−n,n]

d X

sup [Dij u(·, x)]C α/2 ([−n,n])

i,j=1 x∈B(0,n) d X

sup [Dij u(t, ·)]C α (B(0,n))

i,j=1 t∈[−n,n]

Parabolic Equations in Rd+ with Dirichlet Boundary Conditions

180

≤C[Dt u − ∆u]α + an for each n ∈ N and some sequence (an ) converging to 0. Letting n tend to ∞, we conclude that [Dt u]α + [Dx2 u]α ≤ C[Dt u − ∆u]α . Note that this estimate can be straightforwardly extended to functions u taking values in C: it is enough to apply it to the real and imaginary parts of u. Step 4. Here, we prove estimate (7.1.1) when A = ∆. For this purpose, we use a trick commonly used, which consists of adding a new variable. More precisely, for every 1+α/2,2+α u ∈ Cb (Rd+1 ) we consider the function v : Rd+2 → C defined by v(t, x, xd+1 ) = ixd+1 u(t, x)e . By Step 3, we know that [Dt v]α + [Dx2 v]α ≤ C[Dt v − ∆v]α , where ∆ is now the Laplacian on Rd+1 . As it is immediately seen, Dt v(·, ·, xd+1 ) − ∆v(·, ·, xd+1 ) = eixd+1 (Dt u−∆u+u) for every xd+1 ∈ R. Hence, [Dt v−∆v]α ≤ C||Dt u−∆u−u||C α/2,α (Rd+1 ) . b We observe that 2|Dij u(t, x)| = |Dij v(t, x, π) − Dij v(t, x, 0)| ≤ π α [Dij v]α for every (t, x) ∈ Rd+1 and i, j = 1, . . . , d, so that ||Dij u||∞ ≤ 2−1 π α [Dij v]α . Similarly, ||Dt u||∞ ≤ 2−1 π α [Dt v]α and, consequently, ||Dt u||∞ +

d X

||Dij u||∞ ≤ 2−1 π α ([Dt v]α + [Dx2 v]α ).

i,j=1

Finally, notice that [Dij v(·, ·, 0)]α = [Dij u]α for every i, j = 1, . . . , d and [Dt v(·, ·, 0)]α = [Dt u]α . From these estimates and Proposition 1.2.6, we deduce that ||u||C 1+α/2,2+α (Rd+1 ) ≤C([Dt v]α + [Dx2 v]α + ||u||∞ ) ≤ C(||Dt u − ∆u + u||C α/2,α (Rd+1 ) + ||u||∞ ). b

b

(7.1.6) Using (7.1.6) and the interpolation estimate ||u||C α/2,α (Rd+1 ) ≤ ε||u||C 1+α/2,2+α (Rd+1 ) + Cε ||u||∞ , b

b

which is a straightforward consequence of Proposition 1.2.5, we obtain ||u||C 1+α/2,2+α (Rd+1 ) ≤C(||Dt u − ∆u||C α/2,α (Rd+1 ) + ||u||C α/2,α (Rd+1 ) ) b

b

b

≤C(||Dt u − ∆u||C α/2,α (Rd+1 ) + ε||u||C 1+α/2,2+α (Rd+1 ) + Cε ||u||∞ ) b

b

for every ε > 0 and some positive constant Cε , independent of u and blowing up as ε → 0+ . Taking ε sufficiently small, we immediately get (7.1.1). Step 5. To extend (7.1.1) to every operator A, one can use the same arguments as in Chapter 6 first considering the case when A = Tr(QD2 ) for some constant and positive definite matrix Q and then freezing the coefficients to handle the more general case. Since there are no sensible differences with the proof of Theorem 6.2.1 we omit the details.  The following property will be used in the proof of Theorem 7.2.1. Lemma 7.1.2 There exists a positive constant C such that   d−1 X [Did u]α ≤ C [Dt u]α + [Dij u]α + [Ddd u]α i,j=1 1+α/2,2+α

for every u ∈ Cb

(Rd+,T ) and i = 1, . . . , d.

Semigroups of Bounded Operators and Second-Order PDE’s

181

Proof Throughout the proof, we denote by C a positive constant, which is independent of u and may vary from line to line. We split the proof into two steps. (h) (h) (h) Step 1. Let ∆0 and ∆i be the operators defined by (∆0 ζ)(t, x) = ζ(t, x)−ζ(t−h2 , x) (h) and (∆i ζ)(t, x) = ζ(t, x + hei ) − ζ(t, x) for every (t, x) ∈ Rd+,T , h > 0, and every function (h)

(h)

ζ : Rd+,T → R. Using the operators ∆0 and ∆i we can define some new operators that will be used in the proof. More precisely, for every i, j, k ∈ {1, . . . , d} and h ≥ 0 we set (h)

(h)

(h)

(h)

◦ ∆j ,

∆i,j = ∆i

(h)

(h)

(h)

∆i,j,k = ∆i,j ◦ ∆k ,

(h)

(h)

∆0,j = ∆0 ◦ ∆j .

(h)

As it is immediately seen, the operator ∆i,j,k is invariant with respect to each permutation of the triplet (i, j, k). Moreover, Z

(h)

h

h

Z

[Dij u(·, · + rei + sej + hek ) − Dij u(·, · + rei + sej )] ds

dr

∆i,j,k u =

0

0

1+α/2,2+α

for every i, j, k = 1, . . . , d, h > 0 and u ∈ Cb

(Rd+,T ). Therefore,

(h)

||∆i,j,k u||∞ ≤ sup[Dij u(t, ·)]Cbα (Rd+ ) h2+α

(7.1.7)

t≤T

for every h > 0, i, j, k = 1, . . . , d. Arguing similarly, it can be easily shown that (h)

||∆0,j u||∞ ≤ sup[Dt u(t, ·)]Cbα (Rd+ ) h2+α ,

(7.1.8)

t≤T

(h)

||Dij u − h−2 ∆i,j u||∞ ≤ 2 sup[Dij u(t, ·)]Cbα (Rd+ ) hα

(7.1.9)

t≤T

1+α/2,2+α

for every h > 0, i, j = 1, . . . , d and u ∈ Cb Step 2. Here, we set

(Rd+,T ).

  d X (h) (h) [[u]]α,i := sup h−2−α ||∆0,i u||∞ + ||∆i,d,k u||∞ h>0

k=1

1+α/2,2+α

for every u ∈ Cb (Rd+,T ), i = 1, . . . , d, and prove that [Did u]α ≤ C[[u]]α,i for some positive constant, independent of u (and of T ) and every i = 1, . . . , d. For this purpose, we fix i, k = 1, . . . , d, n ∈ N and split (h)

(h/n)

∆k Did u =Did u(·, · + hek ) − h−2 n2 ∆i,d + h−2 n2

n X

(h/n)

[∆i,d

(h/n)

u(·, · + hek ) + h−2 n2 ∆i,d (h/n)

u(·, · + rhn−1 ek ) − ∆i,d

u − Did u

u(·, · + (r − 1)hn−1 ek )].

r=1

Since (h/n)

∆i,d

(h/n)

u(·, · + rhn−1 ek ) − ∆i,d

(h/n)

u(·, · + (r − 1)hn−1 ek ) = ∆i,d,k u(·, · + (r − 1)hn−1 ek ),

taking (7.1.7) and (7.1.9) into account, we can estimate (h)

(h/n)

||∆k Did u||∞ ≤4n−α hα sup[Did u(t, ·)]Cbα (Rd+ ) + h−2 n3 ||∆i,d,k u||∞ t≤T

≤4n

−α α

h sup[Did u(t, ·)]Cbα (Rd+ ) + hα n1−α [[u]]α,i t≤T

182

Parabolic Equations in Rd+ with Dirichlet Boundary Conditions

for every h > 0. This shows that, if x, y ∈ Rd+ are such that y − x = (yk − xk )ek for some k = 1, . . . , d, then   −α 1−α |Did u(t, y) − Did u(t, x)| ≤ 4n sup[Did u(t, ·)]Cbα (Rd+ ) + n [[u]]α,i |x − y|α t≤T

for every t ≤ T . Now, for every x, y ∈ Rd+ and k = 1, . . . , d, we set x(0) = x, x(k) = x(k−1) + (yk − xk )ek , split Did u(t, y) − Did u(t, x) =

d X

(Did u(t, x(k) ) − Did u(t, x(k−1) ))

k=1

and conclude that  |Did u(t, y) − Did u(t, x)| ≤

 4dn−α sup[Did u(t, ·)]Cbα (Rd+ ) + dn1−α [[u]]α,i |x − y|α t≤T

so that, taking n sufficiently large, it follows that sup[Did u(t, ·)]Cbα (Rd+ ) ≤ C[[u]]α,i .

(7.1.10)

t≤T

To estimate the α/2-H¨ older seminorms of Did u, we fix s < t ≤ T and set h = Then, arguing as above and using (7.1.8) and (7.1.9), we get (h)

√

t − s.

(h)

|Did u(t, ·) − Did u(s, ·)| ≤|Did u(t, ·) − h−2 ∆i,d u(t, ·)| + |Did u(s, ·) − h−2 ∆i,d u(s, ·)| (h)

(h)

+ h−2 |(∆0 ◦ ∆i,d u)(t, ·)| α

(h)

≤2 sup[Did u(t, ·)]Cbα (Rd+ ) |t − s| 2 + 2h−2 ||∆0,i u||∞ t≤T

α

α

≤2 sup[Did u(t, ·)]Cbα (Rd+ ) |t − s| 2 + 2[[u]]α,i |t − s| 2 , t≤T

(h)

where we have also used the estimate ||∆h0 ◦ ∆i,d u||Cb (Rd+,T ) ≤ 2||∆h0,i u||Cb (Rd+,T ) . Using (7.1.10), we have so proved that supx∈Rd [Did u(·, x)]C α/2 ((−∞,T ]) ≤ C[[u]]α,i . The estimate +

[Did u]α ≤ C[[u]]α,i now follows for every i = 1, . . . , d. So, using (7.1.7) and (7.1.8), we deduce that   d X (h) (h) [Did u]α ≤C sup h−2−α ||∆0,i u||∞ + ||∆i,d,k u||∞ h>0

 ≤C [Dt u]α +

k=1 d X

(h) ||∆i,d,k u||∞



k=1

 =C [Dt u]α +

d−1 X

(h) ||∆i,d,k u||∞

+

(h) ||∆i,d,d u||∞



k=1

  d−1 X ≤C [Dt u]α + [Dik u]α + [Ddd u]α k=1

and the assertion follows at once.



To complete this section, we prove the following lemma which will be used in the proof of the existence part of Theorem 7.2.1.

Semigroups of Bounded Operators and Second-Order PDE’s

183

Lemma 7.1.3 Let K+ be the function defined by K+ (t, x) = −2χ(0,∞)×Rd+ (t, x)Dd K(t, x) for each (t, x) ∈ R × Rd+ , where K is the Gauss-Weierstrass kernel defined by (5.1.1). Then, the following properties hold true: (i) K+ is infinitely many times differentiable in R × Rd+ ; (ii) Dt K+ = ∆K+ on R × Rd+ ; (iii) the function (t, x0 ) 7→ tγ |x0 |β K+ (t, x0 , xd ) belongs to L1 ((0, ∞) × Rd−1 ) provided that 1 − 2γ − β > 0. In such a case, Z ∞ Z dt tγ |x0 |β K+ (t, x0 , xd ) dx0 = Cβ,γ xdβ+2γ (7.1.11) Rd−1

0

for each xd > 0, where −2γ−1 − d 2

Cβ,γ = 2

π



   1 − 2γ − β β+d−1 Γ Γ ωd−1 ; 2 2

(iv) the functions K+ , Dt K+ , Dj K+ , Dij K+ are integrable on Rd (with respect to t and x0 ) for every xd > 0 and their L1 -norms are bounded on (ε, ∞) for every ε > 0. Proof (i) & (ii). By Lemma 5.1.2 we know that K+ ∈ C ∞ ((0, ∞) × Rd+ ). Moreover, a direct computation shows that Dt K+ = ∆K+ on (0, ∞) × Rd+ . To conclude the proof, it suffices to observe that all the derivatives of K+ vanish as (t, x) tends to (0, x0 ), for each x0 ∈ Rd+ . This implies that K+ ∈ C ∞ (R × Rd+ ) and it solves the heat equation in the whole R × Rd+ . √ (iii) The usual change of variables y 0 = (2 t)−1 x0 and, then, the change of variable s = x2d /4t show that Z ∞ Z dt tγ |x0 |β K+ (t, x0 , xd ) dx0 0 Rd−1 Z ∞ Z x2 |x0 |2 xd d γ−1− d 2 e− 4t dt t |x0 |β e− 4t dx0 = d (4π) 2 0 Rd−1 Z ∞ Z 2 β−1 xd β 0 2 3 2 = d xd t 2 +γ− 2 e− 4t dt |y 0 |β e−|y | dy 0 π2 0 Rd−1 Z ∞ Z β 0 2 1 d β+2γ −γ − 2 −γ− 2 − 2 −s =4 π xd s e dt |y 0 |β e−|y | dy 0 . 0

Rd−1

Hence, the integral is finite provided that −2γ − β − 1 > −2, i.e., 1 − 2γ − β > 0, and (7.1.11) follows. (iv) It is a straightforward consequence of (iii). Indeed, an easy computation shows that 1 Dh K+ (t, x) = − t−1 xh K+ (t, x) + δhd x−1 d K+ (t, x), 2 1 1 Dij K+ (t, x) = − t−1 δij K+ (t, x) + t−2 xi xj K+ (t, x), 2 4 1 −1 1 −1 Did K+ (t, x) = − t xi xd K+ (t, x) + t−2 xi xd K+ (t, x), 2 4 3 1 Ddd K+ (t, x) = − t−1 K+ (t, x) + t−2 x2d K+ (t, x) 2 4

(7.1.12)

for each t > 0, x ∈ Rd+ , i, j = 1, . . . , d − 1 and h = 1, . . . , d. Hence, applying formula (7.1.11)

Parabolic Equations in Rd+ with Dirichlet Boundary Conditions

184

we conclude that the L1 (Rd )-norm of each first-order (resp. second-order) spatial derivative −2 of K+ (·, ·, xd ) is bounded from above by a constant times x−1 d (resp. xd ) for every xd > 0. 1 d Since Dt K+ = ∆K+ it follows that the L (R ) norm of Dt K+ (·, ·, xd ) is bounded from above by a constant times x−2  d . The assertion follows.

7.2

An Auxiliary Boundary Value Problem

In this section, we deal with the boundary value problem ( Dt u(t, x) = Au(t, x) + g(t, x), t ∈ (−∞, T ), x ∈ Rd+ , u(t, x0 , 0) = ψ(t, x0 ), t ∈ (−∞, T ), x0 ∈ Rd−1 ,

(7.2.1)

under Hypotheses 7.0.1, with 0 > c0 = supx∈Rd c(x), and prove the following important result. 1+α/2,2+α

α/2,α

Theorem 7.2.1 For each pair of functions ψ ∈ Cb (RTd−1 ) and g ∈ Cb (Rd+,T ), problem (7.2.1) admits a unique solution u ∈ C 1+α/2,2+α (Rd+,T ) which satisfies the estimate ||u||C 1+α/2,2+α (Rd b

+,T )

≤ C? (||g||C α/2,α (Rd b

+,T )

+ ||ψ||C 1+α/2,2+α (Rd−1 ) ) b

(7.2.2)

T

for some positive constant C? , independent of u, g and ψ. The following proposition, yields the uniqueness of the solution to problem (7.2.1). Proposition 7.2.2 Suppose that u ∈ Cb1,2 (Rd+,T ) solves the boundary value problem (7.2.1). Then, ||u||∞ ≤ |c0 |−1 ||g||∞ + ||ψ||∞ . (7.2.3) Proof We introduce the function v = u + c−1 0 ||g||∞ − ||ψ||∞ , which is nonpositive on d (−∞, T ] × ∂R+ and satisfies the differential equation Dt v − Av ≤ 0 on RdT . The function ϕa : Rd → R, defined by ϕa (x) = a + |x|2 for every x ∈ Rd , satisfies the estimate Aϕa (x) ≤ `1 + `2 |x| + c0 a + c0 |x|2 for every x ∈ Rd+ and positive constants `1 and `2 . Using Young’s inequality, we can estimate `2 |x| ≤ `3 − c0 |x|2 for every x ∈ Rd+ , so that we can choose a large enough such that Aϕa ≤ 0 on Rd+ . The function vn = v − n−1 ϕa is negative on (−∞, T ] × ∂Rd+ and Dt vn − Avn ≤ 0 on Rd+,T . Since vn diverges to −∞ as |x| tends to ∞, uniformly with respect to t ∈ (−∞, T ], it attains its maximum value. The same argument as in the proof of Theorem 4.1.4 shows that the maximum is attained on (−∞, T ] × ∂Rd+ , so that vn is nonpositive on Rd+,T . Letting n tend to ∞ we deduce that v ≤ 0 on Rd+,T , i.e., u ≤ |c0 |−1 ||g||∞ + ||ψ||∞ . Applying the same argument with u being replaced by −u, we get also the estimate −u ≤ |c0 |−1 ||g||∞ + ||ψ||∞ and (7.2.3) follows.  Proof of Theorem 7.2.1 when A = ∆ − I Since it is rather long, we split the proof into four steps. α/2,α Step 1. Here, we prove that, for every ge ∈ Cb (Rd+,T ) there exists a function w ∈ 1+α/2,2+α

Cb (Rd+,T ) such that Dt w = ∆w − w + ge on Rd+,T . By setting ge(t, ·) = ge(T, ·) for every t > T , we can extend ge to the whole Rd+1 (by taking the even extension with respect α/2,α to the last variable xd ) with a function, still denoted by ge, which belongs to Cb (Rd+1 )

Semigroups of Bounded Operators and Second-Order PDE’s g ||C α/2,α (Rd and satisfies the estimate ||e g ||C α/2,α (Rd+1 ) ≤ 2||e

+,T )

b

b

Cc∞ (R),

185

. We also consider a sequence

(ϑn ) ⊂ such that χ(−n,n) ≤ ϑn ≤ χ(−2n,2n) for every n ∈ N. For each n ∈ N we set Z t wn (t, x) = es−t (T (t − s)e gn (s, ·))(x) ds, t ∈ R, x ∈ Rd , −∞

where {T (t)} denotes the Gauss-Weierstrass semigroup and gen = ϑn ge. Note that g ||C α/2,α (Rd ) for some positive constant C0 , independent of ge and ||e gn ||C α/2,α (Rd+1 ) ≤ C0 ||e +,T

b

b

n. Since gen is supported in (−2n, 2n) × Rd , the integral defining function wn converges for every t ∈ R. Adapting the arguments in the proof of Theorem 5.4.2, we can easily show that 1+α/2,2+α wn ∈ Cb (Rd+1 ) and solves the equation Dt w = ∆w − w + gen on Rd+1 . Moreover, g ||C α/2,α (Rd ) , C1 being indeby (7.1.1) it satisfies the estimate ||wn ||C 1+α/2,2+α (Rd+1 ) ≤ C1 ||e +,T

b

b

pendent of n and ge. Thus, applying Arzel`a-Ascoli theorem to the sequences (wn ), (Dt wn ), (Dj wn ) and (Dij wn ) (i, j = 1, . . . , d), we conclude that, up to a subsequence, wn converges 1+α/2,2+α in C 1,2 ([−r, r] × B(0, r)) (for every r > 0) to a function w ∈ Cb (Rd+1 ), which solves the equation Dt w = ∆w − w + ge and satisfies the estimate ||w||C 1+α/2,2+α (Rd+1 ) ≤ b C2 ||e g ||C α/2,α (Rd+,T ) , the constant C2 being independent of ge. 1+α/2,2+α

Step 2. In view of Step 1, u ∈ Cb

(Rd+,T ) solves the boundary value problem 1+α/2,2+α

(7.2.1) if and only if the function v = u − w ∈ Cb (

Dt v(t, x) = ∆v(t, x) − v(t, x), v(t, x0 , 0) = ψ(t, x0 ) − w(t, x0 , 0),

(Rd+,T ) solves problem

t ∈ (−∞, T ], t ∈ (−∞, T ],

x ∈ Rd+ , x0 ∈ Rd−1 ,

(7.2.4)

where w is the solution to the equation Dt w − ∆w + w = ge on Rd+,T provided by Step 1 and ge is the even extension (with respect to the last variable xd ) of g. The function w satisfies the estimate ||w||C 1+α/2,2+α (Rd ) ≤ C2 ||g||C α/2,α (Rd ) . (7.2.5) +,T

b

b

+,T

To ease the notation, we set ψb = ψ − w(·, ·, 0). We will prove that the function v : → R, defined by

Rd+,T

Z

b y 0 ) ds dy 0 es−t K+ (t − s, x0 − y 0 , xd )ψ(s,

v(t, x) = Rd

1+α/2,2+α

for every (t, x) ∈ Rd+,T , belongs to Cb (7.2.4) and satisfies the estimate ||v||C 1+α/2,2+α (Rd b

+,T )

(Rd+,T ), solves the boundary value problem

b 1+α/2,2+α d−1 + ||g|| α/2,α d ≤ C3 (||ψ|| C (R ) C (R b

T

b

+,T )

)

(7.2.6)

for some positive constant C3 , independent of v, ψ and g. An easy application of the dominated convergence theorem shows that v ∈ C 1,2 (RdT ) and solves the equation Dt v = ∆v − v. To prove estimate (7.2.6) and to show that v(·, ·, 0) = ψb on Rd−1 T , we perform the change of variables t − s = x2d r, x0 − y 0 = xd z 0 , to write v in the much more convenient form Z 2 b − rx2 , x0 − xd z 0 ) dr dz 0 v(t, x) = e−rxd K+ (r, z 0 , 1)ψ(t (7.2.7) d Rd

Parabolic Equations in Rd+ with Dirichlet Boundary Conditions

186

for each t ∈ (−∞, T ] and x ∈ Rd+ . By applying the dominated convergence theorem and taking Lemma 7.1.3(iii) into account, we easily see that Z b 0 , x0 ) dr dz 0 = ψ(t b 0 , x0 ) lim 0 v(t, x) = K+ (r, z 0 , 1)ψ(t 0 (t,x)→(t0 ,x0 ,0)

Rd

for every (t0 , x00 ) ∈ Rd−1 T . Hence, the boundary condition in (7.2.4) is satisfied. In view of Proposition 1.2.6, to prove (7.2.2) it suffices to show that Dt v, Dij v (i, j = α/2,α 1, . . . , d) belong to Cb (Rd+,T ) and ||v||∞ + ||Dt v||C α/2,α (Rd

+,T

b

≤C(||g||C α/2,α (Rd b

+,T )

d X

+ )

||Dij v||C α/2,α (Rd

+,T )

b

i,j=1

+ ||ψ||C 1+α/2,2+α (Rd−1 ) ), T

where, here above and throughout the rest of the proof, C denotes a positive constant, which depends at most on d and α, and may vary from line to line. b ∞ for each (t, x) ∈ As it is immediately seen, from (7.2.7) it follows that |v(t, x)| ≤ ||ψ|| d R+,T . α/2,α

Step 3. We now estimate the Cb (Rd+,T )-norm of Dt v and Dij v for i, j = 1, . . . , d − 1. Since the arguments are just the same, we show in details how to estimate Dt v. Differentiating under the integral sign, from (7.2.7) it follows that Z 2 b − rx2 , x0 − xd z 0 ) dr dz 0 Dt v(t, x) = e−rxd K+ (r, z 0 , 1)Dt ψ(t d Rd

b ∞. for each t ∈ (−∞, T ] and x ∈ Rd+ . From this formula, we deduce that ||Dt v||∞ ≤ ||Dt ψ|| Moreover, for each t1 < t2 ≤ T and x ∈ Rd+ we can estimate |Dt v(t2 , x) − Dt v(t1 , x)| Z 2 b 2 − rx2 , x0 − xd z 0 ) − Dt ψ(t b 1 − rx2 , x0 − xd z 0 )| dr dz 0 ≤ e−rxd K+ (r, z 0 , 1)|Dt ψ(t d d d R Z α b x)] α/2 K+ (r, z 0 , 1) dr dz 0 ≤ sup [Dt ψ(·, |t − t1 | 2 C ((−∞,T ]) 2 b

x∈Rd−1

Rd

b α/2,0 d−1 |t2 − t1 | α2 , ≤||Dt ψ|| C ) (R b

T

where we have taken advantage of (7.1.11), so that b [Dt v(·, x)]C α/2 ((−∞,T ]) ≤ ||Dt ψ|| C α/2,0 (Rd−1 )

(7.2.8)

T

for each x ∈ Rd+ . Similarly, for each t ≤ T and xd ≥ 0 it holds that b [Dt v(t, ·, xd )]Cbα (Rd−1 ) ≤ ||Dt ψ|| C 0,α (Rd−1 )

(7.2.9)

T

for each t ≤ T and xd ≥ 0. To estimate the α-H¨older seminorm of the function Dt v(t, x0 , ·), we observe that |Dt v(t, x0 , xd ) − Dt v(t, x0 , x ed )| Z 2 2 b − rx2 , x0 − xd z 0 )| dr dz 0 ≤ |e−rxd − e−rexd |K+ (r, z 0 , 1)|Dt ψ(t d Rd

Semigroups of Bounded Operators and Second-Order PDE’s

187

Z

b − rx2 , x0 − xd z 0 ) − Dt ψ(t b − re K+ (r, z 0 , 1)|Dt ψ(t x2d , x0 − x ed z 0 )| dr dz 0 d Z α 2 2 α b 2 r 2 K+ (r, z 0 , 1) dr dz 0 ≤C|xd − x ed | ||Dt ψ||∞ Rd Z α 2 2 α b x)] α/2 2 |x − x e | + sup [Dt ψ(·, r 2 K+ (r, z 0 , 1) dr dz 0 d Cb ((−∞,T ]) d d−1 d x∈R R Z α 0 α b ·)]C α (Rd−1 ) |xd − x ed | + sup[Dt ψ(t, |z | K+ (r, z 0 , 1) dz 0 b +

Rd

t≤T

Rd

α 2

b α/2,α d−1 (|x2 − x ≤C||Dt ψ|| ed |α ) e2d | + |xd − x d (R C )

(7.2.10)

T

b

for each t ≤ T , xd , x ed ≥ 0, with xd < x ed . If further 3xd ≤ x ed , i.e., xd + x ed ≤ 2|xd − x ed |, then |x2d − x e2d | ≤ 2|xd − x ed |2 so that from (7.2.10) it follows that b α/2,α d−1 |xd − x |Dt v(t, x0 , xd ) − Dt v(t, x0 , x ed )| ≤ C||Dt ψ|| ed |α . C (R )

(7.2.11)

T

Formula (7.2.10) is of no help when xd < x ed < 3xd . In such a case, we rewrite Dt v in the form Z Z b y 0 ) dy 0 Dt v(t, x) = ds es−t K+ (t − s, x0 − y 0 , xd )Dt ψ(s, R

Rd−1

for every t ∈ (−∞, T ] and x ∈ Rd+ . Differentiating with respect to the variable xd , taking (7.1.12) into account and observing that Z Dd K+ (s, y 0 , xd ) ds dy 0 = 0, xd > 0, Rd

we get Z Dd Dt v(t, x) =

es−t ds

Z

b y 0 ) dy 0 Dd K+ (t − s, x0 − y 0 , xd )Dt ψ(s,

Rd−1

R

Z

Z

b y 0 ) − Dt ψ(t, b x0 )) dy 0 ds Dd K+ (t − s, x0 − y 0 , xd )(Dt ψ(s, Rd−1 Z ∞ Z 1 −s b − s, x − y 0 ) − Dt ψ(t, b x0 )) dy 0 e ds s−1 K+ (s, y 0 , xd )(Dt ψ(t = − xd 2 d−1 Z 0∞ ZR −1 −s b − s, x0 − y 0 ) − Dt ψ(t, b x0 )) dy 0 + xd e ds K+ (s, y 0 , xd )(Dt ψ(t

=

e

s−t

R

0

Rd−1

for every t ∈ (−∞, T ] and x ∈ Rd+ . Hence, estimating b − s, x0 − y 0 ) − Dt ψ(t, b x0 )| ≤ ||Dt ψ|| b α/2,α d−1 (|s|α/2 + |y 0 |α ) |Dt ψ(t C (R ) T

b

we obtain 1 b α/2,α d−1 |Dd Dt v(t, x)| ≤ xd ||Dt ψ|| Cb (RT ) 2 +

Z

∞

Z

α

(s 2 −1 + s−1 |y 0 |α )K+ (s, y 0 , xd ) dy 0

ds 0

b x−1 d ||Dt ψ||Cbα/2,α (Rd−1 ) T

Z

∞

Rd−1

Z ds

0

α

(s 2 + |y 0 |α )K+ (s, y 0 , xd ) dy 0 .

Rd−1

Taking (7.1.11) once more into account, we conclude that b α/2,α d−1 xα−1 , |Dd Dt v(t, x)| ≤ Cα00 ||Dt ψ|| C (R ) d b

T

t ∈ (−∞, T ], x ∈ Rd+ .

188

Parabolic Equations in Rd+ with Dirichlet Boundary Conditions

Hence, observing that the condition xd + x ed > 2|xd − x ed | is equivalent to 2xd > |xd − x ed |, we can estimate b α/2,α d−1 xα−1 |xd − x |Dt v(t, x0 , xd ) − Dt v(t, x0 , x ed )| ≤C||Dt ψ|| ed | (R C ) d T

b

b α/2,α d−1 |xd − x ≤C||Dt ψ|| ed |α . (R C )

(7.2.12)

T

b

From (7.2.11) and (7.2.12) it follows that b α/2,α d−1 [Dt v(t, x0 , ·)]Cbα (R+ ) ≤ C||Dt ψ|| (R C ) b

(7.2.13)

T

for every t ≤ T and x0 ∈ Rd−1 . So, (7.2.8), (7.2.9) and (7.2.13) imply that Dt v belongs to α/2,α b 1+α/2,2+α d−1 . Cb (Rd+,T ) and ||Dt v||C α/2,α (Rd ) ≤ C||ψ|| (R C ) +,T

b

b

T

Step 4. We now complete the proof estimating the last derivatives Did v. Pd−1 Recalling that Dt v = ∆v − v, we can write Ddd v = Dt v − j=1 Djj v + v and, from Step α/2,α b 1+α/2,2+α d−1 . 3, we conclude that Ddd v ∈ Cb (Rd+,T ) and ||Ddd v||C α/2,α (Rd ) ≤ C||ψ|| C (R ) b

+,T

b

T

Estimating the parabolic H¨ older norm of second-order derivatives Did v (i = 1, . . . , d−1) trying to follow the same technique as above is not straightforward, since computing Did v and using the change of variables which led to (7.2.7), a singularity at xd = 0 seems to appear. To overcome this difficulty, we take advantage of Lemma 7.1.2. First of all, we observe that Z b − s, x0 − y 0 ) ds dy 0 Did v(t, x) = e−s Dd K+ (s, y 0 , xd )Di ψ(t Rd

for every (t, x) ∈ Rd+,T . Noticing that ∞

Z

−s

Z

|Dd K+ (s, y 0 , xd )| dy 0 ≤

e

Rd−1

0

1 , 2

1/2,0 (1+α)/2,0 (Rd+,T ) ∩ C 0,1 (Rd+,T ) with and Di ψb ∈ Cb (Rd+,T ) ∩ C 0,1 (Rd+,T ) ,→ Cb

b 1/2,0 d ||Di ψ|| C (R

+,T )

b

b 0,1 d + ||Di ψ|| C (R

+,T )

b 1+α/2,2+α d ≤ C||ψ|| C (R+,T )

(see Lemma 1.2.4), using Lemma 7.1.3 we can estimate Z b − s, x0 − y 0 ) − Di ψ(t, b x0 )| ds dy 0 |Did v(t, x)| ≤ |Dd K+ (s, y 0 , xd )||Di ψ(t d R Z b x0 ) ds dy 0 + e−s Dd K+ (s, y 0 , xd )Di ψ(t, Rd Z √ 1 b∞ b 1+α/2,2+α d−1 ≤C||ψ|| |Dd K+ (s, y 0 , xd )|( s + |y 0 |) ds dy 0 + ||Di ψ|| Cb (RT ) 2 Rd b 1+α/2,2+α d−1 ≤C||ψ|| Cb

(RT

)

for every (t, x) ∈ Rd+,T . Lemma 7.1.3 can be also used to check that Did v belongs to C α/2,α ((−∞, T )×Rd−1 ×(ε, ∞)) for each ε > 0. As a byproduct, the function vε = v(·, ·, ·+ε) 1+α/2,2+α belongs to Cb (Rd+,T ) for every ε > 0 and, using Lemma 7.1.2, we can estimate [Did vε ]α = sup[Did vε (t, ·)]Cbα (Rd+ ) + sup [Did vε (·, x)]C α/2 ((−∞,T )) t≤T

x∈Rd +

Semigroups of Bounded Operators and Second-Order PDE’s  ≤C ||Dt v||C α/2,α (Rd b

+,T

+ )

d−1 X

189 

||Dij v||C α/2,α (Rd

+,T

b

i,j=1

+ ||Ddd v||C α/2,α (Rd ) b

+,T )

b 1+α/2,2+α d−1 . ≤C||ψ|| (R C ) T

b

Since the constant C is independent of ε, we can let ε tend to 0 in the first and last side α/2,α of such a chain of inequalities and conclude that Did v ∈ Cb (Rd+,T ) and [Did v]α ≤ b 1+α/2,2+α d−1 . Estimate (7.2.6) follows immediately, taking Proposition 1.2.4(i) into C||ψ|| (RT

Cb

)

account. Finally, (7.2.5) and (7.2.6) yield estimate (7.2.2) and conclude the proof.



As a consequence of Theorem 7.2.1, with A = ∆ − I, we can now prove the following a priori estimates Theorem 7.2.3 There exists a positive constant C0 , which depends only on d, α and, in a continuous way on the H¨ older norms of the coefficients of the operator A and the ellipticity constant µ in Hypothesis 7.0.1, such that ||u||C 1+α/2,2+α (Rd

+,T )

b

≤C0 (1 − c−1 0 )||Dt u − Au||C α/2,α (Rd b

+,T )

+ C0 ||u(·, ·, 0)||C 1+α/2,2+α (Rd−1 ) . T

b

(7.2.14) Proof The weaker estimate ||u||C 1+α/2,2+α (Rd

+,T )

b

≤C0 (||Dt u − Au||C α/2,α (Rd b

+,T )

+ ||u||∞ + ||u(·, ·, 0)||C 1+α/2,2+α (Rd−1 ) ) b

T

(7.2.15) can be obtained arguing as in Chapter 6, first considering the case when A = Tr(QD2 ) for some constant and positive definite matrix Q and then freezing the coefficients to handle the more general case. Since there are no sensible differences with the proof of Theorem 6.2.1 we omit the details. Noting that each function u ∈ C 1+α/2,2+α (Rd+,T ) trivially solves the boundary value problem (7.2.1) with g = Dt u − Au and ψ = u(·, ·, 0), from (7.2.3) it follows that ||u||∞ ≤ |c0 |−1 ||Dt u − Au||∞ + ||u(·, ·, 0)||∞ . Replacing this estimate into (7.2.15), we immediately get (7.2.14).



By applying the continuity method, using Theorems 7.2.1 and 7.2.3 we can complete the proof of Theorem 7.2.1. Proof of Theorem 7.2.1: the general case For σ ∈ [0, 1], let Lσ be the linear 1+α/2,2+α operator defined by Lσ = (Dt u − Aσ u, u(·, ·, 0)) for every u ∈ X = Cb (Rd+,T ), where Aσ = σA + (1 − σ)(∆ − I). Clearly, each operator Lσ is bounded from X into α/2,α Y = Cb (Rd+,T ) × C 1+α/2,2+α (Rd−1 T ) (endowed with the classical norm, i.e., ||(g, ψ)||Y = ||g||C α/2,α (Rd ) + ||ψ||C 1+α/2,2+α (Rd−1 ) ). Let c0σ denote the potential term of the operator Aσ . b

+,T

T

Then, c0σ = σ − 1 + σc ≤ c0 ∨ (−1) for every σ ∈ [0, 1]. Hence, from (7.2.14) it follows that ||Lσ u||Y ≥ K0 ||u||X for a constant K0 independent of σ. Since the operator L0 is invertible by Theorem 7.2.1, applying Theorem 6.1.1 we conclude that also the operator L1 is invertible and we are done. 

Parabolic Equations in Rd+ with Dirichlet Boundary Conditions

190

7.3

Proof of Theorem 7.0.2 and a Corollary

Proof of Theorem 7.0.2 Throughout the proof, we denote by C a positive constant, independent of the functions that we consider, which may vary from line to line. Moreover, we use the notation P(f, g, ψ) to denote the Cauchy problem (7.0.1). Without loss of generality we can assume that c0 < 0. Indeed, suppose that c0 ≥ 0. Then, observe that the function u solves problem P(f, g, ψ) if and only if the function (t, x) 7→ v(t, x) = e−(c0 +1)t u(t, x) solves the Cauchy problem  0 t ∈ (0, T ], x ∈ Rd+ ,   Dt v(t, x) = A v(t, x) + gb(t, x), b x0 ), v(t, x0 , 0) = ψ(t, t ∈ (0, T ], x0 ∈ Rd−1 ,   v(0, x) = f (x), x ∈ Rd+ , where A0 = A − (c0 − 1)I has potential term which is not greater than −1 on Rd + , gb(t, x) = b x0 ) = e−(c0 +1)t ψ(t, x0 ) for every (t, x) ∈ e−(c0 +1)t g(t, x) for every (t, x) ∈ [0, T ]×Rd+ and ψ(t, d−1 [0, T ] × R . Moreover, ||b g ||C α/2,α ((0,T )×Rd ) ≤ C||g||C α/2,α ((0,T )×Rd ) , +

b

+

b

b 1+α/2,2+α ||ψ|| ≤ C||ψ||C 1+α/2,2+α ((0,T )×Rd−1 ) , C ((0,T )×Rd−1 ) b

b

||u||C 1+α/2,2+α ((0,T )×Rd ) ≤ C||v||C 1+α/2,2+α ((0,T )×Rd ) . +

b

+

b

Hence, if v satisfies (7.0.2), then u satisfies (7.0.2) as well, with possibly a different constant C. So, in the rest of the proof, we assume that c0 < 0. As a first step, we observe that 1+α/2,2+α u ∈ Cb ((0, T ) × Rd+ ) solves problem P(f, g, ψ) if and only if the function v1 = u − f solves the Cauchy problem P(0, gf , ψf ), where gf = g + Af and ψf = ψ − f (·, 0). The α/2,α function gf belongs to Cb ((0, T ) × Rd+ ) and ||gf ||C α/2,α ((0,T )×Rd ) ≤ ||g||C α/2,α ((0,T )×Rd ) + C||f ||C 2+α (Rd ) . b

+

b

+

b

+

Similarly, ψf belongs to C 1+α/2,2+α ((0, T ) × Rd−1 ) and ||ψf ||C 1+α/2,2+α ((0,T )×Rd−1 ) ≤ C(||ψ||C 1+α/2,2+α ((0,T )×Rd−1 ) + ||f ||C 2+α (Rd ) ). +

b

We extend gf (0, ·) to Rd by setting gf (0, x) = gf (0, x0 , −xd ) for every x0 ∈ Rd−1 and xd < 0. Since, by assumptions, Af (·, 0) + g(0, ·, 0) = 0, clearly gf (0, ·) belongs to Cbα (Rd ) and ||gf (0, ·)||Cbα (Rd ) ≤ 2||gf (0, ·)||Cbα (Rd+ ) ≤ C(||g||C α/2,α ((0,T )×Rd ) + ||f ||C 2+α (Rd ) ). b

b

+

1+α/2,2+α

By Theorem 6.2.6, there exists a unique function v2 ∈ Cb ((0, T ) × Rd ) which d vanishes on {0} × R and such that Dt v2 = Av2 + gf (0, ·) on (0, T ] × Rd . The function v2 also satisfies the estimate ||v2 ||C 1+α/2,2+α ((0,T )×Rd ) ≤ C||gf (0, ·)||Cbα (Rd ) ≤ C(||f ||C 2+α (Rd ) + ||g||C α/2,α ((0,T )×Rd ) ). b

b

+

b

+

(7.3.1)

Semigroups of Bounded Operators and Second-Order PDE’s

191

Note that v1 solves the Cauchy problem P(0, gf , ψf ) if and only if the function v3 = 1+α/2,2+α v1 − v2 ∈ Cb ((0, T ) × Rd+ ) solves the Cauchy problem  t ∈ (0, T ], x ∈ Rd+ ,   Dt v3 (t, x) = Av3 (t, x) + gf (t, x) − gf (0, x), 0 0 0 v3 (t, x , 0) = ψf (t, x ) − v2 (t, x , 0), t ∈ (0, T ], x0 ∈ Rd−1 , (7.3.2)   d v3 (0, x) = 0, x ∈ R+ . Since ψ(0, ·) ≡ f (·, 0) on Rd−1 and v2 vanishes on {0} × Rd+ , it follows that ψf (0, ·) − v2 (0, ·, 0) ≡ 0 on Rd−1 . Similarly, Dt ψf (0, ·)−Dt v2 (0, ·, 0) ≡ Dt ψ(0, ·)−Af (·, 0)−g(0, ·, 0) ≡ 0. Therefore, we can extend the function ψf − v2 (·, ·, 0) to RTd−1 by setting ψf (t, x0 ) − v2 (t, x0 , 0) = 0 for each t < 0 and x0 ∈ Rd−1 , obtaining a function which belongs to d C 1+α/2,2+α (Rd−1 T ). Similarly, we can extend gf −gf (0, ·) to R+,T in the trivial way obtaining α/2,α

a function which belongs to Cb (Rd+,T ). Hence, by Theorem 7.2.1, there exists a unique 1+α/2,2+α d function v3 ∈ C (R+,T ) which solves the Cauchy problem (7.3.2). Note that the condition v3 (0, ·) ≡ 0 is satisfied since v3 vanishes on Rd+,0 . Indeed, by Proposition 7.2.2 (with T = 0) Dt v3 = Av3 on Rd+,0 and v3 (t, x0 , 0) = 0 for every (t, x) ∈ R0d−1 . The quoted proposition also shows that ||v3 ||C 1+α/2,2+α ((0,T )×Rd ) ≤C(||ψf − v2 (·, ·, 0)||C 1+α/2,2+α ((0,T )×Rd−1 ) +

b

b

+ ||gf − gf (0, ·)||C α/2,α ((0,T )×Rd+ ) ) ≤C(||g||C α/2,α ((0,T )×Rd ) + ||f ||C 2+α (Rd ) + ||ψ||C 1+α/2,2+α ((0,T )×Rd−1 ) ). b

+

b

+

(7.3.3) We thus conclude that the function u = v2 + v3 + f , which belongs to C 1+α/2,2+α ((0, T ) × Rd+ ), is a solution (actually, the unique solution by Corollary 4.1.5) of the Cauchy problem P(f, g, ψ). Moreover, from (7.3.1) and (7.3.3), estimate (7.0.2) follows at once.  Applying (7.0.2) we obtain the following corollary. Corollary 7.3.1 There exists a positive constant C, which depends on α, d and, in a continuous way, on µ and the Cbα (Rd+ )-norm of the coefficients of the operator A, such that ||u||C 1+α/2,2+α ((0,T )×Rd ) ≤ C(||u(0, ·)||C 2+α (Rd+ ) + ||Dt u − Au||C α/2,α ((0,T )×Rd+ ) b

+

+ ||u||C 1+α/2,2+α ((0,T )×∂Rd+ ) ). Remark 7.3.2 The conditions ψ(0, ·) = f (·, 0) and Dt ψ = Af + g(0, ·, 0) on Rd−1 are also necessary for problem (7.0.1) to have a solution u ∈ C 1,2 ([0, T ] × Rd+ ). Indeed, since u(t, ·) = ψ on ∂Rd+ for each t ∈ [0, T ], then f = u(0, ·) should agree with ψ(0, ·) on ∂Rd+ . Moreover, Dt u(t, ·, 0) = Dt ψ(t, ·) on ∂Rd−1 for each t ∈ [0, T ]. This latter condition implies that Dt ψ(t, x0 ) = Au(t, x0 , 0) + g(t, x0 , 0) for each t ∈ [0, T ] and x0 ∈ Rd−1 . Taking t = 0 it follows that Dt ψ(0, ·) = Af (·, 0) + g(0, ·, 0) on Rd−1 . Remark 7.3.3 One could wonder whether, removing the assumption of α/2-H¨older regularity with respect to time on the data g and ψ, (i.e., assuming f ∈ Cb2+α (Rd+ ), g ∈ Cb0,α ([0, T ] × Rd+ ) and ψ ∈ Cb1,2 ([0, T ] × Rd−1 ) ∩ Cb0,2+α ([0, T ] × Rd−1 ) plus the compatibility conditions ψ(0, ·) = f (·, 0) and Dt ψ(0, ·) = Af (·, 0) + g(0, ·, 0) on Rd−1 ), the Cauchy

Parabolic Equations in Rd+ with Dirichlet Boundary Conditions

192

problem (7.0.1) admits a unique solution u ∈ Cb1,2 ([0, T ] × Rd+ ) ∩ Cb0,2+α ([0, T ] × Rd+ ). Unfortunately, this is not the case. Indeed, suppose that g ≡ 0. Writing Z t Dt u(r, x) dr, s, t ∈ [0, T ], x ∈ Rd+ , u(t, x) − u(s, x) = s

and, using the equality Dt u = Au to infer that Dt u ∈ Cb0,α ([0, T ] × Rd+ ), it can be easily proved that ||u(t, ·) − u(s, ·)||Cbα (Rd+ ) ≤ C||Dt u||C 0,α ([0,T ]×Rd ) |t2 − t1 | for every t1 , t2 ∈ [0, T ]. +

b

Therefore, using (1.1.6), we can estimate α

1− α

2 ||u(t, ·) − u(s, ·)||C 2 (Rd ) ≤ C||u(t, ·) − u(s, ·)||C2 α (Rd ) ||u(t, ·) − u(s, ·)||C 2+α (Rd ) b

+

b

+

b

+

and then conclude that the first- and second-order spatial derivatives of u actually belong α/2,α to Cb ((0, T ) × Rd+ ). (We refer the reader to Step 4 in the proof of Theorem 5.4.2 for further details.) This result implies that Au belongs to C α/2,0 ([0, T ] × Rd+ ). Since Dt u = Au on [0, T ] × Rd+ ), we conclude that Dt u ∈ C α/2,0 ([0, T ] × Rd+ ) and, as a byproduct, Dt ψ = Dt u(·, ·, 0) should belong to C α/2,0 ([0, T ] × Rd−1 ).

7.4

More on the Cauchy Problem (7.0.1)

In this section, we state two relevant theorems. The first one is an optimal spatial regularity result for solutions to problem (7.0.1) and the other one shows that the more the coefficients of the operator A and the data f , g and ψ are smooth, the more the classical solution u to problem (7.0.1) is itself smooth. As already remarked, the proof of such results are postponed to Chapter 14. Theorem 7.4.1 Let Hypotheses 7.0.1 be satisfied and fix θ ∈ (0, 1). Suppose that f belongs to Cb2+α−2θ (Rd+ ), satisfies, on ∂Rd+ the conditions f ≡ 0 and Af ≡ 0 (this latter condition if 2 + α − 2θ > 1) and g ∈ C((0, T ] × Rd+ ) is such that supt∈(0,T ] tθ ||g(t, ·)||Cbα (Rd+ ) < ∞ and g(t, ·) = 0 on ∂Rd+ for every t ∈ (0, T ]. Then, the Cauchy problem (7.0.1) admits a unique classical solution u, which is bounded in (0, T ) × Rd+ . In addition, u(t, ·) ∈ Cb2+α (Rd+ ) for every t ∈ (0, T ] and sup tθ (||u(t, ·)||C 2+α (Rd ) + ||Dt u(t, ·)||Cbα (Rd+ ) ) + ||u||C 0,2+α−2θ ([0,T ]×Rd ) b

t∈(0,T ]

+

+

b

  ≤C0 ||f ||C 2+α−2θ (Rd ) + sup tθ ||g(t, ·)||Cbα (Rd+ ) b

+

(7.4.1)

t∈(0,T ]

for some positive constant C0 , independent of u, f and g. In particular, if f and g belong to Cb2+α (Rd+ ) and C([0, T ]×Rd+ ), respectively, f ≡ Af ≡ g(t, ·) = 0 on ∂Rd+ for every t ∈ [0, T ] and supt∈[0,T ] ||g(t, ·)||Cbα (Rd+ ) < ∞, then, u belongs to Cb1,2 ([0, T ] × Rd+ ), u(t, ·) ∈ Cb2+α (Rd+ ) for every t ∈ [0, T ] and   sup ||u(t, ·)||C 2+α (Rd ) + sup ||Dt u(t, ·)||Cbα (Rd+ ) ≤ C1 ||f ||C 2+α (Rd ) + sup ||g(t, ·)||Cbα (Rd+ ) , t∈[0,T ]

b

+

t∈[0,T ]

b

+

t∈(0,T ]

(7.4.2) for some positive constant C1 , independent of u, f and g.

Semigroups of Bounded Operators and Second-Order PDE’s

193

Remark 7.4.2 Fix two functions f ∈ Cb2+α (Rd+ ) and g ∈ C([0, T ] × Rd+ ) such that supt∈[0,T ] ||g(t, ·)||Cbα (Rd+ ) < ∞. Even if the compatibility conditions f ≡ g(0, ·) + Af ≡ 0 on ∂Rd+ might seem more natural than the conditions f ≡ Af ≡ g(0, ·) ≡ 0 on ∂Rd+ , they are not enough to guarantee that the Cauchy problem (7.0.1) admits a solution u ∈ Cb1,2 ([0, T ] × Rd+ ) ∩ Cb0,2+α ([0, T ] × Rd+ ). Indeed, if these conditions were enough, then, for every triplet of functions f ∈ 2+α Cb (Rd+ ), g ∈ C 0,α ([0, T ] × Rd+ ), ψ ∈ Cb0,2+α ([0, T ] × Rd+ ), such that Dt ψ ∈ Cb0,α ([0, T ] × Rd+ ), satisfying the compatibility conditions f (·, 0) = ψ(0, ·) and Dt ψ(0, ·) = Af (·, 0) + g(0, ·, 0) on Rd−1 , the Cauchy problem  t ∈ (0, T ], x ∈ Rd+ ,   Dt u(t, x) = Au(t, x) + g(t, x), 0 0 u(t, x , 0) = ψ(t, x ), t ∈ (0, T ], x0 ∈ Rd−1 , (7.4.3)   d u(0, x) = f (x), x ∈ R+ , would admit a (unique) solution u ∈ Cb1,2 ([0, T ] × Rd+ ) ∩ Cb0,α ([0, T ] × Rd+ ). Such a solution u can be seen as the sum of ψ and the function v ∈ Cb1,2 ([0, T ]×Rd+ )∩Cb0,α ([0, T ]×Rd+ ), solution of the Cauchy problem (7.4.3) with f and g being replaced with the functions fe = f −ψ(0, ·) and ge = g +Aψ −Dt ψ. Note that such functions belong to Cb2+α (Rd+ ) and Cb0,α ([0, T ]×Rd+ ), respectively, and satisfy the compatibility conditions fe(x0 , 0) = Afe(x0 , 0) + ge(0, x0 , 0) = 0 for every x0 ∈ Rd−1 . Remark 7.3.3 leads us to a contradiction. Theorem 7.4.3 Fix k ∈ N and let Hypotheses 7.0.1 be satisfied with the coefficients of the (k+α)/2,k+α ((0, T )× operator A in Cbk+α (Rd+ ). Further, suppose that f ∈ Cb2+k+α (Rd+ ), g ∈ Cb (k+2+α)/2,k+2+α d−1 d ((0, T ) × R ) satisfy the compatibility conditions R+ ), ψ ∈ Cb  k  X    Dtk ψ(0, x0 ) = (Ak f )(x0 , 0) + (Ak−j Dtj g(0, ·))(x0 , 0), if k is even,   j=0 (7.4.4) k−1 X   k−1 0 k−1 0 k−j−1 j 0   D ψ(0, x ) = (A f )(x , 0) + (A Dt g(0, ·))(x , 0), if k is odd,   t j=0

1+α/2,2+α

for every x0 ∈ Rd−1 . Then, the solution u ∈ Cb ((0, T ) × Rd+ ) to problem (7.0.1) (k+2+α)/2,k+2+α d actually belongs to Cb ((0, T ) × R+ ) and there exists a positive constant C, independent of u, f , g and ψ, such that ||u||C (k+2+α)/2,k+2+α ((0,T )×Rd ) +

b

≤C(||f ||C k+2+α (Rd ) + ||g||C (k+α)/2,k+α ((0,T )×Rd ) + ||ψ||C (k+2+α)/2,k+2+α ((0,T )×Rd−1 ) ). b

7.5

+

b

+

(7.4.5)

b

The Associated Semigroup

In this subsection, we prove that the Cauchy problem  t ∈ (0, ∞),   Dt u(t, x) = Au(t, x), 0 u(t, x , 0) = 0, t ∈ (0, ∞),   u(0, x) = f (x),

x ∈ Rd+ , x0 ∈ Rd−1 , x ∈ Rd+ ,

(7.5.1)

194

Parabolic Equations in Rd+ with Dirichlet Boundary Conditions

is governed in Cb (Rd+ ) by a semigroup of bounded operators. Note that, if f ∈ Cb (Rd+ ), then f is not defined on ∂Rd+ (and even if it is defined on Rd+ in general it does not identically vanish on ∂Rd+ ). Thus, we cannot expect that a solution u to (7.5.1) is continuous on {0} × ∂Rd+ . Hence, we need to slightly modify the definition of classical solution. Definition 7.5.1 A classical solution to problem (7.5.1) is a function u ∈ C([0, ∞)×Rd+ )∩ C((0, ∞) × Rd+ ) ∩ C 1,2 ((0, ∞) × Rd+ ) which solves the equation Dt u = Au on (0, ∞) × Rd+ and satisfies the boundary and initial conditions in (7.5.1). Theorem 7.5.2 Let u ∈ C 1,2 ((0, T ) × Rd+ ) ∩ C((0, T ) × Rd+ ) be a classical solution to α/2,α the differential equation Dt u = Au + g, corresponding to g ∈ Cloc ((0, T ) × Rd+ ), such 1+α/2,2+α that u(t, ·) = 0 on ∂Rd+ for every t ∈ (0, T ). Then, u ∈ Cloc ((0, T ) × Rd+ ) and for every 0 < τ0 < τ1 < T0 < T and every 0 < r1 < r2 there exists a positive constant C = C(µ, τ0 , τ1 , T0 , r1 , r2 ) such that ||u||C 1+α/2,2+α ((τ1 ,T0 )×B+ (0,r1 )) ≤ C(||u||C((τ0 ,T0 )×B+ (0,r2 )) + ||g||C α/2,α ((τ0 ,T0 )×B+ (0,r2 )) ), where B+ (0, r) = B(0, r) ∩ Rd+ . The proof of this theorem is deferred to Chapter 8 (see Theorem 8.3.2). Using Theorem 7.5.2, we can prove the following result. Theorem 7.5.3 For each f ∈ Cb (Rd+ ) there exists a unique bounded classical solution u of the Cauchy problem (7.5.1). Further, ||u(t, ·)||Cb (Rd+ ) ≤ ec0 t ||f ||Cb (Rd+ ) , In particular, if A = ∆, then Z u(t, x) = k(t, x, y)f (y) dy, Rd +

t > 0.

(7.5.2)

t ∈ (0, ∞), x ∈ Rd+ ,

(7.5.3)

where k(t, x, y) =

1

 d

(4πt) 2

e−

|x−y|2 4t

− e−

|x0 −y 0 |2 +|xd +yd |2 4t



,

t > 0, x, y ∈ Rd+ .

Proof The uniqueness part and estimate (9.4.3) follow from Corollary 4.1.5 (see the proof of Corollary 4.1.5), so we concentrate on the existence part. We fix f ∈ BU C(Rd+ ) and a sequence (gn ) ⊂ Cb2+α (Rd+ ) converging to f uniformly on Rd+ and such that supn∈N ||gn ||∞ < ∞. Consider a nondecreasing function θ ∈ C ∞ ([0, ∞)) such that 0 ≤ θ ≤ 1 in [0, ∞), θ(s) = 0 for s ∈ [0, 1] and θ(s) = 1 for s ≥ 2, and set fn (x) = θ(nxd )gn (x) for every x ∈ Rd+ and n ∈ N. Then the sequence (fn ) ⊂ Cb2+α (Rd+ ) converges to f locally uniformly on Rd+ , supn∈N ||fn ||∞ < ∞ and fn ≡ Afn ≡ 0 on ∂Rd+ for every n ∈ N. By Theorem 7.0.2, for every n ∈ N, the Cauchy problem (7.5.1), with f being replaced by fn , admits a unique classical solution, un which, in addition, belongs to 1+α/2,2+α Cb ((0, T ) × Rd+ ). Moreover, by Theorem 7.5.2, for every 0 < τ0 < T0 and r > 0 we can determine a positive constant C, independent of n such that ||un ||C 1+α/2,2+α ((τ0 ,T0 )×B+ (0,r)) ≤ C sup ||fn ||∞ . n∈N

Semigroups of Bounded Operators and Second-Order PDE’s

195

By Arzel` a-Ascoli theorem and a compactness argument, we can show that there ex1+α/2,2+α ists a subsequence (unk ) converging to some function u ∈ Cloc ((0, ∞) × Rd+ ) in 1,2 C ([τ0 , T0 ] × B+ (0, r)) for every τ0 , T0 and r as above. In particular, u is bounded in each strip (0, T0 ) × Rd+ and, clearly, it vanishes on (0, ∞) × ∂Rd+ and Dt u = Au on (0, ∞) × Rd+ . To prove that u solves the Cauchy problem (7.5.1), we just need to show that it can be extended by continuity on {0}×Rd+ setting u(0, ·) = f . This is an easy task if f ∈ BU C(Rd+ ) and vanishes on ∂Rd+ . Indeed, in this case, we can also assume that the sequence (fn ) converges to f uniformly on Rd+ . By the estimate ||un −um ||C ([0,T ]×Rd ) ≤ CT ||fn −fm ||∞ , which b

+

holds true for every m, n ∈ N and some positive constant CT , independent of m and n (see Corollary 4.1.5), we easily infer that un converges to u uniformly on Rd+ and, consequently, u can be extended by continuity on {0} × Rd+ by setting u(0, ·) = f . For a general function f ∈ Cb (Rd+ ), we adapt the argument used in the proof of Proposition 6.4.1. Given M > 0, we consider a function ϑM ∈ Cc∞ (Rd ) such that χB(0,M ) ≤ ϑM ≤ χB(0,2M ) . By the uniqueness of the solution to problem (7.5.1), we can (1)

(2)

(1)

(2)

split unk = uk + uk for every k ∈ N, where uk and uk are the classical solutions to the Cauchy problem (7.5.1) with f = ϑM fnk and f = (1 − ϑM )fnk , respectively. The above (1) arguments show that uk converges uniformly to the unique classical solution uϑM f of the Cauchy problem (7.5.1) with f being replaced by f ϑM . (2) As far as the function uk is concerned, we claim that (2)

|uk (t, ·)| ≤ sup ||fnk ||∞ (ec0 t − uϑM (t, ·)),

t ≥ 0, k ∈ N,

k∈N

where uϑM is the classical solution to the Cauchy problem (7.5.1) subject to the initial condition uϑM (0, ·) = ϑM . Indeed, fix k ∈ N. Then the function ζk± , defined by ζk± (t, x) = (2) e−c0 t (±uk (t, x) − supn∈N ||fn ||∞ (ec0 t − uϑM (t, x))) for every (t, x) ∈ [0, ∞) × Rd+ , satisfies the differential inequality Dt ζk ≤ (A − c0 )ζk in (0, ∞) × Rd+ . Moreover, it is nonpositive on {0} × Rd+ and on (0, ∞) × ∂Rd+ . Hence, by the maximum principle, ζk± is nonpositive in [0, ∞) × Rd+ , and the claim follows in this case. From the claim we conclude that (1)

|unk (t, ·) − f | ≤ |uk − f | + sup ||fn ||∞ (ec0 t − uϑM (t, ·)),

t > 0.

(7.5.4)

n∈N

Hence, letting k tend to ∞ in both the sides of (7.5.4), we obtain that |u(t, x) − f (x)| ≤ |uϑM f (t, x) − f (x)| + sup ||fn ||∞ [ec0 t − uϑM (t, x)]

(7.5.5)

n∈N

for every t > 0 and x ∈ Rd+ . Fix x0 ∈ B+ (0, M ). Letting (t, x) tend to (0, x0 ) in both sides of (7.5.5), we conclude that u(t, x) tends to f (x0 ). By the arbitrariness of M , u can be extended by continuity to {0} × Rd+ , by setting u(0, x) = f (x) for x ∈ Rd . Function u is the solution to problem (7.5.1) that we were looking for. In particular, since every subsequence of (un ) admits a subsequence which converges to a solution of the Cauchy problem (7.5.1), we infer that the sequence (un ) itself converges to u in C 1,2 ([τ0 , T0 ] × B+ (0, r)) for every τ0 , T0 , r > 0 with τ0 < T0 . To complete the proof, we consider the particular case when A = ∆. If f ∈ Cb (Rd+ ) vanishes on ∂Rd+ , then the odd extension fo of f with respect to the last variable belongs to Cb (Rd ) and ||fo ||Cb (Rd ) = ||f ||C (Rd ) . Hence, we can apply Theorem 5.4.2 and conclude that b

+

the function v = T (·)fo , where {T (t)} denotes the Gauss-Weierstrass semigroup, belongs to Cb ([0, ∞) × Rd ) ∩ C 1,2 ((0, ∞) × Rd ), solves the equation Dt v = ∆v and v(0, ·) = fo .

196

Parabolic Equations in Rd+ with Dirichlet Boundary Conditions

Function v can be represented by Z Z √ |y|2 v(t, x) = e− 4 fo (x + ty) dy =

k(t, x, y)f (y) dy

Rd +

Rd

√ 2 for every t ∈ [0, ∞) and x ∈ Rd . Since the function y 7→ e−|y| /4 fo (x0 + ty 0 , yd ) is odd with respect to the last variable for every x0 ∈ Rd−1 , it follows immediately that v vanishes on (0, ∞) × ∂Rd+ . Thus, v is the classical solution to problem (7.5.1). For a general f ∈ Cb (Rd+ ), we fix a bounded (with respect to the sup-norm) sequence (fn ) ∈ Cb2+α (Rd+ ) converging to f locally uniformly on Rd+ and such that un vanishes on [0, ∞) × ∂Rd+ . From the first part of the proof the sequence (un ) converges to u pointwise on (0, ∞) × Rd . Since Z un (t, x) = k(t, x, y)fn (y) dy, (t, x) ∈ (0, ∞) × Rd+ , n ∈ N, Rd +

applying the dominated convergence theorem, formula (7.5.3) follows at once.



As a byproduct of Theorem 7.5.3, we deduce the following corollary. Corollary 7.5.4 There exists a semigroup {T (t)} of bounded operators associated with the realization of operator A with homogeneous Dirichlet boundary conditions in Cb (Rd+ ). Moreover, if (fn ) ⊂ Cb (Rd+ ) is a sequence which converges to a function f ∈ Cb (Rd ), locally uniformly on Rd+ , then T (t)fn converges to T (t)f locally uniformly on Rd+ , as n tends to ∞, for every t > 0. Proof The semigroup can be defined as follows: for every t > 0 and f ∈ Cb (Rd+ ), T (·)f is the unique classical solution to problem (7.5.1). The semigroup property follows from the uniqueness of the classical solution to such a problem. To prove the last assertion of the corollary, we can argue as in the proof of Theorem 7.5.3. Suppose that (fn ) ∈ Cb (Rd+ ) is a bounded sequence converging to a function f ∈ Cb (Rd+ ), locally uniformly on Rd+ . For every n ∈ N, set un = T (·)fn and K = supn∈N ||fn ||∞ . Since each function un satisfies the assumptions of Theorem 7.5.2, the sequence (un ) is bounded in C 1+α/2,2+α ((τ0 , T0 ) × B+ (0, r)) for every 0 < τ0 < T0 and r > 0. Hence, it 1+α/2,2+α admits a subsequence (unk ) converging to some function u ∈ Cloc ((0, ∞) × Rd+ ) in C 1,2 ([τ0 , T0 ] × B+ (0, r)) for every τ0 , T0 and r as above. Due, to Corollary 4.1.5, u is bounded in each strip (0, T0 )×Rd+ . Moreover, it vanishes on (0, ∞)×∂Rd+ and Dt u = Au on (0, ∞) × Rd+ . To show that u = T (·)f , we fix M > 0 and consider a function ϑM ∈ Cc∞ (Rd ) such that χB(0,M ) ≤ ϑM ≤ χB(0,2M ) . We split unk = T (·)(ϑM fnk ) + T (·)((1 − ϑM )fnk ) for every k ∈ N. Since ϑM fnk converges uniformly in Rd+ to ϑM f , T (·)(ϑM fnk ) converges to T (·)(ϑM f ) uniformly on (0, T ) × Rd+ for every T > 0. On the other hand, as in the proof of Theorem 7.5.3, by the maximum principle we see that |T (t)((1 − ϑM )fnk )| ≤ + K(ec0 t − T (t)ϑM ) for every t ≥ 0 and k ∈ N. Summing up, we have proved that +

|T (t)fnk − f | ≤ |T (t)(ϑM fnk ) − f | + K(ec0 t − T (t)ϑM ),

t > 0.

Letting k tend to ∞ in both the sides of the previous estimate, we get +

|u(t, x) − f (x)| ≤ |(T (t)(ϑM f ))(x) − f (x)| + K[ec0 t − (T (t)ϑM )(x)],

τ > 0, x ∈ Rd+ .

From this formula, it follows that u(t, x) tends to f (x0 ) as (t, x) tends to (0, x0 ), for every x0 ∈ B+ (0, M ). By the arbitrariness of M , we deduce that u = T (·)f . Finally, since every subsequence of (T (t)fn ) admits a subsequence which converges to T (t)f , the sequence (T (t)fn ) itself converges to T (t)f locally uniformly in Rd+ for every t > 0. 

Semigroups of Bounded Operators and Second-Order PDE’s

7.6

197

Notes

For the proof of the main theorem 7.0.2 the most ideas are taken from [21, Chapters 5 and 9].

7.7

Exercises 1+α/2,2+α

1. Prove that if u ∈ Cb a, b ∈ R with a < b, then Z u(t, x) =

(Rd+1 ) has support contained in [a, b] × Rd for some

t

(T (t − s)(Dt u(s, ·) − ∆u(s, ·))(x) ds,

−∞

where {T (t)} is the Gauss-Weierstrass semigroup. 2. Let u ∈ C 3 (C1 (0, 0)) ∩ C(C1 (0, 0)) solve the homogeneous heat equation. Further, suppose that Dj u belongs to C((−1, 0] × B(0, 1)) for every j = 1, . . . , d and let ψ ∈ Cc∞ ((−1, 1) × B(0, 1)) satisfy the condition ψ(0, 0) = 1. (i) Prove that the function vγ = γu2 +ψ|∇x u|2 solves the equation Dt vγ = ∆vγ +Ψγ on C1 (0, 0), where Ψγ = 2(γ +ψ∆ψ+|∇x ψ|2 −ψDt ψ)|∇x u|2 + 2ψ 2 |Dx2 u|2 + 8ψ

d X

2 Dij uDi ψDj u.

i,j=1

(ii) Prove that Ψγ ≥ (2γ + 2ψ∆ψ + 2|∇x ψ|2 − 8|∇x ψ|2 − 2ψDt ψ)|∇x u|2 and deduce that γ can be fixed large enough such that Ψγ ≥ 0 on C1 (0, 0). (iii) Show that there exists a positive constant β0 , independent of u, such that |∇x u(0, 0)| ≤ β0 ||u||C(C1 (0,0)) . 3.

(i) Prove that, if u is as in the previous exercise with C1 (0, 0) being replaced by Cr (0, 0), then |∇x u(0, 0)| ≤ κr−1 ||u||C(Cr (0,0)) for some positive constant κ. (ii) By properly applying the above result to the function Dj u (j = 1, . . . , d) instead of u, prove that, if u ∈ C 4 (Cr (0, 0)) with spatial derivatives up to the second-order which are continuous on (r2 , 0] × B(0, r), then |Dij u(0, 0)| ≤ β1 r−2 ||u||C(Cr (0,0)) for every i, j = 1, . . . d and some positive constant β1 , independent of u. (iii) More generally, prove that if u ∈ C k (Cr (0, 0)) has spatial derivatives up to the (k − 2)-th order which are continuous on (r2 , 0] × Br , then the value at the origin of each (k − 2)-th order spatial derivatives can be bounded in modulus by βh r−h ||u||C(Cr (0,0)) for some positive constant βh , independent of u.

4. Complete the proof of Theorem 7.1.1 (see Step 5). 5. Prove estimates (7.1.8) and (7.1.9).

198

Parabolic Equations in Rd+ with Dirichlet Boundary Conditions

1+α/2,2+α 6. Prove that, if ψb belongs to Cb ((−∞, T ] × Rd−1 ), then Di ψb belongs to α/2,α b 1+α/2,2+α b α/2,α ≤ C||ψ|| Cb ((−∞, T )×Rd−1 ) and ||Di ψ|| ((−∞,T ]×Rd−1 ) ((−∞,T )×Rd−1 ) Cb Cb b for each i = 1, . . . , d − 1 and a positive constant C, independent of ψ.

g ||C α/2,α (Rd 7. Prove the estimate ||wn ||C 1+α/2,2+α (Rd+1 ) ≤ C1 ||e b

b

+,T )

in the proof of Theorem

7.2.1 with a constant C1 independent of n and ge. 8. Prove that the function Did v (i = 1, . . . , d) in the proof of Theorem 7.2.1 belongs to α/2,α Cb ((−∞, T ) × Rd−1 × (ε, ∞)) for every ε > 0. 9. Prove that Z

∞

e 0

−s

Z

Dd K+ (s, y 0 , xd ) dy 0 ≤

Rd−1

[Hint: recall that Dd K+ = Ddd K = Dt K − the heat kernel.] 10. Prove Theorem 7.2.3.

Pd−1 j=1

1 . 2

Djj K on (0, ∞) × Rd , where K is

Chapter 8

R

Parabolic Equations in d+ with More General Boundary Conditions

In this chapter, we consider the equation Dt u = Au + g on (0, T ] × Rd+ , subject to more general boundary conditions. More precisely, we deal with the problem  Dt u(t, x) = Au(t, x) + g(t, x), t ∈ (0, T ], x ∈ Rd+ ,    ∂u (8.0.1) a(x0 )u(t, x0 , 0) + (t, x0 , 0) = ψ(t, x0 ), t ∈ (0, T ], x0 ∈ Rd−1 ,  ∂η   u(0, x) = f (x), x ∈ Rd+ , under the following conditions on the coefficients of the operator A = Tr(QD2 ) + hb, ∇i + c, on a and η. Hypotheses 8.0.1 (i) The coefficients qij = qji , bj (i, j = 1, . . . , d) and c are bounded and α-H¨ older continuous in Rd+ for some α ∈ (0, 1); (ii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for all x ∈ Rd+ and ξ ∈ Rd . (iii) the functions a and η belong to Cb1+α (Rd−1 ) and Cb1+α (Rd−1 , Rd ), respectively. Moreover, |η(y)| = 1 for each y ∈ Rd−1 and η0 := supRd−1 ηd < 0, where, as usually, ηd denotes the last component of the function η. To simplify the notation, in what follows we will find it convenient to denote by B the ∂ differential boundary operator aI + . ∂η The main result of this section is the following counterpart of Theorem 7.0.2. Theorem 8.0.2 Let Hypotheses 8.0.1 be satisfied. Then, for every f ∈ Cb2+α (Rd+ ), g ∈ α/2,α Cb ((0, T ) × Rd+ ) and ψ ∈ C (1+α)/2,1+α ((0, T ) × Rd−1 ) such that Bf = ψ(0, ·) on Rd−1 , there exists a unique solution u ∈ Cb0,1 ((0, T ] × Rd+ ) ∩ C([0, T ] × Rd+ ) ∩ C 1,2 ((0, T ] × Rd+ ) 1+α/2,2+α to problem1 (8.0.1). In addition, u belongs to Cb ((0, T ) × Rd+ ) and there exists a positive constant C, independent of f , g and ψ, such that ||u||C 1+α/2,2+α ((0,T )×Rd ) +

b

≤C(||f ||C 2+α (Rd ) + ||g||C α/2,α ((0,T )×Rd ) + ||ψ||C (1+α)/2,1+α ((0,T )×Rd−1 ) ). b

+

b

+

(8.0.2)

b

As a consequence of this theorem, we prove the last type of local Schauder estimates, up to a portion of the boundary, for solutions to parabolic equations on ∂Rd+ , which satisfy zero- or first-order boundary conditions. We also prove some other regularity results for the 1 We call bounded classical solution to problem (8.0.1) a solution which satisfies the previous smoothness properties.

199

Parabolic Equations in Rd+ with More General Boundary Conditions

200

solution to the above Cauchy problem, in the spirit of the results proved in Sections 6.3 and 6.5. The chapter is split into sections as follows. In Section 8.1, we prove an a priori estimate 1+α/2,2+α for functions u ∈ Cb ((0, T ) × Rd+ ). Such estimate allows to bound from above the 1+α/2,2+α d Cb ((0, T ) × R+ )-norm of a function u in terms of the Cb2+α (Rd+ )-norm of u(0, ·), α/2,α (1+α)/2,1+α the Cb ((0, T ) × Rd+ )-norm of the function Dt u − Au and the Cb ((0, T ) × Rd+ )norm of the function Bu. From this estimate, (8.0.2) follows at once. To prove the a priori estimate, we use the same idea as in Chapter 6, considering first the particular case when A = ∆ and the coefficients of the operator B are constant. Then, freezing the coefficients of the operators A and B around a point x0 ∈ Rd+ and localizing the problem in a ball centered at x0 , we cover the general case. Next, in Section 8.2, we prove Theorem 8.0.2, first solving the Neumann Cauchy problem for the Laplacian, which is easier to analyze, since we can determine an explicit formula for its solution, and next using the method of continuity together with the a priori estimate of Section 8.1. Then, in Section 8.3, we prove Schauder estimates up to a portion of ∂Ω for solutions to parabolic equations associated with a second-order uniformly elliptic operators on Rd+ , which identically vanish on ∂Rd+ or satisfy a first-order condition on the ∂Rd+ . Such estimates are of two types depending on the fact that u is smooth or not at t = 0. Section 8.4 is devoted to prove further regularity results for the classical solution to the Cauchy problem (8.0.1). Finally, in Section 8.5, under Hypotheses 8.0.1 we introduce a semigroup of bounded operators in Cb (Rd+ ), associated with the pair (A, B). The restriction of such a semigroup to BU C(Rd+ ) is strongly continuous. In the construction of the semigroup, which is obtained by an approximation procedure, a crucial role is played by the estimates of Section 8.3. β/2,β Notation. Throughout the chapter, we set Zβ = {ψ ∈ Cb ([0, T ] × Rd−1 ) : ψ(0, ·) ≡ β/2,β 0}, if β < 2 and Zβ = {ψ ∈ Cb ([0, T ] × Rd−1 ) : ψ(0, ·) ≡ Dt ψ(0, ·) ≡ 0} otherwise. It is β/2,β a Banach space when endowed with the norm of Cb ([0, T ] × Rd−1 ).

8.1

A Priori Estimates

The first main step in the proof of Theorem 8.0.2 is represented by the a priori estimate in the next proposition. Proposition 8.1.1 There exists a positive constant   b = C d, α, µ, max ||qij ||α , max ||bj ||α , ||c||α , ||a||1+α , ||η||1+α , |η0 | C i,j=1...,d

j=1,...,d

such that2 b ||u||C 1+α/2,2+α ((0,T )×Rd ) ≤C(||u(0, ·)||C 2+α (Rd ) + ||Dt u − Au||C α/2,α ((0,T )×Rd ) b

+

b

+

+ ||Bu||C (1+α)/2,1+α ((0,T )×Rd−1 ) )

b

+

(8.1.1)

b

1+α/2,2+α

for every function u ∈ Cb is continuous on (0, ∞)6 . 2 Here,

((0, T )×Rd+ ). Moreover, the function C(d, α, µ, ·, ·, ·, ·, ·, ·)

|| · ||α = || · ||C α (Rd ) , || · ||1+α = || · ||C 1+α (Rd−1 ) . b

+

b

Semigroups of Bounded Operators and Second-Order PDE’s

201

To prove Proposition 8.1.1, we need two preliminary lemmas. Lemma 8.1.2 Let Q be a symmetric matrix. Then, there exists an invertible matrix P , which leaves Rd+ and ∂Rd+ invariant, such that P QP T is the identity matrix. Moreover, −1/2 −1/2 P T ed = Ked , where λmax ≤ K ≤ λmin and λmin , λmax denote the minimum and maximum eigenvalues of the matrix Q. Proof Denote by λ1 , . . . , λd the eigenvalues of Q, ordered from the smallest to the largest one. Further, let P1 be an orthogonal matrix such that P1 QP1T = diag(λ1 , . . . , λd ) −1/2 −1/2 and set P2 = diag(λ1 , . . . , λd ). Then, clearly, P2 P1 QP1T P2T = I. Finally, let ξ = −1 T −1 P2 (P1 ) ed and let P3 be a rotation such that P3T transforms ed into ||ξ||−1 ξ. Set P = P3 P2 P1 : this is the matrix we were looking for. Indeed, clearly, P T ed = P1T P2 P3T ed = Pd p Ked , where K = ||P2−1 (P1T )−1 ed ||−1 and P2−1 (P1T )−1 ed = j=1 λj ((P1T )−1 ed )j , where √ T −1 ((P the j-th component of the vector (P1T )−1 ed , so that λ1 ≤ K −1 ≤ √ 1 ) ed )j denotes λd , since (P1T )−1 is an isometry. Next, we observe hP x, ed i = hx, P T ed i = Khx, ed i for every x ∈ Rd and, from this chain of equalities, it follows immediately that P leaves Rd+  and ∂Rd+ invariant. This completes the proof. Lemma 8.1.3 For every ψ ∈ Z1+α , the Cauchy problem  d   Dt u(t, x) = ∆u(t, x), t ∈ (0, T ), x ∈ R+ , u(t, x0 , 0) = ψ(t, x0 ), t ∈ [0, T ], x0 ∈ Rd−1 ,   u(0, x) = 0, x ∈ Rd+ .

(8.1.2)

(1+α)/2,1+α

1+α/2,2+α

((0, T )×Rd+ ). Moreover, admits a unique solution u ∈ Cloc ((0, T )×Rd+ )∩Cb for every β ∈ [0, α], there exists a positive constant Cβ , independent of ψ and u, such that ||u||C (1+β)/2,1+β ([0,T ]×Rd ) ≤ Cβ ||ψ||C (1+β)/2,1+β ([0,T ]×Rd−1 ) .

(8.1.3)

+

b

1+α/2,2+α

Proof The uniqueness of the solution in Cloc ((0, T )×Rd+ )∩C (1+α)/2,1+α ((0, T )× d R+ ) of the Cauchy problem (8.1.2) follows from the maximum principle in Theorem 4.1.1. We split the rest of the proof into three steps. Step 1. Here, we prove that estimate (8.1.3) holds true if ψ ∈ Z2+α . In such a case, by 1+α/2,2+α Theorem 7.2.1 the Cauchy problem (8.1.2) admits a unique solution u ∈ Cb ((0, T )× d R+ ). Going through the proof of that theorem in the case when A = ∆ − I, one can easily realize that Z t Z u(t, x0 , xd ) = ds K+ (t − s, x0 − y 0 , xd )ψ(s, y 0 ) dy 0 . (8.1.4) 0

Rd−1

Indeed, if we extend ψ in the trivial way to (−∞, T ] × Rd−1 , then we obtain a 1+α/2,2+α function ψb which belongs to Cb ((−∞, T ) × Rd−1 ) and satisfies the condition b ||ψ||C 1+α/2,2+α ((−∞,T )×Rd−1 ) = ||ψ||C 1+α/2,2+α ([0,T ]×Rd−1 ) . By the quoted theorem, the Cauchy b b problem ( Dt u(t, x) = ∆u(t, x), t ∈ (−∞, T ], x ∈ Rd+ , b x0 ), u(t, x0 , 0) = ψ(t, t ∈ (−∞, T ], x0 ∈ Rd−1 , admits a unique solution u ∈ C 1+α/2,2+α (Rd+,T ), given by (8.1.4) for each t ∈ (−∞, T ] and x ∈ Rd+ , which in addition satisfies the estimate ||u||C 1+α/2,2+α (Rd+,T ) ≤ C∗ ||ψ||C 1+α/2,2+α ([0,T ]×Rd−1 )

Parabolic Equations in Rd+ with More General Boundary Conditions

202

for some constant C∗ , independent of u and ψ. Such a theorem also shows that u vanishes b ·) = 0 if t ≤ 0. Therefore, u solves the Cauchy problem (8.1.2). on (−∞, 0] × Rd+ since ψ(t, To prove that u satisfies estimate (8.1.3), we observe that the right-hand side of (8.1.4) extends with a bounded operator Γ from Cb ((−∞, T ]×Rd−1 ) into Cb (Rd+,T ), as a straightforward computation reveals. Thus, thanks to Proposition A.5.15 we deduce that Γ is bounded (1+β)/2,1+β (1+β)/2,1+β from Cb ((−∞, T ] × Rd−1 ) into Cb (Rd+,T ) for every β ∈ [0, α] and this yields estimate (8.1.3). Step 2. Now, we show that each function in Z1+α is the uniform limit of a sequence in (1+α)/2,1+α Z2+α , which is bounded in Z1+α . For this purpose, we fix ψ ∈ Cb ((0, T ) × Rd−1 ) and extend it to Rd with the function ψ∗ , defined as follows:  (t, x) ∈ (−∞, 0) × Rd−1 ,   0, ψ(t, x), (t, x) ∈ [0, T ] × Rd−1 , ψ∗ (t, x) =   ψ(T, x), (t, x) ∈ (T, ∞) × Rd−1 . (1+α)/2,1+α

Clearly, ψ∗ ∈ Cb

(Rd ) and there exists a positive constant C∗,β such that

||ψ∗ ||C (1+β)/2,1+β (Rd ) ≤ C∗,β ||ψ||C (1+β)/2,1+β ([0,T ]×Rd−1 ) b

b

for every β ∈ [0, α]. Then, we consider a nonnegative function φ ∈ C ∞ (R) supported on [0, 1], with L1 (R)-norm equal to one, a nonnegative function ϑ ∈ Cc∞ (Rd ), supported on B(0, 1), with L1 (Rd−1 )-norm equal to one, and, for every n ∈ N, we set Z ψn (t, x) = φ(s)ϑ(y)ψ∗ (t − n−1 s, x − n−1 y) ds dy, t ∈ R, x ∈ Rd−1 . Rd 1+α/2,2+α

Clearly, each function ψn belongs to Cb

(Rd ), vanishes on (−∞, 0] × Rd−1 and

||ψn ||C (1+β)/2,1+β (Rd ) ≤ ||ψ∗ ||C (1+β)/2,1+β (Rd ) ≤ C∗,β ||ψ||C (1+β)/2,1+β ([0,T ]×Rd−1 ) b

b

b

for every β ∈ [0, α]. Moreover, the sequence (ψn ) converges to ψ∗ uniformly on Rd as n tends to ∞. The restriction of the sequence (ψn ) to [0, T ] × Rd−1 is the sequence of elements of Z2+α we were looking for. Step 3. Now, we can complete the proof. We fix ψ ∈ Z1+α and consider the sequence (ψn ) ⊂ Z2+α , defined in Step 2. By Step 1, for each n ∈ N, the Cauchy problem (8.1.2) 1+α/2,2+α admits a unique solution un ∈ Cb ((0, T ) × Rd+ ) and it satisfies the estimates ||un ||∞ ≤ C0 ||ψn ||∞ ≤ C0 sup ||ψn ||∞ ,

(8.1.5)

n∈N

||un − um ||∞ ≤ Cβ ||ψn − ψm ||∞ ,

(8.1.6)

||un ||C (1+β)/2,1+β ([0,T ]×Rd ) ≤ Cβ C∗,β ||ψ||C (1+β)/2,1+β ([0,T ]×Rd−1 ) +

b

(8.1.7)

b

for every m, n ∈ N and β ∈ [0, α]. Inequality (8.1.6) implies that the sequence (un ) converges uniformly on [0, T ] × Rd+ to a function u ∈ Cb ([0, T ] × Rd+ ). On the other hand, estimate (8.1.7) shows that the restrictions to [0, T ] × B(0, r) of the sequences (un ) and (Dj un ) (j = 1, . . . , n) are equibounded and equicontinuous. Thus, the Arzel`a-Ascoli theorem implies that, up to a subsequence, un and Dj un (j = 1, . . . , n) converge to some functions v and (1+β)/2,β Dj v, which belong to Cb ([0, T ] × B(0, r)). Moreover, ||v||C (1+β)/2,β ([0,T ]×B(0,r)) + ||Dj v||C (1+β)/2,β ([0,T ]×B(0,r)) ≤ CC∗,β ||ψ||C (1+β)/2,1+β ([0,T ]×Rd−1 ) b

b

b

(8.1.8)

Semigroups of Bounded Operators and Second-Order PDE’s

203

for every β ∈ [0, α]. Since un converges to u uniformly on [0, T ]×Rd+ , we conclude that v ≡ u and Dj v = Dj u for each j = 1, . . . , d. Further, since the constant in (8.1.8) is independent (1+α)/2,1+α of r, we also infer that u ∈ Cb ([0, T ] × Rd+ ) and ||u||C (1+β)/2,1+β ([0,T ]×Rd ) ≤ CC∗,β ||ψ||C (1+β)/2,1+β ([0,T ]×Rd−1 ) +

b

b

for every β ∈ [0, α]. In particular, u(0, ·) = 0 and u(t, ·) = ψ(t, ·) on ∂Rd+ . To prove that u solves the differential equation Dt u = ∆u, we take advantage of the interior Schauder estimates in Theorem 6.2.9 together with estimate (8.1.5), which show e and that for every ε ∈ (0, T /2) and r, R > 0, with r < R, there exist positive constants C b independent of ψ, such that C, e n || b ||un ||C 1+α/2,2+α ((2ε,T −2ε)×B(0,r)) ≤ C||u ≤ C||ψ|| ∞. C([ε,T −ε]×B(0,R)) The same argument as above can be used to prove the convergence of the sequence (un ) in C 1+α/2,2+α ((2εT −2ε)×B(0, r)) to some function u, so that u belongs to C 1+α/2,2+α ((2εT − 1+α/2,2+α 2ε)×B(0, r)). Due to the arbitrariness of ε and R, we conclude that u ∈ Cloc ((0, T )× d d R+ ). Clearly, Dt u = Au on (0, T ) × R+ since each function un solves the same differential equation in that domain.  Proof of Proposition 8.1.1 Throughout the proof we denote by C a positive constant, possibly depending on d and α, but independent of the functions that we consider as well as of µ and the coefficients of the operators A and B, which may vary from line to line. We split the rest of the proof into some steps. ∂ Step 1. Here, we assume that A = ∆ and B = aI + , a being a positive constant and ∂η d η be a unit vector of R with ηd < 0. Let u be as in the statement of the proposition. Denote by v the unique solution in 1+α/2,2+α Cb ((0, T ) × Rd ) of the Cauchy problem ( Dt v(t, x) = ∆v(t, x) + g(t, x), t ∈ (0, T ], x ∈ Rd , v(0, x) = f (x), x ∈ Rd , α/2,α

where g ∈ Cb ((0, T ) × Rd ) and f ∈ Cb2+α (Rd ) are defined by g(t, ·) = Eα (Dt u(t, ·) − ∆u(t, ·)), for every t ∈ [0, T ], and f = E2+α u(0, ·), where the extension operators Eα and E2+α are defined in Proposition B.4.1. Thanks to Theorem 5.4.2, we can estimate ||v||C 1+α/2,2+α ((0,T )×Rd ) ≤C(||f ||C 2+α (Rd ) + ||g||Cbα ((0,T )×Rd ) ) b

b

≤C(||u(0, ·)||C 2+α (Rd ) + ||Dt u − ∆u||C α/2,α ((0,T )×Rd ) ). b

b

+

(8.1.9)

+

Clearly, the function w = u − v belongs to C 1+α/2,2+α ((0, T ) × Rd+ ) and solves the Cauchy problem  d   Dt w(t, x) = ∆w(t, x), t ∈ (0, T ], x ∈ R+ , w(t, x0 , 0) = w0 (t, x0 ), t ∈ [0, T ], x0 ∈ Rd−1 ,   w(0, x) = 0, x ∈ Rd+ , where w0 is the restriction to [0, T ] × ∂Rd+ of the function u − v. Theorem 7.0.2 shows that ||w||C 1+α/2,2+α ((0,T )×Rd ) ≤ C||w0 ||C 1+α/2,2+α ((0,T )×Rd−1 ) . b

+

b

(8.1.10)

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204

To estimate the norm of the function w0 , we begin by observing that, by Proposition 6.2.12, the function w is infinitely many times differentiable on (0, ∞) × Rd+ . This implies that the ∂w (1+α)/2,1+α belongs to C 1,2 ((0, T ) × Rd+ ). It also belongs to Cb function ζ = aw + ((0, T ) × ∂η d R+ ) and solves the Cauchy problem  t ∈ (0, T ], x ∈ Rd+ ,   Dt ζ(t, x) = ∆ζ(t, x), ζ(t, x0 , 0) = Bw0 (t, x0 ), t ∈ (0, T ], x0 ∈ Rd−1 ,   ζ(0, x) = 0. Therefore, estimates (8.1.3) and (8.1.9) show that ||ζ||C (1+α)/2,1+α ((0,T )×Rd ) ≤Cα ||Bw0 ||C (1+α)/2,1+α ((0,T )×Rd−1 ) +

b

b

≤Cα (||Bu||C (1+α)/2,1+α ((0,T )×Rd−1 ) + ||Bv||C (1+α)/2,1+α ((0,T )×Rd−1 ) ) b

b

≤C(||Bu||C (1+α)/2,1+α ((0,T )×Rd−1 ) + a||v||C 1+α/2,2+α ((0,T )×Rd ) ) b

b

≤C(1 + a)(||Bu||C (1+α)/2,1+α ((0,T )×Rd−1 ) + ||u(0, ·)||C 2+α (Rd ) b

b

+ ||Dt u − ∆u||C α/2,α ((0,T )×Rd ) ).

+

(8.1.11)

+

b

Next, we observe that d X

ηi ηj Dij w −

i,j=1

d−1 X

ηi ηj Dij w = −ηd2 Ddd w + 2

i,j=1

d X

ηi ηd Did w,

i=1

so that Ddd w = −

=−

d d−1 d 1 X 1 X 2 X η η D w + η η D w + ηi Did w i j ij i j ij ηd2 i,j=1 ηd2 i,j=1 ηd i=1 d d−1 1 X ∂w 1 X 2 ∂ η D + ηi ηj Dij w + Dd w i i ηd2 i=1 ∂η ηd2 i,j=1 ηd ∂η

d d d−1 1 X a X 2 2a 1 X =− 2 ηi Di ζ + 2 ηj Dj w + Dd ζ − Dd w + 2 ηi ηj Dij w. ηd i=1 ηd j=1 ηd ηd ηd i,j=1

Pd−1 Since Ddd w = Dt w − j=1 Djj w, it follows that w0 solves the equation Dt w0 = A0 w0 + φ on [0, T ] × Rd−1 , where d−1 d−1 d−1 1 X a X a X 0 2 A0 = ∆ + 2 ηi ηj Dij + 2 ηj Dj =: Tr(Q D ) + 2 ηj D j , ηd i,j=1 ηd j=1 ηd i=1

φ=−

d 1 X 2 a ηj Dj ζ(·, ·, 0) + Dd ζ(·, ·, 0) − Dd w(·, ·, 0) ηd2 j=1 ηd ηd

and Q0 is the diffusion matrix of the operator A0 . Note that we can estimate hQ0 (x)ξ, ξi = |ξ|2 +

1 ηd2

X d−1 i=1

2 ξi ηi

≥ |ξ|2 ,

x ∈ Rd+ , ξ ∈ Rd .

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205

α/2,α

Since w0 vanishes at t = 0 and φ belongs to Cb ((0, T ) × Rd−1 ), the assumptions of Theorem 6.2.1 are satisfied and, using also (8.1.11), we deduce that ||w0 ||C 1+α/2,2+α ((0,T )×Rd−1 ) b

≤C||φ||C α/2,α ((0,T )×Rd−1 ) b

≤Cηd−2 (1 + a)(||ζ||C (1+α)/2,1+α ((0,T )×Rd ) + ||w||C (1+α)/2,1+α ((0,T )×Rd ) ) +

b

+

b

≤Cηd−2 (1 + a)(||Bu||C (1+α)/2,1+α ((0,T )×Rd−1 ) + ||u(0, ·)||C 2+α (Rd ) b

+

b

+ ||Dt u − ∆u||C α/2,α ((0,T )×Rd ) + ||w||C (1+α)/2,1+α ((0,T )×Rd ) ). +

b

+

b

Replacing this estimate in the right-hand side of (8.1.10), we conclude that ||w||C 1+α/2,2+α ((0,T )×Rd ) +

b

≤Cηd−2 (1

+ a)(||Bu||C (1+α)/2,1+α ((0,T )×Rd−1 ) + ||u(0, ·)||C 2+α (Rd ) b

+

b

+ ||Dt u − ∆u||C α/2,α ((0,T )×Rd ) + ||w||C (1+α)/2,1+α ((0,T )×Rd ) ). +

b

(8.1.12)

+

b

To get rid of the norm of w from the right-hand side of (8.1.12), we use Proposition 1.2.5, which implies that ||w||C (1+α)/2,1+α ((0,T )×Rd ) ≤ ε||w||C 1+α/2,2+α ((0,T )×Rd ) + C(1 ∧ ε)−1−α ||w||∞ +

b

(8.1.13)

+

b

for every ε ∈ (0, 1) and use the maximum principle in Proposition 4.1.9, together with (8.1.9), to estimate ||w||∞ ≤2e2T (1 ∨ |ηd |−1 ∨ T )||Bϕ||∞ ≤Ce2T (1 ∨ |ηd |−1 ∨ T )(||Bu||∞ + ||v||C 1+α/2,2+α ([0,T ]×Rd ) ) b

2T

≤Ce

−1

(1 ∨ |ηd |

∨ T )(||Bu||∞ + ||u(0, ·)||C 2+α (Rd ) + ||Dt u − ∆u||C α/2,α ((0,T )×Rd ) ). b

+

b

+

(8.1.14) Replacing (8.1.14) in (8.1.13) and (8.1.13) in (8.1.12), we obtain that ||w||C 1+α/2,2+α ((0,T )×Rd ) +

b

≤Cηd−2 (1 + a)ε||w||C 1+α/2,2+α ((0,T )×Rd ) +

b

+ K(α, d, ε, a, ηd , T )(||Bu||C (1+α)/2,1+α ((0,T )×Rd−1 ) + ||u(0, ·)||C 2+α (Rd ) b

b

+

+ ||Dt u − ∆u||C α/2,α ((0,T )×Rd ) ) b

+

for each ε > 0, where K(d, α, ε, a, ηd , T ) = Ce2T ηd−2 (1 + a)(1 ∧ ε)−1−α (1 ∨ |ηd |−1 ∨ T ). Choosing Cηd−2 (1 + a)ε = 1/2, the term ε||w||C 1+α/2,2+α ((0,T )×Rd ) can be moved to the leftb

+

hand side of the previous inequality and this yields estimate (8.1.1) in this particular case with b = Ce2T η −2 (1 + a)(1 ∧ 2−1 C −1 ηd2 (1 + a)−1 )−1−α (1 ∨ |ηd |−1 , T ). C d Step 2. Here, we consider the case when A = Tr(QD2 u) and Q is a symmetric and constant matrix with positive eigenvalues. We use Lemma 8.1.2 to determine an invertible matrix P , which leaves Rd+ and ∂Rd invariant and satisfies the conditions P QP T = I

Parabolic Equations in Rd+ with More General Boundary Conditions

206

−1/2

and P T ed = Ked , where λmax ≤ K ≤ µ−1/2 and µ, λmax denote the minimum and maximum eigenvalues of Q. Next, we introduce the function v : [0, T ] × Rd+ → R, defined by v(t, x) = u(t, P −1 x) for every (t, x) ∈ [0, T ] × Rd+ . As it is easily seen, Dt v − ∆v = Dt u(·, P −1 ·) − Au(·, P −1 ·) in (0, T ] × Rd+ , v(0, ·) = u(0, P −1 ·) in Rd+ and ∂v B 0 v := av(·, ·, 0) + (·, ·, 0) = (Bu)(·, P −1 (·, 0)) on [0, T ] × ∂Rd+ , where η∗ = P η. Note ∂η∗ that hη∗ , νi = −hP η, ed i = −hη, P T ed i = −Khη, ed i = Khη, νi, so that hη∗ , νi is positive −1 and bounded from below and above by λ−1 hη, νi, respectively. Therefore, max hη, νi and µ we can apply Step 1 and conclude that ||v||C 1+α/2,2+α ((0,T )×Rd ) +

b

≤C(d, α, a, µ, λmax , T )(||v(0, ·)||C 2+α (Rd ) + ||Dt v − ∆v||C α/2,α ((0,T )×Rd ) +

b

+

b

0

+ ||B v||C (1+α)/2,1+α ((0,T )×Rd−1 ) ),

(8.1.15)

b

where

√ Ce2T λmax (1 + a)(1 ∨ λmax |ηd |−1 ∨ T ) C(d, α, a, µ, λmax , T ) = . 2 −1 )1+α ηd2 (1 ∧ 2−1 C −1 λ−1 max ηd (1 + a)

Note that ∇v(0, x) = (P −1 )T ∇u(0, P −1 x) and D2 v(t, x) = (P −1 )T D2 u(t, P −1 x)P −1 for every (t, x) ∈ [0, T ] × Rd+ , so that 2+α 2 )||u(0, ·)||C 2+α (Rd ) . ||v(0, ·)||C 2+α (Rd ) ≤ C(1 ∨ λmax b

+

b

+

Similarly, α 2 ||Dt v − ∆v||C α/2,α ((0,T )×Rd ) ≤ C(1 ∨ λmax )||Dt u − Au||C α/2,α ((0,T )×Rd ) , +

b

+

b

1+α 2

||B 0 v||C (1+α)/2,1+α ((0,T )×Rd−1 ) ≤ C(1 ∨ λmax )||Bu||C (1+α)/2,1+α ((0,T )×Rd−1 ) . b

b

Finally, since u(t, x) = v(t, P x) for every (t, x) ∈ [0, T ] × Rd+ , we can estimate ||u||C 1+α/2,2+α ((0,T )×Rd ) ≤ C(1 ∨ µ−

2+α 2

+

b

)||v||C 1+α/2,2+α ((0,T )×Rd ) . +

b

Replacing all these estimates in (8.1.15) we obtain (8.1.1), with √ 2T 2+α 2+α Ce λmax (1 + a)(1 ∨ λmax |ηd |−1 ∨ T ) 2 b = (1 ∨ λmax C )(1 ∨ µ− 2 ) 2 . 2 −1 )1+α ηd (1 ∧ 2−1 C −1 λ−1 max ηd (1 + a) Step 3. Now, we can deal with the general case, freezing the coefficients. We fix r ∈ (0, 1], x0 ∈ Rd+ and choose a function ϑx0 ,r ∈ Cc∞ (Rd ) such that χB(x0 ,r/2) ≤ ϑx0 ,r ≤ χB(x0 ,r) . For every β ∈ (0, 2 + α], we also denote by Kβ a positive constant such that ||ϑx0 ,r ||C β (Rd ) ≤ b

Kβ r−β . Clearly, the function v = ϑx0 ,r u belongs to C 1+α/2,2+α ((0, T ) × Rd+ ) if u does. Therefore, applying estimate (8.1.1) with A and B being replaced by Bx0 = a(x00 ) +

Ax0 = Tr(Q(x0 )D2 ), and observing that λmax (Q(x0 )) ≤ Cq0 := C

∂ , ∂η(x00 )

max ||qij ||∞ , a(x0 ) ≤ ||a||∞ , |ηd (x0 )| ≥ |η0 |

i,j=1,...,d

for every x0 ∈ Rd+ , we get ||v||C 1+α/2,2+α ((0,T )×Rd ) ≤C∗ (||v(0, ·)||C 2+α (Rd ) + ||Dt v − Ax0 v||C α/2,α ((0,T )×Rd ) b

+

b

+

b

+

Semigroups of Bounded Operators and Second-Order PDE’s

207

+ ||Bx0 v||C (1+α)/2,1+α ((0,T )×Rd−1 ) ),

(8.1.16)

b

where 2+α 2

C∗ = (1 ∨ q0

− 2+α 2

)(1 ∨ µ

√ Ce2T q0 (1 + ||a||∞ )(1 ∨ q0 |η0 |−1 ∨ T ) . ) 2 −1 )1+α η0 (1 ∧ 2−1 C −1 η02 q−1 0 (1 + ||a||∞ )

Since v = u on [0, T ] × B+ (x0 , r/2), we can estimate ||u||C 1+α/2,2+α ((0,T )×B+ (x0 ,r/2)) ≤ ||v||C 1+α/2,2+α ((0,T )×Rd ) . +

b

Moreover, ||v(0, ·)||C 2+α (Rd ) ≤ Cr−2−α ||u(0, ·)||C 2+α (Rd ) . +

b

(8.1.17)

+

b

Repeating the same arguments as in the proof of (6.2.13), we can estimate ||Dt u − Ax0 u||C α/2,α ((0,T )×B+ (x0 ,r)) b

≤||Dt u − Au||C α/2,α ((0,T )×Rd+ ) + Cqα ||u||C 0,2 ([0,T ]×Rd ) +

b

+ (bα + ||c||Cbα (Rd+ ) )C||u||C α/2,1+α ((0,T )×Rd ) + Cq0 rα +

b

d X

||Dij u||C α/2,α ((0,T )×Rd ) , (8.1.18) +

b

i,j=1

where we have set B+ (x0 , r) = B(x0 , r) ∩ Rd+ , qα = maxi,j=1,...,d ||qij ||Cbα (Rd+ ) and bα = maxj=1,...,d ||bj ||Cbα (Rd+ ) . Therefore, ||Dt v − Ax0 v||C α/2,α ((0,T )×Rd ) +

b

≤||Dt u − Ax0 u||C α/2,α ((0,T )×B+ (x0 ,r)) + r−2−α (1 + q0 + b0 + ||c||∞ )C||u||C 1,2 ([0,T ]×Rd ) b

+

b

+ Cr−1 qα ||u||C (1+α)/2,1+α ((0,T )×Rd ) + Cr−1−α qα ||u||C 0,1 ([0,T ]×Rd ) +

b

+

b

+ Cr−2−α qα ||u||C α/2,α ((0,T )×Rd ) + Cr−1−α bα ||u||C α/2,α ((0,T )×Rd ) +

b

+

b

≤||Dt u − Au||C α/2,α ((0,T )×Rd ) + Cq0 rα ||u||C 1+α/2,2+α ((0,T )×Rd ) +

b

+r

−2−α

+

b

(1 + qα + bα + ||c||Cbα (Rd+ ) )C||u||C 1,2 ([0,T ]×Rd ) ,

(8.1.19)

+

b

where we have used Proposition 1.2.4, which shows that ||f ||C α/2,1+α ((0,T )×Rd ) + ||f ||C (1+α)/2,1+α ((0,T )×Rd ) ≤ C||f ||C 1,2 ([0,T ]×Rd ) +

b

+

b

b

+

for every function f ∈ Cb1,2 ([0, T ] × Rd+ ), the constant C being independent also of T , and the fact that r ∈ (0, 1). Finally, we observe that ||Bx0 v||C (1+α)/2,1+α ((0,T )×Rd−1 ) b

≤||Bx0 u||C (1+α)/2,1+α ([0,T ]×∂0 B+ (x0 ,r)) + Cr−1−α ||Bx0 u||∞ + Cr−α |||∇Bx0 u|||∞ b

+ Cr−1 ||Bx0 u||C α/2,α ((0,T )×Rd−1 ) + Cr−1 ||u||C (1+α)/2,1+α ((0,T )×Rd−1 ) b

b

+ Cr−2−α ||u||∞ + Cr−1−α |||∇u|||∞ + Cr−2 ||u||C α/2,α ((0,T )×Rd−1 ) b

≤||Bu||C (1+α)/2,1+α ((0,T )×Rd−1 ) + ||Bu − Bx0 u||C (1+α)/2,1+α ([0,T ]×∂0 B+ (x0 ,r)) b

Parabolic Equations in Rd+ with More General Boundary Conditions

208

+ (1 + ||a||∞ )Cr−2−α ||u||C 0,1 ([0,T ]×Rd ) +

b

+ (1 + ||a||∞ )Cr−2 ||u||C α/2,α ((0,T )×Rd ) + Cr−1 ||u||C (1+α)/2,1+α ((0,T )×Rd ) +

b

+

b

≤||Bu||C (1+α)/2,1+α ((0,T )×Rd−1 ) + ||Bu − Bx0 u||C (1+α)/2,1+α ([0,T ]×∂0 B+ (x0 ,r)) b

+ (1 + ||a||∞ )Cr−2−α ||u||C 1,2 ([0,T ]×Rd ) +

b

and ||Bu − Bx0 u||C (1+α)/2,1+α ([0,T ]×∂0 B+ (x0 ,r)) ≤Cr(||a||Cb1 (Rd−1 ) + ||η||Cb1 (Rd−1 ;Rd ) )||u||C 1+α/2,2+α ((0,T )×Rd ) +

b

+ (||a||C 1+α (Rd−1 ) + ||η||C 1+α (Rd−1 ;Rd ) )C||u||C 1,2 ([0,T ]×Rd ) , b

b

+

b

where ∂0 B+ (x0 , r) := {x0 ∈ Rd−1 : |x0 − x00 |2 + x20,d < r2 }. Therefore, taking into account that r ∈ (0, 1), the previous two estimates yield ||Bx0 v||C (1+α)/2,1+α ((0,T )×Rd−1 ) b  ≤C ||Bu||C (1+α)/2,1+α ((0,T )×Rd−1 ) b

+ (||a||Cb1 (Rd−1 ) + ||η||Cb1 (Rd−1 ;Rd ) )r||u||C 1+α/2,2+α ((0,T )×Rd ) +

b

+ (1 + ||a||C 1+α (Rd−1 ) + ||η||C 1+α (Rd−1 ;Rd ) )r b

−2−α

b

 ||u||C 1,2 ([0,T ]×Rd ) . b

(8.1.20)

+

Replacing (8.1.17)–(8.1.20) into (8.1.16) we deduce that ||u||C 1+α/2,2+α ([0,T ]×B+ (x0 ,r)) b ≤C∗∗ ||Dt u − Au||C α/2,α ((0,T )×Rd+ ) + ||Bu||C (1+α)/2,1+α ((0,T )×Rd−1 ) b

α

+ r ||u||C 1+α/2,2+α ((0,T )×Rd ) + r

−2−α

+

b

 ||u(0, ·)||C 2+α (Rd ) + r−2−α ||u||C 1,2 ([0,T ]×Rd ) , +

b

b

+

where C∗∗ = CC∗ (1 + qα + bα + ||c||Cbα (Rd+ ) + ||a||C 1+α (Rd−1 ) + ||η||C 1+α (Rd−1 ;Rd ) ). Since the b b constant in the previous estimate is independent of x0 , we can take the supremum as x0 ∈ Rd+ and conclude that ||u||C 1+α/2,2+α ((0,T )×Rd ) + b  ≤C∗∗ ||Dt u − Au||C α/2,α ((0,T )×Rd+ ) + ||Bu||C (1+α)/2,1+α ((0,T )×Rd−1 )

b  + rα ||u||C 1+α/2,2+α ((0,T )×Rd ) + r−2−α ||u(0, ·)||C 2+α (Rd ) + r−2−α ||u||C 1,2 ([0,T ]×Rd ) . +

b

+

b

b

+

Choosing r such that 2C∗∗ rα = 1, we can move rα ||u||C 1+α/2,2+α ((0,T )×Rd ) to the left-hand b

+

side of the previous inequality thus obtaining ||u||C 1+α/2,2+α ((0,T )×Rd ) + b  ≤C∗∗∗ ||u(0, ·)||C 2+α (Rd ) + ||Dt u − Au||C α/2,α ((0,T )×Rd+ ) + ||u||C 1,2 ([0,T ]×Rd ) + b + b  + ||Bu||C (1+α)/2,1+α ((0,T )×Rd−1 ) . (8.1.21) b

Semigroups of Bounded Operators and Second-Order PDE’s

209

To get rid of the Cb1,2 -norm of u from the right-hand side of the previous estimate, we apply estimate (1.2.7), taking also (4.1.5) into account, and deduce that 2

||u||C 1,2 ([0,T ]×Rd ) ≤2ε||u||C 1+α/2,2+α ((0,T )×Rd ) + CT (ε ∧ 1)− α ||u||∞ +

b

+

b

≤2ε||u||C 1+α/2,2+α ((0,T )×Rd ) +

b

2

+ CCT (ε ∧ 1)− α (||Dt u − Au||∞ + ||Bu||∞ + ||u(0, ·)||∞ ), where the function T 7→ CT is continuous on [0, ∞). Replacing this estimate into the righthand side of (8.1.21) we conclude the proof. 

8.2

Proof of Theorem 8.0.2

The second main step in the proof of Theorem 8.0.2 is the following proposition. α/2,α

Proposition 8.2.1 For every g ∈ Cb ((0, T ) × Rd+ ), vanishing on {0} × ∂Rd+ , and every (1+α)/2,1+α d−1 ψ∈C ((0, T ) × R ), vanishing on {0} × Rd−1 , the Cauchy problem  Dt u(t, x) = ∆u(t, x) + g(t, x),    ∂u (t, x0 , 0) = ψ(t, x0 ),  ∂x d   u(0, x) = 0, 1+α/2,2+α

admits a unique solution u ∈ Cb

t ∈ (0, T ],

x ∈ Rd+ ,

t ∈ (0, T ],

x0 ∈ Rd−1 ,

(8.2.1)

x ∈ Rd+ ,

((0, T ) × Rd+ ).

Proof To begin with, we fix some notation. Throughout the proof, we denote by C a positive constant, independent of the functions that we are considering as well as of the time and spatial variables, which may vary from line to line. Moreover, we simply write || · ||β,γ instead of the usual notation || · ||C β,γ ([0,T ]×Rd ) . For further use, observe that, since +

b

|x0 |β e−

|x0 |2 4t

β

≤ Cβ t 2 e−

|x0 |2 8t

t > 0, x0 ∈ Rd−1 , β > 0,

,

we can estimate d

t 2 |K(t, x)| + t +t

d+3 2

d+1 2

|Dj K(t, x)| + t

|Dtj K(t, x)| + t

d+4 2

d+2 2

|Dij K(t, x)| + t |x0 |2 − 8t

|Dtt K(t, x)| ≤ Ce

d+2 2

|Dt K(t, x)| (8.2.2)

for every t > 0, x ∈ Rd and i, j = 1, . . . , d. Finally, as in Lemma 7.1.3, we set K+ (t, x) = −2Dd K(t, x) for every t > 0 and x ∈ Rd+ . As formula (7.1.11) shows, Z ∞ Z sγ ds |y 0 |β K+ (s, y 0 , xd ) dy 0 ≤ Cxdβ+2γ (8.2.3) 0

Rd−1

for each xd > 0 and β, γ ∈ R such that 2γ + β < 1. Let us introduce the function ge : [0, T ] × Rd → R, obtained extending the function g, even with respect to the last variable, i.e., ge(t, x0 , xd ) = g(t, x0 , −xd ) for evα/2,α ery t ∈ [0, T ], x0 ∈ Rd−1 and xd < 0. Clearly, ge belongs to Cb ((0, T ) × Rd ) and

Parabolic Equations in Rd+ with More General Boundary Conditions

210

||e g ||C α/2,α ((0,T )×Rd ) ≤ ||g||C α/2,α ((0,T )×Rd ) . By Theorem 5.4.2, there exists a unique function b

b

1+α/2,2+α

u1 ∈ Cb ((0, T ) × Rd ) such that Dt u1 = ∆u1 + ge on [0, T ] × Rd and u1 (0, ·) = 0. As it is easily seen also the function (t, x0 , xd ) 7→ u1 (t, x0 , −xd ) solves the same problem, so that, by uniqueness, u1 (t, x0 , xd ) = u1 (t, x0 , −xd ) for every t ∈ [0, T ], x0 ∈ Rd−1 and xd ∈ R. As a byproduct, Dd u1 vanishes on [0, T ] × ∂Rd+ . Next, we show that the function u2 : [0, T ] × Rd+ → R, defined by t

Z

Z

u2 (t, x) = −2

K(t − s, x0 − y 0 , xd )ψ(s, y 0 ) dy 0

ds Rd−1

0

1+α/2,2+α

for every t ∈ [0, T ], x ∈ Rd+ belongs to Cb problem  Dt u2 (t, x) = ∆u2 (t, x),    ∂u 2 (t, x0 , 0) = ψ(t, x0 ),  ∂x d   u2 (0, x) = 0,

((0, T ) × Rd+ ) and solves the Cauchy

t ∈ (0, T ],

x ∈ Rd+ ,

t ∈ (0, T ],

x0 ∈ Rd−1 , x ∈ Rd+ .

Once this is done, we can conclude that the function u1 + u2 is the unique solution to the problem (8.2.1) we were looking for. The uniqueness follows straightforwardly from estimate (8.1.1). Clearly u2 (0, ·) = 0. Moreover, applying the dominated convergence theorem, we can differentiate under the integral sign in (0, T )×Rd+ , twice with respect to the spatial variables and once with respect to the time variable, and conclude that Z t Z Dj u2 (t, x) = − 2 ds Dj K(t − s, x0 − y 0 , xd )ψ(s, y 0 ) dy 0 Rd−1

0

Z

t

=−2

Z ds

Z

t

Dij u2 (t, x) = − 2

Z ds

t

=−2

Z ds

t

Ddd u2 (t, x) = − 2

Z ds

Dt u2 (t, x) = − 2

Dj K(t − s, y 0 , xd )Di ψ(s, x0 − y 0 ) dy 0 , Ddd K(t − s, y 0 , xd )ψ(s, x0 − y 0 ) dy 0 ,

Rd−1

0

Z

Dij K(t − s, y 0 , xd )ψ(s, x0 − y 0 ) dy 0

Rd−1

0

Z

(j < d),

Rd−1

0

Z

K(t − s, x0 − y 0 , xd )Dj ψ(s, y 0 ) dy 0 ,

Rd−1

0

t

Z ds

Dt K(t − s, y 0 , xd )ψ(s, x0 − y 0 ) dy 0

Rd−1

0

for every (t, x) ∈ [0, T ]×Rd+ , i = 1, . . . , d−1 and j = 1, . . . , d. In particular, since Dt K = ∆K on (0, ∞) × Rd , it is immediate to check that u2 solves the homogeneous heat equation on (0, T ) × Rd+ . Performing the change of variables t − s = x2d r, x0 − y 0 = xd z 0 as in the proof of Theorem 7.2.1, we can write Z ∞ Z b − rx2 , x0 − xd z 0 ) dz 0 Dd u2 (t, x) = dr K+ (r, z 0 , 1)ψ(t (8.2.4) d 0

Rd−1

for every (t, x) ∈ (0, T ) × Rd+ , where ψb : (−∞, T ] × Rd+ → R is obtained extending

Semigroups of Bounded Operators and Second-Order PDE’s

211

in the trivial way ψ for negative times. Since ψ(0, ·) = 0, the function ψb belongs to (1+α)/2,1+α Cb ((−∞, T ) × Rd−1 ) and b (1+α)/2,1+α ≤ ||ψ||(1+α)/2,1+α . ||ψ|| ((−∞,T )×Rd−1 ) C

(8.2.5)

b

Using formula (8.2.4), estimate (8.2.3) and the dominated convergence theorem, it is easy to conclude that Dd u2 can be extended by continuity to [0, T ] × Rd+ . Moreover, Z ∞ Z dr K+ (r, z 0 , 1)ψ(t0 , x00 ) dz 0 = ψ(t0 , x00 ) Dd u2 (t0 , x00 , 0) = Rd−1

0

for every (t0 , x00 ) ∈ [0, T ] × Rd−1 . Let us now prove that u2 and its first-order derivatives are bounded. For this purpose, using (8.2.2) we deduce that Z |u2 (t, x)| ≤2||ψ||∞

t

Z

0

0

0

K(t − s, x − y , xd ) dy ≤ C||ψ||∞

ds Rd−1

0

Z

t

√ 1 r− 2 dr = C T ||ψ||∞

0

and, √ similarly, using the second expression above for Dj u2 , we can show that |Dj u2 (t, x)| ≤ C T ||Di ψ||∞ for every (t, x) ∈ (0, T ) × Rd+ and j = 1, . . . , d − 1. The boundedness of the spatial derivative Dd u2 , follows immediately from formula (8.2.4) and estimate (8.2.3). Next, we prove that the second-order derivatives Dij u2 (i = 1, . . . , d − 1, j = 1, . . . , d) α/2,α belong to Cb ((0, T ) × Rd+ ). To prove that, for every i, j = 1, . . . , d − 1 and t ∈ (0, T ], the function Dij u2 (t, ·) is α-H¨ older continuous on Rd+ , uniformly with respect to the time variable, we write Z t Z Dij u2 (t, x) = −2 ds Dj K(t − s, x0 − y 0 , xd )[Di ψ(s, y 0 ) − Di ψ(s, x0 )] dy 0 Rd−1

0

for every t ∈ [0, T ] and x ∈ Rd+ and then split Dij u2 (t, x e) − Dij u2 (t, x) Z t Z = ds [Dj K(s, x0 − y 0 , xd ) − Dj K(s, x e0 − y 0 , x ed )] Rd−1 \B(x0 ,2|x−e x|)

0

× [Di ψ(t − s, y 0 ) − Di ψ(t − s, x e0 )] dy 0 Z

t

+2

Z ds Rd−1 \B(x0 ,2|x−e x|)

0

Z

t

−2

Z ds B(x0 ,2|x−e x|)

0

Z +2 4 X

Z

Dj K(s, x e0 − y 0 , x ed )[Di ψ(t − s, y 0 ) − Di ψ(t − s, x e0 )] dy 0 Dj K(s, x0 − y 0 , xd )[Di ψ(t − s, y 0 ) − Di ψ(t − s, x0 )] dy 0

ds 0

=:

t

Dj K(s, x0 − y 0 , xd )[Di ψ(t − s, x e0 ) − Di ψ(t − s, x0 )] dy 0

B(x0 ,2|x−e x|)

Ij (t, x, x e)

j=1

for every x, x e ∈ Rd+ and t ∈ (0, T ]. Using (8.2.2), we get |Dj K(s, x0 − y 0 , xd ) − Dj K(s, x e0 − y 0 , x ed )|

Parabolic Equations in Rd+ with More General Boundary Conditions Z 1 ≤|x − x e| |∇Dj K(s, λe x0 + (1 − λ)x0 − y 0 , xd )| dλ

212

0

−d 2 −1

1

Z

≤C|x − x e|s

e−

|λx e 0 +(1−λ)x0 −y 0 | 8s

dλ.

0

If y 0 ∈ / B(x0 , 2|x − x e|), then |y 0 − λe x0 − (1 − λ)x0 | = |y 0 − x0 + λ(x0 − x e0 )| ≥ |y 0 − x0 | − |x0 − x e0 | ≥

1 0 |y − x0 |. 2

|y 0 −x0 |2

d

Thus, |Dj K(s, x0 − y 0 , xd ) − Dj K(s, x e0 − y 0 , x ed )| ≤ C|x − x e|s− 2 −1 e− 8s . Using this inequality, we can estimate the term I1 as follows: Z t Z |y 0 −x0 |2 − d+2 2 I1 (t, x, x e) ≤C||Di ψ||0,α |x − x e| s ds |y 0 − x0 |α e− 8s dy 0 Rd−1 \B(x0 ,2|x−e x|) Z ∞ |z 0 |2 0 α 0 − d+2 − 8s 2

0

Z

|z | dz

≤C||Di ψ||0,α |x − x e|

s

Rd−1 \B(0,2|x−e x|)

e

ds.

0

Performing the change of variable r = |z 0 |2 (8s)−1 , we obtain that Z ∞ Z ∞ |z 0 |2 d+2 d−2 d s− 2 e− 8s ds = 8 2 |z 0 |−d r 2 e−r dr = C|z 0 |−d . 0

0

Hence, I1 (t, x, x e) ≤C||Di ψ||0,α |x − x e|

Z

|z 0 |α−d dz 0

Rd−1 \B(0,2|x−e x|)

Z =C||Di ψ||0,α |x − x e|

∞

|z 0 |α−2 dz 0

2|x−e x|

=C||Di ψ||0,α |x − x e|α . Next, we observe that I2 (t, x, x e) = 0. Indeed, Z Dj K(s, x0 − y 0 , xd )[Di ψ(t − s, x e0 ) − Di ψ(t − s, x0 )] dy 0 Rd−1 \B(x0 ,2|x−e x|) Z 0 0 =[Di ψ(t − s, x e ) − Di ψ(t − s, x )] Dj K(s, z 0 , xd ) dz 0 Rd−1 \B(0,2|x−e x|)

and the last integral term vanishes, since the domain is symmetric with respect to the j-th variable and the function under the integral sign is odd with respect to that variable. Let us consider the term I3 . Taking (8.2.2) into account, it is easy to show that I3 (t, x, x e) ≤ C||Di ψ||0,α

Z

t

s−

d+1 2

Z ds B(x0 ,2|x−e x|)

0

|y 0 − x e0 |α e−

|y 0 −x e 0 |2 8s

dy 0

and, since B(x0 , 2|x − x e|) ⊂ B(e x0 , 3|x − x e|), it follows that I3 (t, x, x e) ≤C||Di ψ||0,α

Z

t

s−

d+1 2

=C||Di ψ||0,α 0

|z 0 |α e−

|z 0 |2 8s

dz 0

B(0,3|x−e x|)

0

Z

Z ds

3|x−e x|

ρd+α−2 dρ

Z 0

∞

s−

d+1 2

e−

|z 0 |2 8s

dz 0 ds

Semigroups of Bounded Operators and Second-Order PDE’s Z 3|x−ex| ≤C||Di ψ||0,α ρα−1 dρ

213

0

=C||Di ψ||0,α |x − x e|α . The term I4 is completely similar and even easier to estimate, since we do not need to replace the ball B(x0 , 2|x − x e|) with the ball B(e x0 , 3|x − x e|). Summing up, we have proved d that Dij u2 (t, ·) is α-H¨ older continuous on R+ , uniformly with respect to t ∈ [0, T ]. α/2,0 Let us prove that Dij u2 belongs to Cb ([0, T ] × Rd+ ). For this purpose, we rewrite this term in the following form for every (t, x) ∈ [0, T ] × Rd+ and j = 1, . . . , d, i = 1, . . . , d − 1: Z t Z b y 0 ) − Di ψ(t, b y 0 )] dy 0 Dij u2 (t, x) = − 2 ds Dj K(t − s, x0 − y 0 , xd )[Di ψ(s, d−1 −∞ R Z ∞ Z b y 0 ) dy 0 . ds Dj K(s, x0 − y 0 , xd )Di ψ(t, −2 Rd−1

0

Next, we fix x ∈ Rd+ , t1 , t2 ∈ [0, T ], with t1 < t2 , and split Z t2 Z b y 0 ) − Di ψ(t b 1 , y 0 )] dy 0 ds Dj K(t1 − s, x0 − y 0 , xd )[Di ψ(s, Rd−1

−∞ t2

Z

Z

=

b y 0 ) − Di ψ(t b 1 , y 0 )] dy 0 Dj K(t1 − s, x0 − y 0 , xd )[Di ψ(s,

ds 2t1 −t2 Rd−1 Z 2t1 −t2 Z

+

b y 0 ) − Di ψ(t b 1 , y 0 )] dy 0 Dj K(t1 − s, x0 − y 0 , xd )[Di ψ(s,

ds Rd−1

−∞

and Z

t2

Z

b y 0 ) − Di ψ(t b 2 , y 0 )] dy 0 Dj K(t2 − s, x0 − y 0 , xd )[Di ψ(s,

ds Rd−1

−∞ Z t2

Z

=

b y 0 ) − Di ψ(t b 2 , y 0 )] dy 0 Dj K(t2 − s, x0 − y 0 , xd )[Di ψ(s,

ds 2t1 −t2 Z 2t1 −t2

+

Rd−1

Z

b y 0 ) − Di ψ(t b 1 , y 0 )] dy 0 Dj K(t2 − s, x0 − y 0 , xd )[Di ψ(s,

ds Rd−1

−∞ Z 2t1 −t2

+

Z

b 1 , y 0 ) − Di ψ(t b 2 , y 0 )] dy 0 Dj K(t2 − s, x0 − y 0 , xd )[Di ψ(t

ds Rd−1

−∞

so that Dij u2 (t2 , x) − Dij u2 (t1 , x) Z t1 Z b y 0 ) − Di ψ(t b 1 , y 0 )] dy 0 =2 ds Dj K(t1 − s, x0 − y 0 , xd )[Di ψ(s, Rd−1

2t1 −t2 Z t2

−2

Z

b y 0 ) − Di ψ(t b 2 , y 0 )] dy 0 Dj K(t2 − s, x0 − y 0 , xd )[Di ψ(s,

ds Rd−1

2t1 −t2 Z 2t1 −t2

−2

Z

[Dj K(t2 − s, x0 − y 0 , xd ) − Dj K(t1 − s, x0 − y 0 , xd )]

ds Rd−1

−∞

b y 0 ) − Di ψ(t b 1 , y 0 )] dy 0 × [Di ψ(s, Z −2

2(t2 −t1 )

Z ds

0

Rd−1

b 2 , y 0 ) − Di ψ(t b 1 , y 0 )] dy 0 Dj K(t2 − s, x0 − y 0 , xd )[Di ψ(t

Parabolic Equations in Rd+ with More General Boundary Conditions

214 =

4 X

Hj (t1 , t2 , x).

j=1

Taking (8.2.2) into account, we can easily show that Z tk Z α 2 |tk − s| ds Hk (t1 , t2 , x) ≤2||Di ψ||α/2,0 ≤C||Di ψ||α/2,0

|Dj K(tk − s, y 0 , xd )| dy 0

Rd−1

2t1 −t2 Z tk

|tk − s|

α−2 2

α

ds = C||Di ψ||α/2,0 |t2 − t1 | 2

2t1 −t2

for k = 1, 2. Next, we note that |Dj K(t2 − s, x0 − y 0 , xd ) − Dj K(t1 − s, x0 − y 0 , xd )| Z 1 |Dtj K(λt2 + (1 − λ)t1 − s, x0 − y 0 , xd )| dλ ≤|t2 − t1 | 0

and λt2 + (1 − λ)t1 − s ≥ t1 − s for every λ ∈ [0, 1], so that, taking again (8.2.2) into account, we deduce that H3 (t1 , t2 , x) ≤C||ψ||α/2,0 (t2 − t1 )

2t1 −t2

Z

Z (t1 − s) ds α 2

Rd−1

−∞

Z

2t1 −t2

≤C||ψ||α/2,0 (t2 − t1 )

dy 0

Z

α

−∞ 2t1 −t2

0

1

e

− 8(λt

|y 0 |2

2 +(1−λ)t1 −s)

(λt2 +(1−λ)t1 −s)

d+3 2

1

(t1 − s) 2 ds

(λt2 + (1 − λ)t1 − s)−2 dλ

0

Z ≤C||ψ||α/2,0 (t2 − t1 )

Z

(t1 − s)

α−4 2

α

ds = C||ψ||α/2,0 (t2 − t1 ) 2 .

−∞

Finally, we consider the term H4 , which we rewrite as follows: H4 (t1 , t2 , x) Z 2(t2 −t1 ) Z =−2 ds 0

Z

2(t2 −t1 )

Z

+2

ds

b 1 , y 0 ) − Di ψ(t b 1 , x0 )] dy 0 Dj K(t2 − s, x0 − y 0 , xd )[Di ψ(t

Rd−1

0

Z

b 2 , y 0 ) − Di ψ(t b 2 , x0 )] dy 0 Dj K(t2 − s, x0 − y 0 , xd )[Di ψ(t

Rd−1

2(t2 −t1 )

−2

b 2 , x0 ) − Di ψ(t b 1 , x0 )] ds [Di ψ(t

2(t2 −t1 )

Z

=−2

ds

b 2 , y 0 ) − Di ψ(t b 2 , x0 )] dy 0 Dj K(t2 − s, x0 − y 0 , xd )[Di ψ(t

Rd−1

0

Z

Dj K(t2 − s, x0 − y 0 , xd ) dy 0

Rd−1

0

Z

Z

2(t2 −t1 )

Z

+2

ds

b 1 , y 0 ) − Di ψ(t b 1 , x0 )] dy 0 . Dj K(t2 − s, x0 − y 0 , xd )[Di ψ(t

Rd−1

0

Note that, using (8.2.2), we can show that Z 2(t2 −t1 ) Z 0 0 0 0 0 b b ds D K(t − s, x − y , x )[D ψ(t , y ) − D ψ(t , x )] dy j 2 d i j i j 0 Rd−1 Z 2(t2 −t1 ) Z |z 0 |2 d+1 − ≤C||Di ψ||0,α s− 2 ds |z 0 |α e 8(t2 −s) dz 0 Rd−1

0

Z ≤C||Di ψ||0,α

2(t2 −t1 )

− α−2 2

s 0

α

ds = C||Di ψ||0,α (t2 − t1 ) 2

dλ

Semigroups of Bounded Operators and Second-Order PDE’s

215

α

for j = 1, 2. Therefore, |H4 (t1 , t2 , x)| ≤ C||Di ψ||0,α (t2 − t1 ) 2 . Combining all the above α estimates, we conclude that |Dij u2 (t2 , x) − Dij u2 (t1 , x)| ≤ C||Di ψ||α/2,α (t2 − t1 ) 2 . In particular, taking t1 = 0 and recalling that Dij u2 (0, ·) = 0, we deduce that Dij u2 is bounded on (0, T ) × Rd+ . Since the function Dij u2 (t, ·) can be extended by continuity to Rd+ for every α/2,0 t ∈ [0, T ], the previous estimate shows that u2 ∈ Cb ([0, T ] × Rd+ ). Summing up, we have α/2,α d proved that Dij u2 ∈ Cb ((0, T ) × R+ ) as it has been claimed. Next, we consider the function Did u2 and perform the change of unknowns t − s = x2d r, x0 − y 0 = xd z 0 , to write Z ∞ Z b − rx2 , x0 − xd z 0 ) dz 0 dr K+ (r, z 0 , 1)Di ψ(t Did u2 (t, x) = d Rd−1

0

for every (t, x) ∈ [0, T ] × Rd+ . Thanks to (8.2.3) and (8.2.5), we can estimate α

|Did u2 (t2 , x02 , xd ) − Did u2 (t1 , x01 , xd )| ≤ C||Di ψ||α/2,α (|t1 − t2 | 2 + |x02 − x01 |α ) for every t1 , t2 ∈ [0, T ], x1 , x2 ∈ Rd−1 and xd > 0. Thus, the function Did u2 (·, ·, xd ) is αH¨ older continuous on [0, T ] × Rd−1 , with respect to the parabolic distance, for every xd > 0. Moreover, since Did u2 (0, ·) = 0, from the previous estimate it follows that Did u2 is bounded on (0, T ) × Rd+ and ||Did u2 (·, ·, xd )||C α/2,α ((0,T )×Rd−1 ) ≤ C||Di ψ||α/2,α

(8.2.6)

b

for every (t, x) ∈ [0, T ]×Rd+ . On the other hand, since the integral of the function K+ (·, ·, xd ) over (0, ∞) × Rd−1 is independent of xd > 0, differentiating under the integral sign, we deduce that Z t Z Z ∞ Z ds Dd K+ (t − s, y 0 , xd ) dy 0 = ds Dd K+ (s, y 0 , xd ) dy 0 = 0 −∞

Rd−1

0

Rd−1

for t ∈ [0, T ]. Therefore, for t ∈ [0, T ], x0 ∈ Rd−1 and r > 0, we can estimate |Didd u2 (t, x0 , r)| Z t Z b y 0 ) − Di ψ(t, b x0 )] dy 0 = ds Dd K+ (t − s, x0 − y 0 , r)[Di ψ(s, Rd−1 Z−∞   Z ∞ 1 r b − s, y 0 ) − Di ψ(t, b x0 )] dy 0 ds = − K+ (s, x0 − y 0 , r)[Di ψ(t r 2s 0 Rd−1   Z t Z α r 1 + |K+ (s, y 0 , r)|[s 2 + |y 0 |α ] dy 0 . ≤||Di ψ||α/2,α ds r 2s d−1 0 R Using (8.2.3), it is easy to check that |Didd u2 (t, x0 , r)| ≤ Crα−1 ||Di ψ||α/2,α , so that Z z2 |Did u2 (t, x0 , z2 ) − Did u2 (t, x0 , z1 )| ≤ C||Di ψ||α/2,α rα−1 dr ≤ C||Di ψ||α/2,α |z2 − z1 |α z1

(8.2.7) for each t ∈ [0, T ], x0 ∈ Rd−1 and 0 < z1 < z2 . From (8.2.6) and (8.2.7), we conclude that α/2,α Did u2 ∈ Cb ((0, T ) × Rd+ ). Let us now consider the time derivative of u2 . Since the integral of the function Dt K(·, x) over (0, ∞) vanishes for every x ∈ Rd+ , using the dominated convergence theorem we can write Z ∞ Z b − s, y 0 ) − ψ(t, b y 0 )] dy 0 Dt u2 (t, x) = −2 ds Dt K(s, x0 − y 0 , xd )[ψ(t 0

Rd−1

216

Parabolic Equations in Rd+ with More General Boundary Conditions

for every (t, x) ∈ (0, T ) × Rd+ . Therefore, we can estimate |Dt u2 (t, x e) − Dt u2 (t, x)| Z ∞ Z 0 0 0 0 0 0 0 b b ds [Dt K(s, y − x e ,x ed ) − Dt K(s, y − x , xd )][ψ(t − s, y ) − ψ(t, y )] dy =2 d−1 0 R Z |x−ex|2 Z b − s, y 0 ) − ψ(t, b y 0 )| dy 0 ≤2 ds |Dt K(s, y 0 − x e0 , x ed ) − Dt K(s, y 0 − x0 , xd )||ψ(t d−1 0 R Z Z ∞ b − s, y 0 ) − ψ(t, b y 0 )| dy 0 ds |Dt K(s, y 0 − x e0 , x ed ) − Dt K(s, y 0 − x0 , xd )||ψ(t +2 |x−e x|2

Rd−1

=:L1 (t, x, x e) + L2 (t, x, x e)

(8.2.8)

for every t ∈ [0, T ] and x, x e ∈ Rd+ . Using (8.2.2) we infer that L1 (t, x, x e) ≤||ψ||(1+α)/2,1+α

|x−e x|2

Z

1+α 2

s

Z

Z ds Rd−1

0 |x−e x|2

≤C||ψ||(1+α)/2,0

s

α−d−1 2

Z

e−

ds

|z 0 |2 8s

dz 0

Rd−1

0 |x−e x|2

Z

(|Dt K(s, z 0 , x ed )| + |Dt K(s, z 0 , xd )|) dz 0

=C||ψ||(1+α)/2,0

s

α−2 2

ds

0

=C||ψ||(1+α)/2,0 |x − x e|α .

(8.2.9)

As far as the term L2 is concerned, we use (8.2.2) to estimate |Dt K(s, y 0 − x e0 , x ed ) − Dt K(s, y 0 − x0 , xd )| Z 1 d Dt K(s, y 0 − λe x0 − (1 − λ)x0 , λe xd + (1 − λ)xd ) dλ = 0 dλ Z 1 ≤|x − x e| |∇Dt K(s, y 0 − λe x0 − (1 − λ)x0 , λe xd + (1 − λ)xd )| dλ 0

1

Z e −x ≤C|x e|

s−

d+3 2

e−

|y 0 −λx e 0 −(1−λ)x0 |2 8s

dλ

0

and, using this estimate, we get Z |L2 (t, x, x e)| ≤C||ψ||(1+α)/2,0 |x − x e| =C||ψ||(1+α)/2,0 |x − x e|

∞

s |x−e x|2 Z ∞

s |x−e x|2 Z ∞

=C||ψ||(1+α)/2,0 |x − x e|

α−d−2 2

α−d−2 2

α−d−2 2

|x−e x|2

=C||ψ||(1+α)/2,0 |x − x e|

Rd−1 Z 1

ds

Z

1

Z

|y 0 −λx e 0 −(1−λ)x0 |2 8s

dλ

0

Z

e−

dλ

ds

e−

|y 0 −λx e 0 −(1−λ)x0 |2 8s

dy 0

Rd−1

e−

|z 0 |2 8s

dz 0

Rd−1

∞

s |x−e x|2

dy 0

0

s Z

Z ds

α−3 2

ds = C||ψ||(1+α)/2,0 |x − x e|α .

(8.2.10)

From (8.2.8)–(8.2.10), it follows that the function Dt u2 (t, ·) is α-H¨older continuous on Rd+ , α/2,0 uniformly with respect to t ∈ [0, T ]. To prove that Dt u2 belongs also to Cb ([0, T ] × Rd+ ), we rewrite Dt u2 as follows: Z t Z b y 0 ) − ψ(t, b y 0 )] dy 0 Dt u2 (t, x) = − 2 ds Dt K(t − s, x0 − y 0 , xd )[ψ(s, −∞

Rd−1

Semigroups of Bounded Operators and Second-Order PDE’s

217

for every (t, x) ∈ (0, T ) × Rd+ . Further, we fix t1 , t2 ∈ [0, T ] with t1 < t2 and split Dt u2 (t2 , x) − Dt u2 (t1 , x) Z Z t1 b y 0 ) − ψ(t b 1 , y 0 )] dy 0 ds Dt K(t1 − s, x0 − y 0 , xd )[ψ(s, =2 Rd−1

2t1 −t2 Z t2

−2

Z

b y 0 ) − ψ(t b 2 , y 0 )] dy 0 Dt K(t2 − s, x0 − y 0 , xd )[ψ(s,

ds Rd−1

2t1 −t2 Z 2t1 −t2

−2

Z

[Dt K(t2 − s, x0 − y 0 , xd ) − Dt K(t1 − s, x0 − y 0 , xd )]

ds Rd−1

−∞

b y 0 ) − ψ(t b 1 , y 0 )] dy 0 × [ψ(s, Z

2t1 −t2

−2

Z Rd−1

−∞

=

4 X

b 1 , y 0 ) − ψ(t b 2 , y 0 )] dy 0 Dt K(t2 − s, x0 − y 0 , xd )[ψ(t

ds Mj (t1 , t2 , x).

j=1

Arguing as in the case of the term H1 , we can easily show that |Mj (t1 , t2 , x)| ≤ α C||ψ||(1+α)/2,0 |t2 −t1 | 2 for j = 1, 2. Next, using (8.2.2) and observing that λt2 +(1−λ)t1 −s ≥ t1 − s for every λ ∈ [0, 1], we can prove that Z |Dt K(t2 − s, x0 − y 0 , xd ) − Dt K(t1 − s, x0 − y 0 , xd )| dy 0 Rd−1

Z ≤C(t2 − t1 )

1

Z

(λt2 + (1 − λ)t1 − s)−

dλ 0

d+4 2

e

− 8(λt

|y 0 |2

2 +(1−λ)t1 −s)

dy 0

Rd−1 − 52

≤C(t2 − t1 )(t1 − s)

.

Therefore, Z

2t1 −t2

|M3 (t1 , t2 , x)| ≤ C||ψ||(1+α)/2,0 (t2 − t1 )

(t1 − s)

α−4 2

α

ds = C||ψ||(1+α)/2,0 (t2 − t1 ) 2 .

−∞

Finally, we consider the term M4 . Changing the order of integration, we rewrite such a term as follows: Z e 1 , y 0 ) − ψ(t e 2 , y 0 )] dy 0 . M4 (t1 , t2 , x) = −2 K(2(t2 − t1 ), x0 − y 0 , xd )[ψ(t Rd−1

Hence, using (8.2.2) we deduce that |M4 (t1 , t2 , x)| ≤||ψ||(1+α)/2,0 (t2 − t1 )

1+α 2

Z

K(2(t2 − t1 ), y 0 , xd ) dy 0

Rd−1 α

=C||ψ||(1+α)/2,0 (t2 − t1 ) 2 . Combining all the above estimates, we conclude that α

|Dt u2 (t2 , x) − Dt u2 (t1 , x)| ≤ C||ψ||(1+α)/2,0 (t2 − t1 ) 2 . Since Dt u2 (0, ·) = 0, from the previous estimate it also follows that the function Dt u2 α/2,α is bounded on (0, T ) × Rd+ . We have so proved that Dt u2 ∈ Cb ((0, T ) × Rd+ ). Since

Parabolic Equations in Rd+ with More General Boundary Conditions Pd−1 α/2,α Ddd u2 = Dt u2 − j=1 Djj u2 , we infer that also Ddd u2 belongs to Cb ((0, T ) × Rd+ ). This completes the proof. 

218

Now, we can prove Theorem 8.0.2. Proof of Theorem 8.0.2 The uniqueness part of the proof and estimate (8.0.2) follow from estimate (8.1.1). Hence, we just need to prove the existence part. We fix f , g and ψ as in the statement and, to begin with, we show that we can confine ourselves to considering the case when f identically vanishes on Rd+ and ψ(0, ·), g(0, ·) vanish on Rd−1 and on Rd+ , 1+α/2,2+α respectively. For this purpose, we determine a function ζ ∈ Cb ((0, T ) × Rd+ ) such d that ζ(0, ·) = f , Dt ζ(0, ·) − Aζ(0, ·) = g(0, ·) on R+ . Function u solves the Cauchy problem (8.0.1) if and only if v = u − ζ solves the problem  Dt v(t, x) = Av(t, x) + gζ (t, x), t ∈ (0, T ], x ∈ Rd+ ,    ∂v (t, x0 , 0) = ψζ (t, x0 ), t ∈ (0, T ], x0 ∈ Rd−1 , a(x0 )v(t, x0 , 0) +  ∂η   v(0, x) = 0, x ∈ Rd+ , ∂ζ α/2,α (·, ·, 0). Clearly, gζ ∈ Cb ((0, T )×Rd+ ) ∂η (1+α)/2,1+α and ψζ ∈ Cb ((0, T ) × Rd−1 ), this latter one due to the compatibility condition satisfied by f and ψ. Moreover, gζ (0, ·) and ψζ (0, ·) identically vanish on Rd+ and on Rd−1 , respectively. The function ζ can be defined as the unique solution of the Cauchy problem ( Dt ζ(t, x) = Aζ(t, x) + h(t, x), t ∈ (0, T ], x ∈ Rd , (8.2.11) ζ(0, x) = (E2+α f )(x), x ∈ Rd , where gζ = g−Dt ζ +Aζ and ψζ = ψ−aζ(·, ·, 0)−

where h solves the Cauchy problem ( Dt h(t, x) = Ah(t, x), h(0, x) = Eα g(0, ·),

t ∈ (0, T ],

x ∈ Rd , x ∈ Rd+ ,

(8.2.12)

and Eα ∈ L(Cbα (Rd+ ), Cbα (Rd )), E2+α ∈ L(Cb2+α (Rd+ ), Cb2+α (Rd )) are the extension operators defined in Proposition B.4.1. Note that by Proposition 6.4.1, the Cauchy problem (8.2.12) admits a unique solution h ∈ Cb ([0, T ] × Rd ) ∩ C 1,2 ((0, T ) × Rd ). Actually, h belongs to C α/2,α ((0, T ) × Rd ). Indeed, by Theorem 6.4.3, we know that α

||h(t, ·)||Cbα (Rd ) + t1− 2 ||h(t, ·)||Cb2 (Rd ) ≤ C1 ||Eα g(0, ·)||Cbα (Rd ) ≤ C2 ||g||C α/2,α ((0,T )×Rd ) b

+

for every t ∈ (0, T ] and some positive constants C1 and C2 , independent of t and g. Hence, we infer that ||Dt h(t, ·)||∞ ≤ C3 tα/2−1 ||g||C α/2,α ((0,T )×Rd ) for every t ∈ (0, T ] and some b

+

positive constant C3 , independent of t and g, so that Z t α |h(t, x) − h(s, x)| = Dt h(r, x) dr ≤ C4 ||g||C α/2,α ((0,T )×Rd ) |t − s| 2 b s

for every s, t ∈ (0, T ], with s < t, and x ∈ Rd , where the constant C4 is independent of g, s, α/2,α t and x. From these last two estimates, it follows immediately that h ∈ Cb ((0, T ) × Rd ). Now, we can apply Theorem 6.2.6 to infer that the Cauchy problem (8.2.11) admits a

Semigroups of Bounded Operators and Second-Order PDE’s

219

1+α/2,2+α

unique solution ζ ∈ Cb ((0, T ) × Rd ). Clearly, ζ(0, ·) = f . Moreover, Dt ζ(0, ·) = Aζ(0, ·) + h(0, ·) = Aζ(0, ·) + g(0, ·) on Rd+ so that Dt ζ(0, ·) − Aζ(0, ·) = g(0, ·). To prove the solvability of problem (8.0.1) when f ≡ 0, g(0, ·) ≡ 0 and ψ(0, ·) ≡ 0, we use the method of continuity. For every σ ∈ [0, 1], we introduce the elliptic operator Aσ and the boundary differential operator Bσ defined by   ∂ ∂ − (1 − σ) . Aσ = σA + (1 − σ)∆, Bσ = σ aI + ∂η ∂xd Each operator Aσ = Tr(Qσ D2 ) + hb, ∇i + c is elliptic and hQσ (x)ζ, ζi = σhQ(x)ζ, ζi + (1 − σ)|ζ|2 ≥ (1 ∧ µ)|ζ|2 for every ζ ∈ Rd and σ ∈ [0, 1]. Moreover, the diffusion coefficients of the operator Aσ are clearly bounded and α-H¨ older on Rd+ and their H¨older norms can be bounded from above by a positive constant, independent of σ. Further, the function ησ = ση − (1 − σ)ed belongs to Cb1+α (Rd−1 , Rd ), satisfies the condition supRd−1 [σηd − (1 − σ)] < 0, and its C 1+α -norm can be bounded from above by a positive constant independent of σ. Thus, we can apply estimate (8.1.1), which shows that ||u||C 1+α/2,2+α ((0,T )×Rd ) ≤C(||Dt u − Aσ u||C α/2,α ((0,T )×Rd ) + ||Bσ u||C α/2,1+α ((0,T )×Rd−1 ) ) b

+

b

+

b

(8.2.13) 1+α/2,2+α

for every u ∈ Cb ((0, T )×Rd+ ) such that u(0, ·) ≡ 0 and Dt u(0, ·) ≡ 0, every σ ∈ [0, 1] and some positive constant C, independent of σ. 1+α/2,2+α We now introduce the Banach spaces Xα = {u ∈ Cb ((0, T ) × Rd+ ) : u(0, ·) ≡ 1+α/2,2+α 0, Dt u(0, ·) ≡ 0}, endowed with the norm of Cb ((0, T ) × Rd+ ), Yα = {g ∈ α/2,α α/2,α d Cb ((0, T ) × R+ ) : g(0, ·) ≡ 0}, endowed with the norm of Cb ((0, T ) × Rd+ ), and (1+α)/2,1+α Z1+α = {ψ ∈ Cb ((0, T ) × Rd+ ) : ψ(0, ·) ≡ 0}, endowed with the norm of (1+α)/2,1+α Cb ((0, T )×Rd+ ). For each σ ∈ [0, 1], we also introduce the operator Tσ : Xα → Yα × Z1+α (this latter space being endowed with the norm ||(g, ψ)||Yα ×Z1+α = ||g||Yα + ||ψ||Z1+α ), defined by T u = (Dt u − Aσ u, Bσ u) for every u ∈ Xα . Estimate (8.2.13) can be rewritten in terms of the operator Tσ as ||u||Xα ≤ C||Tσ u||Yα ×Z1+α . By Proposition 8.2.1, the operator T0 is invertible, so that all the operators Tσ are invertible as well. In particular, operator α/2,α T1 is invertible and this is equivalent to saying that, for every g ∈ Cb ((0, T ) × Rd+ ) and (1+α)/2,1+α ψ ∈ Cb ((0, T ) × Rd−1 ) such that g(0, ·) ≡ 0 and ψ(0, ·) ≡ 0, the Cauchy problem 1+α/2,2+α (8.0.1) is uniquely solvable in Cb ((0, T ) × Rd+ ). Thus, the proof is complete.  Remark 8.2.2 In Chapter 14 (see Theorem 8.4.1), using tools from the semigroup theory, we will prove an optimal spatial regularity result for solutions to problem (8.0.1).

8.3

Interior Schauder Estimates for Solutions to Parabolic Equations in Domains: Part III

As a byproduct of the results in this chapter, we prove some Schauder estimates up to a portion of ∂Rd+ , satisfied by the solutions to the differential equation Dt u = Au + g in (0, T ]×Rd+ , Such estimates are of local type, i.e., they allow to estimate the parabolic H¨older

220

Parabolic Equations in Rd+ with More General Boundary Conditions

norm of u and its derivatives in every compact subset of (0, T ] × Rd+ (or even of [0, T ] × Rd+ ) in terms of the sup-norm of the solution itself in a larger compact set. We assume the following conditions on the coefficients of the operator A = Tr(QD2 ) + hb, ∇i + c. Hypotheses 8.3.1 (i) The coefficients qij = qji , bj and c (i, j = 1, . . . , d) of the operator α A belong to Cloc (Rd+ ) for some α ∈ (0, 1); (ii) there exists a positive continuous function µ : Rd+ → R such that hQ(x)ξ, ξi ≥ µ(x)|ξ|2 for every ξ ∈ Rd and x ∈ Rd+ ; ∂ 1+α 1+α (Rd+ ; Rd ), belong to Cloc (Rd+ ) and to Cloc ∂η respectively. Moreover, a(x) ≥ 0 and ηd (x) < 0 for every x ∈ Rd+ , where ηd denotes the last component of the function η.

(iii) the coefficients of the operator B = aI +

Now, we state precisely the first result of this section. α/2,α

Theorem 8.3.2 Let g ∈ Cloc ((0, T ) × Rd+ ) be a given function and u ∈ C 1,2 ((0, T ) × Rd+ ) be a solution to the differential equation Dt u = Au + g. Assume that one of the following condition is satisfied: (i) Hypotheses 8.3.1(i)–(ii) are satisfied, u ∈ C((0, T ) × Rd+ ) and it identically vanishes on (0, T ) × ∂Rd+ ; (ii) Hypotheses 8.3.1 are satisfied, u ∈ C 0,1 ((0, T ) × Rd+ ) and Bu identically vanishes on (0, T ) × ∂Rd+ . 1+α/2,2+α

Then, u ∈ Cloc ((0, T ) × Rd+ ), for every 0 < τ0 < τ1 < T0 < T and every pair of bounded open sets Ω1 ⊂ Ω2 ⊂ Rd+ , such that d(Ω1 , Ω \ Ω2 ) > 0, there exists a positive constant C1 which depends on τ0 , τ1 , T0 Ω1 , Ω2 and, in a continuous way on the α-H¨ older norm of the coefficients of the operator A over Ω2 (and also the C 1+α -norm over Ω2 of the coefficients of the operator B under condition (ii)), the supremum over Ω2 of the function µ, such that ||u||C 1+α/2,2+α ((τ1 ,T0 )×Ω1 ) ≤ C1 (||u||C([τ0 ,T0 ]×Ω2 ) + ||g||C α/2,α ((τ0 ,T0 )×Ω2 ) ).

(8.3.1)

In particular, the constant C blows up as (τ1 − τ0 ) ∧ (T − T0 ) ∧ d(Ω1 , Ω \ Ω2 ) tends to zero. 1+α/2,2+α 2+α Finally, if u(0, ·) ∈ Cloc (Rd+ ), then u ∈ Cloc ([0, T ) × Rd+ ) and for every T0 ∈ (0, T ) there exists a positive constant C2 , which depends on T0 , Ω1 , Ω2 and, in a continuous way on the α-H¨ older norm of the coefficients of the operator A over Ω2 (and also the C 1+α norm over Ω2 of the coefficients of the operator B under condition (ii)), the supremum over Ω2 of the function µ, such that ||u||C 1+α/2,2+α ((0,T0 )×Ω1 ) ≤ C2 (||u(0, ·)||C 2+α (Ω2 ) + ||u||C([0,T0 ]×Ω2 ) + ||g||C α/2,α ((τ0 ,T0 )×Ω2 ) ). (8.3.2) In particular, the constant C2 blows up as (T − T0 ) ∧ d(Ω1 , Ω \ Ω2 ) tends to zero. Proof The main step of the proof of (8.3.1) is the case Ω1 = B+ (x0 , r) and Ω2 = B+ (x0 , 2r), where x0 in an arbitrary point of ∂Rd+ , r is a positive number and B+ (x0 , σ) = B(x0 , σ) ∩ Rd+ for every σ > 0. This is the content of Steps 1 and 2. Indeed, once this estimate is proved, a covering argument will allow us, in Step 3, to obtain estimate (8.3.1)

Semigroups of Bounded Operators and Second-Order PDE’s

221

in its full generality. Finally, in Step 4 we prove estimate (8.3.2). Throughout the proof, we denote by C a positive constant, which may vary from line to line and depends at most on τ0 , τ1 , T0 , Ω1 , Ω2 and, in a continuous way, on the supremum of µ over Ω2 and on the α-H¨ older norm of the coefficients of the operator A over Ω2 . Step 1. Here, we assume that condition (i) is satisfied. Fix 0 < τ0 < τ1 < T0 < T , r > 0 and let Ω1 and Ω2 be above. As in the proof of Theorem 6.2.9, we introduce the real sequences (rn ) and (tn ), defined by rn = (2 − 2−n )r and tn = τ0 + 2−n (τ1 − τ0 ), for every n ∈ N ∪ {0}. Note that the previous sequences are, respectively, increasing and decreasing. Next, we consider two functions ϕ, ϑ ∈ C ∞ (R) which satisfy the conditions χ[2,∞) ≤ ϕ ≤ χ[1,∞) and χ(−∞,1] ≤ ϑ ≤ χ(−∞,2] . For every n ∈ N ∪ {0}, we set     |x − x0 | − rn t − tn+1 , ϑn (x) = ϑ 1 + ϕn (t) = ϕ 1 + tn − tn+1 rn+1 − rn for every t ∈ R and x ∈ Rd+ . As it is easily seen, ϕn (t) = 1 if t ≥ tn and vanishes if t ≤ tn+1 . Similarly, ϑn (x) = 1 if x ∈ B(x0 , rn )+ and ϑn (x) = 0 if x ∈ / B(x0 , rn+1 )+ . Moreover, we e 2 ) + heb, ∇i + e c, where qeij = %qij + (1 − %)δij , ebj = %bj introduce the operator Ae = Tr(QD (i, j = 1, . . . , d), e c = %c, and % ∈ Cc∞ (Rd ) is compactly supported in Rd+ with 0 ≤ % ≤ 1 e extended in the trivial and % ≡ 1 in B(x0 , 2r)+ . Clearly, the coefficients of the operator A, way to the whole Rd+ and still denoted in the same way, belong to Cbα (Rd+ ) and coincide Pd with the coefficients of the operator A in B(x0 , 2r)+ . Moreover, i,j=1 qeij (x)ξi ξj ≥ µ0 |ξ|2 for all x, ξ ∈ Rd , where µ0 = 1 ∧ inf x∈Rd+ ∩supp% µ(x) > 0. Each function vn = uϕn ϑn belongs to Cb1,2 ([0, T ] × Rd+ ), vanishes at t = 0 and on [0, T ] × ∂Rd+ . Moreover, it satisfies the equation Dt vn = Avn + gn in (0, ∞) × Rd+ , where gn = ϕn ϑn g − ϕn u(Aϑn − cϑn ) − 2ϕn hQ∇ϑn , ∇x ui+ϕ0n ϑn u. Since the coefficients of the operator A satisfy Hypotheses 7.0.1, Theorem 7.0.2 guarantees that the Cauchy problem  t ∈ [0, T0 ], x ∈ Rd+ ,   Dt v(t, x) = Av(t, x) + gn (t, x), v(t, x) = 0, t ∈ [0, T0 ], x ∈ ∂Rd+ ,   v(0, x) = 0, x ∈ Rd+ , admits a unique solution in C 1,2 ([0, T0 ] × Rd+ ) ∩ Cb ([0, T ] × Rd+ ) which actually belongs to 1+α/2,2+α 1+α/2,2+α Cb ((0, T0 ) × Rd+ ). By uniqueness, function vn belongs to Cb ((0, T0 ) × Rd+ ) and ||vn ||C 1+α/2,2+α ((0,T0 )×Rd ) ≤ C||gn ||C α/2,α ((0,T0 )×Rd ) for every n ∈ N, where the conb

+

b

+

stant C is independent of n. In particular, taking n = 0, we deduce that u belongs to C 1+α/2,2+α ((τ1 , T0 ) × B+ (x0 , r)). The arbitrariness of τ1 and r imply that u ∈ C 1+α/2,2+α ((τ0 , T ) × B+ (x0 , 2r)). The same arguments as in the proof of Theorem 6.2.9 permit us to conclude that ||u||C 1+α/2,2+α ((τ1 ,T0 )×B+ (x0 ,r)) ≤ c(||u||C([τ0 ,T0 ]×B+ (x0 ,2r)) + ||g||C α/2,α ((τ0 ,T0 )×B+ (x0 ,2r)) ), i.e., estimate (8.3.1) with Ω1 = B+ (x0 , r) and Ω2 = B+ (x0 , 2r) follows. Step 2. We now assume that u ∈ C 0,1 ((0, T ) × Rd+ ) and Bu = 0 on ∂Rd+ . We consider the same sequence (ϕn ) as in Step 1 and modify the sequence (ϑn ), introducing a new ∂ϑn sequence (ϑn ) of compactly supported function such that = 0 identically vanishes on ∂η ∂ϑn,1 (0, T ) × ∂Rd+ . We can do it by setting ϑn = ϑn,1 − ϑn,2 , where ∂η     2|x − x0 | − rn 2|x − x0 | − rn − rn+1 ϑn,1 (x) = ϑ 1 + , ϑn,2 = ϑ 1 + . rn+1 − rn rn+1 − rn

222

Parabolic Equations in Rd+ with More General Boundary Conditions

Note that ϑn = 1 in B(x0 , rn ) and it vanishes outside B(x0 , rn+1 ). Moreover, there exists a positive constant K, independent of x0 and n such that ||ϑn ||Cbm (Rd ) ≤ K2(m+1)n for every m = 0, 1, 2, 3. Define the function vn as in Step 1. The choice of the cut-off function ϑn implies that vn ∈ C 1,2 ([0, T0 ] × ∂Rd+ ) and solves the Cauchy problem  t ∈ [0, T0 ], x ∈ Rd+ ,   Dt v(t, x) = Av(t, x) + gn (t, x), Bv(t, x) = 0, t ∈ [0, T0 ], x ∈ ∂Rd+ ,   v(0, x) = 0, x ∈ Rd . Since this problem admits a unique solution in C 1,2 ([0, T0 ]×∂Rd+ ) and this function actually belongs to C 1+α/2,2+α ((0, T0 ) × Rd+ ) (see Theorem 8.0.2). Hence, vn ∈ C 1+α/2,2+α ((0, T0 ) × Rd+ ). Moreover,  d X ζn ≤2 C ||vn+1 ||C α/2,α ((0,T0 )×Rd ) + ||Dj vn+1 ||C α/2,α ((0,T0 )×Rd ) 5n

b

b

j=1

+

 + ||g||C α/2,α ((τ0 ,T0 )×B+ (x0 ,2r)) ≤25n C εζn+1 + ε−(1+α) ||vn+1 ||∞ + ||g||C α/2,α ((τ0 ,T0 )×B+ (x0 ,2r))




for every ε > 0, where ζn = ||vn ||C 1+α/2,2+α ((0,T0 )×Rd ) . Hence, repeating the same arguments b

+

as in Step 1, taking η ∈ (0, 2−5(2+α) ) and ε = 2−5n C −1 η, estimate (8.3.1) follows with the above choices of I1 , I2 , Ω1 and Ω2 . Step 3. Fix two bounded open sets Ω1 ⊂ Ω2 ⊂ Rd+ such that d = d(Ω1 , Rd+ \ Ω2 ) > 0. We first assume that ∂Ω1 ∩ ∂Rd+ 6= ∅ and denote by r any positive number which is less than the supremum of the distances of the points of Ω from ∂Rd+ and introduce the sets Ω0j,r = {x ∈ Ωj : d(x, ∂Rd+ ∩ ∂Ω1 ) > r},

Ω00j,r = {x ∈ Ωj : d(x, ∂Rd+ ∩ ∂Ω1 ) < r},

for j = 1, 2. Observe that Ω01,2r ⊂ Ω02,r and d(∂Ω01,2r , ∂Ω02,r ) > 0. Indeed, suppose that this is not the case. Then, there exists x ∈ ∂Ω01,2r ∩ ∂Ω02,r . Since ∂Ω01,2r ⊂ {x ∈ Ω1 : d(x, ∂Rd+ ∩ ∂Ω1 ) = 2r} ∩ {x ∈ ∂Ω1 : d(x, ∂Rd+ ∩ ∂Ω1 ) ≥ 2r}, ∂Ω02,r ⊂ {x ∈ Ω2 : d(x, ∂Rd+ ∩ ∂Ω1 ) = r} ∩ {x ∈ ∂Ω2 : d(x, ∂Rd+ ∩ ∂Ω1 ) ≥ r}, it follows that if x ∈ {x ∈ Ω1 ∩ ∂Ω2 : d(x, ∂Rd+ ∩ ∂Ω1 ) = 2r} ∪ {x ∈ ∂Ω1 ∩ ∂Ω2 : d(x, ∂Rd+ ∩ ∂Ω1 ) ≥ 2r} Both the two cases lead us to a contradiction. Indeed, in the first case, x ∈ Ω1 and x ∈ Rd+ \ Ω2 and this contradicts the conditions d(Ω1 , Rd+ \ Ω2 ) > 0. Similarly, in the second case x ∈ Rd+ \ Ω2 . Hence, d(x, Ω1 ) ≥ d(Ω1 , Rd+ \ Ω2 ) = d > 0. Taking a sequence (xn ) ∈ Ω1 which converges to x as n tends to ∞, we get 0 = lim |x − xn | ≥ d(x, Ω1 ) ≥ d, n→∞ which, clearly, is a contradiction. We can thus apply the interior estimates in Theorem 6.2.9 to infer that ||u||C 1+α/2,2+α ((τ1 ,T0 )×Ω01,r ) ≤C(||u||C([τ0 ,T0 ]×Ω0

2,r )

+ ||g||C α/2,α ((τ0 ,T0 )×Ω02,r ) )

≤C(||u||C([τ0 ,T0 ]×Ω2 ) + ||g||C α/2,α ((τ0 ,T0 )×Ω2 ) ). Now, we set δ = dist(Ω1 , Rd+ \ Ω2 ) and observe that [ Ω001,δ/2 ⊂ B+ (x0 , δ/2) ⊂ Ω002,δ . x0 ∈∂Ω1 ∩∂Rd +

(8.3.3)

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223

The first inclusion is trivial. As far as the other one is concerned, we observe that, if x ∈ B+ (x0 , δ/2), then, clearly, d(x, ∂Ω1 ∩ ∂Rd+ ) < δ/2. To prove that x ∈ Ω2 , we observe that d(∂Ω1 , Rd+ \ Ω2 ) ≥ d > δ, whereas d(x, ∂Ω1 ) ≤ |x − x0 | < δ/2. Hence, x ∈ Ω2 . Next, we note that d(Ω001,δ/2 , Rd+ \ Ω001.δ ) > 0. By contradiction, let us assume that (xn ) ⊂ Ω001,δ/2 and (yn ) ⊂ Rd+ \ Ω002,δ two sequences such that |xn − yn | tends to zero as n tends to ∞. We claim that yn ∈ Rd+ \ Ω002,δ at least for n sufficiently large. Assume, by contradiction that yn ∈ Ω2 \ Ω002,δ . Then, d(yn , ∂Ω1 ∩ ∂Rd+ ) ≥ δ. On the other hand, d(xn , ∂Ω1 ∩ ∂Rd+ ) ≤ δ/2. Note that, up to a subsequence, xn converges to a point x ∈ Rd+ and, consequently, also yn converges to x. Letting n tend to ∞ in the above estimates the contradiction follows. Since Ω01,δ is a compact set, we can find out m ∈ N and points x1 , . . . , xm ∈ K1 such m [ that Ω01,δ/2 ⊂ B+ (xj , δ/2). By Step 1, with r = δ, we get j=1

||u||C 1+α/2,2+α ((τ1 ,T0 )×B+ (xk ,δ/2)) ≤C(||u||C([τ0 ,T0 ]×B+ (xk ,δ)) + ||g||C α/2,α ((τ0 ,T0 )×B+ (xk ,δ)) ) ≤C(||u||C([τ0 ,T0 ]×Ω2 ) + ||g||C α/2,α ((τ0 ,T0 )×Ω2 ) )

(8.3.4)

for every j, k = 1, . . . , m. Since, ||u||C 1+α/2,2+α ((τ1 ,T0 )×Ω00

) 1,δ/2

≤

m X

||u||C 1+α/2,2+α ((τ1 ,T0 )×B+ (xk ,δ/2)) ,

k=1

from (8.3.4) we conclude that ||u||C 1+α/2,2+α ((τ1 ,T )×Ω00

1,δ/2

)

≤ C(||u||C([τ0 ,T0 ]×Ω2 ) + ||g||C α/2,α ((τ0 ,T0 )×Ω2 ) ).

(8.3.5)

Combining (8.3.3) (with r = δ/4) and (8.3.5), the assertion follows at once in this case. e 2 the open Suppose now that ∂Ω1 ∩ ∂Rd+ = ∅. Then, δ := d(∂Ω1 , ∂Rd+ ) > 0. Denote by Ω d e 2 . Moreover, d(∂Ω1 , ∂Ω2 ) > 0. set of all x ∈ Ω2 such that d(x, ∂R+ ) > δ/2. Clearly, Ω1 ⊂ Ω Indeed, suppose that there exists x ∈ ∂Ω1 ∩ ∂Ω2 . Then, x ∈ ∂Ω1 , so that d(x, ∂Rd+ ) ≥ δ. e 2 it thus follows that x ∈ ∂Ω2 ∩ Rd+ . But this contradicts the condition Since x ∈ ∂ Ω d d(Ω1 , R+ \ Ω2 ) > 0. Hence, applying Theorem 6.2.9, we deduce that ||u||C 1+α/2,2+α ((τ1 ,T0 )×Ω1 ) ≤C(||u||Cb ((τ0 ,T0 )×Ω e 2 ) + ||g||C α/2,α ((τ0 ,T0 )×Ω e2)) ≤C(||u||C([τ0 ,T0 ]×Ω2 ) + ||g||C α/2,α ((τ0 ,T0 )×Ω2 ) ) and the assertion follows. Step 4. Here, we prove estimate (8.3.2). It suffices to consider the case Ω1 = B+ (x0 , r) and Ω2 = B+ (x0 , 2r) since then, the arguments in Step 3 can be repeated with no changes. For every n ∈ N the function vn in Step 1 belongs to Cb1,2 ([0, T ] × Rd ), Moreover, vn (0, ·) = u(0, ·)ϑn belongs to Cb2+α (Rd ). Hence, by Theorems 7.0.2 and 8.0.2 the function 1+α/2,2+α vn belongs to Cb ((0, T0 ) × Rd ) and ||vn ||C 1+α/2,2+α ((0,T0 )×Rd ) ≤ C(||ϑn u(0, ·)||C 2+α (Rd ) + ||gn ||C α/2,α ((0,T )×Rd ) ) b

b

b

for every n ∈ N ∪ {0}. In particular, taking n = 0, we deduce that u belongs to C 1+α/2,2+α ((0, T0 ) × B+ (x0 , r)). The arbitrariness of r imply that function u belongs to C 1+α/2,2+α ((0, T0 ) × B+ (x0 , σ)) for every σ > 0. Since ||ϑn u(0, ·)||C 2+α (Rd ) ≤ 23n C||u(0, ·)||C 2+α (B+ (x0 ,2r)) for every n ∈ N, the same argub ments used in Steps 1 and 2 show that ζn ≤ 24n C(εζn+1 + ε−(1+α) ||vn+1 ||∞ + ||u(0, ·)||C 2+α (B+ (x0 ,2r)) + ||g||C α/2,α ((0,T0 )×B+ (x0 ,2r)) )

224

Parabolic Equations in Rd+ with More General Boundary Conditions

if condition (i) is satisfied and ζn ≤ 25n C(εζn+1 + ε−(1+α) ||vn+1 ||∞ + ||u(0, ·)||C 2+α (B+ (x0 ,2r)) + ||g||C α/2,α ((0,T0 )×B+ (x0 ,2r)) ) otherwise. Here, ζm = ||vm ||C 1+α/2,2+α ((0,T0 )×B+ (x0 ,rm )) for every m ∈ N. Starting from these two estimates, formula (8.3.2) can be easily proved. 

8.4

More on the Cauchy Problem (8.0.1)

In this section, we state two relevant theorems. The first one is an optimal spatial regularity result for solutions to problem (8.0.1) and the other one shows that the more the coefficients of the operators A, B and the data f , g and ψ are smooth, the more the classical solution u to problem (7.0.1) is itself smooth. Theorem 8.4.1 Let Hypotheses 8.0.1 be satisfied. Suppose that f belongs to Cb2+α−2θ (Rd+ ), satisfies the condition Bf ≡ 0 on ∂Rd+ if 2 + α − 2θ > 1 and g ∈ C((0, T ] × Rd ) is such that supt∈(0,T ] tθ ||g(t, ·)||Cbα (Rd ) < ∞ for some θ ∈ (0, 1). Then, the Cauchy problem (8.0.1) admits a unique classical solution u, which is bounded in [0, T ] × Rd . In addition, u(t, ·) ∈ Cb2+α (Rd+ ) for every t ∈ (0, T ] and sup tθ (||u(t, ·)||C 2+α (Rd ) + ||Dt u(t, ·)||Cbα (Rd+ ) ) + ||u||C 0,2+α−2θ ([0,T ]×Rd ) b

t∈(0,T ]

+

+

b

  ≤C0 ||f ||C 2+α−2θ (Rd ) + sup tθ ||g(t, ·)||Cbα (Rd+ ) , b

+

t∈(0,T ]

for some positive constant C0 , independent of u, f and g. In particular, if f and g belong to Cb2+α (Rd+ ) and C([0, T ] × Rd+ ), respectively, and supt∈[0,T ] ||g(t, ·)||Cbα (Rd+ ) < ∞, then, u belongs to Cb1,2 ([0, T ] × Rd+ ), u(t, ·) ∈ Cb2+α (Rd+ ) for every t ∈ [0, T ] and   sup ||u(t, ·)||C 2+α (Rd ) + sup ||Dt u(t, ·)||Cbα (Rd+ ) ≤ C1 ||f ||C 2+α (Rd ) + sup ||g(t, ·)||Cbα (Rd+ ) t∈[0,T ]

b

+

b

t∈[0,T ]

+

t∈(0,T ]

for some positive constant C1 , independent of u, f and g. Proof It suffices to adapt the arguments used in the proof of Theorem 7.4.1 (see Section 14.2.1), with no relevant changes.  Theorem 8.4.2 Fix k ∈ N and let Hypotheses 8.0.1 be satisfied with the coefficients of the operator A in Cbk+α (Rd+ ) and the coefficients of the operator B in Cbk+2+α (Rd+ ) and in Cbk+2+α (Rd+ ; Rd ), respectively. Further, suppose that f ∈ Cb2+k+α (Rd+ ), g ∈ (k+α)/2,k+α Cb ((0, T ) × Rd+ ), ψ ∈ C (k+1+α)/2,k+1+α ((0, T ) × Rd−1 ) satisfy the compatibility conditions  Bf (0, x0 , 0) = ψ(0, x0 ),       k  X   k−j j 0  Dtk ψ(0, x0 ) = B (Ak f )(x0 , 0) + (A Dt g(0, ·))(x , 0) , if k is odd, j=0     k−1  X   k−1 0 k−1 0 k−j−1 j 0  (A Dt g(0, ·))(x , 0) , if k is even, f )(x , 0) +   Dt ψ(0, x ) = B (A j=0

Semigroups of Bounded Operators and Second-Order PDE’s

225

1+α/2,2+α

for every x0 ∈ Rd−1 . Then, the solution u ∈ Cb ((0, T ) × Rd+ ) to problem (8.0.1) (k+2+α)/2,k+2+α actually belongs to Cb ((0, T ) × Rd+ ) and there exists a positive constant C, independent of u, f , g and ψ, such that ||u||C (k+2+α)/2,k+2+α ((0,T )×Rd ) +

b

≤C(||f ||C k+2+α (Rd ) + ||g||C (k+α)/2,k+α ((0,T )×Rd ) + ||ψ||C (k+1+α)/2,k+1+α ((0,T )×Rd−1 ) ). b

+

+

b

b

Proof The assertion can be obtained adapting the arguments used in the proof of Theorem 7.4.3 (see Section 14.2.1). Hence, the details are omitted. 

8.5

The Associated Semigroup

In this subsection, under Hypothesis 8.0.1, we prove that the Cauchy problem  t ∈ (0, ∞), x ∈ Rd+ ,   Dt u(t, x) = Au(t, x), Bu(t, x) = 0, t ∈ (0, ∞), x ∈ ∂Rd+ ,   u(0, x) = f (x), x ∈ Rd+ ,

(8.5.1)

is governed in Cb (Rd+ ) (resp. in Cb (Rd+ )) by a semigroup of bounded operators. For this purpose, we begin by showing that for every f ∈ Cb (Rd+ ) (resp. f ∈ Cb (Rd+ )) the Cauchy problem (8.5.1) admits a unique classical solution. In the proof of Theorem 8.5.2, we will use the following lemma. Lemma 8.5.1 For every f ∈ BU C(Rd+ ) there exists a sequence (fn ) ⊂ Cb2+α (Rd+ ) such that Bfn = 0 for every n ∈ N, and fn converges to f uniformly in Rd+ as n tends to ∞ Proof We fix f ∈ BU C(Rd+ ), extend it to Rd even with respect to the last variable and denote by fe the so extended function. Next, for every n ∈ N, we introduce the function hn defined by Z hn (x) = ϕn (y)fe (x − y) dy, x ∈ Rd+ , Rd

where ϕn (x) = nd ϕ(nx) and ϕ ∈ Cc∞ (Rd ) has L1 (Rd )-norm equal to one. Each function hn belongs to Cb2+α (Rd+ ) and converges to f uniformly on Rd+ . We observe that every function h ∈ Cb2+α (Rd+ ) can be approximated in the sup-norm of Rd+ by a sequence of function (fn ) ⊂ Cb2+α (Rd+ ) such that Bfn = 0 on ∂Rd+ . We can do it as follows. We introduce a function % ∈ Cb2+α ([0, ∞)) such that χ[0,1] ≤ % ≤ χ[0,2] and set %n (x) = %(nx) for every x ∈ [0, ∞). For every n ∈ N, we define the function fn by setting fn = h − (EBh)%n , where E is the operator in Proposition B.4.12(i), which satisfies the ∂Eψ = ψ on ∂Rd+ for every ψ ∈ Cb (Rd−1 ) (note that, by Hypothesis 8.0.1(iii), condition ∂η the function ηd belongs to Cb1+α (Rd−1 ) and its supremum over Rd+ is negative). It follows that BE(Bh) = Bh on ∂Rd+ , so that Bfn = 0 on ∂Rd+ . Moreover, fn converges to h uniformly on Rd+ . Indeed, by the definition of E (see the proof of Proposition B.4.12), we get Z 0 |fn (x) − h(x)| = ζ(xd )%n (xd ) ψ(x − xd y)ϑ(y) dy , x ∈ Rd+ , Rd−1

226

Parabolic Equations in Rd+ with More General Boundary Conditions

where ϑ ∈ Cc∞ (Rd−1 ) and ζ ∈ C ∞ ([0, ∞)) satisfy the conditions ||ϑ||L1 (Rd−1 ) = 1, and ζ(s) = s for every s ∈ [0, 1/2] and ζ(s) = 0 for every s ≥ 1. Hence, ||fn − h||∞ ≤ sup |ζ(s)%n (s)|||ψ||∞ ≤ s∈[0,∞)

sup

|ζ(s)|||ψ||∞ ≤

s∈[0,2n−1 ]

2 ||ψ||∞ n

for n ≥ 4, so that fn converges to h uniformly on Rd+ . Now, we are almost done. Given f ∈ BU C(Rd ) and n ∈ N, we consider a function hn ∈ Cb2+α (Rd+ ) such that ||hn − f ||∞ ≤ n−1 . Then, we introduce a function fn ∈ Cb2+α (Rd+ ) such that Bfn = 0 on ∂Rd+ and ||fn − hn || ≤ n−1 . Summing up ||fn − f ||∞ ≤ ||fn − hn ||∞ + ||hn − f ||∞ ≤ 2n−1 so that (fn ) is the sequence that we are looking for.  Theorem 8.5.2 Let Hypotheses 8.0.1 be satisfied. Then, the following properties are satisfied. (i) for every f ∈ Cb (Rd+ ), the Cauchy problem (8.5.1) admits a unique solution u ∈ C([0, ∞) × Rd+ ) ∩ C 1+α/2,2+α ((0, ∞) × Rd+ ), which is bounded with respect to the supnorm in (0, T ) × Rd+ for every T > 0. (ii) for every f ∈ Cb (Rd+ ), the Cauchy problem (8.5.1) admits a unique solution u ∈ C([0, ∞) × Rd+ ) ∩ C 1+α/2,2+α ((0, ∞) × Rd+ ), which is bounded with respect to the supnorm in (0, T ) × Rd+ for every T > 0. In both the cases, for every T > 0 there exists a positive constant CT , independent of u and f , such that ||u||Cb ((0,T )×Rd+ ) ≤ CT ||f ||∞ . (8.5.2) Proof (i) To begin with, we assume that f ∈ BU C(Rd+ ) and we consider a sequence (fn ) ∈ Cb2+α (Rd+ ) which converges to f uniformly on Rd+ , with the properties in Lemma 8.5.1. By Theorem 8.0.2, for every n ∈ N, the Cauchy problem (8.5.1), with f being replaced 1+α/2,2+α by fn , admits a unique solution un which belongs to Cb ((0, T ) × Rd+ ) for every T > 0. Moreover, Proposition 4.1.9 (see estimate (4.1.5)) shows that for every T > 0 there exists a positive constant KT , independent of n, such that ||un ||C ([0,T ]×Rd ) ≤ KT ||f ||∞ for b

+

every n ∈ N. From Theorem 8.3.2 and this estimate, we deduce that, for every m ∈ N, there exist two positive constants C1 and C2 , depending on m, but being independent of n, such that ||un ||C 1+α/2,2+α ((m−1 ,m)×B+ (0,m)) ≤ C1 ||un ||C([(2m)−1 ,m]×B+ (0,2m)) ≤ C2 ||f ||∞ . Fix m ∈ N. By a compactness argument, up to a subsequence we can assume that un converges to a function u(m) ∈ C 1+α/2,2+α ((m−1 , m) × B+ (0, m)). Note that u(m) = u(m+1) on [m−1 , m] × B+ (0, m). Indeed, we can determine a common subsequence of (un ) which converges to u(m) and to u(m+1) in [m−1 , m] × B+ (0, m). Hence, we can glue the functions 1+α,2+α u(m) (m ∈ N) and determine a function u ∈ Cloc ((0, ∞) × Rd+ ). Since Bun = 0 on d d [0, ∞)) × ∂R+ , it follows that Bu = 0 on (0, ∞) × ∂R+ . To conclude the proof, we need to show that u can be extended by continuity on [0, ∞) × Rd+ and u(0, ·) = f on Rd+ . For this purpose, we use estimate (4.1.5) to get ||un − um ||C ([0,T ]×Rd ) ≤ C3 ||fn − fm ||∞ for some b

+

positive constant C3 , independent of m and n. Since fn converges to f uniformly on Rd+ , (un ) is a Cauchy sequence in Cb ([0, T ] × Rd+ ) for every T > 0. Hence, it converges uniformly

Semigroups of Bounded Operators and Second-Order PDE’s

227

in [0, T ] × Rd+ to a function v ∈ Cb ([0, T ] × Rd+ ), which satisfies the condition v(0, ·) = f , since un (0, ·) = fn on ∂Rd+ . Clearly, v = u on (0, ∞) × Rd+ . Estimate (8.5.2) now follows from (4.1.5) To complete the proof, we consider the case when f is not uniformly continuous over Rd+ . We consider a sequence (fn0 ) ∈ BU C(Rd+ ) which converges to f locally uniformly on Rd+ and such that ||fn0 ||∞ ≤ ||f ||∞ . We can determine such a sequence, as in the first part of Lemma 8.5.1 by setting Z fn0 (x) = ϕn (y)fe (x − y) dx, x ∈ Rd+ , n ∈ N, Rd

where ϕn (x) = nd ϕ(nx) and ϕ ∈ Cc∞ (Rd ) has L1 (Rd )-norm equal to one. We still denote by u0n the solution to problem (8.5.1), with f being replaced by fn0 . Repeating the same arguments as in the first part of the proof, we can show that there exists a function u ∈ 1+α Cloc ((0, ∞)×Rd+ ) such that u0n converges to u in C 1+α/2,2+α ((a, b)×Ω) for every 0 < a < b and every bounded open set Ω ⊂ Rd+ . Since each function un satisfies estimate (8.5.2), with f being replaced by fn0 , and ||fn0 ||∞ ≤ ||f ||∞ , it follows immediately that function u satisfies (8.5.2). We now distinguish two cases. The potential is nonpositive on Rd+ . We fix r > 0 and a function ϑ ∈ Cc∞ (Rd+ ), such that χB+ (0,r) ≤ ϑ ≤ χB+ (0,2r) . For every n ∈ N, we split fn = ϑfn + (1 − ϑ)fn . Since ϑfn and (1 − ϑ)fn belong to BU C(Rd+ ), by the previous result, the Cauchy problem (8.5.1), with f being replaced by ϑfn (resp. (1 − ϑ)fn ), admits a unique solution vn (resp. wn ) which 1+α/2,2+α belongs to Cloc ((0, ∞) × Rd+ ) ∩ C([0, ∞) × Rd+ ) and is bounded in (0, T ) × Rd+ for every T > 0. Clearly, by uniqueness it holds that u0n = vn + wn for every n ∈ N. Since fn converges to f locally uniformly on Rd+ , the function vn converges uniformly on [0, ∞) × Rd+ to a function v which is clearly continuous in this set and satisfies the condition v(0, ·) = ϑf . As far as the function wn is concerned, we claim that |wn | ≤ ||fn ||∞ (1 − uϑ ) in (0, ∞) × Rd+ , where by uϑ we have denoted the solution to problem (8.5.1) with f being replaced by ϑ. To prove the claim, we observe that the function zn = wn − ||fn ||∞ (1 − uϑ ) satisfies the differential equations Dt zn − Azn = ||fn ||∞ c ≤ 0 on (0, ∞) × Rd+ , Bzn = −a||fn ||∞ ≤ 0 on (0, ∞) × ∂Rd+ , and zn (0, ·) = (1 − ϑ)(fn − ||fn ||∞ ) ≤ 0. Hence, we can apply Corollary 4.1.8 and conclude that zn ≤ 0 on (0, ∞) × Rd+ , i.e., wn ≤ ||fn ||∞ (1 − uϑ ) on (0, ∞) × Rd+ . Replacing fn with −fn , we conclude that −wn ≤ ||fn ||∞ (1 − uϑ ). The claim is proved. Using the claim we can show that |u(t, x) − f (x)| = lim |un (t, x) − f (x)| ≤ |v(t, x) − f (x)| + ||f ||∞ (1 − uϑ (t, x)), n→∞

for every (t, x) ∈ (0, T ) × Rd+ . Fix x0 ∈ B+ (0, r). Then ϑ(x0 ) = 1 and, consequently, lim (t,x)→(0,x0 )

|u(t, x) − f (x)| ≤

|v(t, x) − f (x)|   + C1 ||f ||∞ 1 − lim uϑ (t, x) = 0. lim

(t,x)→(0,x0 )

(t,x)→(0,x0 )

Hence, u can be extended by continuity at (0, x0 ) and u(0, x0 ) = f (x0 ). The arbitrariness of r > 0 yields the continuity of u on [0, ∞) × Rd+ . The general case. Denote by c0 the supremum over Rd+ of the function c. Then, the 1+α/2,2+α function (t, x) 7→ u0n (t, x) = e−c0 t un (t, x) belongs to Cb ((0, T )×Rd+ ) for every T > 0 and solves the Cauchy problem (8.5.1) with f being replaced by fn and with the operator A being replaced by the operator A−c0 . Clearly, u0n converges locally uniformly on (0, ∞)×Rd+

228

Parabolic Equations in Rd+ with More General Boundary Conditions

to the function u∗ , defined by u∗ (t, x) = e−c0 t u(t, x) for every (t, x) ∈ (0, ∞) × Rd+ . Since the potential of the operator A − c0 is nonpositive on Rd+ , function u∗ can be extended by continuity up to t = 0, setting u∗ (0, ·) = f . Consequently, the function u can be extended by continuity to {0} × Rd+ by setting u(0, ·) = f . (ii) The proof is completely similar to the second part of the proof of property (i). In the case when c ≤ 0, the sequence (fn0 ) now converges locally uniformly on Rd to f . Since each function u0n satisfies estimate (8.5.2), with f replaced by fn0 and ||fn0 ||∞ ≤ ||f ||∞ , the convergence of u0n to u in (0, ∞) × Rd+ implies that function u satisfies estimate (8.5.2). Next, we fix an open set Ω ⊂ Rd+ , with closure contained in Rd+ , and a function ϑ ∈ ∞ C (Rd+ ) such that χΩ ≤ ϑ ≤ 1 on Rd+ . Then, we split u0n = vn + wn . Since fn ϑ converges uniformly in Rd+ to f ϑ, the function vn converges uniformly on (0, 1) × Rd+ to a function v. Now, repeating verbatim the arguments in the last part of the proof of property (i) the assertion follows.  As a consequence of the previous theorems, we can define a semigroup associated with the pair (A, B) in BU C(Rd ), in Cb (Rd+ ) and in Cb (Rd+ ). Theorem 8.5.3 Under Hypotheses 8.0.1, we can associate a semigroup of bounded operator {T (t)} with the pair (A, B) in Cb (Rd+ ) (resp. Cb (Rd+ )). Moreover, each operator T (t) maps BU C(Rd+ ) into itself and the restriction of the semigroup to BU C(Rd+ ) gives rise to a strongly continuous semigroup. Finally, if (fn ) ⊂ Cb (Rd+ ) is a sequence which converges to a function f ∈ Cb (Rd+ ), locally uniformly in Rd+ , then, for every t > 0, the function T (t)fn converges to T (t)f locally uniformly in Rd+ . Proof For every t > 0 and f ∈ Cb (Rd+ ) (resp. f ∈ Cb (Rd+ )) let us set T (t)f = u(t, ·), 1+α/2,2+α where u is the solution in C([0, ∞) × Rd+ ) ∩ Cloc ((0, ∞) × Rd+ ) (resp. in C([0, ∞) × 1+α/2,2+α ((0, ∞) × Rd+ )) of the Cauchy problem (8.5.1). Estimate (8.5.2) shows Rd+ ) ∩ Cloc that each operator bounded from Cb (Rd+ ) into Cb (Rd+ ). Moreover, by Theorem 8.0.2, T (t)g belongs to Cb2+α (Rd+ ) for every t > 0 and g ∈ Cb2+α (Rd+ ) such that Bg = 0 on ∂Rd+ . Hence, T (t)g ∈ BU C(Rd+ ). Take f ∈ BU C(Rd+ ) and a sequence (fn ) ⊂ Cb2+α (Rd+ ) converging to f uniformly in Rd+ (see Lemma 8.5.1). As n tends to ∞, T (·)fn converges to T (·)f , uniformly on [0, σ] × Rd+ for every σ > 0, and each function T (·)fn is uniformly continuous on [0, σ] × Rd+ for every σ as above. Hence, the function T (t)f is uniformly continuous on Rd+ for every t > 0 and T (t)f tends to f uniformly in Rd+ as t tends to zero. As a byproduct, the restriction of the semigroup {T (t)} to BU C(Rd ) is a strongly continuous semigroup. The last assertion of the theorem follows arguing as in the second part of the proof of Theorem 8.5.2. 

8.6

Exercises

1. Prove Theorem 8.4.1. 2. Prove Theorem 8.4.2.

Chapter 9 Parabolic Equations in Bounded Smooth Domains Ω

In this chapter, we complete the analysis of parabolic Cauchy problems with bounded coefficients, dealing with the Cauchy-Dirichlet problem  t ∈ [0, T ], x ∈ Ω,   Dt u(t, x) = Au(t, x) + g(t, x), u(t, x) = h(t, x), t ∈ [0, T ], x ∈ ∂Ω, (9.0.1)   u(0, x) = f (x), x ∈ Ω, and the Cauchy problem    Dt u(t, x) = Au(t, x) + g(t, x), Bu(t, x) = h(t, x),   u(0, x) = f (x),

t ∈ [0, T ], x ∈ Ω, t ∈ [0, T ], x ∈ ∂Ω,

(9.0.2)

x ∈ Ω,

set in a bounded, smooth domain1 Ω ⊂ Rd , A is the second-order differential operator, defined by Aζ(x) =

d X i,j=1

qij (x)Dij ζ(x) +

d X

bj (x)Dj ζ(x) + c(x)ζ(x),

x ∈ Ω,

j=1

on smooth enough functions ζ : Ω → R and B is the first-order differential operator Bζ(x) = a(x)ζ(x) +

∂ζ (x) ∂η

on smooth enough functions ζ : ∂Ω → R. In Section 9.1, under suitable assumptions on the coefficients of the operators A, B and on Ω, we prove optimal Schauder estimates for solutions to the Cauchy problems (9.0.1) and (9.0.2). More precisely, we prove that if f ∈ C 2+α (Ω), g ∈ C α/2,α ((0, T ) × Ω), h ∈ C 1+α/2,2+α ([0, T ] × ∂Ω) (resp. h ∈ C (1+α)/2,2+α ([0, T ] × ∂Ω)) satisfy the due compatibility conditions, then problem (9.0.1) (resp. problem (9.0.2)) admits a unique solution u ∈ C 1+α/2,2+α ((0, T ) × Ω). Through local charts, each of the previous problems is split into two problems: one “far away” from the boundary of Ω, to which the results in Theorem 6.2.1 can be applied, and the other one “close” to the boundary of Ω, to which we can apply the results in Theorems 7.0.2 and 8.0.2. A partition of the unity allows us to glue the solutions to these new problems and provide us a solution to problem (9.0.1) (resp. (9.0.2)). In Section 9.2 we conclude the analysis of interior Schauder estimates for solutions of parabolic problems, providing suitable estimates up to the boundary of Ω. This result is based on the interior Schauder estimates in Chapters 6 and 8. Section 9.3 contains further regularity results for the classical solutions to the Cauchy 1 By

domain, we mean an open connected set.

229

230

Parabolic Equations in Bounded Smooth Domains Ω

problems (9.0.1) and (9.0.2), where by classical solution to problem (9.0.1) (resp. (9.0.2)) we mean a solution which is continuous on ([0, T ] × Ω) ∪ (0, T ] × Ω (resp. continuous on [0, T ] × Ω) and belongs to C 1,2 ((0, T ] × Ω) (resp. to C 0,1 ((0, T ] × ∂Ω) ∩ C 1,2 ((0, T ] × Ω)). Finally, as in all the previous chapters, in Section 9.4 we show that we can associate a semigroup of boundary operators in C(Ω) and in Cb (Ω) with the operator A with homogeneous Dirichlet boundary conditions (resp. with homogeneous first-order boundary conditions).

9.1

Optimal Schauder Estimates for Solutions to Problems (9.0.1) and (9.0.2)

Throughout this section, we assume the following conditions on Ω and on the coefficients of the operator A. Hypotheses 9.1.1 (i) Ω is a bounded domain of class C 2+α for some α ∈ (0, 1) (see Definition B.2.1); (ii) the coefficients qij = qji , bj (i, j = 1, . . . , d) and c of operator A belong to C α (Ω); (iii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for all x ∈ Ω and ξ ∈ Rd ; (iv) the coefficients of the operator B belong to C 1+α (∂Ω) and C 1+α (∂Ω, Rd ), respectively. Moreover, a(x) ≥ 0, |η(x)| = 1 and hη(x), ν(x)i > 0 for each x ∈ ∂Ω, where ν(x) denotes the unit outward normal vector to ∂Ω at x ∈ ∂Ω. The main result of this section are the following counterparts of Theorems 6.2.6, 7.0.2 and 8.0.2 for problems (9.0.1) and (9.0.2). Theorem 9.1.2 Let Hypotheses 9.1.1(i)–(iii) be satisfied. Then, for every f ∈ C 2+α (Ω), g ∈ C α/2,α ((0, T ) × Ω), h ∈ C 1+α/2,2+α ([0, T ] × ∂Ω) such that f ≡ h(0, ·) and Dt h(0, ·) ≡ g(0, ·) + Af on ∂Ω, there exists a unique solution u ∈ C 1,2 ((0, T ] × Ω) ∩ C([0, T ] × Ω) to the Cauchy problem (9.0.1). In addition, u ∈ C 1+α/2,2+α ((0, T ) × Ω) and there exists a positive constant C, which depends on d, α, Ω and, in a continuous way, on µ and the C α (Ω)-norm of the coefficients of the operator A, such that ||u||C 1+α/2,2+α ((0,T )×Ω) ≤ C(||f ||C 2+α (Ω) + ||g||C α/2,α ((0,T )×Ω) + ||h||C 1+α/2,2+α ([0,T ]×∂Ω) ). (9.1.1) Theorem 9.1.3 Under Hypotheses 9.1.1, for every f ∈ C 2+α (Ω), g ∈ C α/2,α ((0, T ) × Ω) and h ∈ C (1+α)/2,1+α ([0, T ] × ∂Ω), such that Bf ≡ h(0, ·) on ∂Ω, there exists a unique solution u ∈ C 1,2 ((0, T ] × Ω) ∩ C 0,1 ([0, T ] × Ω) to the Cauchy problem (9.0.2). In addition, u ∈ C 1+α/2,2+α ((0, T ) × Ω) and there exists a positive constant C, which depends on d, α, Ω and, in a continuous way, on µ, on the C α (Ω)-norm of the coefficients of the operator A and on the C 1+α (∂Ω)- and C 1+α (∂Ω; Rd )-norm of the coefficients of the operator B, such that ||u||C 1+α/2,2+α ((0,T )×Ω) ≤ C(||f ||C 2+α (Ω) + ||g||C α/2,α ((0,T )×Ω) + ||h||C (1+α)/2,1+α ([0,T ]×∂Ω) ). (9.1.2) In the proof of Theorem 9.1.2 we will take advantage of the following lemmas.

Semigroups of Bounded Operators and Second-Order PDE’s

231

Lemma 9.1.4 Let Ω be an open set of class C k for some k > 0. Then, there exists a finite covering {Uj : j = 1, . . . , N − 1} of ∂Ω such that each Uj is an open subset of Rd . Moreover, for every j = 1, . . . , N − 1 there exists a bijective function ψj : Ui → B(0, r) of class C k , with inverse of class C k , such that ψj (Uj ∩ Ω) = B+ (0, r) := B(0, r) ∩ Rd+ and ψj (Uj ∩ ∂Ω) = B(0, r) ∩ ∂Rd+ . Finally, the following properties are satisfied. (i) There exist an open set UN , compactly contained in Ω, such that {Uj : j = 1, . . . , N } is a covering of Ω, and a family of functions {ϑi : i = 1, . . . , N } such that supp(ϑj ) ⊂ Uj PN for every j = 1, . . . , N and j=1 ϑ2j = 1 on Ω. 0 0 (ii) There exist an open set UN , compactly contained in Ω, such that {U1 , . . . UN −1 , UN } is a covering of Ω, and two families of functions {φi : i = 1, . . . , N } and {ζi : i = 1, . . . , N } such that supp(φj ) and supp(ζj ) are contained in Uj for every j = 1, . . . , N − 1, PN ∂ζj 0 = 0 on ∂Ω. supp(φN ), supp(ζN ) ⊂ UN , j=1 φj ζj = 1 on Ω and ∂η

Proof See Corollary B.1.3 Proposition B.2.13 and Remark B.2.4.



Lemma 9.1.5 Under Hypotheses 9.1.1(i)–(iii), the following properties are satisfied. (i) Let Ω0 and Ω00 be two open subsets of Ω such that Ω0 b Ω00 b Ω. Then, there exists e with coefficients in C α (Rd ), such that the diffusion a uniformly elliptic operator A, b e is the trivial extension to Rd of the matrix is symmetric for every x ∈ Rd and Au function Au for every function u ∈ C 2 (Rd ), with support contained in Ω0 . (ii) Let Uk and ψk (k = 1, . . . , N − 1) be as in Lemma 9.1.4. For every function ζ : Ω ∩ Uk → R, denote by ζ ] the function defined by ζ ] = ζ ◦ ψk−1 : B+ (0, r) → R. Then, for every smooth enough function ϑ : Uk → R with compact support in Uk , there exists a uniformly elliptic operator Abk , with coefficients in Cbα (Rd+ ), such that the diffusion matrix is symmetric for every x ∈ Rd+ and, if u : Uk ∩ Ω → R is a smooth enough function and v is the trivial extension to Rd+ of the function (uϑ)] , then Abk v is the trivial extension outside B+ (0, r) of the function (A(uϑ))] , defined on B+ (0, r). (iii) Under the same notation as in (ii), there exists a first-order boundary differential ∂ operator Bbk = a(k) I + (k) with coefficients in Cb1+α (Rd−1 ) and in Cb1+α (Rd−1 ; Rd ), ∂b η η (k) , ed i < 0 and Bbk v is the trivial extension outside respectively, such that inf Rd−1 hb d B(0, r) ∩ ∂R+ of the function (B(uϑ))] . If Ω is of class C k+2+α and the coefficients of the operator A belong to C k+α (Ω) for some k ∈ N, then the coefficients of operator Ae (resp. Abk ) belong to Cbk+α (Rd ) (resp. to Cbk+α (Rd+ )). Similarly, if the coefficients of the operator B belong to C k+1+α (∂Ω) and C k+1+α (∂Ω; Rd ), respectively, then the coefficients of the operator Bbk belong to Cbk+1+α (Rd−1 ) and Cbk+1+α (Rd−1 ; Rd ), respectively. Proof Throughout the proof, given a function ϑ defined on a subset of Rd , we denote by ϑ its trivial extension to Rd . Moreover, we denote by ψk,j (j = 1, . . . , d) the components of the function ψk . (i) We fix a function % ∈ Cc∞ (Rd ) such that χΩ0 ≤ % ≤ χΩ00 and define the coefficients qeij , bej (i, j = 1, . . . , d) and e c of the operator Ae as follows: qeij = %qij + (1 − %)δij , ebj = %bj and e c = %c for every i, j = 1, . . . , d. It is easy to check that the so defined functions belong

232

Parabolic Equations in Bounded Smooth Domains Ω

to Cbα (Rd ) and d X

qeij (x)ξi ξj =%(x)

i,j=1

d X

qij (x)ξi ξj + (1 − %(x))|ξ|2

i,j=1

≥[%(x)µ + 1 − %(x)]|ξ|2 ≥ (1 ∧ µ)|ξ|2 e is the trivial for every x, ξ ∈ Rd , so that the operator Ae is uniformly elliptic. Moreover, Au d 0 extension to R of the function Au, if u is compactly supported in Ω . Finally, we note that, if the coefficients of the operator A belong to C k+α (Ω) for some k ∈ N, then the coefficients of the operator Ae belong to Cbk+α (Rd ). (ii) Let ϑ and u be as in the statement. A straightforward computation shows that A(uϑ) = (Ak (uϑ)] ) ◦ ψk , where Ak = Tr(Q(k) D2 ) + hb(k) , ∇i + c(k) and its coefficients are defined as follows: (k)

(k)

qij = [(Jac ψk )Q(Jac ψk )∗ ]]ij ,

bj

= [Tr(QD2 ψk,j ) + hb, ∇ψk,j i]] ,

c(k) = c]

for every i, j = 1, . . . , d. It is easy to check that the coefficients of the operator Ak belong to C α (B+ (0, r)) if the coefficients of the operator A belong to C α (Ω). Further, we note that hQ(k) (y)ξ, ξi =hQ(ψk−1 (y))(Jac ψk (ψk−1 (y)))∗ ξ, (Jac ψk (ψk−1 (y)))∗ ξi ≥µ|(Jac ψk (ψk−1 (y)))∗ ξ|2 for every y ∈ B+ (0, r) and ξ ∈ Rd . Since the matrix Jac ψk (x) is invertible at every point x ∈ Uxk , it follows that |det(Jac ψk (x))| > 0 for every x ∈ Uxk . Thus, the function y 7→ ||(Jac ψk (ψk−1 (y)))−1 || is bounded in B(0, r0 ) by a positive constant M0 , where r0 ∈ (0, r) is chosen in such a way that ψk (supp(ϑ)) ⊂ B(0, r0 ). As a byproduct, we deduce that |(Jac ψk (ψk−1 (y)))∗ ξ| ≥ M0−1 |ξ| for every y ∈ B(0, r0 ), ξ ∈ Rd , and this implies that there exists a positive constant µk such that hQ(k) (y)ξ, ξi ≥ µk |ξ|2 for every ξ ∈ Rd and y ∈ B+ (0, r0 ). Now, we extend the coefficients of the operator Ak to Rd+ arguing as in (i). The only difference is the definition of the function % ∈ Cc∞ (Rd ), which now is chosen in such a way that χB(0,r0 ) ≤ % ≤ χB(0,r00 ) , where r00 is arbitrarily fixed in the interval (r0 , r). Finally, it is easy to check that, if the coefficients of the operator A belong to C k+α (Ω) for some k ∈ N, then the coefficients of the operator Aek belong to Cbk+α (Rd+ ). ∂(uϑ) ∂(uϑ)] (iii) A straightforward computation shows that = ◦ ψk on ∂Ω, where ∂η ∂η (k) (k) −1 η ◦ ψk = (Jac ψk )η on Uk ∩ ∂Ω. We recall that ν(x) = −|∇ψk,d (x)| ∇ψk,d (x) for every x ∈ Uk ∩ ∂Ω. Hence, hη

(k)

]

, ed i = h((Jac ψk )η) , ed i =

] d  X ∂ψk,d j=1

∂xj

= −|(∇ψk,d )] |hν ] , η ] i

and, observing that |(∇ψk,d )] | nowhere vanishes on Uk ∩ ∂Ω, we conclude that hη (k) , ed i < 0 on Uk ∩ ∂Ω. It thus follows that (B(uϑ))] = a] (uϑ)] +

∂(uϑ)] ∂η (k)

on B(0, r) ∩ ∂Rd+ . Clearly, a] and η (k) belong to C 1+α (B 0 (0, r)) and to C 1+α (B 0 (0, r); Rd ), respectively, where B 0 (0, r) denotes the ball of Rd−1 centered at zero and with radius

Semigroups of Bounded Operators and Second-Order PDE’s

233

r. We now extend these coefficients to Rd−1 with functions b a(k) and ηb(k) which be1+α 1+α d−1 d−1 d long to Cb (R ) and Cb (R ; R ), respectively, in such a way that the condition η (k) , ed i < 0 is satisfied. For this purpose, we set b a(k) = ϕa ◦ ψk−1 (·, 0) + 1 − ϕ, inf Rd−1 hb ηb(k) = ϕ(Jac ψk )η ◦ ψk−1 (·, 0) + ϕ − 1, where ϕ : Rd−1 → R is a smooth function compactly supported in B 0 (0, r) which is equal to one in B 0 (0, re), where re < r is chosen in such a way that ψk−1 (·, 0)(B 0 (0, re)) ⊂ supp ϑ. It is easy to check that, if Ω is of class C k+2+α for some k ∈ N, then the functions a(k) and ηb(k) belong to C k+1+α (Rd−1 ) and to C k+1+α (Rd−1 ; Rd ), respectively.  Proposition 9.1.6 There exist two positive constants C1 and C2 which depends on d, α, Ω and, in a continuous way, on µ and the C α (Ω)-norm of the coefficients of the operator A, (the constant C2 depends also, in a continuous way, on the C 1+α (∂Ω)- and C 1+α (∂Ω; Rd )norm of the coefficients of the operator B) such that ||u||C 1+α/2,2+α ((0,T )×Ω) ≤ C1 ||u(0, ·)||C 2+α (Ω) + ||Dt u − Au||C α/2,α ((0,T )×Ω)
+ ||u||C 1+α/2,2+α ([0,T ]×∂Ω)

(9.1.3)

and ||u||C 1+α/2,2+α ((0,T )×Ω) ≤ C2 ||u(0, ·)||C 2+α (Ω) + ||Dt u − Au||C α/2,α ((0,T )×Ω) + ||Bu||C (1+α)/2,1+α ([0,T ]×∂Ω) ).

(9.1.4)

for every u ∈ C 1+α/2,2+α ((0, T ) × Ω). Proof Fix u ∈ C 1+α/2,2+α ((0, T ) × Ω) and let {Uj : j = 1, . . . , N } be the covering of Ω in Lemma 9.1.4. We also consider a partition of the unity {ϑi : i = 1, . . . , N } associated PN to this covering (see Proposition B.1.1). We split u = k=1 uk , where uk = uϑk and analyze separately the cases k < N and k = N . We begin by this latter case, since it is easier. Throughout the proof, given a function % defined in a subset of Rd (resp. Rd+1 ), we denote by % its trivial extension to Rd . Moreover, by C, we denote a positive constant, which may vary from line to line and depends at most on d, Ω and, in a continuous way, on µ, on the C α (Ω)-norm of the coefficients of the operator A and on the C 1+α (∂Ω)- and C 1+α (∂Ω; Rd )-norm of the coefficients of the operator B. The function uN . As it is easily seen, the function uN belongs to C 1+α/2,2+α ((0, T ) × Ω) and its support is contained in [0, T ] × Ω0 where Ω0 is an open set whose closure is contained in Ω. By Lemma 9.1.5, we can determine an uniform elliptic operator, with coefficients which e N is the trivial extension to [0, T ] × Rd of the function AuN . belong to Cbα (Rd ), such that Au Note that Dt uN − AuN = ϑN (Dt u − Au) + u(A − c)ϑN + 2hQ∇u, ∇ϑN i. Therefore, the function Dt uN − AuN belongs to C α/2,α ((0, T ) × Ω) and ||Dt uN − AuN ||C α/2,α ((0,T )×Ω)  ≤ ||Dt u − Au||C α/2,α ((0,T )×Ω) ||ϑN ||Cbα (Rd ) + ||u||C α/2,α ((0,T )×Ω) ||(A − c)ϑ||C α (Ω) +2

N X

 ||qij ||C α (Ω) ||Dj u||C α/2,α ((0,T )×Ω) ||Di ϑN ||Cbα (Rd )

i,j=1

≤C(||Dt u − Au||C α/2,α ((0,T )×Ω) + ||u||C α/2,1+α ((0,T )×Ω) ). e N . Similarly, uN (0, ·) belongs Clearly, the same estimate is satisfied by function Dt uN − Au to Cb2+α (Rd ) and ||uN (0, ·)||C 2+α (Rd ) ≤ C||u(0, ·)||C 2+α (Rd ) . b

b

234

Parabolic Equations in Bounded Smooth Domains Ω

From Theorem 6.2.1 we infer that ||uN ||C 1+α/2,2+α ((0,T )×Ω) ≤ C(||u(0, ·)||C 2+α (Ω) + ||Dt u − Au||C α/2,α ((0,T )×Ω) b

+ ||u||C α/2,1+α ((0,T )×Ω) ).

(9.1.5)

The function uk (k = 1, . . . , N −1). Let us fix such an index k and consider the diffeomorphism ψk in Lemma 9.1.4. Due to the smoothness of ψk , the function vk : [0, T ]×B+ (0, r) → R, defined by vk (t, y) = uk (t, ψk−1 (y)) for every t ∈ [0, T ] and y ∈ B+ (0, r), belongs to C 1+α/2,2+α ([0, T ] × B+ (0, r)). By Lemma 9.1.5, there exists a uniformly elliptic operator bk , with coefficients in C α (Rd+ ), such that Abk vk is the trivial extension to [0, T ] × Rd+ of A b the function (t, y) 7→ (Auk )(t, ψk−1 (y)). Arguing as in the case of the function uN , it can be easily checked that ||Dt vk − Abk vk ||C α/2,α ((0,T )×Rd+ ) ≤ C(||Dt u − Au||C α/2,α ((0,T )×Ω) + ||u||C α/2,1+α ((0,T )×Ω) ). (9.1.6) Similarly, vk (0, ·) = (u(0, ·)ϑk ) ◦ ψk−1 . Hence, ||vk (0, ·)||C 2+α (Rd ) ≤ C||u(0, ·)||C 2+α (Ω) .

(9.1.7)

||vk ||C 1+α/2,2+α ([0,T ]×∂Rd+ ) ≤ C||u||C 1+α/2,2+α ([0,T ]×∂Ω) .

(9.1.8)

b

+

Finally,

Hence, from (9.1.6)–(9.1.8) and Corollary 7.3.1 it follows that ||vk ||C 1+α/2,2+α ((0,T )×Rd ) ≤ C(||u(0, ·)||C 2+α (Ω) + ||Dt u − Au||C α/2,α ((0,T )×Ω) +

b

+ ||u||C 1+α/2,2+α ([0,T ]×∂Ω) ) + ||u||C α/2,1+α ((0,T )×Ω) ). Coming back to function uk , observing it vanishes outside (0, T ) × Ω ∩ Uk and uk = vk ◦ ψk in (0, T ) × Ω ∩ Uk , we infer that ||uk ||C 1+α/2,2+α ((0,T )×Ω) ≤ C(||u(0, ·)||C 2+α (Ω) + ||Dt u − Au||C α/2,α ((0,T )×Ω) + ||u||C 1+α/2,2+α ([0,T ]×∂Ω) + ||u||C α/2,1+α ((0,T )×Ω) ).

(9.1.9)

Similarly, by Lemma 9.1.5 it follows easily that Bbk vk is the trivial extension to ∂Rd+ of   ∂ϑk the function (ϑk Bu) ◦ ψk−1 (·, 0) + u ◦ ψk−1 (·, 0) defined in the ball of Rd−1 , centered ∂η at zero, with radius r. Hence, ||Bbk vk ||C (1+α)/2,1+α ((0,T )×Rd−1 ) ≤C(||Bu||C (1+α)/2,1+α ([0,T ]×∂Ω) + ||u||C (1+α)/2,1+α ([0,T ]×∂Ω) ) b

≤C(||Bu||C (1+α)/2,1+α ([0,T ]×∂Ω) + ||u||C (1+α)/2,1+α ((0,T )×Ω) ). (9.1.10) Therefore, taking Proposition 8.1.1 and observing that C (1+α)/2,1+α ((0, T ) × Ω) is continuously embedded into C α/2,1+α ((0, T ) × Ω), we conclude that ||vk ||C 1+α/2,2+α ((0,T )×Rd ) ≤ C(||u(0, ·)||C 2+α (Ω) + ||Dt u − Au||C α/2,α ((0,T )×Ω) b

+

+ ||Bu||C 1+α/2,2+α ([0,T ]×∂Ω) + ||u||C (1+α)/2,1+α ((0,T )×Ω) ).

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Coming back to function uk , observing it vanishes outside (0, T ) × Ω ∩ Uk and uk = vk ◦ ψk in (0, T ) × Ω ∩ Uk , we infer that ||uk ||C 1+α ((0,T )×Ω) ≤ C(||u(0, ·)||C 2+α (Ω) + ||Dt u − Au||C α/2,α ((0,T )×Ω) + ||Bu||C (1+α)/2,1+α ([0,T ]×∂Ω) + ||u||C (1+α)/2,1+α ((0,T )×Ω) ).

(9.1.11)

Conclusion. From (9.1.5) and (9.1.9) it follows that ||u||C 1+α/2,2+α ((0,T )×Ω) ≤

N X

||uk ||C 1+α/2,2+α ((0,T )×Ω)

k=1

≤C ||u(0, ·)||C 2+α (Ω) + ||Dt u − Au||C α/2,α ((0,T )×Ω) + ||u||C 1+α/2,2+α ([0,T ]×∂Ω) + ||u||C α/2,1+α ((0,T )×Ω) ).

(9.1.12)

Using Proposition 1.2.5, we can estimate ||u||C α/2,1+α ((0,T )×Ω) ≤ C(ε||u||C 1+α/2,2+α ((0,T )×Ω) + ε−1−α ||u||∞ ) for every ε ∈ (0, 1). Replacing this estimate in the right-hand side of (9.1.12) and, then, taking ε sufficiently small, we can move the C 1+α/2,2+α ((0, T ) × Ω)-norm of u from the right- to the left-hand side of (9.1.12), thus obtaining estimate (9.1.3). Arguing similarly, from (9.1.5) and (9.1.11) the proof of estimate (9.1.4) can be completed.  To prove Theorem 9.1.2 one needs also the following lemma. Lemma 9.1.7 The following properties are satisfied. 1+α/2,2+α

(i) Let u ∈ Cb ((0, T ) × Rd ) be such that u(0, ·) ≡ 0. Then, there exists a positive constant C, independent of u and λ ≥ 1, such that α

λ1− 2 ||u||C α/2,α ((0,T )×Rd ) + λ

1−α 2

b

d X i=1

||Di u||C α/2,α ((0,T )×Rd ) b

≤C||Dt u − Au + λu||C α/2,α ((0,T )×Rd ) .

(9.1.13)

b

1+α/2,2+α

(ii) Let u ∈ Cb ((0, T ) × Rd+ ) be such that u(0, ·) ≡ 0 on Rd+ and u (resp. Bu) vanishes on [0, T ] × ∂Rd+ for every t ∈ [0, T ]. Then, there exists a positive constant C, independent of u and λ ≥ 1, such that α

λ1− 2 ||u||C α/2,α ((0,T )×Rd ) + λ +

b

1−α 2

d X i=1

≤C||Dt u − Au + λu||C α/2,α ((0,T )×Rd ) . b

||Di u||C α/2,α ((0,T )×Rd ) b

+

(9.1.14)

+

Proof We limit ourselves to proving estimate (9.1.13), since the proof of (9.1.14) can be obtained adapting the same arguments. √ Let v : [0, T ]×Rd+1 → R be the function defined by v(t, y, x) = cos( λy)u(t, x) for every 1+α/2,2+α t ∈ [0, T ], y ∈ R and x ∈ Rd . As it is immediately seen, v belongs to Cb ((0, T ) × Rd+1 ). Let us also denote by A0 the operator defined on smooth enough functions ζ : Rd+1 → d X R by A0 ζ(y, x) = Dyy ζ + qij (x)Dij ζ(x, y) for every (y, x) ∈ Rd+1 . It is easy to check i,j=1

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that A0 is an elliptic operator with ellipticity constant given by the minimum between 1 and the ellipticity constant µ of the operator A. By Theorem 6.2.1 there exists a positive constant C1 , independent of v, such that ||v||C 1+α/2,2+α ((0,T )×Rd+1 ) ≤ C1 ||Dt v − A0 v||C α/2,α ((0,T )×Rd+1 ) .

(9.1.15)

b

b

√ We observe that Dt v(t, y, x) − A0 v(t, y, x) = (Dt u(t, x) − Au(t, x) + λu(t, x)) cos( λy) for each t ∈ [0, T ], y ∈ R, x ∈ Rd , so that α

||Dt v − A0 v||C α/2,α ((0,T )×Rd+1 ) ≤||Dt u − Au + λu||C α/2,α ((0,T )×Rd ) +C2 λ 2 ||Dt u − Au + λu||∞ b

b

α

≤C3 λ 2 ||Dt u − Au + λu||C α/2,α ((0,T )×Rd ) ,

(9.1.16)

b

(recall that λ ≥ 1) for some positive constants C2 and C3 , independent of λ and u. Moreover, λ||u||C α/2,α ((0,T )×Rd ) + b

d d X √ X α ||Dij u||∞ λ sup ||Di u(·, x)||C α/2 ((0,T )) + C∗ λ 2 d i=1 x∈R

≤||Dyy v(·, 0, ·)||C α/2,α ((0,T )×Rd ) + b

+

d X

sup

d+1 i=1 (y,x)∈R

i,j=1

d X

sup d

i,j=1 (t,x)∈[0,T ]×R

[Dij v(t, ·, x)]Cbα (R)

[Dyi v(·, y, x)]C α/2 ((0,T ))

≤||v||C 1+α/2,2+α ((0,T )×Rd+1 ) ,

(9.1.17)

b

where C∗ denotes the α-H¨ older seminorm of the function cosine. Replacing (9.1.16) and (9.1.17) in (9.1.15), we conclude that α

λ1− 2 ||u||C α/2,α ((0,T )×Rd ) + λ b

1−α 2

d X

sup ||Di u(·, x)||C α/2 ((0,T )) + C∗

d i=1 x∈R

d X

||Dij u||∞

i,j=1

≤C4 ||Dt u − Au + λu||C α/2,α ((0,T )×Rd )

(9.1.18)

b

1−α

1

eα ||ζ|| 2−α the constant C4 > 0 being independent of u, λ. Since ||ζ||C 1+α (Rd ) ≤ C ||ζ||C2−α 2 d Cbα (Rd ) b b (R ) 2 d e for every ζ ∈ Cb (R ) and some positive constant Cα , independent of ζ (see Proposition 1.1.4), from (9.1.18) we obtain that d X

sup ||Dj u(t, ·)||Cbα (Rd ) ≤ C5 λ−

j=1 t∈[0,T ]

1−α 2

||Dt u − Au + λ, u||C α/2,α ((0,T )×Rd ) .

(9.1.19)

b

the constant C5 being independent of u and λ. From (9.1.18) and (9.1.19), estimate (9.1.13) follows at once.  We have all the tools to prove Theorem 9.1.2. Proof of Theorem 9.1.2 The uniqueness of the solution u, which belongs to C([0, T ]× Ω) ∩ C 1,2 ((0, T ] × Ω), follows from Corollary 4.1.5. Let us prove the existence part. For this purpose, we consider the operator Eα0 in Corollary B.4.10 and begin by observing that u ∈ C 1+α/2,2+α ((0, T ) × Ω) solves the Cauchy problem (9.0.1) if and only if the function v = u − Eα0 h − f + Eα0 h(0, ·) solves the equation Dt v = Av + g1 on [0, T ] × Ω, where g1 = g + Af − Dt Eα0 h + AEα0 h − AEα0 h(0, ·), and vanishes

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on parabolic boundary of (0, T ]×Ω. Note that the function g1 belongs to C α/2,α ((0, T )×Ω). For this reason, in the rest of the proof we will assume that f and h identically vanish on ∂Ω and on [0, T ] × ∂Ω, respectively. The strategy of the proof is the following: for each λ ≥ 1 we will determine a bounded α/2,α 1+α/2,2+α operator S(λ) mapping Cb ((0, T ) × Ω) into Cb ((0, T ) × Ω) with the following properties: (i) S(λ)ζ vanishes on the parabolic boundary of (0, T ]×Ω for each ζ ∈ C α/2,α ((0, T )×Ω); (ii) Dt S(λ)ζ = AS(λ)ζ − λS(λ)ζ + ζ − R(λ)ζ for each ζ ∈ C α/2,α ((0, T ) × Ω) and some bounded operator R(λ) mapping C α/2,α ((0, T ) × Ω) into itself and with norm which vanishes as λ tends to ∞. Properties (i) and (ii) are all that we need to conclude the proof. Indeed, thanks to (ii) we can fix λ large enough such that the operator I − R(λ) is invertible in C α/2,α ((0, T ) × Ω). Thus, given g ∈ C α/2,α ((0, T ) × Ω) we can determine ζ in the same space such that ζ − R(λ)ζ = g. Function u(t, x) = eλt (S(λ)ζ)(t, x) solves the equation Dt u − Au = g, by (ii), and vanishes on the parabolic boundary of (0, T ) × Ω, by (i). This accomplishes the proof. To prove properties (i) and (ii), we introduce the covering {U1 , . . . , UN } of Ω and the associated partition of the unity {ϑ2i : i = 1, . . . , N } defined in Lemma 9.1.4(i). Moreover, for fixed λ ≥ 1 and ζ ∈ C α/2,α ((0, T ) × Ω) we consider the Cauchy problems  d b b   Dt v(t, y) = Ak v(t, y) − λv(t, y) + ζk (t, y), t ∈ [0, T ], y ∈ R + , v(t, y) = 0, t ∈ [0, T ], y ∈ ∂Rd+ ,   v(0, y) = 0, y ∈ Rd+ for k = 1, . . . , N − 1 and ( Dt w(t, x) = AN w(t, x) − λw(t, x) + ζN (t, x), w(0, x) = 0,

t ∈ [0, T ],

x ∈ Rd , x ∈ Rd ,

(9.1.20)

(9.1.21)

where the operators Abk (k = 1, . . . , N − 1) and AN are defined in Lemma 9.1.5, ζbk (k = 1, . . . , N − 1) is the trivial extension to [0, T ] × Rd+ of the function (t, y) 7→ ϑk (ψk−1 (y))ζ(t, ψk−1 (y)) and ζN is the trivial extension to [0, T ] × Rd of the function α/2,α ((0, T ) × Rd+ ) and to ϑN ζ. Clearly, ζbk (k = 1, . . . , N − 1) and ζN belong to Cb α/2,α d Cb ((0, T )×R ), respectively. Moreover, there exists a positive constant C1 , independent of ζ (as all the other constants that appear in the proof), such that ||ζbk ||C α/2,α ((0,T )×Rd ) + ||ζN ||C α/2,α ((0,T )×Rd ) ≤ C1 ||ζ||C α/2,α ((0,T )×Ω) b

+

(9.1.22)

b

for every k = 1, . . . , N − 1. Since the coefficients of the operators Abk and AN satisfy Hypotheses 7.0.1 and 6.0.1, respectively, Theorems 6.2.1 and 7.0.2 show that problems 1+α/2,2+α (9.1.20) (k = 1, . . . , N −1) and (9.1.21) admit unique solutions vk ∈ Cb ((0, T )×Rd+ ) 1+α/2,2+α d and uN ∈ Cb ((0, T ) × R ). Thus, we can introduce the operators Sk (λ) (k = 1, . . . , N ), defined on functions ζ ∈ C α/2,α ((0, T ) × Ω) as follows: ˆ (Sk (λ)ζ)(t, x) = vk (t, ψk (x)) for each t ∈ [0, T ] and x ∈ Ω ∩ Uk , if k < N ; ˆ SN (λ)ζ = uN .

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Parabolic Equations in Bounded Smooth Domains Ω

A change of variables shows that the function Sk (λ)ζ (k = 1, . . . , N − 1) belongs to C 1+α/2,2+α ((0, T ) × (Ω ∩ Uk )) and solves the Cauchy problem  in [0, T ] × (Ω ∩ Uk ),   Dt uk = Auk − λuk + ζk , uk (t, x) = 0, on [0, T ] × (Uk ∩ ∂Ω),   uk (0, x) = 0, on {0} × (Ω ∩ Uk ), where ζk = ϑk ζ. Since ϑk is compactly supported in Uk , the function ϑk Sk (λ)ζ (k = 1, . . . , N − 1) belongs to C 1+α/2,2+α ((0, T ) × Ω). Based on all the above remarks, we can define a linear operator S(λ) : C α/2,α ((0, T ) × PN Ω) → C 1+α/2,2+α ((0, T ) × Ω) setting S(λ)ζ = k=1 ϑk Sk (λ)ζ for every ζ ∈ C α/2,α ((0, T ) × Ω). As it is immediately seen, the function S(λ)ζ vanishes on the parabolic boundary of (0, T ] × Ω for every ζ ∈ C α/2,α ((0, T ) × Ω), so that property (i) is satisfied. Moreover, Dt S(λ)ζ − AS(λ)ζ + λS(λ)ζ =

N X

ϑk (Dt Sk (λ)ζ − ASk (λ)ζ + λSk (λ)ζ)

k=1

−

N X

(Aϑk − cϑk )Sk (λ)ζ − 2

k=1

N X

hQ∇x Sk (λ)ζ, ∇ϑk i

k=1

=ζ − R(λ)ζ, where R(λ)ζ :=

N X

(Aϑk − cϑk )Sk (λ)ζ + 2

k=1

N X

hQ∇x Sk (λ)ζ, ∇ϑk i.

k=1

Clearly, R(λ) is a bounded operator mapping C α/2,α ((0, T ) × Ω) into itself. Moreover, from Lemma 9.1.7 and estimate (9.1.22) we deduce that ||SN (λ)ζ||C α/2,α ((0,T )×Rd ) + b

d X

||Di SN (λ)ζ||C α/2,α ((0,T )×Rd ) ≤ C2 λ

α−1 2

b

i=1

||ζ||C α/2,α ((0,T )×Ω) b

(9.1.23) and ||vk ||C α/2,α ((0,T )×Rd ) + b

+

d X i=1

||Di vk ||C α/2,α ((0,T )×Rd ) ≤ C3 λ +

b

α−1 2

||ζ||C α/2,α ((0,T )×Ω)

for every k = 1, . . . , N − 1 and λ ≥ 1. It follows that ||Sk (λ)ζ||C α/2,α ((0,T )×Ω∩Uk ) +

d X

||Di Sk (λ)ζ||C α/2,α ((0,T )×Ω∩Uk ) ≤ C4 λ

α−1 2

||ζ||C α/2,α ((0,T )×Ω)

i=1

(9.1.24) for every k = 1, . . . , N − 1. From (9.1.23) and (9.1.24), we easily conclude that ||R(λ)||L(C α/2,α ((0,T )×Ω)) ≤ Cλ Property (ii) follows immediately. Finally, estimate (9.1.1) follows from (9.1.3).

α−1 2

,

λ ≥ 1. 

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239

Proof of Theorem 9.1.3 The proof is analogous to that of Theorem 9.1.2. Hence, we skip some details. The uniqueness of the classical solution to problem (9.0.2) follows from Proposition 4.1.9. As far as the existence part is concerned, we introduce the 0 covering {U1 , . . . , UN −1 , UN } of ∂Ω and the function φk , ζk (k = 1, . . . , N ) in Lemma 9.1.4. We also introduce the operator Eα in Proposition B.4.14(i) and note that a function u ∈ C 1+α/2,2+α ((0, T ) × Ω) solves problem (9.0.2) if and only if the function v = u − Eα h − f + Eα h(0, ·), which belongs to C 1+α/2,2+α ((0, T ) × Ω), solves problem (9.0.2) with f ≡ 0 on Ω and h = 0 on [0, T ] × ∂Ω. Hence, in the rest of the proof, we confine ourselves to this case. We fix λ > 1, ϕ ∈ C α/2,α ((0, T ) × Ω) and denote by uN the unique solution in 1+α/2,2+α C ((0, T ) × Rd ) (see Theorem 6.2.1) of the Cauchy problem ( Dt w(t, x) = AN w(t, x) − λw(t, x) + ϕN (t, x), t ∈ [0, T ], x ∈ Rd , w(0, x) = 0, x ∈ Rd , where ϕN is the trivial extension to [0, T ] × Rd of the function φN ϕ. 1+α/2,2+α Next, for k = 1, . . . , N −1, we denote by vk the unique solution in Cb ((0, T )×Rd+ ) of the Cauchy problem  b bk (t, y), t ∈ [0, T ], y ∈ Rd + ,   Dt v(t, y) = Ak v(t, y) − λv(t, y) + ϕ Bbk v(t, y) = 0, t ∈ [0, T ], y ∈ ∂Rd+ ,   v(0, y) = 0, y ∈ Rd+ , where Abk and Bbk are the elliptic operator and the boundary operator in Lemma 9.1.5(iii) and ϕ bk is the trivial extension to [0, T ]×Rd+ of the function (t, y) 7→ φk (ψk−1 (y))ϕ(t, ψk−1 (y)). The existence of such a solution is guaranteed by Theorem 8.0.2. It is easy to check that the function uk , defined by uk (t, x) = vk (t, ψk (x)) for every (t, x) ∈ [0, T ] × Ω ∩ Uk , belongs to C 1+α/2,2+α ((0, T ) × Ω ∩ Uk ) and it solves the Cauchy problem  in [0, T ] × (Ω ∩ Uk ),   Dt uk = Auk − λuk + ϕk , Buk (t, x) = 0, on [0, T ] × (Uk ∩ ∂Ω),   uk (0, x) = 0, on {0} × (Ω ∩ Uk ), where ϕk = φk ϕ. For every k ∈ {1, . . . , N }, we define the operator Sk0 ∈ L(C α/2,α ((0, T ) × Ω), C 1+α/2,2+α ((0, T ) × Ω)) by setting ˆ Sk0 (λ)ϕ = ζk uk for k = 1, . . . , N − 1, where uk is the trivial extension to [0, T ] × Ω of the function uk ; 0 ˆ SN (λ)ϕ = ζN uN .

∂ζk 0 = 0 on ∂Ω. Moreover, BSN (λ)ϕ = 0 on ∂Ω, since ∂η 0 function SN (λ)ϕ is compactly supported in Ω. PN For every λ > 1, denote by S(λ) the operator defined by S(λ) = k=1 Sk0 (λ). Clearly, S(λ)ϕ ∈ C 1+α/2,2+α ((0, T ) × Ω) and BS(λ)ϕ identically vanishes on ∂Ω. Moreover,

Note that BSk0 (λ)ϕ = ζk Buk + uk

Dt S(λ)ϕ − AS(λ)ϕ + λS(λ)ϕ =

N X

ζk φk ϕ −

k=1

=ϕ − R0 (λ)ϕ.

N X

(Aζk − cζk )uk − 2

k=1

N X k=1

hQ∇x uk , ∇ζk i

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Parabolic Equations in Bounded Smooth Domains Ω

Using Lemma 9.1.7, it can be easily checked that R0 (λ) is bounded in C α/2,α ((0, T )×Ω) and α−1 ||R0 (λ)||L(C α/2,α ((0,T )×Ω)) ≤ Cλ 2 for λ > 1 and a positive constant C, independent of λ. Hence, for λ sufficiently large, the operator norm of R0 (λ) can be made less than 1/2, so that the operator I −R0 (λ) is invertible in C α/2,α ((0, T )×Ω)). The function eλ· S(λ)(I −R0 (λ))−1 g is the solution to problem (9.0.2) we were looking for. Finally, estimate (9.1.2) follows straightforwardly from (9.1.4).  Remark 9.1.8 In Chapter 14 (see Theorems 14.3.8 and 14.3.9), using tools from the semigroup theory, we will prove an optimal spatial regularity result for solutions to problems (9.0.1) and (9.0.2).

9.2

Interior Schauder Estimates for Solutions to Parabolic equations in Domains: Part IV

As a byproduct of the results in this chapter and in Chapter 8, we prove some Schauder estimates up to a portion of ∂Ω, satisfied by the solutions to the differential equation Dt u = Au + g in (0, T ] × Ω, Ω ⊂ Rd being an open connected set of class C 2+α . In this section Ω is allowed to be unbounded. The notion of smooth open sets has been given in Definition B.2.1 for bounded sets, but, clearly, the boundedness assumption on Ω can be removed. We assume the following conditions on the coefficients of the operator A. Hypotheses 9.2.1 (i) The coefficients qij = qji , bj and c (i, j = 1, . . . , d) of the operator α A belong to Cloc (Ω); (ii) there exists a positive continuous function µ : Ω → R such that hQ(x)ξ, ξi ≥ µ(x)|ξ|2 for every x ∈ Ω and ξ ∈ Rd . ∂ 1+α 1+α belong to Cloc (Ω) and to Cloc (Ω; Rd ), ∂η respectively. Moreover, a(x) ≥ 0 and ηd (x) < 0 for every x ∈ Ω, where ηd denotes the last component of the function η.

(iii) the coefficients of the operator B = aI +

Now, we state precisely the first result of this subsection. α/2,α

Theorem 9.2.2 Let g ∈ Cloc ((0, T ) × Ω) be a given function and u ∈ C 1,2 ((0, T ) × Ω) be a solution to the differential equation Dt u = Au + g. Assume that one of the following condition is satisfied. (i) Hypotheses 9.2.1(i)–(ii) are satisfied, u ∈ C((0, T ) × Ω) and it identically vanishes on (0, T ) × ∂Ω; (ii) Hypotheses 9.2.1 are satisfied, u ∈ C((0, T )×Ω) and Bu identically vanishes on (0, T )× ∂Ω. 1+α/2,2+α

Then, u ∈ Cloc ((0, T )×Ω), for every 0 < τ0 < τ1 < T0 < T and every pair of bounded open sets Ω1 ⊂ Ω2 ⊂ Ω, such that d(Ω1 , Ω \ Ω2 ) > 0, there exists a positive constant C1 which depends on τ0 , τ1 , T0 Ω1 , Ω2 and, in a continuous way on the α-H¨ older norm of the

Semigroups of Bounded Operators and Second-Order PDE’s

241

coefficients of the operator A over Ω2 (and also on the C 1+α -norm over Ω2 of the coefficients of the operator B under condition (ii)), the supremum over Ω2 of the function µ, such that ||u||C 1+α/2,2+α ((τ1 ,T0 )×Ω1 ) ≤ C1 (||u||C([τ0 ,T0 ]×Ω2 ) + ||g||C α/2,α ((τ0 ,T0 )×Ω2 ) ).

(9.2.1)

In particular, the constant C blows up as (τ1 − τ0 ) ∧ (T − T0 ) ∧ d(Ω1 , Ω \ Ω2 ) tends to zero. 2+α Finally, if u(0, ·) ∈ Cloc (Ω) and condition (i) (resp. condition (ii)) is satisfied, then u ∈ 1+α/2,2+α Cloc ([0, T ) × Ω) and for every T0 ∈ (0, T ) there exists a positive constant C2 , which depends on T0 , Ω1 , Ω2 and, in a continuous way on the α-H¨ older norm of the coefficients of the operator A over Ω2 (and also on the C 1+α -norm over Ω2 of the coefficients of the operator B under condition (ii)), the supremum over Ω2 of the function µ, such that ||u||C 1+α/2,2+α ((0,T0 )×Ω1 ) ≤ C2 (||u(0, ·)||C 2+α (Ω2 ) + ||u||C([0,T0 ]×Ω2 ) + ||g||C α/2,α ((τ0 ,T0 )×Ω2 ) ). (9.2.2) In particular, the constant C2 blows up as (T − T0 ) ∧ d(Ω1 , Ω \ Ω2 ) tends to zero. Proof Fix τ0 , τ1 , T , Ω1 , Ω2 as in the statement of the theorem and consider a finite covering {Uj : j = 1, . . . , N − 1} of ∂Ω ∩ Ω2 . Note that there exists δ > 0 such that {Uj : j = 1, . . . , N − 1} is a covering of {x ∈ Rd : d(x, ∂Ω ∩ Ω2 ) ≤ δ}. Indeed, suppose that this is not the case. Then, there exists a sequence (xn ) ⊂ Rd such that d(xn , ∂Ω∩Ω2 ) ≤ n−1 SN −1 and xn ∈ / j=1 Uj . In particular, the sequence (xn ) is bounded. Hence, up to a subsequence, xn converges to a point x ∈ ∂Ω ∩ Ω2 . Hence, x ∈ Uj for some j ∈ {1, . . . , N − 1} and, since Uj is an open set, xn ∈ Uj for n large, which is a contradiction. We split the rest of the proof into three steps. In the first two ones, we prove (9.2.1) under conditions (i) and (ii). Finally, in Step 3, we prove estimate (9.2.2). Step 1. Here, we assume condition (i). We set UN = {x ∈ Ω3 : d(x, ∂Ω ∩ Ω2 ) > δ} for some bounded open set Ω3 such that Ω2 b Ω3 . The argument in the beginning of the proof show that {Uj : j = 1, . . . , N } is a covering of Ω2 . Proposition B.1.1 shows that there exists a partition of the unity {ϑi : i = 1, . . . , N }, subordinated to this covering of Ω2 . Fix k ∈ I = {j ∈ {1, . . . , N } : Ω1 ∩ Uj 6= ∅}. We first suppose that k < N . Then, there are two possibilities: (a) Ω2 ∩ Uk 6= Uk ∩ Ω; (b) Ω2 contains Ω ∩ Uk . In the first case, d(Ω1 ∩ Uk , (Ω ∩ Uk ) \ (Ω2 ∩ Uk )) ≥ d(Ω1 , Ω \ Ω2 ) = d. Hence, if ψk : Uk → B+ (0, r) is the diffeomorphism of class C 2+α in Lemma 9.1.4, then d(ψk (Ω1 ∩ Uk ), B+ (0, r) \ ψk (Ω2 ∩ Uk )) ≥ d. Moreover, the function vk : [0, T ] × Rd+ → R, which is defined by vk (t, y) = (ϑk u)(t, ψk−1 (y)) =: uj (t, ψk−1 (y)) for every (t, y) ∈ [0, T ] × B+ (0, r) and, then, trivially extended to [0, T ] × Rd+ , belongs to Cb ([0, T ] × Rd+ ) ∩ C 1,2 ((0, T ) × Rd+ ), solves the differential equation Dt vk = Abk vk + gbk on (0, T ) × Rd+ and vanishes on (0, T ) × ∂Rd+ . Here, Abk is the operator in Lemma 9.1.5 and gbk is the trivial extension to (0, T ) × Rd+ of the function (t, y) 7→ (f ϑk − (Aϑk − cϑk )u − 2hQ∇ϑk , ∇ui)(t, ψk−1 (y)). From Theorem 8.3.2, it follows that vk belongs to C 1+α/2,2+α ((τ1 , T0 ) × ψj ((Ω1 ∩ Uj ))) and ||vk ||C 1+α/2,2+α ((τ1 ,T0 )×ψk ((Ω1 ∩Uk ))) ≤C(||vk ||C([τ0 ,T0 ]×ψk (Ω2 ∩Uk )) + ||gk ||C α/2,α ((τ0 ,T0 )×ψk ((Ω2 ∩Uk ))) ≤C(||u||C([τ0 ,T0 ]×Ω2 ) + ||g||C α/2,α ((τ1 ,T0 )×Ω2 ) )

242

Parabolic Equations in Bounded Smooth Domains Ω

Coming back to the function uk , we conclude that ||uk ||C 1+α/2,2+α ((τ1 ,T0 )×Ω1 ) ≤||uk ||C 1+α/2,2+α ((τ1 ,T0 )×(Ω1 ∩Uk )) ≤C(||u||C([τ0 ,T0 ]×Ω2 ) + ||g||C α/2,α ((τ0 ,T0 )×Ω2 ) ).

(9.2.3)

Suppose now that condition (b) holds. Then, we observe that the support of function uk is contained in Ω ∩ supp(ϑk ) and supp(ϑk ) is a compact subset of Uk . It thus follows that we can find an open set Ω4 such that supp(ϑk ) ⊂ Ω4 b Uk and, consequently, so that ψk (Ω4 ) is compactly embedded into B(0, r), so that there exists r0 < r such that ψk (Ω3 ) ⊂ B(0, r0 ). Therefore, d(ψk (Ω4 ∩ Uk ), Rd+ \ B+ (0, r)) ≥ d(B+ (0, r0 ), Rd+ \ B+ (0, r)) = r − r0 > 0. We can apply again Theorem 8.3.2 and conclude, using condition (b), that ||uk ||C 1+α/2,2+α ((τ1 ,T0 )×Ω1 ) =||uk ||C 1+α/2,2+α ((τ1 ,T0 )×(Ω1 ∩Uk )) ≤||uk ||C 1+α/2,2+α ((τ1 ,T0 )×(Ω4 ∩Uk )) ≤C(||u||C([τ0 ,T0 ]×Ω2 ) + ||g||C α/2,α ((τ0 ,T0 )×Ω2 ) ).

(9.2.4)

Finally, we assume that k = N . Then, we can determine an open set Ω4 compactly supported in Ω such that uN vanishes outside Ω4 . Note that Ω1 ∩Ω4 is compactly supported in Ω2 ∩Ω4 . Hence, we can apply the interior Schauder estimates in Theorem 6.2.9 and obtain that ||uN ||C 1+α/2,2+α ((τ1 ,T0 )×Ω1 ) =||uN ||C 1+α/2,2+α ((τ1 ,T0 )×(Ω1 ∩Ω4 )) ≤C(||u||C([τ0 ,T0 ]×Ω2 ) + ||g||C α/2,α ((τ0 ,T0 )×Ω2 ) ).

(9.2.5)

PN Since u = k=1 uk , from (9.2.3)–(9.2.5) estimate (9.2.1) follows immediately. Step 2. Now we prove estimate (9.2.1) assuming that condition (ii) in the statement of the theorem is satisfied. Arguing as in the proof Proposition B.2.13, we can determine, ∂ζk for every k = 1, . . . , N − 1 a function ζk ∈ Cc2+α (Uk ) such that = 0 on ∂Ω, and a ∂η PN −1 function ϑk ∈ C 2+α (Uk ) such that k=1 ϑk ζk = 1 on ∂Ω2 ∩ Ω. We can determine δ > 0 PN −1 d such that k=1 ϑk ζk ≥ 1/2 in Ωδ = {x ∈ R : d(x, Ω2 ∩ ∂Ω) < δ}. We now consider 0 the open set Ωδ = {x ∈ Ω5 : d(x, Ω2 ∩ ∂Ω) > δ/2} for some bounded open set Ω5 such that Ω2 b Ω5 b Ω. Note that Ω2 ⊂ Ωδ ∪ Ω0δ . Corollary B.1.3 shows that there exist two functions χ1 ∈ Cc∞ (Ωδ ) and χ2 ∈ Cc∞ (Ωδ0 ) such that 0 ≤ χj ≤ 1 on Rd (j = 1, 2) and PN −1 χ1 + χ22 = 1 on Λδ := Ωδ ∪ Ω0δ . Then, the function χ = i=1 χ1 ζi ϑi + χ22 is smooth on Rd and χ ≥ 21 χ1 +χ22 ≥ 12 (χ1 +χ22 ) = 21 on Λδ . Hence, if we define the functions φi (i = 1, . . . , N ) and ζN by setting φi = ϑi χ1 ϕ2 /χ (i = 1, . . . , N − 1), φN = χ2 ϕ2 /χ, and ζN = χ2 , where PN ϕ2 ∈ Cc∞ (Λδ ) is a smooth function such that ϕ2 ≡ 1 on Ω2 , then k=1 φk ζk = 1 on Ω2 . Now, for every k = 1, . . . , N , we consider the function vk , obtained extending in the trivial way to [0, T0 ] × Rd+ the function (t, y) 7→ ζk (ψk−1 (y))u(t, ψk−1 (y)). This function belongs to C 1,2 ((0, T ) × Rd+ ) ∩ C 0,1 ((0, T ) × Rd+ ), Dt vk − Abk vk = gbk on (0, T ) × Rd+ and Bbk v = 0 on (0, T ) × ∂Rd+ , where Ak and gbk are as in Step 1 and Bbk is the boundary operator in Lemma 9.1.5(iii). Since the coefficients of the operator Abk and Bbk satisfy Hypotheses 8.3.1, 1+α/2,2+α we can apply Theorem 8.3.2 and conclude that vk ∈ Cloc ((0, T ) × Rd+ ). Moreover, the arguments in Step 1 can be applied to show that ||uk ||C 1+α/2,2+α ((τ1 ,T0 )×Ω1 ) ≤ C(||u||C([τ0 ,T0 ]×Ω2 ) + ||g||C α/2,α ((τ0 ,T0 )×Ω2 ) ),

Semigroups of Bounded Operators and Second-Order PDE’s 243 PN where uk = ζk u for k = 1, . . . , N . Since u = k=1 φk uk , estimate (9.2.1) follows at once. 2+α Step 3. Here, we assume that u(0, ·) ∈ Cloc (Ω) and prove that function u belongs to 1+α/2,2+α Cloc ([0, T )×Ω) and satisfies (9.2.2) when condition (i) or condition (ii) is satisfied. To fix the ideas, we assume that condition (ii) is satisfied (but the same argument can be applied in the case when condition (i) is satisfied). For every k = 1, . . . , N −1, the function vk , defined 2+α (Rd+ ). in Step 1 belongs to C 1,2 ((0, T ) × Rd+ ) ∩ C 0,1 ((0, T ) × Rd+ ) and vk (0, ·) belongs to Cloc Since ||vk (0, ·)||C 2+α (ψk (Ω2 ∩Uk )) ≤ C||u(0, ·)||C 2+α (Ω2 ) , we can repeat the arguments in Step 1+α/2,2+α

1, taking Theorem 8.3.2 into account and infer that vk ∈ Cloc

([0, T ) × Rd+ ) and

||vk ||C 1+α/2,2+α ((0,T0 )×ψk ((Ω1 ∩Uk ))) ≤C(||u(0, ·)||C 2+α (Ω2 ) + ||u||C([0,T0 ]×Ω2 ) + ||g||C α/2,α ((0,T0 )×Ω2 ) ). Coming back to the function uk , we easily conclude that ||uk ||C 1+α/2,2+α ((0,T0 )×Ω1 ) ≤ C(||u(0, ·)||C 2+α (Ω2 ) + ||u||C([0,T0 ]×Ω2 ) + ||g||C α/2,α ((0,T0 )×Ω2 ) ). (9.2.6) Similarly, ||uN ||C 1+α/2,2+α ((0,T0 )×Ω1 ) ≤ C(||u(0, ·)||C 2+α (Ω2 ) + ||u||C([0,T0 ]×Ω2 ) + ||g||C α ((0,T0 )×Ω2 ) ). (9.2.7) PN Since u = k=1 φk uk , from (9.2.6), (9.2.7) estimate (9.2.2) follows immediately. 

9.3

More on the Cauchy Problems (9.0.1) and (9.0.2)

The following results are the natural counterpart of those in Chapters 6–8. We begin by considering the counterparts of Theorem 7.4.1. Theorem 9.3.1 Let Hypotheses 9.1.1(i)–(iii) be satisfied and fix θ ∈ (0, 1). Suppose that f belongs to C 2+α−2θ (Ω), satisfies, on ∂Ω the conditions f ≡ 0 and Af ≡ 0 (this latter condition if 2+α −2θ > 1) and g ∈ C((0, T ]×Ω) is such that supt∈(0,T ] tθ ||g(t, ·)||C α (Ω) < ∞ and g(t, ·) = 0 on ∂Rd+ for every t ∈ (0, T ]. Then, the Cauchy problem (9.0.1) admits a unique classical solution u, which is bounded in (0, T ) × Rd+ . In addition, u(t, ·) ∈ C 2+α (Ω) for every t ∈ (0, T ] and sup tθ (||u(t, ·)||C 2+α (Ω) + ||Dt u(t, ·)||C α (Ω) ) + ||u||C 0,2+α−2θ ([0,T ]×Ω)

t∈(0,T ]

  ≤C0 ||f ||C 2+α−2θ (Ω) + sup tθ ||g(t, ·)||C α (Ω) t∈(0,T ]

for some positive constant C0 , independent of u, f and g. In particular, if f and g belong to Cb2+α (Ω) and C([0, T ] × Ω), respectively, f ≡ Af ≡ g(t, ·) = 0 on ∂Rd+ for every t ∈ [0, T ] and supt∈[0,T ] ||g(t, ·)||C α (Ω) < ∞, then, u belongs to C 1,2 ([0, T ] × Ω), u(t, ·) ∈ C 2+α (Ω) for every t ∈ [0, T ] and   sup ||u(t, ·)||C 2+α (Ω) + sup ||Dt u(t, ·)||C α (Ω) ≤ C1 ||f ||C 2+α (Ω) + sup ||g(t, ·)||C α (Ω) , t∈[0,T ]

t∈[0,T ]

for some positive constant C1 , independent of u, f and g.

t∈(0,T ]

244

Parabolic Equations in Bounded Smooth Domains Ω

Theorem 9.3.2 Let Hypotheses 8.0.1 be satisfied. Suppose that f belongs to Cb2+α−2θ (Ω), satisfies the condition Bf ≡ 0 on ∂Ω if 2 + α − 2θ > 1 and g ∈ C((0, T ] × Ω) is such that supt∈(0,T ] tθ ||g(t, ·)||Cbα (Ω) < ∞ for some θ ∈ (0, 1). Then, the Cauchy problem (8.0.1) admits a unique classical solution u, which is bounded in [0, T ] × Ω. In addition, u(t, ·) ∈ Cb2+α (Ω) for every t ∈ (0, T ] and sup tθ (||u(t, ·)||C 2+α (Ω) + ||Dt u(t, ·)||Cbα (Ω) ) + ||u||C 0,2+α−2θ ([0,T ]×Ω) b

t∈(0,T ]

b





≤C0 ||f ||C 2+α−2θ (Ω) + sup tθ ||g(t, ·)||Cbα (Ω) , b

t∈(0,T ]

for some positive constant C0 , independent of u, f and g. In particular, if f and g belong to Cb2+α (Ω) and C([0, T ] × Ω), respectively, and supt∈[0,T ] ||g(t, ·)||Cbα (Ω) < ∞, then, u belongs to Cb1,2 ([0, T ] × Ω), u(t, ·) ∈ Cb2+α (Ω) for every t ∈ [0, T ] and   sup ||u(t, ·)||C 2+α (Ω) + sup ||Dt u(t, ·)||Cbα (Ω) ≤ C1 ||f ||C 2+α (Ω) + sup ||g(t, ·)||Cbα (Ω) b

t∈[0,T ]

b

t∈[0,T ]

t∈(0,T ]

for some positive constant C1 , independent of u, f and g. Proof It suffices to adapt the arguments, used in Section 14.2.1, to prove Theorem 7.4.1.  The following results can be proved adapting the arguments in the proof of the forthcoming Theorem 13.1.11, taking Theorems 7.4.3 and 8.4.2 into account. For this reason, we do not enter into details. Theorem 9.3.3 Fix k ∈ N and let Hypotheses 9.1.1(i)–(iii) be satisfied with the coefficients of the operator A in C k+α (Ω). Further, suppose that f ∈ C 2+k+α (Ω), g ∈ C (k+α)/2,k+α ((0, T ) × Ω), h ∈ C (k+2+α)/2,k+α ((0, T ) × ∂Ω) satisfy the compatibility conditions  k  X   Dk h(0, x) = (Ak f )(x) +  (Ak−j Dtj g(0, ·))(x), if k is even,   t j=0

k−1 X   k−1 k−1   Dt h(0, x) = (A f )(x) + (Ak−j−1 Dtj g(0, ·))(x), if k is odd,   j=0

for every x ∈ ∂Ω. Then, the solution u ∈ C 1+α/2,2+α ((0, T ) × Ω) to problem (9.0.1) actually belongs to C (k+2+α)/2,k+2+α ((0, T ) × Ω) and there exists a positive constant C, independent of u, f , g and h, such that ||u||C (k+2+α)/2,k+2+α ((0,T )×Ω) b

≤C(||f ||C k+2+α (Rd ) + ||g||C (k+α)/2,k+α ((0,T )×Ω) + ||h||C (k+2+α)/2,k+2+α ((0,T )×Ω) ). b

+

b

Theorem 9.3.4 Fix k ∈ N and let Hypotheses 9.1.1 be satisfied with the coefficients of the operator A in C k+α (Ω) and the coefficients of the operator B in C k+2+α (Ω) and in C k+2+α (Ω; Rd ), respectively. Further, suppose that f ∈ C 2+k+α (Ω), g ∈ C (k+α)/2,k+α ((0, T ) × Ω), h ∈ C (k+1+α)/2,k+1+α ((0, T ) × ∂Ω) satisfy the compatibility con-

Semigroups of Bounded Operators and Second-Order PDE’s ditions  Bf (x) = h(0, x),       k  X   k−j j  Dtk h(0, x) = B (Ak f )(x) + (A Dt g(0, ·))(x) ,

245

if k is odd,

j=0

    k−1  X   k−1 k−1 k−j−1 j  D h(0, x) = B (A f )(x) + (A D g(0, ·))(x) , if k is even,  t  t j=0

for every x ∈ ∂Ω. Then, the solution u ∈ C 1+α/2,2+α ((0, T ) × Ω) to problem (9.0.2) actually belongs to C (k+2+α)/2,k+2+α ((0, T ) × Ω) and there exists a positive constant C, independent of u, f , g and h, such that ||u||C (k+2+α)/2,k+2+α ((0,T )×Ω) ≤C(||f ||C k+2+α (Ω) + ||g||C (k+α)/2,k+α ((0,T )×Ω) + ||h||C (k+1+α)/2,k+1+α ((0,T )×Ω) ). b

9.4

The Associated Semigroup

We conclude this chapter by proving, under Hypotheses 9.1.1, that the Cauchy-Dirichlet problem  t ∈ (0, ∞), x ∈ Ω,   Dt u(t, x) = Au(t, x), u(t, x) = 0, t ∈ (0, ∞), x ∈ ∂Ω, (9.4.1)   u(0, x) = f (x), x ∈ Ω, and the Cauchy problem    Dt u(t, x) = Au(t, x), Bu(t, x) = 0,   u(0, x) = f (x),

t ∈ (0, ∞), t ∈ (0, ∞),

x ∈ Ω, x ∈ ∂Ω, x ∈ Ω,

(9.4.2)

are governed in C(Ω) (resp. in Cb (Ω)) by a semigroup of bounded operators. Theorem 9.4.1 Let Hypotheses 9.1.1 be satisfied. Then, the following assertions are satisfied. (i) For each f ∈ Cb (Ω) (resp. f ∈ C(Ω)) there exists a unique bounded classical solution u ∈ C([0, ∞)×Ω)∩C 1+α/2,2+α ((0, ∞)×Ω) (resp. C([0, ∞)×Ω)∩C 1+α/2,2+α ((0, ∞)× Ω)) of the Cauchy problem (9.4.1), which is bounded with respect to the sup-norm in (0, T ) × Ω for every T > 0. (ii) For each f ∈ Cb (Ω) (resp. f ∈ C(Ω)) there exists a unique bounded classical solution u ∈ C([0, ∞)×Ω)∩C 1+α/2,2+α ((0, ∞)×Ω) (resp. C([0, ∞)×Ω)∩C 1+α/2,2+α ((0, ∞)× Ω)) of the Cauchy problem (9.4.2), which is bounded with respect to the sup-norm in (0, T ) × Ω for every T > 0. Further, in both the cases, for every T > 0 there exists a positive constant CT , independent of u and f , such that ||u||C((0,T )×Ω) ≤ CT ||f ||Cb (Ω) . (9.4.3)

246

Parabolic Equations in Bounded Smooth Domains Ω

Proof Repeating the same arguments as in the proofs of Theorem 7.5.3, Theorem 8.5.2 and using Theorem 9.1.2, Theorem 9.1.3 and Theorem 9.2.2 the assertions follows.  As in Chapter 8, see Theorem 8.5.3, we can define the semigroups associated with the Cauchy problems (9.4.1) and (9.4.2) in Cb (Ω) and C(Ω). We omit here the proof, since it is similar to that of Theorem 8.5.3. Corollary 9.4.2 Under Hypotheses 9.1.1, we can associate a semigroup of bounded operators {T (t)} to (9.4.1) and a semigroup of bounded operators {TB (t)} to (9.4.2) in Cb (Ω) (resp. C(Ω)). Moreover, each operator T (t) (resp. TB (t)) gives rise to a strongly continuous semigroup on C(Ω). Finally, if (fn ) ⊂ Cb (Ω), which converges to a function f ∈ Cb (Ω), locally uniformly in Ω, then, for every t > 0, the function T (t)fn (resp. TB (t)fn ) converges to T (t)f (resp. TB (t)f ) locally uniformly in Ω.

9.5

Exercises

1. Let Ω1 ⊂ RM and Ω2 ⊂ RN be two open sets and let u ∈ Cb (Ω1 ), v ∈ Cbθ (Ω2 ) for some θ ∈ (0, 1). Set f (x, y) = u(x)v(y) for every (x, y) ∈ Ω1 × Ω2 . Prove that the function f (x, ·) belongs to Cbθ (Ω2 ) for every x ∈ Ω1 and sup [f (x, ·)]C θ (Ω2 ) = ||u||∞ [v]C θ (Ω2 ) . x∈Ω1

b

b

2. Prove estimate (9.1.14). 3. Prove Theorems 9.3.1 and 9.3.2. 4. Prove Theorems 9.3.3 and 9.3.4. 5. Give the details of the proof of Theorem 9.4.1 and Corollary 9.4.2.

Part III

Elliptic Equations

Chapter 10 Elliptic Equations in

R

d

In this chapter, we begin the analysis of elliptic equations, considering the equation λu − Au = f

(10.0.1)

in the whole space Rd , when A is the second-order elliptic operator defined by Aψ(x) =

d X

qij (x)Dij ψ(x) +

i,j=1

d X

bj (x)Dj ψ(x) + c(x)ψ(x)

j=1

=Tr(Q(x)D2 ψ(x)) + hb(x), ∇ψ(x)i + c(x)ψ(x),

(10.0.2)

with Q(x) = (qij (x)) for x ∈ Rd and λ ∈ C has a sufficiently large real part. We deal both with classical solutions, i.e., with solutions in Cb2 (Rd ; C), and with solutions in the Sobolev space W 2,p (Rd ; C), considering the case p ∈ (1, ∞). We also deal with the limiting case f ∈ L∞ (Rd ; C) and f ∈ Cb (Rd ; C). Slightly different assumptions on the coefficients of the operator A are assumed in these cases, but in all of them the operator A is assumed to be uniformly elliptic, i.e., we assume that the minimum and the maximum eigenvalues of the symmetric matrix Q(x) are positive for every x ∈ Rd , bounded on Rd and the infimum over Rd of the minimum eigenvalue of Q(x) is positive. In Section 10.1, we study equation (10.0.1) when f ∈ Cbα (Rd ; C) for some α ∈ (0, 1). The strategy that we follow is the same as in Chapter 6; the starting point is the case when A is the Laplacian (considered in Subsection 10.1.1). In this situation, we show that, for every λ > 0 (actually, this is true for every λ ∈ C with positive real part) the unique classical solution to equation (10.0.1) is the function u : Rd → C defined by Z ∞ u(x) = e−λt (T (t)f )(x) dt, x ∈ Rd , 0

i.e., the Laplace transform of the Gauss-Weierstrass semigroup deeply studied in Chapter 5. This is in the sprit of Chapters 2 and 3 which exploit the interplay between semigroups and the resolvent equation λx − Ax = y when A is the generator of a strongly continuous semigroup or a sectorial operator, associated to an analytic semigroup. Based on the analysis of the Laplace equation, in Subsection 10.1.2, we consider the more general operator A in (10.0.2) whose coefficients are assumed to be bounded and α-H¨older continuous over Rd . The optimal Schauder estimate ||u||C 2+α (Rd ;C) ≤ Cλ ||λu − Au||Cbα (Rd ;C) b

for functions u ∈ Cb2+α (Rd ; C), λ ∈ C, with real part greater than c0 , where we recall that c0 = sup c(x), x∈Rd

and the results proved for the Laplace equation allows us to apply the continuation method 249

250

Elliptic Equations in Rd

to show the existence and uniqueness result of a solution u ∈ Cb2+α (Rd ; C) of the equation λu − Au = f for every λ ∈ C, with Re λ > c0 and every f ∈ Cbα (Rd ; C) (see Theorem 10.1.4). Next, in Subsection 10.1.3 we prove that, the more the coefficients of the operator A and the datum f are smooth, the more the solution u to the equation λu − Au = f is itself smooth. This result is obtained in a simple and elegant way from the corresponding result proved in Chapter 6 for solutions to parabolic Cauchy problems on Rd . To conclude the subsection, we prove interior Schauder estimates for solutions to the equation λu − Au = f in a domain Ω of Rd (see Theorems 10.1.9 and 10.1.11). Also in this case, the proof can be easily obtained from analogous results proved for parabolic equations. In Section 10.2, we study the elliptic equation (10.0.1) when f ∈ Lp (Rd ; C) (p ∈ (1, ∞)) in the context of Sobolev spaces. The strategy is the same used in Section 10.1: first, we study the prototype equation, i.e., the Laplace equation, and then we consider the more general operator A. The main tool to study the Laplace equation in Sobolev spaces is the celebrated Calder´ on-Zygmund inequality that allows to estimate the second-order derivatives of a function u ∈ W 2,p (Rd ; C) from above in terms of the Lp (Rd ; C)-norms of u and its Laplacian. This property is straightforward to prove in the particular case p = 2, where two integration by parts are enough to get the result. Things are much more difficult for the other values of p ∈ (1, ∞). In these situations, we introduce the Newtonian potential N , which has the property that, for functions f smooth and compactly supported on Rd , the function N f solves the Poisson equation ∆u = f . Using the Marcinkiewicz’s interpolation theorem, we show that the operator Dij N , defined on functions f ∈ Cc∞ (Rd ; C) can be extended with a bounded operator on Lp (Rd ; C) for every i, j = 1, . . . , d. From this result, Calder´onZygmund inequality follows easily. Using this inequality, in Subsection 10.2.2 we solve the Laplace equation for every λ ∈ C, with positive real part, and f ∈ Lp (Rd ; C) (p ∈ (1, ∞)). Subsection 10.2.3 is devoted to study equation (10.0.1) in its full generality. As in Section 10.1, two are the ingredients needed to prove that such an equation is uniquely solvable in W 2,p (Rd ; C) for every f ∈ Lp (Rd ; C) and every λ ∈ C with real part greater than λp for a suitable λp ∈ R: (i) the a priori estimate p (10.0.3) |λ|||u||Lp (Rd ;C) + |λ||||∇u|||Lp (Rd ;C) + ||D2 u||Lp (Rd ;C) ≤ C||λu − Au||Lp (Rd ;C) for every u ∈ W 2,p (Rd ; C), λ in a suitable right-halfplane of C and some positive constant C, independent of u and λ, (ii) the continuation method. Estimate (10.0.3) is proved starting from the corresponding estimate proved in Subsection 10.2.2 in the case of the Laplacian, using with the classical technique, already used in Chapter 6, of freezing the coefficients of operator A (see Proposition 10.2.17). In the particular case when A is in divergence form, i.e., when Aψ = div(Q∇ψ) + hb, ∇ψi + cψ on functions ψ ∈ W 2,p (Rd ; C), we can better characterize the constant λp in terms of the drift coefficient b and the potential c of operator A. 2,p Next, we prove some interesting results. We show that, if u ∈ Wloc (Rd ; C)∩Lq (Rd ; C), for q d 2,q d some p, q ∈ (1, ∞), is such that Au ∈ L (R ; C), then u ∈ W (R ; C) and, as a byproduct, that, if u solves the equation λu − Au = f for some λ ∈ C and f ∈ Lp (Rd ; C) ∩ Lq (Rd ; C), then u ∈ W 2,p (Rd ; C) ∩ W 2,q (Rd ; C) (see Proposition 10.2.24 and Corollary 10.2.26). Finally, in Subsection 10.2.4, which is the counterpart of Subsection 10.1.3, we prove further regularity results for solutions of the equation (10.0.1), interior Lp -estimates for solutions of the elliptic equation λu − Au = f on domains of Rd . In the particular case when the coefficients of the operator A are constant over Rd , we also consider a variational

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∂f when f ∈ Lp (Rd ; C), η is a nontrivial vector of Rd ∂η and λ ∈ C has positive real part. We prove that such equation admits a unique solution u ∈ W 1,p (Rd ; C) and provide an estimate of its W 1,p (Rd ; C)-norm which involves only the Lp (Rd ; C)-norm of f . This result represents one of the key tools used in Chapter 12 to study elliptic equations in smooth domain with general boundary conditions. In the last section of this chapter (see Section 10.3) we analyze equation (10.0.1) when the datum f belongs to L∞ (Rd ; C) or to Cb (Rd ; C). The second case can be obtained as a straightforward consequence of the first one. The main step in the solvability in L∞ (Rd ; C) of equation (10.0.1) is the proof of the a priori estimate p d ep ||λu − Au||∞ (10.0.4) |λ|||u||∞ + |λ||||∇u|||∞ + |λ| 2p sup ||D2 u||Lp (B(x0 ,|λ|−1/2 );C) ≤ C

version of the equation λu − Au =

x0 ∈Rd

for every p ∈ (1, ∞), λ ∈ C, with sufficiently large real part, and some positive constant ep . Such an estimate is obtained taking advantage of (10.0.3) and the classical method C of freezing the coefficients of the operator A. Using (10.0.4) and a compactness argument, we show that for every λ in a suitable right-halfplane of the complex plane and for every T 2,p f ∈ L∞ (Rd ; C), there exists a unique solution u ∈ W 1,∞ (Rd ; C) ∩ p 0,

and thus conclude that (Reλ)||u||Cbα (Rd ;C) ≤ Cα,α ||f ||Cbα (Rd ;C) . Next, we introduce the function v : [0, ∞) × Rd → C defined by v(t, x) = eλt u(x) for every t > 0 and x ∈ Rd . Clearly, v ∈ Cb1,2 ([0, T ] × Rd ; C) for every T > 0 and it is immediate to check that it solves the Cauchy problem ( Dt v(t, x) = ∆v(t, x) + g(t, x), t > 0, x ∈ Rd , v(0, x) = v0 (x), x ∈ Rd , where v0 = u and g(t, x) = eλt f (x) for every (t, x) ∈ [0, ∞) × Rd . Applying Theorem 5.4.5 (which clearly holds also for complex-valued functions) with T = 1, α = β = θ, we deduce that v(1, ·) belongs to Cb2+α (Rd ; C) and  
1 ||v(1, ·)||C 2+α (Rd ;C) ≤ C1 ||u||Cbα (Rd ;C) + eRe λ ||f ||Cbα (Rd ;C) ≤ C2 eRe λ + ||f ||Cbα (Rd ;C) , b Re λ the constants C1 and C2 being independent of f and λ. Estimate (10.1.1) follows at once. To complete the proof, we observe that u is the unique bounded classical solution to the equation λu − ∆u = f in view of Corollary 4.2.2(i). 

10.1.2

More general elliptic operators

In this section, we prove optimal Schauder estimate for solutions to the elliptic equation (10.0.1). To begin with, as a byproduct of the a priori parabolic estimates in Section 6.2, we prove some a priori elliptic estimates. The following ones are the standing assumptions of this subsection. Hypotheses 10.1.2 (i) The coefficients qij = qji , bj (i, j = 1, . . . , d) and c are bounded and α-H¨ older continuous in Rd for some α ∈ (0, 1); (ii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for all x, ξ ∈ Rd . Theorem 10.1.3 Under Hypotheses 10.1.2 there exist two positive constants C and Cλ such that ||u||C 2+α (Rd ;C) ≤ Cλ ||λu − Au||Cbα (Rd ;C) (10.1.2) b

for each λ ∈ C such that Reλ > c0 := supx∈Rd c(x) and ||u||C 2+α (Rd ;C) ≤ C ||u||∞ + ||Au||Cbα (Rd ;C)




(10.1.3)

b

for every u ∈ Cb2+α (Rd ; C). Constants C and Cλ depend, in a continuous way, on the ellipticity constant of the operator A and on the Cbα -norms of its coefficients. Cλ depends also on λ and, as a function of λ, it is locally bounded in the halfplane {λ ∈ C : Reλ > c0 }.

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Proof We preliminarily observe that estimate (10.1.3) follows from (10.1.2). Indeed, from this last inequality with λ, fixed positive and larger than c0 , it follows that ||u||C 2+α (Rd ;C) ≤ Cλ (λ||u||Cbα (Rd ;C) + ||Au||Cbα (Rd ;C) ).

(10.1.4)

b

Next, we use estimate (1.1.2) with (θ, β, α) = (0, α, 2+α), applied to the real and imaginary parts of u, and Young inequality to show that ||u||Cbα (Rd ;C) ≤ ε||u||C 2+α (Rd ;C) + Cε ||u||∞ for b every ε > 0 and some positive constant Cε , independent of u, blowing up as ε tends to 0+ . Replacing this inequality in (10.1.4) and then choosing ε such that Cλ λε < 1, we can complete the proof of (10.1.3). To prove (10.1.2), we fix λ as in the statement and introduce the function v : [0, 1]×Rd → C, defined by v(t, x) = eλt ϑ(t)u(x), where ϑ ∈ Cc∞ (R) is a smooth function which satisfies the condition χ[1,∞) ≤ ϑ ≤ χ[1/2,∞) . Clearly, v belongs to Cb0,2+α ([0, 1]×Rd ; C)∩Cb1,2 ([0, 1]× Rd ; C) and v(0, ·) ≡ 0, so that we can apply estimate (6.3.1) to infer that ||u||C 2+α (Rd ;C) = ||v||C 0,2+α ([0,1]×Rd ;C) ≤ K||Dt v − Av||C 0,α ([0,1]×Rd ;C) , b

b

b

where the constant K depends, in a continuous way on µ and on the Cbα (Rd )-norms of the coefficients of the operator A. Since Dt v(t, x) − Av(t, x) = eλt ϑ(t)(λu − Au) + eλt ϑ0 (t)u(x) for every (t, x) ∈ [0, 1] × Rd , from the previous estimate we deduce that ||u||C 2+α (Rd ;C) ≤ KeRe λ (1 + ||ϑ0 ||∞ )(||u||Cbα (Rd ;C) + ||λu − Au||Cbα (Rd ;C) ).

(10.1.5)

b

To get rid of the Cbα (Rd ; C) norm of u from the right-hand side of (10.1.5), we can argue as in the proof of estimate (10.1.3).  Now, we can prove the existence-uniqueness result for classical solutions of the equation (10.0.1). Theorem 10.1.4 For each λ ∈ C, such that Reλ > c0 , and each f ∈ Cbα (Rd ; C), there exists a unique bounded classical solution to the elliptic equation λu − Au = f . Moreover, u ∈ Cb2+α (Rd ; C) and ||u||C 2+α (Rd ;C) ≤ C||f ||Cbα (Rd ;C) (10.1.6) b

for some positive constant C, depending in a continuous way, on λ, the Cbα -norms of the coefficients of the operator A, as well as on the ellipticity constant µ, but being independent of f . Proof The proof follows straightforwardly from estimate (10.1.2), Theorem 10.1.1 and + the continuity method. We first prove it for λ ∈ (c+ 0 , ∞). For every σ ∈ [0, 1] and λ > c0 , 2+α d α d we consider the operator Tσ : Cb (R ; C) → Cb (R ; C), defined by Tσ u = λu − Aσ u for every u ∈ Cb2+α (Rd ; C), where Aσ = σA + (1 − σ)∆. Clearly, each operator Aσ satisfies Hypotheses 10.1.2 and the Cbα (Rd )-norms of its coefficients can be bounded from above uniformly with respect to σ, whereas it ellipticity constant can be bounded from below by µ ∧ 1. Moreover, supRd (σc) = σc0 ≤ c+ 0 for every σ ∈ [0, 1]. Therefore, applying estimate (10.1.2) we infer that ||u||C 2+α (Rd ;C) ≤ Cλ ||Tσ u||Cbα (Rd ;C) . Since operator T0 is invertible (see b Theorem 10.1.1), then all the operators Tσ are invertible. In particular, T1 is invertible and this implies that the equation λu − Au = f is uniquely solvable in Cb2+α (Rd ; C) for every f ∈ Cbα (Rd ; C). To complete the proof, we denote by R(λ, A) the inverse of the operator T1 . Estimate (10.1.2) shows that ||R(λ, A)||L(Cbα (Rd ;C)) ≤ Cλ for every λ ∈ Σc0 := {λ ∈ C : Reλ > c0 } ∩ ρ(A), where the function λ 7→ Cλ is locally bounded in Σc0 . This and Remark A.4.5 imply that Σc0 lies in the resolvent set of the operator A, i.e., the equation λu − Au = f is uniquely solvable in Cb2+α (Rd ; C) for every f ∈ Cbα (Rd ; C) and λ ∈ Σc0 . 

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Remark 10.1.5 In Chapter 14, see Corollary 14.1.4, we will write explicitly how the constant C in (10.1.6) depends on λ.

10.1.3

Further regularity results and interior estimates

We begin this subsection by proving that the more the coefficients of the operator A and f are smooth, the more the solution u of the equation λu − Au = f is itself smooth. For this purpose, we introduce the following set of assumptions. Hypotheses 10.1.6 (i) The coefficients qij = qji , bj (i, j = 1, . . . , d) and c belong to Cbk+α (Rd ) for some α ∈ (0, 1) and k ∈ N; (ii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for all x, ξ ∈ Rd . To simplify the notation, in this subsection we set Σc0 = {λ ∈ C : Reλ > c0 }. Under the above assumptions we can prove the following result. Theorem 10.1.7 Let Hypotheses 10.1.6 be satisfied. Then, for every λ ∈ Σc0 and f ∈ Cbk+α (Rd ; C), the equation λu − Au = f admits a unique solution u ∈ Cbk+2+α (Rd ; C). Moreover, there exists a positive constant Ck , depending in a continuous way on µ, on λ and on the Cbk+α (Rd )-norms of the coefficients of the operator A, such that ||u||C k+2+α (Rd ;C) ≤ Ck ||f ||C k+α (Rd ;C) . b

b

Proof The assertion follows from Theorem 6.5.2(ii). Indeed, consider a function ϑ ∈ Cb∞ ([0, ∞)) such that χ[1,∞) ≤ ϑ ≤ χ[1/2,∞) . Then, the function v : [0, 1] × Rd → C, defined by v(t, x) = eλt ϑ(t)u(x) for every (t, x) ∈ [0, 1] × Rd , solves the parabolic equation Dt u = Au + g, where g(t, x) = eλt ϑ(t)(λu(x) − Au(x)) + eλt ϑ0 (t)u(x),

(t, x) ∈ [0, 1] × Rd .

Moreover, v(0, ·) ≡ 0. Therefore, if u ∈ Cbj+α (Rd ; C) for some j ≤ k, then function g belongs to Cb0,j+α ([0, 1] × Rd ) and ||g||C 0,j+α ([0,1]×Rd ;C) ≤ eλ ||λu − Au||C j (Rd ;C) + eλ ||ϑ0 ||∞ ||u||C j+α (Rd ;C) . b

b

b

Thus, Theorem 6.5.2(ii) shows that v ∈ Cb0,j+2+α ([0, 1] × Rd ; C) and 0 ||v||C 0,j+2+α ([0,1]×Rd ;C) ≤ Cj,λ (||λu − Au||C j+α (Rd ;C) + ||u||C j+α (Rd ;C) ) b

b

b

0 Cj,λ

for some positive constant which depends in a continuous way on λ, µ and the j+α d Cb (R )-norms of the coefficients of the operator A. This estimate implies that 0 ||u||C j+2+α (Rd ;C) ≤ Cj,λ (||λu − Au||C j+α (Rd ;C) + ||u||C j+α (Rd ;C) ). b

b

(10.1.7)

b

Based on this estimate and Theorem 10.1.4 we can make a bootstrap argument work to complete the proof. Indeed, by Theorem 10.1.4, we know that u ∈ Cb2+α (Rd ; C) and ||u||C 2+α (Rd ;C) ≤ C||λu − Au||Cbα (Rd ;C) so that we can take j = 1 in (10.1.7) to infer that b

e1,λ ||λu − Au|| 1+α d , ||u||C 3+α (Rd ;C) ≤ C C (R ;C) b

b

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e1,λ depends in a continuous way on λ, µ and the C j+α (Rd )-norms of where the constant C b the coefficients of the operator A. Iterating this procedure, in a finite number of step we get the assertion.  We now pass to proving elliptic interior Schauder estimates. For this purpose, we assume the following set of assumptions. Hypotheses 10.1.8 α Cloc (Ω);

(i) The coefficients qij = qji , bj (i, j = 1, . . . , d) and c belong to

(ii) there exists a positive function µ : Ω → R, with positive infimum over each compact subset of Ω, such that hQ(x)ξ, ξi ≥ µ(x)|ξ|2 for all ξ ∈ Rd and x ∈ Ω. 2 Theorem 10.1.9 Let Hypotheses 10.1.8 be satisfied. If u ∈ Cloc (Ω; C) is such that λu − 2+α α Au ∈ Cloc (Ω; C) for some λ ∈ C, then u ∈ Cloc (Ω; C) and, for every Ω0 b Ω00 b Ω, there exists a positive constant C, which depend in a continuous way on λ, on the infimum of µ over Ω00 , on the distance between Ω0 and Ω00 and on the coefficients of the operator A, such that

||u||C 2+α (Ω0 ;C) ≤ C(||λu − Au||C α (Ω00 ;C) + ||u||L∞ (Ω00 ;C) ). Proof The proof follows straightforwardly from Theorem 6.2.9. Indeed, it suffices to apply that theorem to the function v : [0, 1] × Rd → C, defined by v(t, x) = eλt u(x) for every (t, x) ∈ [0, 1] × Ω, which belongs to C 1,2 ([0, 1] × Ω; C) and solves the equation Dt v − Av = eλt (λu − Au), choosing K1 = Ω0 , K2 = Ω00 and e.g., I1 = (1/2, 3/4), I2 = (1/3, 5/6).  Also in this situation, if the coefficients of the operator A are smoother we can prove further regularity results of the function u. More precisely, let us assume the following set of assumptions. Hypotheses 10.1.10 (i) The coefficients qij = qji , bj (i, j = 1, . . . , d) and c belong to k+α Cloc (Ω) for some α ∈ (0, 1) and k ∈ N; (ii) there exists a positive function µ : Ω → R, with positive infimum over each compact subset of Ω, such that hQ(x)ξ, ξi ≥ µ(x)|ξ|2 for all ξ ∈ Rd and x ∈ Ω. Then, the following result holds true. Theorem 10.1.11 Let Hypotheses 10.1.10 be satisfied. If u ∈ C 2 (Ω; C) is such that λu − k+α k+2+α Au ∈ Cloc (Ω; C) for some λ > c0 , then u ∈ Cloc (Ω; C) and, for every Ω0 b Ω00 b Ω, there exists a positive constant C, which depend on the distance between Ω0 and Ω00 and, in a continuous way, on λ, the infimum of µ over Ω00 , and on the coefficients of the operator A, such that ||u||C k+2+α (Ω0 ;C) ≤ C(||λu − Au||C k+α (Ω00 ;C) + ||u||C(Ω00 ;C) ). Proof It suffices to argue as in the proof of Theorem 10.1.9, taking Proposition 6.2.11(ii) into account. 

Elliptic Equations in Rd

256

R ; C) (p ∈ (1, ∞))

Solutions in Lp (

10.2

d

In this section, we study the elliptic equation λu− Au = f , when the operator A is given by (10.0.2) and f belongs to Lp (Rd ; C) for some p ∈ (1, ∞). As in the previous section, we first consider the case when A is the Laplacian. Then, using the continuity method, we deal with the more general elliptic operators.

10.2.1

The Calder´ on-Zygmund inequality

b = C(d, b p) such In this subsection, we will prove that there exists a positive constant C that d X b ||Dij u||Lp (Rd ;C) ≤ C||∆u|| u ∈ W 2,p (Rd ; C), (10.2.1) Lp (Rd ;C) , i,j=1

for every p ∈ (1, ∞). Estimate (10.2.1) is the well-known Calder´on-Zygmund inequality, and it represents the key tool to study the elliptic equation λu − ∆u = f when f ∈ Lp (Rd ; C). Remark 10.2.1 Estimate (10.2.1) is straightforward to prove if p = 2. Indeed, integrating twice by parts, it is easy to show that Z Rd

|∆u|2 dx =

d Z X i,j=1

Dii uDjj u dx = −

Rd

d Z X i,j=1

Diij uDj u dx =

Rd

d Z X i,j=1

|Dij u|2 dx

Rd

for every u ∈ Cc∞ (Rd ; C). Since this space is dense in W 2,2 (Rd ; C) (see Theorem 1.3.4), the b 2) = 1. Calder´ on-Zygmund inequality follows immediately with C(d, The case p 6= 2 demands more efforts. To begin with, we introduce the Poisson kernel Γ : Rd \ {0} → R, defined by  1  log(|x|),  2π Γ(x) = 1   |x|2−d , ωd d(2 − d)

d = 2, x ∈ Rd \ {0},

d ≥ 3,

where ωd denotes the surface measure of B(0, 1) ⊂ Rd . Lemma 10.2.2 The Poisson kernel Γ belongs to C ∞ (Rd \ {0}). Moreover, for every multiindex β there exists a positive constant Cβ such that |Dβ Γ(x)| ≤ Cβ |x|2−d−|β| ,

x ∈ Rd \ {0}.

(10.2.2)

In particular, the functions Γ and Di Γ are locally integrable over Rd for every i = 1, . . . , d, whereas the derivatives of higher-order are not locally integrable over Rd . Finally, ∆Γ ≡ 0 on Rd \ {0}. The easy proof of the previous lemma is left to the reader. Now, we introduce the Newtonian potential N . In the rest of the section we assume that p is arbitrarily fixed in (1, ∞).

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Definition 10.2.3 For every f ∈ Lp (Rd ; C), the Newtonian potential N f is defined as the convolution of the Poisson kernel and f , i.e., by Z (N f )(x) = Γ(x − y)f (y) dy, x ∈ Rd . Rd

As the following proposition shows, if f is smooth enough, then N f is a smooth solution of the Poisson equation ∆u = f . Proposition 10.2.4 For every f ∈ Cc2 (Rd ; C), the function N f belongs to C 2 (Rd ; C) and ∆N f = f on Rd . Proof Fix f ∈ Cc2 (Rd ; C). Since Γ ∈ L1loc (Rd ), the function N f is well defined. Moreover, it is smooth on Rd and Z (Dβ N f )(x) = Γ(x − y)Dβ f (y) dy, x ∈ Rd , |β| ≤ 2. Rd

Therefore, for every fixed x ∈ Rd and r > 0, such that the support of f is contained in B(x, r), we can write Z Z (∆N f )(x) = Γ(x − y)∆f (y) dy = Γ(y − x)∆f (y) dy Rd Rd Z = lim Γ(y − x)∆f (y) dy. (10.2.3) ρ→0+

B(x,r)\B(x,ρ)

Integrating by parts and recalling that ∆Γ ≡ 0 on Rd \ {0}, we deduce that Z (∆N f )(x) = lim+ Γ(y − x)∆f (y) dy ρ→0

B(x,r)\B(x,ρ)

 = lim+ − ρ→0

Z Γ(y − x) ∂B(x,ρ)

∂f (y) dσ(y) + ∂ν

Z ∂B(x,ρ)

 ∂Γ (y − x)f (y) dσ(y) , ∂ν (10.2.4)

where ν = ν(y) denotes the unit outward normal vector to ∂B(x, ρ) at y and dσ(y) is the surface measure on ∂B(x, ρ). We now observe that the first integral in the last side of (10.2.4) vanishes as ρ tends to 0+ . Indeed, using (10.2.2) we can estimate Z Z Γ(y − x) ∂f (y) dσ(y) ≤||f ||C 1 (Rd ;C) Γ(y − x) dσ(y) ≤ C||f ||Cb1 (Rd ;C) ρ2 b ∂ν ∂B(x,ρ) ∂B(x,ρ) (10.2.5) for ρ sufficiently small. As far as the second integral in the last-side of (10.2.4) is concerned, we claim that it converges to f (x) as ρ tends to 0+ . Indeed, since ∂Γ (y − x) = |y − x|−1 h∇Γ(y − x), y − xi, ∂ν we deduce that ∂Γ 1 (y − x) = |y − x|1−d , ∂ν dωd

y ∈ ∂B(x, ρ).

Elliptic Equations in Rd

258 Therefore, Z

∂B(x,ρ)

Z 1 ∂Γ 1−d (y − x)f (y) dy − f (x) = |y − x| f (y) dσ(y) − f (x) ∂ν dωd ∂B(x,ρ) Z 1 = f (y) dy − f (x) d−1 dωd ρ ∂B(x,ρ) Z 1 ≤ |f (y) − f (x)| dy (10.2.6) dωd ρd−1 ∂B(x,ρ)

and the last side of the previous chain of inequalities vanishes as ρ tends to 0+ since f is continuous at x. From (10.2.3)-(10.2.6), we immediately deduce that (∆N f )(x) = f (x).  We now extend the Newtonian potential to smooth functions defined on domains Ω ⊂ Rd by setting Z (N f )(x) = Γ(x − y)f (y) dy = (N (f χΩ ))(x), x ∈ Ω. Ω

The same arguments used in the proof of Proposition 10.2.4 can be used to prove the following result. Proposition 10.2.5 Let Ω be a bounded set of class C 1 . Then, Z Z Z ∂u ∂Γ (x − y)u(y) dσ(y) − Γ(x − y) (y) dσ(y) u(x) = Γ(x − y)∆u(y) dy + ∂ν ∂ν ∂Ω Ω ∂Ω for every x ∈ Ω and u ∈ C 2 (Ω; C). In particular, if u and its normal derivative both vanish on ∂Ω, then u = N ∆u. Proposition 10.2.6 For every f ∈ Cc∞ (Rd ; C) it holds that d X

||Dij N f ||2L2 (Rd ;C)

i,j=1

Z = Rd

|(∆N f )(y)|2 dy = ||f ||2L2 (Rd ;C) .

Proof Fix f ∈ Cc∞ (Rd ; C) and r > 0 such that supp(f ) ⊂ B(0, r). Then, for every x ∈ Rd \ B(0, r) it holds that Z Z (Dβ N f )(x) = Γ(x − y)Dβ f (y) dy = Dβ Γ(x − y)f (y) dy. B(0,r)

B(0,r)

Therefore, using (10.2.2) we deduce that Z |f (y)| Cβ |(Dβ N f )(x)| ≤ Cβ dy ≤ ||f ||L1 (B(0,r);C) . d−2+|β| |x − y| (|x| − r)d−2+|β| B(0,r)

(10.2.7)

Taking Proposition 10.2.4 into account and arguing as in the proof of Remark 10.2.1, we can write for R > r Z Z 2 2 ||f ||L2 (Rd ;C) = |f | dy = |∆N f |2 dy B(0,R)

=

d X i,j=1

B(0,R)

Z (Dii N f )(Djj N f ) dy B(0,R)

Semigroups of Bounded Operators and Second-Order PDE’s =−

d Z X

=

Z

+

j=1 d X

|Dij N f |2 dy −

B(0,R)

i,j=1

d Z X

B(0,R)

i,j=1 d X

(Di N f )(Dijj N f ) dy +

i,j=1

Djj N f

∂B(0,R)

259 ∂N f dσ ∂ν

Z (Di N f )(Dij N f )νj dσ ∂B(0,R) 3

d Z X

X ∂N f dσ =: Ij (R). (Djj N f ) ∂ν ∂B(0,R) j=1

j=1

(10.2.8)

Using (10.2.7), we can show that I2 (R) vanishes as R tends to ∞. Indeed, |I2 (R)| ≤

r Z X

|Di N f ||Dij N f | dσ ≤K1 (R − r)

∂B(0,R)

i,j=1

1−2d

Z dσ ∂B(0,R)

=K2 (R − r)1−2d Rd−1 for some positive constants K1 and K2 , which are independent of R and r. Similarly, I3 (R) ≤ K3 (R − r)1−2d Rd−1 for some positive constant K3 , independent of R and r. By monotone convergence lim

R→∞

d Z X i,j=1

B(0,R)

Z

d Z X i,j=1

|∆N f |2 dy =

lim

R→∞

|Dij N f |2 dy = Z

|Dij N f |2 dy,

Rd

|∆N f |2 dy

Rd

B(0,R)

and both the two limits are finite. Letting R tend to ∞ in (10.2.8), the assertion follows.  Now we prove the following additional properties of the operator N . Theorem 10.2.7 For every i, j = 1, . . . , d, the operator Pij , defined on Cc∞ (Rd ; C) by Pij f = Dij N f for every f ∈ Cc∞ (Rd ; C), can be (uniquely) extended with a bounded operator on Lp (Rd ; C). Things are easier in the case p = 2. Indeed, as it has been proved in Proposition 10.2.6, d Z X i,j=1

Rd

|Pij f |2 dx =

Z Rd

|f |2 dx,

f ∈ Cc2 (Rd ; C).

(10.2.9)

Fix f ∈ L2 (Rd ; C), i, j = 1, . . . , d, and consider a sequence (fn ) ⊂ Cc2 (Rd ; C) converging in L2 (Rd ; C) to f . From (10.2.9) it follows that Z Z 2 |Pij fm − Pij fn | dx ≤ |fm − fn |2 dx, m, n ∈ N. Ω

Rd

Hence, (Pij fn ) is a Cauchy sequence in L2 (Rd ; C), so that it converges on L2 (Rd ; C). This is enough to infer that Pij can be extended to L2 (Rd ; C) with a bounded operator, which satisfies the estimate ||Pij ||L(L2 (Rd ;C)) ≤ 1. The case p 6= 2 demands more efforts and some technical results are needed, which we state and prove here below. Proposition 10.2.8 For every f ∈ L1 (Rd ; C) and t > 0, there exist two measurable and disjoint subsets of Rd , say F and G, such that

260

Elliptic Equations in Rd

(i) |f | ≤ t almost everywhere on F ; (ii) there exists at most a countable set K ⊂ N such that G =

[

Qk , where each Qk is a

k∈K

cube with edges parallel to the coordinate axis and such that Qi ∩ Qj = ∂Qi ∩ ∂Qj for every i, j. Moreover, Z 1 t< |f | dx ≤ 2d t, k ∈ K, |Qk | Qk where |Qk | denotes the Lebesgue measure of the cube Qk . Proof Clearly, we can assume that f is real and almost everywhere positive on Rd . We fix t > 0 and begin by considering a partition {Rk : k ∈ N} of Rd consisting of cubes with edges parallel to the coordinate axis and such that Z 1 f dx ≤ t, k ∈ N. |Rk | Rk Note that it is possible to determine such a partition. Indeed, Z 1 1 f dx ≤ ||f ||L1 (Rd ) , |Rk | Rk |Rk | so that it suffices to choose the edge ` of the cube Rk such that `d ≥ t−1 ||f ||L1 (Rd ) . As a second step, we take a cube Rj of the previous family and split each of its edge into 2 parts, thus obtaining 2d subcubes Rj,h h = 1, . . . , 2d . We compute the average of f over each of these subcubes. If the average over the cube Rj,h is larger than t, then the cube Rj,h will be a subset of G. On the other hand, if the average over the cube Rj,h is not larger than t, then we divide this cube into 2d subcubes splitting each of its edges into two parts and computing the average of f over each of this new subcubes. Every subcube, which has the property that the average of f over it is larger than t, will be a subset of G. The remaining subcubes are split into 2d subcubes and the above argument repeated. Arguing in this way, we can construct the set G and, then, the set F = Rd \ G. Let us prove that F and G enjoy properties (i) and (ii). Suppose that x ∈ F is a Lebesgue point e k ) of cubes such that Q e k+1 ⊂ Q e k , |Q e k | = 2 d |Q e k+1 | of f . Then, there exists a sequence (Q for every k ∈ N and Z 1 f dy ≤ t, k ∈ N. e k | Qek |Q Since the measure of Qk tends to zero as k tends to ∞ and x is a Lebesgue point of f , from the previous estimate it follows that Z 1 f (x) = lim f dy ≤ t. e k | Qek k→∞ |Q Property (i) follows from observing that the set of all the Lebesgue points of f has a negligible complement over Rd . Finally, to prove property (ii), we consider one of the cubes Q which constitute G. Then by construction, Z 1 f dy. t< |Q| Q

Semigroups of Bounded Operators and Second-Order PDE’s

261

Moreover, Q is one of the 2d subcubes which form the cube Q0 , which has the property that Z 1 f dy ≤ t. |Q0 | Q0 Since |Q0 | = 2d |Q|, we can write 2d 2 t≥ 0 |Q | d

Z

Z

1 f dy = |Q| 0 Q

1 f dy ≥ |Q| 0 Q

Z f dy Q

and property (ii) follows.



To prove Theorem 10.2.7 we also need the following approximation lemma. Lemma 10.2.9 Let f ∈ L1 (Ω; C) ∩ L2 (Ω; C) be a function with null average over the open set Ω. Then, there exists a sequence (fn ) ⊂ Cc∞ (Ω; C) converging to f in L1 (Ω; C) and in L2 (Ω; C), such that each function fn has null average over Ω. Proof Let us fix f as in the statement of the lemma and a sequence (gn ) ⊂ Cc∞ (Ω; C), converging1 to f in L1 (Ω; C) ∩ L2 (Ω; C). Then, in particular, Z Z lim gn dx = f dx = 0. n→∞

Ω

Ω

Let ϑ ∈ Cc∞ (Ω) be a nonnegative function, with integral over Ω equal to one, and set Z fn = gn − ϑ gn dx, n ∈ N. Ω

Clearly, each function fn belongs to Z

k

Z

|fn − f | dx ≤ k Ω

Cc∞ (Ω; C) k

and has null integral over Ω. Moreover,

Z

|gn − f | dx + k Ω

Ω

k  Z gn dy , ϑ dx k

n ∈ N,

Ω

for k = 1, 2, and the last side of the previous inequality vanishes as n tends to ∞.



Proof of Theorem 10.2.7 We fix i, j ∈ {1, . . . , d} and observe that, as it has been remarked after the statement of the theorem, Pij extends with a contraction in L2 (Rd ; C). In particular Pij is an operator of weak type (2, 2). Let us prove that it is also of weak type (1, 1), i.e., let us prove that there exists a positive constant C such that |{x ∈ Rd : |(Pij f )(x)| ≥ t}| ≤ C||f ||L1 (Rd ;C) t−1

(10.2.10)

for every t > 0 and f ∈ L1 (Rd ; C) ∩ L2 (Rd ; C). Once this property is proved, we can apply Marcinkiewicz interpolation theorem (see Theorem A.5.21) to infer that Pij can be uniquely extended to Lp (Rd ; C) with a bounded operator for every p ∈ (1, 2). This will prove the assertion for such values of p. Let us fix t > 0, f ∈ L1 (Rd ; C) ∩ L2 (Rd ; C) and the partition (F, G), as in Proposition 1 It suffices, for instance, to extend f by zero to Rd , take the convolution with a standard sequence of mollifiers, thus obtaining a sequence (ψn ) ⊂ C ∞ (Rd ; C) ∩ L1 (Rd ; C) ∩ L2 (Rd ; C) which converges to f in L1 (Ω; C) and in L2 (Ω; C). Multiplying each function ψn by the function ϑn , where (ϑn ) ⊂ Cc∞ (Ω) is sequence, bounded with respect to the sup norm, which pointwise converges to 1 on Ω, we obtain the sequence we are looking for.

Elliptic Equations in Rd

262

10.2.8, associated to the function |f | and the parameter t. Finally, let us introduce the functions g1 and g2 defined by  in F,   f, Z g2 = f − g1 , g1 = 1  f dx, in Qk (k ∈ K),  |Qk | Qk [ where G = Qk . Note that g1 ∈ L∞ (Rd ; C). Indeed, |g1 | = |f | on F and on this set k∈K

|f | ≤ t almost everywhere. Moreover, if x ∈ Qk , then Z Z 1 ≤ 1 |g1 (x)| = f dy |f | dy ≤ 2d t. |Qk | |Qk | Qk Qk Summing up, g1 belongs to L∞ (Rd ; C) and ||g1 ||∞ ≤ 2d t. Function g1 also belongs to L1 (Rd ; C). Indeed, Z Z Z |g1 | dx = |g1 | dx + |g1 | dx Rd G ZF XZ = |g1 | dx + |g1 | dx F

k∈K

Qk

Z 1 f dx = |g1 | dx + |Qk | |Qk | Qk F k∈K Z Z X ≤ |g1 | dx + |f | dx = ||f ||L1 (Rd ;C) . Z

X

F

k∈K

(10.2.11)

Qk

Since it belongs to L1 (Rd ; C) ∩ L∞ (Rd ; C), function g1 belongs to L2 (Rd ; C) and H¨older inequality reveals that ||g1 ||2L2 (Rd ;C) ≤ ||g1 ||∞ ||g1 ||L1 (Rd ;C) ≤ 2d t||f ||L1 (Rd ;C) .

(10.2.12)

By assumptions, f ∈ L1 (Rd ; C)∩L2 (Rd ; C), so that, by difference, g2 belongs to L1 (Rd ; C)∩ L2 (Rd ; C). Now, we estimate the Lebesgue measure of the set {x ∈ Rd : |(Pij f )(x)| ≥ t}. As it is immediately seen, |{x ∈ Rd : |(Pij f )(x)| ≥ t}| ≤|{x ∈ Rd : |(Pij g1 )(x)| ≥ 2−1 t}| + |{x ∈ Rd : |(Pij g2 )(x)| ≥ 2−1 t}|. We estimate separately the two terms in the right-hand side of the previous inequality. Using Chebyshev inequality and (10.2.12), we get 4 ||Pij g1 ||2L2 (Rd ;C) t2 4 ≤ 2 ||Pij ||L(L2 (Rd ;C)) ||g1 ||2L2 (Rd ;C) t 2d+2 ≤ ||Pij ||L(L2 (Rd ;C)) ||f ||L1 (Rd ;C) . t

|{x ∈ Rd : |(Pij g1 )(x)| ≥ 2−1 t}| ≤

(10.2.13)

As far as the Lebesgue measure of the set {x ∈ Rd : |(Pij g2 )(x)| ≥ 2−1 t} is concerned, we observe that g2 vanishes on F . Therefore, X X g2 = g2 χG = g2 χQk = g2,k , k∈K

k∈K

Semigroups of Bounded Operators and Second-Order PDE’s

263

where the series converges both in L1 (Rd ; C) and in L2 (Rd ; C). By construction, each function g2,k has null average over Qk . Therefore, thanks to Lemma 10.2.9, for every k ∈ K there exists a sequence (e gk,n ) ⊂ Cc∞ (Qk ; C) which converges to g2,k in L2 (Qk ; C) (or, equivalently, 2 d in L (R ; C), since g2,k is supported on Qk ) such that each function gek,n has null average over Qk . Note that, if x ∈ / Qk , then Z (Pij gek,n )(x) = Dij Γ(x − y)e gk,n (y) dy, x ∈ Rd . Qk

Let us denote by yk the center of the cube Qk and by δk its diameter. Then, since the average of gek,n over Qk vanishes, we can write Z (Pij gek,n )(x) = [Dij Γ(x − y) − Dij Γ(x − yk )]e gk,n (y) dy. Qk

From the mean value theorem it follows that |Dij Γ(x − y) − Dij Γ(x − yk )| ≤ |∇Dij Γ(x − ξk )||y − yk | ≤

Cδk Cδk ≤ , |x − ξk |d+1 (d(x, Qk ))d+1

where ξk is a suitable point on the segment joining y and yk , and C is a positive constant, independent of x, y and yk . Thus, Z |(Pij gek,n )(x)| ≤ |Dij Γ(x − y) − Dij Γ(x − yk )||e gk,n (y)| dy Qk Z C ≤ δ |e gk,n (y)| dy. (10.2.14) k (d(x, Qk ))d+1 Qk If, in addition, x ∈ / B(yk , δk ) and y ∈ Qk is such that |x − y| = d(x, Qk ), then we can estimate |x − yk | ≤ |x − y| + |y − yk | ≤ d(x, Qk ) +

δk 1 ≤ d(x, Qk ) + |x − yk |, 2 2

from which it follows that |x − yk | ≤ 2d(x, Qk ). Therefore, from (10.2.14) we deduce that Z C 0 δk |(Pij gek,n )(x)| ≤ |e gk,n (y)| dy, x∈ / B(yk , δk ). (10.2.15) |x − yk |d+1 Qk Since gek,n converges to g2,k in L2 (Rd ; C), Pij gek,n converges to Pij g2,k in L2 (Rd ; C) as n tends to ∞. Up to a subsequence, we can assume that Pij gek,n converges to Pij g2,k almost everywhere on Rd . Moreover, gek,n converges to g2,k in L1 (Qk ; C) as n tends to ∞ since L2 (Rd ; C) ,→ L1 (Qk ; C). Hence, up to passing to a subsequence, we can let n tend to ∞ in both sides of (10.2.15) and conclude that Z C 0 δk |g2,k (y)| dy (10.2.16) |(Pij g2,k )(x)| ≤ |x − yk |d+1 Qk for almost every x ∈ / B(yk , δk ). Integrating (10.2.16) over Rd \ B(yk , δk ) we obtain Z Z 0 |(Pij g2,k )(x)| dx ≤C δk ||g2,k ||L1 (Qk ;C) |x − yk |−d−1 dx Rd \B(yk ,δk )

=C 00 δk ||g2,k ||L1 (Qk ;C)

Rd \B(yk ,δk ) ∞ −2

Z

ρ

δk

dρ

Elliptic Equations in Rd

264

=C 00 ||g2,k ||L1 (Qk ;C) = C 00 ||g2 ||L1 (Qk ;C)

(10.2.17)

for some positive constant C 00 , independent of f . S Now, we introduce the set G∗ = k∈K B(yk , δk ) and F ∗ = Rd \ G∗ . Integrating the function Pij g2 over F ∗ and taking (10.2.11) and (10.2.17) into account, we get Z Z X XZ |Pij g2,k | dx Pij g2,k dx ≤ |Pij g2 | dx = F∗

F∗

≤

|Pij g2,k | dx ≤ C

00

Rd \B(yk ,δk )

k∈K

=C 00

F∗

k∈K

k∈K

XZ Z

XZ k∈K

|g2 | dy

Qk

|g2 | dy ≤ C 00 (||f ||L1 (Rd ;C) + ||g1 ||L1 (Rd ;C) )

G

≤2C 00 ||f ||L1 (Rd ;C) for some positive constant C 00 , independent of f . Thus, using Chebyshev inequality, we can estimate |{x ∈ Rd : |(Pij g2 )(x)| ≥ 2−1 t}| =|{x ∈ F ∗ : |(Pij g2 )(x)| ≥ 2−1 t}| + |{x ∈ G∗ : |(Pij g2 )(x)| ≥ 2−1 t}| ≤

4C 00 ||f ||L1 (Rd ;C) + |{x ∈ G∗ : |(Pij g2 )(x)| ≥ 2−1 t}|. t

(10.2.18)

Observe that ∗

−1

|{x ∈ G : |(Pij g2 )(x)| ≥ 2

X [ |B(yk , δk )| B(yk , δk ) ≤ t}| ≤m(G ) = ∗

k∈K

k∈K

=ωd

X

δkd

= ωd d

d 2

k∈K

X

`dk

d

= ωd d 2

k∈K

X

|Qk |,

k∈K

where `k denotes the length of the edge of the cube Qk . Since, by Proposition 10.2.8, Z 1 |f | dy > t, |Qk | Qk it follows that |Qk | ≤ t−1 ||f ||L1 (Qk ;C) . Thus, we can continue the previous chain of inequalities and obtain that Z XZ d −1 −1 ∗ −1 2 |f | dy = Cd t |f | dy. |{x ∈ G : |(Pij g2 )(x)| ≥ 2 t}| ≤ ωd d t k∈K

Qk

F

From this estimate and (10.2.18) it follows that e −1 ||f ||L1 (Rd ;C) |{x ∈ Rd : |(Pij g2 )(x)| ≥ 2−1 t}| ≤ Ct e independent of t and f . This estimate, together with (10.2.13), for some positive constant C, yield (10.2.10). To complete the proof, we need to consider the case p ∈ (2, ∞). We use a duality argument. We first observe that Z Z Z (Pij f )(x)ψ(x) dx = (Dij N f )(x)ψ(x) dx = (N f )(x)(Dij ψ)(x) dx Rd

Rd

Rd

Semigroups of Bounded Operators and Second-Order PDE’s Z Z = Dij ψ(x) Γ(x − y)f (y) dy dx d d ZR Z R = f (y) dy Γ(x − y)Dij ψ(x) dx d Rd ZR Z = f (y)(Dij N ψ)(y) dy = f (y)(Pij ψ)(y) dy Rd

265

Rd

for every f, ψ ∈ Cc∞ (Rd ; C). Therefore, we can estimate Z Z ≤ ||f ||Lp (Rd ;C) ||Pij ψ|| p0 d , f P ψ dx = (P f )ψ dx ij ij L (R ;C) Rd

Rd

0

where 1/p + 1/p0 = 1. Since p0 ∈ (1, 2), Pij is a bounded operator on Lp (Rd ; C), so that Z Z d (Pij f )ψ dx = d f Pij ψ dx ≤ K||f ||Lp (Rd ;C) ||ψ||Lp0 (Rd ;C) R

R

for some positive constant K, independent of f, ψ ∈ Cc∞ (Rd ; C). By density, we can extend 0 the previous inequality to every ψ ∈ Lp (Rd ; C). This allows us to conclude that Pij f belongs to Lp (Rd ; C) and ||Pij f ||Lp (Rd ;C) ≤ K||f ||Lp (Rd ;C) . We can use again a density argument to infer that Pij extends with a bounded operator mapping Lp (Rd ; C) into itself. The proof is now complete.  We have all the tools to prove the Calder´on-Zygmund inequality. b = C(p, b d) such that Theorem 10.2.10 There exists a positive constant C d X

b ||Dij u||Lp (Rd ;C) ≤ C||∆u|| Lp (Rd ;C)

(10.2.19)

i,j=1

for every function u ∈ W 2,p (Rd ; C). Proof Fix a function f ∈ Cc∞ (Rd ; C) and apply Proposition 10.2.5, with Ω = B(0, r), where r > 0 is chosen sufficiently large to guarantee that supp(f ) ⊂ B(0, r), to write Z Z f (x) = Γ(x − y)∆f (y) dy = Γ(x − y)∆f (y) dy = N ∆f B(0,r)

Rd

for every x ∈ B(0, r). Thanks to Theorem 10.2.7 we can determine a positive constant b = C(p, b d), independent of f , such that ||Dij N ∆f ||Lp (Rd ;C) ≤ C||∆f ||Lp (Rd ;C) , so that C b ||Dij f ||Lp (Rd ;C) = ||Dij N ∆f ||Lp (Rd ;C) ≤ C||∆f ||Lp (Rd ;C) for every i, j ∈ {1, . . . , d}. Since Cc∞ (Rd ; C) is dense in W 2,p (Rd ; C), we can extend the previous estimate to every f ∈ W 2,p (Rd ; C), obtaining the assertion.  Remark 10.2.11 From the Calder´on-Zygmund inequality it follows immediately that the map f 7→ ||f ||Lp (Rd ;C) + ||∆f ||Lp (Rd ;C) is a norm on W 2,p (Rd ; C), which is equivalent to the usual norm of W 2,p (Rd ; C). Remark 10.2.12 Note that the Calder´on-Zygmund inequality fails to hold in the limiting cases p = 1 and p = ∞. In the case p = ∞, we fix ε > 0 and a function η ∈ C ∞ (Rd ) such

Elliptic Equations in Rd

266

that χB(0,1/2) ≤ η ≤ χB(0,1) . We fix i, j ∈ {1, . . . , d} and consider the function uε : Rd → R, defined by uε (x) = η(x)xi xj log(|x|2 + ε) for every x ∈ Rd . As it is easily seen uε is smooth and supported on B(0, 1). Moreover, a straightforward computation reveals that there exists a positive constant C such that ||uε ||Cb1 (Rd ) +||Dkk uε ||∞ ≤ C for every k = 1, . . . , d and every ε ∈ (0, 1). On the other hand, Dij uε (x, y) = η(x) log(|x|2 + ε) + gε (x) for every x ∈ Rd , where gε vanishes at 0. Therefore, ||Dij uε ||∞ ≥ |Dij uε (0)| = | log(ε)|, so that ||Dij uε ||∞ blows up as ε tends to 0+ . This implies that the estimate ||Dij v||∞ ≤ C(||∆v||∞ + ||v||Cb1 (Rd ) ) cannot hold for every v ∈ W 2,∞ (Rd ) and some positive constant C, independent of v. The case p = 1 is postponed to Remark 10.2.33.

10.2.2

The Laplace equation

In this subsection, we consider the equation λu − ∆u = f for f ∈ Lp (Rd ; C), λ ∈ C, with positive real part, and p arbitrarily fixed in (1, ∞). The main result of this section is contained in the following theorem. Theorem 10.2.13 For every λ ∈ C, with positive real part, and every f ∈ Lp (Rd ; C), there exists a unique solution u ∈ W 2,p (Rd ; C) of the Laplace equation λu − ∆u = f . Moreover, there exists a positive constant Cp , independent of u, f and λ, such that √ |λ|||u||Lp (Rd ;C) + λ|||∇u|||Lp (Rd ;C) + ||D2 u||Lp (Rd ;C) ≤ Cp ||f ||Lp (Rd ;C) . (10.2.20) The uniqueness part follows from the following lemma, which is a particular case of the forthcoming Lemma 10.2.22. Lemma 10.2.14 For every u ∈ W 2,p (Rd ; C) it holds that Z Re ∆u|u|p−2 u dx ≤ 0. Rd

Proof of Theorem 10.2.13 Suppose that u ∈ W 2,p (Rd ; C) solves the equation λu − ∆u = 0 for some λ ∈ C, with positive real part. Then, multiplying both sides of the previous equation by u|u|p−2 , integrating by over Rd and taking the real parts of both sides of the so obtained equation, we get Z Z p Reλ |u| dx − Re ∆u u|u|p−2 dx = 0. Rd

Rd

Therefore, (Re λ)||u||pLp (Rd ) ≤ 0 so that u = 0. We now prove the existence part. For this purpose, we fix f ∈ Cc∞ (Rd ; C) and λ ∈ C with positive real part. Then, by Theorem 10.1.1, there exists a unique function u ∈ Cb2+α (Rd ; C) which solves the equation λu − ∆u = f . The function u is given by Z ∞ u(x) = e−λt (T (t)f )(x) dt, x ∈ Rd . (10.2.21) 0

We suppose that Im λ ≥ 0 and observe that, by (5.2.5), we can write Z ∞ e−λ(t−it) (T (t − it)f )(x) dt, x ∈ Rd , u(x) = (1 − i) 0

Semigroups of Bounded Operators and Second-Order PDE’s

267

where Z (T (t − it)f )(x) =

x ∈ Rd ,

K(t − it, y)f (x − y) dy, Rd

2

and K(z, y) = (4πz)−d/2 e−|y| that

/(4z)

for every y ∈ Rd and z ∈ C with positive real part. Note 

||K(z, ·)||L1 (Rd ;C) ≤

Rez |z|

− d2

for every z ∈ C with positive real part. We claim that the function T (t − it)f belongs to Lp (Rd ; C) for every t > 0 and d

||T (t − it)f ||Lp (Rd ,C) ≤ 2 4 ||f ||Lp (Rd ;C) ,

t > 0.

For this purpose we observe that Z |(T (t − it)f )(x)|p dx Rd p Z Z K(t − it, x − y)f (y) dy dx = Rd Rd Z p Z 1 1 ≤ dx |K(t − it, x − y)| p0 |K(t − it, x − y)| p |f (y)| dy Rd

Rd

 p0  Z

Z

Z

p

≤

p



|K(t − it, x − y)| dy |K(t − it, x − y)||f (y)| dy dx Rd Z dp dx |K(t − it, x − y)||f (y)|p dy ≤2 4p0 Rd Rd Z Z dp =2 4p0 |f (y)|p dy |K(t − it, x − y)| dx d Rd ZR dp ≤2 4 |f (y)|p dy. Rd

Rd

Z

Rd

Therefore, Z Z |u(x)|p dx = Rd

p Z ∞ −λ(t−it) (1 − i) e (T (t − it)f )(x) dt dx Rd 0  p0 Z ∞  Z Z ∞ p p −(Re λ+Im λ)t p −(Re λ+Im λ)t 2 ≤2 e dt e |(T (t − it)f )(x)| dt dx Rd 0 Z0 ∞ Z p −p ≤2 2 |λ| p0 e−(Re λ+Im λ)t dt |(T (t − it)f )(x)|p dx ≤2

p 4 (2+d)

|λ|

0 −p

Rd

||f ||pLp (Rd ;C) ,

so that u ∈ Lp (Rd ; C) and ||u||Lp (Rd ;C) ≤ 2(2+d)/4 |λ|−1 ||f ||Lp (Rd ;C) . If Im λ < 0, then we write Z ∞ u(x) = (1 + i) e−λ(t+it) (T (t + it)f )(x) dt, x ∈ Rd , 0

and repeating the same computations as above, we obtain that ||u||Lp (Rd ;C) ≤ 2

2+d 4

|λ|−1 ||f ||Lp (Rd ;C) .

Elliptic Equations in Rd

268

Next, using the formula (10.2.21), we can show that Dj u belongs to Lp (Rd ; C) for every j = 1, . . . , d. Indeed, we know that Z (Dj T (t)f )(x) = Dj K(t, x − y)f (y) dy, t > 0, x ∈ Rd , Rd

and Dj K(t, x) = −(2t)−1 xj K(t, x) for every (t, x) ∈ (0, ∞) × Rd . Therefore, p Z Z Z dx |(Dj T (t)f )(x)|p dx = D K(t, x − y)f (y) dy j Rd Rd Rd Z  p0 Z Z p |Dj K(t, x)| dx |f (y)|p dy ≤ |Dj K(t, y)| dy Rd Rd Rd Z p p = |Dj K(t, y)| dy ||f ||pLp (Rd ;C) = C1p t− 2 ||f ||Lp (Rd ;C) Rd

for some positive constant C1 , independent of t and f . Therefore, p Z Z Z ∞ p −λt |Dj u(x)| dx = Dj e (T (t)f )(x) dt dx Rd Rd 0 p Z Z ∞ −λt e (Dj T (t)f )(x) dt dx = Rd

0

 p0  Z ∞  p −λt p ≤ e dt e |(Dj T (t)f )(x)| dt dx Rd 0 Z ∞ 0 p −p =C1p λ p0 ||f ||pLp (Rd ;C) t− 2 e−λt dt Z

∞

Z

−λt

0

p

=C2,p λ− 2 ||f ||pLp (Rd ;C) , 1/p

so that Dj u belongs to Lp (Rd ; C) and ||Dj u||Lp (Rd ;C) ≤ C2,p λ−1/2 ||f ||Lp (Rd ;C) for some positive constant C2,p , independent of f , u and t. By difference, we deduce that ∆u ∈ Lp (Rd ; C). We can thus conclude that u ∈ W 2,p (Rd ; C). Indeed, fix a smooth function ϑ ∈ Cc∞ (Rd ), compactly supported in B(0, 1) and consider the sequence of functions (ϑn ), where ϑn (x) = ϑ(n−1 x) for every x ∈ Rd and n ∈ N. Then the function un = ϑn u belongs to W 2,p (Rd ; C) for every n ∈ N and it converges to u in Lp (Rd ; C). Moreover, ∆un = ϑn ∆u + 2h∇ϑn , ∇ui + u∆ϑn so that ||∆un − ∆u||Lp (Rd ;C) ≤||(1 − ϑn )∆u||Lp (Rd ;C) + C3 n−1 |||∇u|||Lp (Rd ;C) + C3 n−2 ||u||Lp (Rd ;C) for every n ∈ N and some positive constant C3 , independent of n. This inequality and the dominated convergence theorem show that ∆un converges to ∆u in Lp (Rd ). Moreover, using Remark 10.2.11 we can write ||un − um ||W 2,p (Rd ;C) ≤ ||un − um ||Lp (Rd ;C) + ||∆un − ∆um ||Lp (Rd ;C) for every m, n ∈ N. Therefore, (un ) is a Cauchy sequence in W 2,p (Rd ; C), so that it converges to a function v ∈ W 2,p (Rd ; C). Since it converges to u in Lp (Rd ; C), it follows that u belongs to W 2,p (Rd ; C). Clearly, u solves the equation λu − ∆u = f . To complete the proof, we need to check estimate (10.2.20) and prove the solvability

Semigroups of Bounded Operators and Second-Order PDE’s

269

of equation λu − ∆u = f for every f ∈ Lp (Rd ; C). For this purpose, using the Calder´onZygmund inequality (10.2.19), we infer that d X

b b ||Dij u||Lp (Rd ;C) ≤C||∆u|| Lp (Rd ;C) = C||λu − f ||Lp (Rd ;C)

i,j=1

b b ≤C(|λ|||u|| Lp (Rd ;C) + ||f ||Lp (Rd ;C) ) ≤ 2C||f ||Lp (Rd ;C) . Finally, we apply Theorem 1.3.8 to write 1

1

1

|||∇u|||Lp (Rd ;C) ≤ Cp ||u||L2 p (Rd ;C) ||D2 u||L2 p (Rd ;C) ≤ C p |λ|− 2 ||f ||Lp (Rd ;C) for some positive constant C p , independent of f , u and λ. Estimate (10.2.20) follows. Finally, we fix f ∈ Lp (Rd ; C) and consider a sequence (fn ) ⊂ Cc∞ (Rd ; C) converging to f in Lp (Rd ; C). We denote by un ∈ W 2,p (Rd ; C) the unique function such that λun −∆un = fn . Writing (10.2.20) with u and f being replaced by un − um and fn − fm , respectively, we easily see that (un ) is a Cauchy sequence in W 2,p (Rd ; C), so that it converges to a function u ∈ W 2,p (Rd ; C) which solves the equation λu − ∆u = f . Finally, writing (10.2.20), with (u, f ) being replaced by (un , fn ) and letting n tend to ∞, we conclude that u satisfies (10.2.20). 

10.2.3

More general elliptic operators

Here, we extend Theorem 10.2.13 to the more general elliptic operator A = Tr(QD2 ) + hb, ∇i + c, under the following assumptions on the coefficients of the operator A. Hypotheses 10.2.15 (i) The coefficients qij = qji (i, j = 1, . . . , d) belong to BU C(Rd ), whereas the coefficients bj (j = 1, . . . , d) and c belong to L∞ (Rd ); (ii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for all x, ξ ∈ Rd . The main result of this subsection is the following theorem. We stress that also in this subsection p ∈ (1, ∞) is arbitrarily fixed. Theorem 10.2.16 Under Hypotheses 10.2.15, there exists a positive constant λp such that, for every λ ∈ C with Re λ ≥ λp , and every f ∈ Lp (Rd ; C), the equation λu − Au = f admits a unique solution u ∈ W 2,p (Rd ; C). Moreover, there exists a positive constant Cp , which depends only on p, d and, in a continuous way, on the ellipticity constant µ, on M = max{||qij ||∞ , ||bj ||∞ , ||c||∞ : i, j = 1, . . . , d} and on the modulus of continuity of the diffusion coefficients, such that p |λ|||u||Lp (Rd ;C) + |λ||||∇u|||Lp (Rd ;C) + ||D2 u||Lp (Rd ;C) ≤ Cp ||f ||Lp (Rd ;C) .

(10.2.22)

(10.2.23)

Also the constant λp depends only on p, d and, in a continuous way, on the ellipticity constant µ and M . The proof is based on the continuity method and the main step is the proof of the a priori estimate in the following proposition.

Elliptic Equations in Rd

270

Proposition 10.2.17 Under Hypotheses 10.2.15, there exist two positive constants λp and Cp , which depend only on p and (in a continuous way) on the ellipticity constant of operator A, on the constant M in (10.2.22) and on the modulus of continuity of the diffusion coefficients, such that p |λ|||u||Lp (Rd ;C) + |λ||||∇u|||Lp (Rd ;C) + ||D2 u||Lp (Rd ;C) ≤ Cp ||λu − Au||Lp (Rd ;C) (10.2.24) for every u ∈ W 2,p (Rd ; C) and λ ∈ C with Re λ ≥ λp . Remark 10.2.18 From estimate (10.2.24) it follows easily that there exists a positive constant C∗,p such that ||u||W 2,p (Rd ;C) ≤ C∗,p (||u||Lp (Rd ;C) + ||Au||Lp (Rd ;C) )

(10.2.25)

for every u ∈ W 2,p (Rd ; C). In the proof of Proposition 10.2.17, we will use the following covering lemma. Lemma 10.2.19 There exists[an integer ξ = ξ(d) and, for \ every r > 0, a sequence (xn ) d with the following properties: B(xj , r/2) = R and, if B(xj , r) 6= ∅, then the cardij∈S

j∈N

nality of S does not exceed ξ. Proof Fix r > 0, set ` = d−1/2 r and consider a countable covering {Q(xn , `) : n ∈ N} of cubes with edges of length `, parallel to the coordinate axis, such that Q(xn , `) ∩ Q(xm , `) = ∂Q(xn , `) ∩ ∂Q(xm , `) if m 6= n. Thanks to the choice of `, the cube Q(xn , `) is contained in the ball B(xn , r) for every n ∈ N. This shows that {B(xn , r) : n ∈ \ N} is a covering of Rd . Let us fix a subset S of N such that B(xn , r) 6= ∅ and take x in such an intersection. n∈S

Then, Q(xn , `) ⊂ B(xn , r) ⊂ B(x, 3r/2) for every n ∈ S. Since the cubes Q(xn , `) have interior parts which are pairwise disjoint, it follows that X \ rd d = Q(x , `) |Q(x , `)| = (]S)` = (]S) n n d , d2 n∈S n∈S where | · | denotes the Lebesgue measure and ]S the cardinality of the set S. Moreover, \  d 3 Q(xn , `) ≤ |B(x, 3r/2)| = ωd rd , 2 n∈S

where ωd denotes the Lebesgue measure of the unit ball of Rd . From these two formulas we infer that ]S ≤ 2−d 3d dd/2 ωd and the assertion follows with ξ(d) = [2−d 3d dd/2 ωd ] + 1, where [ · ] denotes the integer part of the quantity in brackets.  Remark 10.2.20 The result of Lemma 10.2.19 can be rephrased saying that there exists ξ = ξ(d) ∈ N and, for every r > 0, a sequence (xn ) ⊂ Rd such that 1≤

∞ X n=1

χB(xn ,r/2) ≤

∞ X

χB(xn ,r) ≤ ξ

n=1

on Rd . This is the form of Lemma 10.2.19 that we will use in the proof of Proposition 10.2.17.

Semigroups of Bounded Operators and Second-Order PDE’s

271

Proof of Proposition 10.2.17 We first consider the case when A = Tr(QD2 ) and Q is a constant diagonal matrix, with positive eigenvalues.√Given a function u ∈ W 2,p (Rd ; C) 2,p d and λ ∈ C with positive √ real part, the function v = u( Q ·) belongs to W (R ; C) and λv − ∆v = (λu − Au)( Q ·). Applying estimate (10.2.20), we deduce that p p bp ||(λu − Au)( Q ·)||Lp (Rd ;C) |λ|||v||Lp (Rd ;C) + |λ||||∇v|||Lp (Rd ;C) + ||D2 v||Lp (Rd ;C) ≤ C b independent of v and λ, and estimate (10.2.24) follows immefor some positive constant C, diately writing u in terms of v. Next, we consider the case when Q is constant and diagonalizable over Rd . We denote by P an orthogonal matrix such that P QP T is a diagonal matrix. If u ∈ W 2,p (Rd ; C), then we can apply estimate (10.2.24) with A = Tr(P QP T D2 ) to the function v = u(P T ·). Writing u in terms of v, estimate (10.2.24) follows with A = Tr(QD2 ). To complete the proof, let us consider the general case when A = Tr(QD2 ) + hb, ∇i + c. For this purpose, we fix x0 ∈ Rd , r > 0 and a smooth function ϑx0 ,r ∈ Cc∞ (Rd ) such that χB(x0 ,r/2) ≤ ϑx0 ,r ≤ χB(x0 ,r) . Further, we denote by L a positive constant, independent of x0 and r, such that r|||∇ϑx0 ,r |||∞ + r2 ||D2 ϑx0 ,r ||∞ ≤ L for every r > 0. Finally, we fix u ∈ W 2,p (Rd ; C) and λ ∈ C with positive real part. Clearly, the function v = ϑx0 ,r u belongs to W 2,p (Rd ; C), so that we can apply estimate (10.2.24), with A = Ax0 = Tr(Q(x0 )D2 ) to the function v, and obtain p |λ|||v||Lp (Rd ;C) + |λ||||∇v|||Lp (Rd ;C) + ||D2 v||Lp (Rd ;C) ≤ Cp∗ ||λv − Ax0 v||Lp (Rd ;C) . (10.2.26) Since ϑx0 ,r ≡ 1 on B(x0 , r/2) and ϑx0 ,r ≤ 1 on Rd , it follows that ||u||W 2,p (B(x0 ,r/2);C) ≤ ||v||W 2,p (Rd ;C) . Moreover, ||λv − Ax0 v||Lp (Rd ;C) ≤||λu − Ax0 u||Lp (B(x0 ,r);C) + ||u||Lp (B(x0 ,r);C) ||Ax0 ϑx0 ,r ||∞ + 2||Q(x0 )||∞ |||∇u|||Lp (B(x0 ,r);C) |||∇ϑx0 ,r |||∞ ML ≤||λu − Ax0 u||Lp (B(x0 ,r);C) + 2 ||u||Lp (B(x0 ,r);C) r 2M L + |||∇u|||Lp (B(x0 ,r);C) . r Replacing these estimates in (10.2.26), we get p |λ|||u||Lp (B(x0 ,r/2);C) + |λ||||∇u|||Lp (B(x0 ,r/2);C) + ||D2 u||Lp (B(x0 ,r/2);C)   ML 2M L ∗ ≤Cp ||λu−Ax0 u||Lp (B(x0 ,r);C) + 2 ||u||Lp (B(x0 ,r);C) + |||∇u|||Lp (B(x0 ,r);C) . (10.2.27) r r Next, we observe that ||λu − Ax0 u||Lp (B(x0 ,r);C) ≤||λu − Au||Lp (B(x0 ,r);C) + ||Ax0 u − Au||Lp (B(x0 ,r);C) ≤||λu − Au||Lp (B(x0 ,r);C) + ||Q − Q(x0 )||L∞ (B(x0 ,r);C) ||D2 u||Lp (B(x0 ,r);C) + |||b|||∞ |||∇u|||Lp (B(x0 ,r);C) + ||c||∞ ||u||Lp (B(x0 ,r);C) ≤||λu − Au||Lp (B(x0 ,r);C) + dω(r)||D2 u||Lp (B(x0 ,r);C) + M |||∇u|||Lp (B(x0 ,r);C) + M ||u||Lp (B(x0 ,r);C) , where ω(r) = max

sup |qij (x) − qij (y)|,

1≤i,j≤d |x−y|≤r

r > 0,

Elliptic Equations in Rd

272

so that, from (10.2.27) we deduce that p |λ|||u||Lp (B(x0 ,r/2);C) + |λ||||∇u|||Lp (B(x0 ,r/2);C) + ||D2 u||Lp (B(x0 ,r/2);C)    ML ≤Cp∗ ||λu − Au||Lp (B(x0 ,r);C) + M + 2 ||u||Lp (B(x0 ,r);C) r    2M L + M+ |||∇u|||Lp (B(x0 ,r);C) + dω(r)||D2 u||Lp (B(x0 ,r);C) . r Using repeatedly the estimate (a+b)p ≤ 2p−1 (ap +bp ) which holds true for every a, b ≥ 0, we get p

|λ|p ||u||pLp (B(x0 ,r/2);C) + |λ| 2 |||∇u|||pLp (B(x0 ,r/2);C) + ||D2 u||pLp (B(x0 ,r/2);C) p ep [(K1 (r, M ))p ||u||p p ≤C L (B(x0 ,r);C) + ||λu − Au||Lp (B(x0 ,r);C)

+ (K2 (r, M ))p |||∇u|||pLp (B(x0 ,r);C) + (ω(r))p ||D2 u||pLp (B(x0 ,r);C) ],

(10.2.28)

where the constants Kj (r, M ) (j = 1, 2), which are independent of u and depend in a ep , depends continuously on µ and continuous way on M , blow up as r tends to 0+ , and C ep is on the sup-norm of the coefficients of the matrix-valued function Q. In particular, C independent of u and r. Thanks to Lemma 10.2.19 and Remark 10.2.20, we can write ||w||pLp (Rd ;C) ≤

∞ X

||w||pLp (B(xn ,r/2);C)

(10.2.29)

n=1

and ∞ X

||w||pLp (B(xn ,r);C) =

n=1

∞ Z X Rd

n=1

χB(xn ,r) |w|p dx ≤ ξ(d)||w||pLp (Rd ;C)

(10.2.30)

for every w ∈ Lp (Rd ; C). Applying estimates (10.2.29) and (10.2.30) to u and its first- and second-order derivatives, and using (10.2.28) we obtain p

|λ|p ||u||pLp (Rd ;C) + |λ| 2 |||∇u|||pLp (Rd ;C) + ||D2 u||pLp (Rd ;C) ≤|λ|p

∞ X

p

||u||pLp (B(xn ,r/2);C) + |λ| 2

n=1

∞ X

|||∇u|||pLp (B(xn ,r/2);C) +

n=1

∞ X

||D2 u||pLp (B(xn ,r/2);C)

n=1

 ∞ ∞ X X e ≤Cp (K1 (r, M ))p ||u||pLp (B(xn ,r);C) + (K2 (r, M ))p |||∇u|||pLp (B(xn ,r);C) n=1

+ (ω(r))

p

∞ X

||D

n=1 2

u||pLp (B(xn ,r);C)

n=1

+

∞ X

||λu −

Au||pLp (B(xn ,r);C)



n=1

ep ξ(d)[(K1 (r, M ))p ||u||p p d + (K2 (r, M ))p |||∇u|||p p d ≤C L (R ;C) L (R ;C) + (ω(r))p ||D2 u||pLp (Rd ;C) + ||λu − Au||pLp (Rd ;C) ]. Observing that (a + b + c)p ≤ 41−1/p (ap + bp + cp ) for every triplet of nonnegative numbers a, b, c, from the previous chain of inequalities, we get 1

|λ|||u||Lp (Rd ;C) + |λ| 2 |||∇u|||Lp (Rd ;C) + ||D2 u||Lp (Rd ;C) bp [K1 (r, M )||u||Lp (Rd ;C) + K2 (r, M )|||∇u|||Lp (Rd ;C) ≤C

Semigroups of Bounded Operators and Second-Order PDE’s + ω(r)||D2 u||Lp (Rd ;C) + ||λu − Au||Lp (Rd ;C) ],

273 (10.2.31)

bp = [4p−1 C ep ξ(d)]1/p . where C To get rid of the Lp -norms of u, |∇u| and |D2 u| from the right-hand side of the preb ω(r) ≤ 1/2. Next, we fix λp > 0 such that vious inequality we fix r > 0 such that C p p b b 4Cp K1 (r, M ) ≤ λp and 4Cp K2 (r, M ) ≤ λp . With these choices of r and λp , from (10.2.31), (10.2.24) follows. This completes the proof.  Proof of Theorem 10.2.16 As it has been already claimed, the proof is based on the continuity method. For this purpose, we introduce the one-parameter family of elliptic operators Aσ (σ ∈ (0, 1)) defined by Aσ = σA + (1 − σ)∆ for every σ ∈ [0, 1]. Denoting by σ qij , bσj and cσ (i, j = 1, . . . , d, σ ∈ [0, 1]) the coefficients of the operator Aσ , it is immediate Pd σ to check that i,j=1 qij (x)ξi ξj ≥ (µ ∧ 1)|ξ|2 for every x, ξ ∈ Rd , the diffusion coefficients are bounded and uniformly continuous over Rd , whereas the other coefficients are measurable σ and bounded on Rd , uniformly with respect to σ. Finally, the modulus of continuity of qij is bounded from above by the modulus of continuity of qij for every i, j = 1, . . . , d. For every λ ∈ C with Re λ ≥ λp and σ ∈ [0, 1], we consider the operator Tσ : W 2,p (Rd ; C) → Lp (Rd ; C), defined by Tσ u = λu − Aσ u for every u ∈ W 2,p (Rd ; C). Esbp , independent of u timate (10.2.24) shows that, we can determine a positive constant C 2,p b and λ, such that ||Tσ u||Lp (Rd ;C) ≥ Cp ||u||W 2,p (Rd ;C) for every u ∈ W (Rd ; C) and σ ∈ [0, 1]. Since, by Theorem 10.2.13, the operator T0 is invertible, all the operators Tσ are invertible, by Theorem 6.1.1. In particular, T1 is invertible and this is equivalent to saying that, for every f ∈ Lp (Rd ; C), the equation λu − Au = f admits a unique solution u ∈ W 2,p (Rd ; C). Clearly, this solution satisfies the estimate (10.2.23).  A better characterization of λp can be provided in the case when A is in divergence form, i.e., when A = div(Q∇) + hb, ∇i + c, under the following assumptions on its coefficients. Hypotheses 10.2.21 (i) The coefficients qij = qji and bj (i, j = 1, . . . , d) belong to Cb1 (Rd ) and W 1,∞ (Rd ), respectively, whereas the coefficient c belongs to L∞ (Rd ); (ii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for all x, ξ ∈ Rd . Lemma 10.2.22 For every u ∈ W 2,p (Rd ; C) it holds that Z Re div(Q∇u)u|u|p−2 dx ≤ 0. Rd

In particular, if u is real-valued, then Z div(Q∇u)u|u|p−2 dx ≤ 0. Rd

Proof In the simplest case p ≥ 2, it suffices to integrate by parts, observing that the 0 function u|u|p−2 belongs to W 1,p (Rd ; C), where 1/p + 1/p0 = 1, see Proposition 1.3.9, and
∇(u|u|p−2 ) = u|u|p−4 (p − 1)Re(u∇u) − iIm(u∇u) , so that Z

p−2

div(Q∇u)u|u| Rd

Z

dx = − (p − 1) hQ(u∇u), Re(u∇u)i|u|p−4 dx d R Z +i hQ(u∇u), Im(u∇u)i|u|p−4 dx Rd

Elliptic Equations in Rd

274

and, consequently, Z Z p−2 Re dx = − (p − 1) div(Q∇u)u|u| hQRe(u∇u), Re(u∇u)i|u|p−4 dx Rd Rd Z − hQIm(u∇u), Im(u∇u)i|u|p−4 dx. Rd

The assertion follows. In the case when p ∈ (1, 2), we fix δ > 0, and a sequence (ϕn ) ∈ Cc∞ (Rd ; C) converging to u in W 2,p (Rd ; C). Next, for every n ∈ N, we multiply the function div(Q∇ϕn ) by ϕn (|ϕn |2 + δ)p/2−1 ∈ Cc∞ (Rd ; C) and integrate by parts over Rd to obtain as above that Z p−2 Re div(Q∇ϕn )ϕn (|ϕn |2 + δ) 2 dx Rd Z p−4 = − (p − 1) hQRe(ϕn ∇ϕn ), Re(ϕn ∇)ϕn i(|ϕn |2 + δ) 2 dx d R Z p−4 − hQIm(ϕn ∇ϕn ), Im(ϕn ∇ϕn )i(|ϕn |2 + δ) 2 dx Rd Z p−4 − δRe hQ∇ϕn , ∇ϕn i(|ϕn |2 + δ) 2 dx Rd

so that Z Z p−2 p−4 2 2 dx ≤ −δRe hQ∇ϕn , ∇ϕn i(|ϕn |2 + δ) 2 dx. Re div(Q∇ϕn )ϕn (|ϕn | + δ) Rd

Rd

Letting δ tend to 0+ , by dominated convergence, we get Z div(Q∇ϕn )ϕn |ϕn |p−2 dx ≤ 0.

(10.2.32)

Rd

Finally, we observe that div(Q∇ϕn ) converges to div(Q∇u) in Lp (Rd ; C) and ϕn |ϕn |p−2 0 converges to u|u|p−2 in Lp (Rd ; C). The first convergence is straightforward. To prove the p0 1 second one, we observe that the function wn := 2 p−1 (|u|p + |ϕn |p ) − ϕn |ϕn |p−2 − u|u|p−2 0

is nonnegative for every n ∈ N. Moreover, it converges to 2p |u|p almost everywhere on Rd as n tends to ∞. Therefore, applying Fatou’s lemma we infer that Z 0 1 1 2p ||u||pLp (Rd ;C) ≤ lim inf wn dx =2 p−1 ||u||pLp (Rd ;C) + 2 p−1 lim ||ϕn ||pLp (Rd ;C) n→∞ n→∞ d R Z 0 ϕn |ϕn |p−2 − u|u|p−2 p dx, − lim sup n→∞

Rd

so that Z lim sup n→∞

0 ϕn |ϕn |p−2 − u|u|p−2 p dx ≤ 0

Rd

and the claimed convergence follows. Hence, estimating Z Z p−2 p−2 |ϕ | dx − div(Q∇u)u|u| dx div(Q∇ϕ )ϕ n n n d R Rd Z Z ≤ |div(Q∇ϕn ) − div(Q∇u)||ϕn |p−1 dx + |div(Q∇u)| ϕn |ϕn |p−2 − u|u|p−2 dx Rd

Rd

Semigroups of Bounded Operators and Second-Order PDE’s

275

p−1 ≤||div(Q∇ϕn ) − div(Q∇u)||Lp (Rd ;C) ||ϕn ||L p (Rd ;C)

+ ||div(Q∇u)||Lp (Rd ;C) ||ϕn |ϕn |p−2 − u|u|p−2 ||Lp0 (Rd ;C) and Z letting n tend to ∞ we conclude that the right-hand side of (10.2.32) converges to div(Q∇u)u|u|p−2 dx. The proof is complete.  Rd

Let us set

 λp = sup x∈Rd

 1 c(x) − div b(x) . p

(10.2.33)

Theorem 10.2.23 For every λ ∈ C with Re λ > λp and every f ∈ Lp (Rd ; C), the equation λu − Au = f admits a unique solution u ∈ W 2,p (Rd ; C). Moreover, estimate (10.2.23) holds true for every λ ∈ C with Re λ ≥ λp + ε and every ε > 0, with a constant Cp which depends on ε, p, d and, in a continuous way, on M 0 = max{||qij ||Cb1 (Rd ) , ||bj ||W 1,∞ (Rd ) , ||c||∞ : i, j = 1, . . . , d}. Proof Fix f ∈ Lp (Rd ; C) and suppose that u ∈ W 2,p (Rd ; C) solves the equation λu − Au = f . Multiplying both sides of this equation by u|u|p−2 and using Lemma 10.2.22, we get Z Z Z Re f u|u|p−2 dx =Reλ |u|p dx − Re div(Q∇u)u|u|p−2 dx Rd Rd Rd Z Z p−2 − Re hb, ∇uiu|u| dx − c|u|p dx Rd Rd Z Z ≥ (Re λ − c)|u|p dx − Re hb, ∇uiu|u|p−2 dx. (10.2.34) Rd

Rd

Since |u|p belongs to W 1,1 (Rd ) and ∇(|u|p ) = 2−1 p(u∇u + u∇u)|u|p−2 , we can write Z Z Z 1 1 Re hb, ∇uiu|u|p−2 dx = Re hb, ∇uiu|u|p−2 dx + Re hb, ∇uiu|u|p−2 dx 2 2 d d d R R Z R Z 1 1 p = hb, ∇(|u| )i dx = − (divb)|u|p dx. p Rd p Rd Replacing this formula in the right-hand side of (10.2.34), we obtain  Z  Z 1 p−1 p Re λ − c + divb |u| dx ≤ Re f u|u|p−2 dx ≤ ||f ||Lp (Rd ;C) ||u||L p (Rd ;C) , p Rd Rd so that (Reλ − λp )||u||Lp (Rd ;C) ≤ ||f ||Lp (Rd ;C) . Based on this estimate, we can show that the equation λu − Au = f is solvable in W 2,p (Rd ; C) for every λ ∈ C with Re λ > λp and estimate (10.2.23) holds true for every such λ’s. Clearly, we have only to address the case when the constant λp in Theorem 10.2.16 (which here we denote by λ∗p ) is larger than the constant in (10.2.33). Denote by Ap the realization of the operator A in Lp (Rd ; C), i.e., the operator Ap : W 2,p (Rd ; C) → Lp (Rd ; C), defined by Ap u = Au for every u ∈ W 2,p (Rd ; C). This operator is closed and its resolvent set contains the halfline {λ ∈ C : Reλ ≥ λ∗p }. By the above estimate, (Re λ − λp )||R(λ, Ap )||L(Lp (Rd ;C)) ≤ 1,

λ ∈ C ∩ ρ(Ap ), Reλ > λp .

(10.2.35)

b ∈ σ(Ap ) with Reλ b > λp , then, by Remark A.4.5, the operator norm of If there existed λ

Elliptic Equations in Rd

276

b which of course cannot be the case due R(λ, Ap ) would blow up as λ ∈ ρ(Ap ) tends to λ, to estimate (10.2.35). Therefore, each λ ∈ C with real greater than λp belongs to ρ(Ap ), i.e., the equation λu − Au = f is uniquely solvable in W 2,p (Rd ; C) for such values of λ and every f ∈ Lp (Rd ; C). To prove that estimate (10.2.23) holds true for every λ ∈ C with (λp ∨0)+ε ≤ Reλ ≤ λ∗p , we fix any such λ, f ∈ Lp (Rd ; C) and set λ∗ = λ∗p +iIm λ. Then, using the resolvent identity we write u = R(λ, Ap )f = R(λ∗ , Ap )f + (λ∗ − λ)R(λ∗ , Ap )R(λ, Ap )f and estimate ||u||Lp (Rd ;C) ≤(1 + |λp − λ∗p |||R(λ, Ap )||L(Lp (Rd ;C)) )||R(λ∗ , Ap )f ||Lp (Rd ;C) ≤Cp |λ∗ |−1 [1 + |λp − λ∗p |(Re λ − λp )−1 ]||f ||Lp (Rd ;C) .

(10.2.36)

Similarly, we can estimate 1

|||∇u|||Lp (Rd ;C) ≤ Cp |λ∗ |− 2 [1 + |λp − λ∗p |(Re λ − λp )−1 ]||f ||Lp (Rd ;C)

(10.2.37)

||D2 u||Lp (Rd ;C) ≤ Cp [1 + |λp − λ∗p |(Re λ − λp )−1 ]||f ||Lp (Rd ;C) .

(10.2.38)

and Since Re λ is positive and less than λ∗p , so that |λ∗ |2 = |λ∗p |2 +|Im λ|2 ≥ |Re λ|2 +|Im λ|2 = |λ| . From (10.2.36)-(10.2.38), estimate (10.2.23) follows for some positive constant Cp,ε , which blows up as ε tends to 0+ . This completes the proof.  2

We now present some interesting consequences of Theorem 10.2.16, which we prove assuming Hypotheses 10.2.21. These results can be proved also under Hypotheses 10.1.2. We refer the reader to Corollaries 12.2.5 and 12.2.6 for further details. 2,p Proposition 10.2.24 Under Hypotheses 10.2.21, let u ∈ Wloc (Rd ; C) be a function such q d that Au and u belong to L (R ; C) for some q ∈ (1, ∞). Then, u ∈ W 2,q (Rd ; C).

Proof Let us set f = λ0 u − Au with λ0 > λq , where λq is defined by (10.2.33). By assumptions, f ∈ Lq (Rd ; C) and by Theorem 10.2.16, there exists a unique function v ∈ W 2,q (Rd ; C) such that λ0 v − Av = f . To prove that v = u and, thus, conclude the proof, 2,r we consider the function w = v − u which clearly belongs to Lq (Rd ; C) ∩ Wloc (Rd ; C), where r = p ∧ q. Moreover, λ0 w − Aw = 0. We multiply both sides of this equation by ϑ ∈ Cc∞ (Rd ; C) and integrate over Rd getting Z Z Z Z 0 = λ0 wϑ dx − div(Q∇w)ϑ dx − hb, ∇wiϑ dx − cwϑ dx. (10.2.39) Rd

Rd

Rd

Rd

Integrating by parts, we can write Z div(Q∇w)ϑ dx = − Rd

d Z X i,j=1

Z qij Dj wDi ϑ dx =

Rd

div(Q∇ϑ)w dx Rd

and Z −

hb, ∇wiϑ dx = Rd

d Z X j=1

Z Dj (bj ϑ)w dx =

Rd

Z

Replacing these two expressions in (10.2.39), we conclude that Z 0= (λ0 ϑ − A∗ ϑ)w dx, Rd

hb, ∇ϑiw dx.

(divb)ϑw dx + Rd

Rd

(10.2.40)

Semigroups of Bounded Operators and Second-Order PDE’s

277

where A∗ ϑ = div(Q∇ϑ) − hb, ∇ϑi + (c − divb)ϑ. By density, we can extend (10.2.40) to every 0 function ϑ ∈ W 2,q (Rd ; C), where 1/q + 1/q 0 = 1. Note that the coefficients of operator A∗ satisfy Hypotheses 10.2.21 and     1 1 λq0 (A∗ ) = sup c(x) − (divb)(x) + 0 (divb)(x) = sup c(x) − (divb)(x) = λq . q q x∈Rd x∈Rd Therefore, we can apply Theorem 10.2.23 which shows that the operator λ0 −A∗ is surjective 0 0 0 from W 2,q (Rd ; C) to Lq (Rd ; C). Since the function |w|q−2 w belongs to Lq (Rd ; C), the ∗ q−2 2,q 0 d equation λ0 ϑ − A Zϑ = |w| w admits a solution ϑ ∈ W (R ; C). From formula (10.2.40) |w|q dx = 0, so that w = 0 or, equivalently, u = v.

we conclude that



Rd

Proposition 10.2.25 Suppose that u ∈ W 2,p (Rd ; C) is such that λu − Au ∈ Lq (Rd ; C) for some λ ∈ C and q ∈ (p, ∞). Then, u belongs to W 2,q (Rd ; C). Proof Fix u and λ as in the statement of the proposition. If p ≥ d/2, then u belongs to Lq (Rd ; C) by the Sobolev embedding theorems (see properties (i) and (ii) in Theorem 1.3.6). Since λu − Au belongs to Lq (Rd ; C), by difference Au belongs to Lq (Rd ; C) as well. We can thus apply Proposition 10.2.24 and conclude that u ∈ W 2,q (Rd ; C). Let us now suppose that p < d/2. Then, by the Sobolev embedding theorems (see −1 − 2d−1 . Clearly, if p1 ≥ q, property (i) in Theorem 1.3.6), u ∈ Lp1 (Rd ; C), where p−1 1 =p then we get the assertion arguing as above. So, let us assume that p1 < q. We use a bootstrap argument to get the assertion. Since u ∈ W 2,p (Rd ; C), the function λu − Au belongs to Lp (Rd ; C). By assumptions, such a function belongs also to Lq (Rd ; C) so that, by H¨ older’s inequality, it belongs to Lr (Rd ; C) for every r ∈ [p, q]. In particular, λu−Au belongs to Lp1 (Rd ; C) and, by difference, Au belongs to Lp1 (Rd ; C). Thus, we can apply Proposition 10.2.24 and conclude that u ∈ W 2,p1 (Rd ; C). We can repeat the previous argument with −1 p being replaced by p1 . If p1 ≥ 2−1 d or p2 ≥ q, where p−1 = p−1 , then we are 2 1 − 2d 2,p2 d done. Otherwise, u ∈ W (R ; C) and we apply again the same argument. We can thus recursively define the numbers p1 < p2 < p3 < . . ., where p−1 = p−1 − 2jd−1 for each j. j Therefore, there exists j ∈ N such that pj < q and pj+1 ≥ q, so that in (j + 1)-steps, we deduce that u ∈ Lq (Rd ; C) and easily conclude the proof.  Corollary 10.2.26 Let q ∈ (1, ∞), f ∈ Lp (Rd ; C) ∩ Lq (Rd ; C) and let λ ∈ C be such that the equation λu − Au = f admits a unique solution u ∈ W 2,p (Rd ; C) and a unique solution v ∈ W 2,q (Rd ; C). Then, u ≡ v. Proof Without loss of generality, we can assume that p < q. We fix λ, f , u and v as in the statement of the corollary. Since λu − Au = f ∈ Lq (Rd ; C), Proposition 10.2.25 shows that u belongs to W 2,q (Rd ; C). By assumptions, the previous equation admits a unique  solution in W 2,q (Rd ; C) and this implies that u ≡ v. To conclude this section, we present some results concerning the regularity of distributional solutions to autonomous elliptic equations. We assume that the operator A is uniformly elliptic, i.e., Hypothesis 10.1.2(ii) is satisfied. The smoothness assumptions on the coefficients of A will be made clear in the statement of the forthcoming theorem. Theorem 10.2.27 Fix p ∈ (1, ∞) and denote by p0 the conjugate exponent to p. The following properties are satisfied:

Elliptic Equations in Rd

278

(i) if qij ∈ Cb1 (Rd ), bi ∈ Cb (Rd ), c ∈ Cb (Rd ) (i, j = 1, . . . , d) and u ∈ Lp (Rd ; C) satisfies the estimate Z ϕ ∈ Cc∞ (Rd ; C), (10.2.41) d uAϕ dx ≤ C||ϕ||W 1,p0 (Rd ;C) , R

then u belongs to W 1,p (Rd ; C); (ii) if qij ∈ Cb2 (Rd ), bi ∈ Cb1 (Rd ) (i, j = 1, . . . , d), div b ∈ Lp (Rd ), c ∈ Cb (Rd ) and f, u ∈ Lp (Rd ; C) are such that Z Z uAϕ dx = f ϕ dx, ϕ ∈ Cc∞ (Rd ; C), (10.2.42) Rd

Rd

then u ∈ W 2,p (Rd ; C). If A = div(Q∇) + hb, ∇i + c then the same result holds under the weaker assumptions qij , bi ∈ Cb1 (Rd ) (i, j = 1, . . . , d). Proof (i) Clearly, estimate (10.2.41) holds true also for real-valued functions ϕ, so that also the real and imaginary parts of the function u ∈ Lp (Rd ; C) satisfy (10.2.41). Hence, throughout the proof of this property, we will deal with real-valued function. Let A0 be the principal part of the operator A, i.e., A0 = Tr(QD2 ). From (10.2.41), we deduce that Z ϕ ∈ Cc∞ (Rd ). (10.2.43) uA0 ϕ dx ≤ C||ϕ||W 1,p0 (Rd ) , Rd

For every h ∈ Rd \ {0}, set τh u = |h|−1 (u(· + h) − u). An explicit computation shows that Z

Z

Z

τh u A0 ϕ dx = Rd

uA0 (τ−h ϕ) dx + Rd

u Rd

d X

(τ−h qij )Dij ϕ(· − h) dx

i,j=1

for every ϕ ∈ Cc∞ (Rd ). Using the boundedness of the diffusion coefficients qij in Rd (i, j = 1, . . . , d) and (10.2.43), we obtain Z ≤ C1 ||ϕ|| 2,p0 d (10.2.44) τ u(λϕ − A ϕ) dx h 0 W (R ) Rd

for every ϕ ∈ Cc∞ (Rd ), λ > 0 and some positive constant C1 = C1 (λ), independent of ϕ and h. Since u ∈ Lp (Rd ), by density (10.2.44) can be extended to every function in 0 ϕ ∈ W 2,p (Rd ). We fix a large λ so that the equation λv − A0 v = f is uniquely solvable in W 2,p (Rd ) for every f ∈ Lp (Rd ) (see Theorem 10.2.16) and denote by ϕ the solution of the previous equation when f = τh u|τh u|p−2 . By (10.2.23), the function ϕ satisfies the estimate ||ϕ||W 2,p0 (Rd ) ≤ C2 ||τh u||p−1 for some positive constant C2 , independent of h. Plugging Lp (Rd ) this particular function into (10.2.44), we obtain that ||τh u||Lp (Rd ) ≤ C3 , with C3 being independent of h. From this estimate, it follows at once that u ∈ W 1,p (Rd ). (ii) As in the proof of property (i), we can assume that u and f are real-valued functions. We first assume that A is in nondivergence form. By property (i), u belongs to W 1,p (Rd ). Therefore, taking (10.2.42) into account, we get Z Z uA0 ϕ dx = f1 ϕ dx, ϕ ∈ Cc∞ (Rd ), Rd

Rd

where A0 is the same operator as in the proof of (i) and f1 = f + hb, ∇ui + u(div b − c) belongs to Lp (Rd ). Hence, Z Z u(λϕ − A0 ϕ) dx = gϕ dx, ϕ ∈ Cc∞ (Rd ), (10.2.45) Rd

Rd

Semigroups of Bounded Operators and Second-Order PDE’s

279

where g = λu − f1 belongs to Lp (Rd ). By density, we can extend (10.2.45) to every function 0 ϕ ∈ W 2,p (Rd ). Now, we fix λ sufficiently large such that the equations λv − A0 v = h1 and 0 ∗ λv − A0 v = h2 are uniquely solvable in W 2,p (Rd ) and W 2,p (Rd ), respectively, for every Pd 0 h1 ∈ Lp (Rd ) and h2 ∈ Lp (Rd ), where A∗0 = Tr(QD2 ) + 2div(Q∇) + ij=1 (Dij qij ) is the formal adjoint to the operator A0 (see again Theorem 10.2.16). To prove that u ∈ W 2,p (Rd ), we introduce the solution v ∈ W 2,p (Rd ) of the equation λv − A∗0 v = g and, using (10.2.45), R 0 we conclude that Rd z(λϕ − A0 ϕ) dx = 0 for every ϕ ∈ W 2,p (Rd ), where z = v − u. Taking 0 ϕ ∈ W 2,p (Rd ) such that λϕ − A0 ϕ = z|z|p−2 and plugging this function into the above integral, we deduce that z = 0. If A is in divergence form, then A∗0 = div(Q∇), so that we do not need to differentiate twice the diffusion coefficients and the assumption qij ∈ Cb1 (Rd ) (i, j = 1, . . . , d) is enough to repeat the same computations as above. 

10.2.4

Further regularity results and interior Lp -estimates

In this subsection, which is the counterpart in the context of Lp -spaces of Subsection 10.1.3 we show that the more f and the coefficients of the operator A are smooth, the more the solution u of the elliptic equation λu − Au = f is itself smooth. We also prove some interesting interior Lp -estimates. Hypotheses 10.2.28 (i) The coefficients qij = qji , bj (i, j = 1, . . . , d) and c belong to W k,∞ (Rd ) for some k ∈ N; (ii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for all x, ξ ∈ Rd . Theorem 10.2.29 Let Hypotheses 10.2.28 be satisfied. If f ∈ W k,p (Rd ; C) for some p ∈ (1, ∞) and λ ∈ C, with Reλ ≥ λp (see Theorem 10.2.16), then the equation λu − Au = f admits a unique solution u ∈ W k+2,p (Rd ; C). Moreover, there exists a positive constant Ck,p , which depends on p, d, k and, in a continuous way, on λ, and the W k,∞ (Rd )-norm of the coefficients of the operator A, such that ||u||W k+2,p (Rd ;C) ≤ Ck,p ||f ||W k,p (Rd ;C) .

(10.2.46)

Proof Throughout the proof, we denote by K a positive constant, independent of the functions u and f that we will consider as well as of the positive parameter r, which may vary from line to line. If the constant depends on λ, then we stress this dependence writing Kλ instead of K. Even if we do not make it explicit, the constants in the proof may depend on p and, in a continuous way, on the W k,∞ (Rd )-norms of the coefficients of the operator A and on the ellipticity constant µ. We argue by induction on k ∈ N. Suppose that k = 1 and fix λ ∈ C, with Re λ > λp , and f ∈ W 1,p (Rd ; C). By Theorem 10.2.16, we know that there exists a unique function u ∈ W 2,p (Rd ; C), which solves the equation λu−Au = f . For every r > 0 and h ∈ {1, . . . , d}, let u(r,h) : Rd → C be the function defined by u(r,h) (x) = r−1 (u(x + reh ) − u(x)) for every x ∈ Rd . As it is easily seen, d X

λu(r,h) − Au(r,h) =f (r,h) +

(r,h)

qij

Dij u(· + reh )

i,j=1

+

d X j=1

(r,h)

bj

Dj u(· + reh ) + c(r,h) u(· + reh ),

(10.2.47)

Elliptic Equations in Rd

280 (r,h)

(r,h)

where the functions f (r,h) , qij , bj (i, j = 1, . . . , d) and c(r,h) are defined in analogy with the definition of the function u(r,h) . Next, we observe that Z r 1 (r,h) |f (x)| = x ∈ Rd , r > 0. Dh f (x + seh ) ds , r 0 and hence ||f r,h ||Lp (Rd ;C) ≤ ||Dh f ||Lp (Rd ;C) . Arguing in the same way, we can show that (r,h)

(r,h)

||qij ||∞ ≤ ||Dh qij ||∞ , ||bj ||∞ ≤ ||Dh bj ||∞ and ||c(r,h) ||∞ ≤ ||Dh c||∞ for every i, j = 1, . . . , d. Therefore, the right-hand side of (10.2.47) (say gr,h ) belongs to Lp (Rd ; C) and ||gr,h ||Lp (Rd ;C) ≤||Dh f ||Lp (Rd ;C) +

d X

||Dh qij ||∞ ||Dij u||Lp (Rd ;C)

i,j=1

+

d X

||Dh bj ||∞ ||Dj u||Lp (Rd ;C) + ||Dh c||∞ ||u||Lp (Rd ;C)

j=1

X  d d X ||qij ||Cb1 (Rd ) + ≤K ||bj ||Cb1 (Rd ) +||c||Cb1 (Rd ) ||u||W 2,p (Rd ;C) + ||Dh f ||Lp (Rd ;C) i,j=1

j=1

  d d X X ≤Kλ 1 + ||qij ||Cb1 (Rd ) + ||bj ||Cb1 (Rd ) + ||c||Cb1 (Rd ) ||f ||W 1,p (Rd ;C) , i,j=1

j=1

where we have used estimate (10.2.24) in the last part of the previous chain of inequalities. Applying again such estimate, we conclude that ||u(r,h) ||W 2,p (Rd ;C) ≤ Kλ ||f ||W 1,p (Rd ;C) for every r > 0. In particular, for every multi-index β with length 2, it follows that ||(Dβ u)(r,h) ||Lp (Rd ;C) = ||Dβ u(r,j) ||Lp (Rd ;C) ≤ Kλ ||f ||W 1,p (Rd ;C) . Thus, applying Proposition 1.3.11 (to the real and imaginary parts of (Dβ u)(r,h) ) we deduce that function Dβ u ∈ W 1,p (Rd ; C) and |||∇Dβ u|||Lp (Rd ;C) ≤ Kλ ||f ||W 1,p (Rd ;C) . From the arbitrariness of β, the assertion follows in the case k = 1. Let us now suppose that the assertion is true for some k ∈ N and prove it with k being replaced by k+1. By assumptions u ∈ W k+2,p (Rd ; C) and ||u||W k+2,p (Rd ;C) ≤ Ck,p ||f ||W k,p (Rd ;C) , the constant Ck,p being independent of u and f . Let us fix a multi-index β with length k. Then, λDβ u − ADβ u = ψβ on Rd , where ψβ =Dβ f +

d X X β  Dγ qij Dβ−γ Dij u γ i,j=1 0 2 for every (x, y) ∈ Ω and n ∈ N, where ϑ is a smooth function, compactly supported in Rd−2 with Lp -norm equal to one. Clearly, each function un belongs to W 2,p (Ω; C) for every 1/p p ∈ (1, ∞). Moreover, ∆un ≡ 0 in Ω and ||un ||Lp (Ω) = ωd for every n ∈ N, if d = 2, and (∆un )(x) = (x1 + ix2 )n

d X j=3

Djj ϑ(x3 , . . . , xd ),

x ∈ Ω,

Elliptic Equations in Rd

284 1/p

and ||un ||Lp (Ω) = ωd ||ϑ||Lp (Rd−2 ) for every n ∈ N, otherwise. From (10.2.55) it would follow that ||un ||W 2,p (B(0,1)) ≤ Cp0 for every n ∈ N and some positive constant Cp0 , independent of n. Clearly, this estimate cannot hold true since the Lp -norms of Dj un and Dij un (i, j = 1, 2) blow up as n tends to ∞. Theorem 10.2.36 Let p ∈ (1, ∞) and Ω0 , Ω00 be open sets such that Ω0 b Ω00 b Ω. Then, there exists a positive constant C, which depends on Ω0 , Ω00 , the function µ, the sup-norm of the coefficients of the operator A and the moduli of continuity of the diffusion coefficients over Ω0 , such that ||u||W 2,p (Ω0 ;C) ≤ C(||u||Lp (Ω00 ;C) + ||Au||Lp (Ω00 ;C) )

(10.2.56)

for every u ∈ W 2,p (Ω00 ; C). Proof Since the operator A has real-valued coefficients, it suffices to prove (10.2.56) for real valued functions. The core of the proof consists in proving estimate (10.2.56) with Ω0 = B(y, r) and Ω00 = B(y, 2r). For this purpose, for every n ∈ N ∪ {0} we set rn = (2 − 2−n )r. Moreover, for every n ∈ N we introduce a function ϑn ∈ Cc∞ (Rd ) such that χB(y,rn ) ≤ ϑn ≤ χB(y,rn+1 ) . Fix u ∈ W 2,p (B(y, 2r)). Then, each function vn = ϑn u, extended in the trivial way outside B(y, 2r), belongs to W 2,p (Rd ). Moreover, since vn vanishes outside B(y, rn+1 ) it follows easily that the trivial extension to Rd of the function A(ϑn u) coincides with A0 (ϑn u), 0 where A0 = Tr(Q0 D2 ) + hb0 , ∇i + c0 and qij = qij ϑn+1 + (1 − ϑn+1 )∆, b0j = bj and c0 = c for every i, j = 1, . . . , d, qij , bj and c denoting the trivial extension of the functions qij , bj and c to Rd . The operator A0 acts on functions v ∈ W 2,p (Rd ) and its coefficients satisfy Hypotheses 10.2.15. Therefore, applying estimate (10.2.25) we can write ||uϑn ||W 2,p (Rd ) ≤C∗,p (||uϑn ||Lp (Rd ) + ||A0 (uϑn )||Lp (Rd ) ) ≤Cp (||u||Lp (B(y,2r)) + ||A(uϑn )||Lp (B(y,2r)) ). Since A(uϑn ) = ϑn Au + 2hQ∇ϑn , ϑui + u(A − c)ϑn for every n ∈ N, we can estimate ||A(uϑn )||Lp (B(y,2r)) ≤||ϑn Au||Lp (B(y,2r)) + 2||Q||L∞ (B(y,2r)) |||∇ϑn |||∞ |||∇u|||Lp (B(y,rn+1 )) + C1 (|||∇ϑn |||∞ + ||D2 ϑn ||∞ )||u||Lp (B(y,2r)) ≤||Au||Lp (B(y,2r)) + C2 2n |||∇u|||Lp (B(y,rn+1 )) + 4n C2 ||u||Lp (B(y,2r)) ≤||Au||Lp (B(y,2r)) + C2 2n |||∇(uϑn+1 )|||Lp (Rd ) + 4n C2 ||u||Lp (B(y,2r)) for some positive constants C1 and C2 , independent of u and n ∈ N. Moreover, using the estimate (1.3.9) we infer that |||∇(uϑn+1 )|||Lp (Rd ) ≤ε||uϑn+1 ||W 2,p (Rd ) + ε−1 C3 ||ϑn+1 u||Lp (Rd ) ≤ε||uϑn+1 ||W 2,p (Rd ) + ε−1 C3 ||u||Lp (B(y,2r)) for every ε > 0 and some positive constant C3 , independent of ε and n. Putting everything together, we conclude that ||uϑn ||W 2,p (Rd ) ≤C4 (1 + 4n + 2n ε−1 )||u||Lp (B(y,2r)) + C4 ||Au||Lp (B(y,2r)) + 2n εC4 ||uϑn+1 ||Lp (Rd ) . We fix ξ ∈ (0, 1/4) and choose ε = 2−n C4−1 ξ to get ||uϑn ||W 2,p (Rd ) ≤ 4n C5 (1 + ξ −1 )||u||Lp (B(y,2r)) + C4 ||Au||Lp (B(y,2r)) + ξ||uϑn+1 ||Lp (Rd )

Semigroups of Bounded Operators and Second-Order PDE’s

285

for some positive constant C5 , independent of n, ξ and u. Multiplying both sides of this inequality by ξ n and summing between 0 and m, we deduce that m X

ξ n ||uϑn ||W 2,p (Rd ) ≤C5 (1 + ξ −1 )||u||Lp (B(y,2r))

n=0

m X

(4ξ)n

n=0

+

m X

ξ n+1 ||uϑn+1 ||W 2,p (Rd ) + C4 ||Au||Lp (B(y,2r))

n=0

m X

ξn

n=0

≤C6 (1 + ξ −1 )||u||Lp (B(y,2r)) +

m X

ξ n+1 ||uϑn+1 ||W 2,p (Rd )

n=0

+ C6 ||Au||Lp (B(y,2r))

(10.2.57)

for a positive constant C6 , which is independent of u, n and ξ. Since ||ϑn u||W 2,p (Rd ) ≤ C7 4n ||u||W 2,p (B(y,2r)) for every n ∈ N and some positive constant C7 , independent of n and u, the series on the first- and last-side of the previous chain of inequalities converge, due to the choice of ξ. Therefore, letting m tend to ∞ in (10.2.57), we conclude that ∞ X

ξ n ||uϑn ||W 2,p (Rd ) ≤C6 (1 + ξ −1 )||u||Lp (B(y,2r)) +

n=0

∞ X

ξ n ||uϑn ||W 2,p (Rd ) + C6 ||Au||Lp (B(y,2r))

n=1

or, equivalently, ||ϑ0 u||W 2,p (Rd ) ≤ C6 (1+ξ −1 )||u||Lp (B(y,2r)) +C6 ||Au||Lp (B(y,2r)) . Since ϑ0 = 1 on B(y, r), estimate (10.2.56) follows in this case. We can now address the general case. We fix Ω0 and Ω00 as in the statement of the theorem and denote by r a positive number such that 2r < dist(Ω0 , ∂Ω00 ). Clearly, Ω0 ⊂ S B(y, r). Since Ω0 is a compact subset of Rd , we can extract a finite subcovering so y∈Ω0 Sk 0 that Ω ⊂ j=1 B(yj , r). From the first part of the proof, we infer that ||u||W 2,p (B(yj ,r)) ≤ Kp ||u||Lp (B(yj ,2r)) + ||Au||Lp (B(yj ,2r)) ) ≤ Kp ||u||Lp (Ω00 ) + ||Au||Lp (Ω00 ) ) for every j = 1, . . . , k and some positive constant Kp independent of u. Thus, ||u||pW 2,p (Ω0 ) ≤

k X

||u||pW 2,p (B(yj ,r)) ≤ Kpp (||u||Lp (Ω00 ) + ||Au||Lp (Ω00 ) )p

j=1

and the assertion follows.



Finally, based on Theorem 10.2.29, we can prove higher-order interior Lp -estimates under the following set of assumptions. Hypotheses 10.2.37 k,∞ Wloc (Ω);

(i) The coefficients qij = qji , bj (i, j = 1, . . . , d) and c belong to

(ii) there exists a positive function µ : Ω → R, with positive infimum over each compact subset of Ω, such that hQ(x)ξ, ξi ≥ µ(x)|ξ|2 for all ξ ∈ Rd and x ∈ Ω. 2,p Theorem 10.2.38 Let Hypotheses 10.2.37 be satisfied. If u ∈ Wloc (Ω; C) is such that Au ∈ k,p k+2,p Wloc (Ω; C), then u ∈ Wloc (Ω; C) and, for every Ω0 b Ω00 b Ω, there exists a positive constant C, which depends on Ω0 , Ω00 , Ω and, in a continuous way on λ, on the infimum of µ over Ω00 and on the coefficients of the operator A, such that

||u||W k+2,p (Ω0 ;C) ≤ C(||u||Lp (Ω00 ;C) + ||Au||W k,p (Ω00 ;C) ).

(10.2.58)

Elliptic Equations in Rd

286

Proof Fix λ ∈ (λp , ∞), u, Ω0 and Ω00 , as in the statement of the theorem, and a function ϑ ∈ Cc∞ (Rd ) such that χΩ0 ≤ ϑ ≤ χΩ∗ , where Ω∗ denotes the set of all x ∈ Ω such that d(x, Ω0 ) < δ, where δ = dist(Ω0 , ∂Ω00 ). Clearly, Ω0 b Ω∗ b Ω00 . Moreover, set f = λu − Au and, for every function ψ : Ω → C, denote by ψ the trivial extension of ψ to Rd . We stress here, without further mentioning it, that the constants that will appear in the proof depend at most on Ω0 , Ω00 and, in a continuous way, on the infimum of µ over Ω00 and on the W 1,∞ (Ω00 )-norm of the coefficients of the operator A. As it is easily seen, the function v = uϑ belongs to W 2,p (Rd ; C) and λv − A0 v = g := f ϑ − 2hQ0 ∇u, ∇ϑi − u(A0 − c0 )ϑ 0 on Rd , where A0 = Tr(Q0 D2 ) + hb0 , ∇i + c0 , Q0 = (qij ), with qij = qij % + (1 − %)∆, b0 = (bj ), with b0j = bj %, and c0 = c% for every i, j = 1, . . . , d, % being a smooth function compactly supported in Ω00 and such that ϑ ≡ 1 on Ω∗ . 2,p Since u ∈ Wloc (Ω; C) and Au ∈ W k,p (Ω; C), function g belongs to W 1,p (Rd ; C) and ||g||W 1,p (Rd ;C) ≤ C(||Au||W 1,p (Ω∗ ,C) + ||u||W 1,p (Ω∗,C) ). By Theorem 10.2.29 v belongs to W 3,p (Rd ; C) and, applying (10.2.56), with Ω0 and Ω00 being replaced, respectively, by Ω∗ and Ω00 , we can estimate

e1,p (||Au||W 1,p (Ω00 ,C) + ||u||Lp (Ω00 ,C) ). ||v||W 3,p (Rd ;C) ≤ C1,p ||g||W 2,p (Rd ;C) ≤ C Since v ≡ u on Ω0 , from the previous chain of inequalities, estimate (10.2.58) follows for k = 1. If k > 1, then we can make a bootstrap argument work to get the assertion. If k = 2, then g ∈ W 2,p (Rd ), so that v ∈ W 4,p (Rd ; C) and e2,p (||Au||W 2,p (Ω00 ,C) + ||u||Lp (Ω00 ,C) ). ||v||W 4,p (Rd ;C) ≤ C2,p ||g||W 2,p (Rd ;C) ≤ C Iterating this procedure, we complete the proof.

10.3

Solutions in L∞ (



R ; C) and in C (R ; C) d

b

d

In this section, we analyze the solvability of the equation λu − Au = f when f ∈ L∞ (Rd ; C) and when f ∈ Cb (Rd ; C), taking advantage of the results of the previous section. The Cb -case will follow as a straightforward consequence of the analysis in the L∞ -space. In the first part of the section, we still assume that Hypotheses 10.2.15 are satisfied and prove the following result. Theorem 10.3.1 There exists λ∞ ∈ R such that, for every f ∈ L∞ (Rd ; C) and every λ ∈ C with Reλ > λ∞ , the equation λu − Au = f admits a unique solution u ∈ Cb1+α (Rd ; C) ∩ 2,p Wloc (Rd ; C), for every p ∈ [1, ∞) and α ∈ (0, 1), such that Au ∈ L∞ (Rd ; C). Moreover, for ep , which depends only on p, d, and, in a every p ∈ (d, ∞), there exists a positive constant C continuous way on µ, the sup-norm of the coefficients of the operator A and on the modulus of continuity of the diffusion coefficients, such that p d ep ||λu − Au||∞ (10.3.1) |λ|||u||∞ + |λ||||∇u|||∞ + |λ| 2p sup ||D2 u||Lp (B(x0 ,|λ|−1/2 );C) ≤ C x0 ∈Rd

Semigroups of Bounded Operators and Second-Order PDE’s

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for every λ ∈ C, with Re λ > λ∞ . Finally, for every α ∈ (0, 1) and the same values of λ, bλ such that there exists a positive constant C bλ ||λu − Au||∞ . ||u||C 1+α (Rd ;C) ≤ C

(10.3.2)

b

This constant depends only on d, α and, in a continuous way on λ, µ, the sup-norm of the coefficients of the operator A and on the modulus of continuity of the diffusion coefficients Proof We split the proof into three steps. In the first one, based on the results of the 2,p previous section, we prove that for every u ∈ Wloc (Rd ; C) (p ∈ (1, ∞)), r ∈ (0, 1], x0 ∈ Rd , γ ∈ [1, ∞), λ ∈ C, with Re λ ≥ λp , λp being the constant appearing in Theorem 10.2.16, it holds that p |λ|||u||Lp (B(x0 ,r);C) + |λ||||∇u|||Lp (B(x0 ,r);C) + ||D2 u||Lp (B(x0 ,r);C)   e ∗ ||λu − Au||Lp (B(x ,(γ+1)r);C) + 1 ||u||Lp (B(x ,(γ+1)r);C) + 1 |||∇u|||Lp (B(x ,(γ+1)r);C) ≤C p 0 0 0 γr2 γr (10.3.3) e ∗ , which depends only on p, d, and, in a continuous way on for some positive constant C p µ, the sup-norm of the coefficients of the operator A and on the modulus of continuity of the diffusion coefficients. Using this inequality, in Step 2, we prove the estimate (10.3.1) for 2,p every u ∈ Wloc (Rd ; C) ∩ Cb1 (Rd ; C) (p > d) and λ ∈ C with Reλ ≥ λp ∨ 1. Finally, in Step 3, we conclude the proof. Step 1. Let us fix x0 ∈ Rd , p, r, γ and λ as above, and introduce a smooth function ϑ such that χB(x0 ,r) ≤ ϑ ≤ χB(x0 ,(γ+1)r) and ||ϑ||∞ + γr|||∇ϑ|||∞ + γ 2 r2 ||D2 ϑ||∞ ≤ K for some positive constant K, independent of γ and r. 2,p Given a function u ∈ Wloc (Rd ; C), we consider the function v = uϑ, which belongs to 2,p d W (R ; C) and solves the elliptic equation λv − Av = ϑf − u(Aϑ − cϑ) − 2hQ∇u, ∇ϑi, where we have set f = λu − Au. Since the right-hand side of the previous equation belongs to Lp (Rd ; C), we can invoke Theorem 10.2.16 to infer that p |λ|||v||Lp (Rd ;C) + |λ||||∇v|||Lp (Rd ;C) + ||D2 v||Lp (Rd ;C) ≤Cp ||ϑf − u(Aϑ − cϑ) − 2hQ∇u, ∇ϑi||Lp (Rd ;C) .

(10.3.4)

Recalling that ϑ is supported in B(x0 , (γ + 1)r) it is easy to estimate ||ϑf − u(Aϑ − cϑ) − 2hQ∇u, ∇ϑi||Lp (Rd ;C)  e1 ||f ||Lp (B(x ,(γ+1)r);C) + ||Aϑ − cϑ||∞ ||u||Lp (B(x ,(γ+1)r);C) ≤C 0 0  + 2||Q||∞ ||∇u|||Lp (B(x0 ,(γ+1)r);C) |||∇ϑ|||∞  e2 ||f ||Lp (B(x ,(γ+1)r);C) + (|||∇ϑ|||∞ + ||D2 ϑ||∞ )||u||Lp (B(x ,(γ+1)r);C) ≤C 0 0  p + |∇u|||L (B(x0 ,(γ+1)r);C) |||∇ϑ|||∞     1 1 1 e ≤C3 ||f ||Lp (B(x0 ,(γ+1)r);C) + + ||u||Lp (B(x0 ,(γ+1)r);C) + |||∇u|||Lp (B(x0 ,(γ+1)r);C) γr γ 2 r2 γr e e e3 ||f ||Lp (B(x ,(γ+1)r);C) + C3 ||u||Lp (B(x ,(γ+1)r);C) + C3 |||∇u|||Lp (B(x ,(γ+1)r);C) (10.3.5) ≤C 0 0 0 γr2 γr for some positive constants C1 , C2 , C3 , which depend at most on d and, in a continuous way on the sup-norm of the coefficients of the operator A. Since p ||v||Lp (Rd ;C) + |λ||||∇v|||Lp (Rd ;C) + ||D2 v||Lp (Rd ;C)

Elliptic Equations in Rd p ≥|λ|||u||Lp (B(x0 ,r);C) + |λ||||∇u|||Lp (B(x0 ,r);C) + ||D2 u||Lp (B(x0 ,r);C) ,

288

from (10.3.4) and (10.3.5), estimate (10.3.3) follows at once. 2,p Step 2. We now fix p > d and u ∈ Wloc (Rd ; C) ∩ Cb1 (Rd ; C). By the Sobolev embedding theorem 1.3.6, we get ||v||L∞ (B(0,1);C) ≤ C1,p (||v||Lp (B(0,1);C) + |||∇v|||Lp (B(0,1);C) )

(10.3.6)

for every function v ∈ W 1,p (B(0, 1); C) and some positive constant C1,p , that we can assume larger than one. The constant C1,p as all the forthcoming constants depend at most only on p, d, α and, in a continuous way on µ, the sup-norm of the coefficients of the operator A and on the modulus of continuity of the diffusion coefficients. Given w ∈ W 1,p (B(x0 , r); C) (p > d) and taking the function v, defined by v(x) = w(x0 + rx) for every x ∈ B(0, 1), in (10.3.6), d we conclude that ||w||L∞ (B(x0 ,r);C) ≤ C1,p r− p (||w||Lp (B(x0 ,r);C) + r|||∇w|||Lp (B(x0 ,r);C) ). Applying this inequality to u and each of the first-order derivatives of u, we infer that p |λ|||u||L∞ (B(x0 ,|λ|−1/2 );C) + |λ||||∇u|||L∞ (B(x0 ,|λ|−1/2 );C) p d ≤C2,p |λ| 2p (||u||Lp (B(x0 ,|λ|−1/2 );C) + |λ||||∇u|||Lp (B(x0 ,|λ|−1/2 );C) ) for every λ ∈ C, with Reλ ≥ λp ∨ 1, so that, using (10.3.3), we deduce that d

p

|λ||||∇u|||L∞ (B(x0 ,|λ|−1/2 );C) + |λ| 2p ||D2 u||Lp (B(x0 ,|λ|−1/2 );C) p d ≤C2,p |λ| 2p [||u||Lp (B(x0 ,|λ|−1/2 );C) + |λ||||∇u|||Lp (B(x0 ,|λ|−1/2 );C) + ||D2 u||Lp (B(x0 ,|λ|−1/2 );C) ] |λ|||u||L∞ (B(x0 ,|λ|−1/2 );C) + d

≤C3,p |λ| 2p [||λu − Au||Lp (B(x0 ,(γ+1)|λ|−1/2 );C) + γ −1 |λ|||u||Lp (B(x0 ,(γ+1)|λ|−1/2 );C) p + γ −1 |λ||||∇u|||Lp (B(x0 ,(γ+1)|λ|−1/2 );C) ]. Now, observing that Z

Z

p

|w| dx ≤||w||∞ B(x0 ,(γ+1)|λ|−1/2 )

d

dx = (γ + 1)d d−1 ωd |λ|− 2 ||w||∞

B(x0 ,(γ+1)|λ|−1/2 )

for every w ∈ L∞ (Rd ; C), we can continue the previous chain of inequalities and conclude that p d |λ|||u||L∞ (B(x0 ,|λ|−1/2 );C) + |λ||||∇u|||L∞ (B(x0 ,|λ|−1/2 );C) + λ 2p ||D2 u||Lp (B(x0 ,|λ|−1/2 );C) p d ≤C4,p (γ + 1) p [||f ||∞ + γ −1 |λ|||u||∞ + γ −1 |λ||||∇u|||∞ ]. (10.3.7) Since d/p < 1, we can fix γ large enough such that C4,p γ −1 (γ + 1)d/p < 1/2. With this choice of γ, we can move the terms containing the L∞ -norm of u and its gradient from the right- to the left-hand side of (10.3.7) to obtain |λ|||u||L∞ (B(x0 ,|λ|−1/2 );C) +

p

d

|λ||||∇u|||L∞ (B(x0 ,|λ|−1/2 );C) + |λ| 2p ||D2 u||Lp (B(x0 ,r);C)

d

≤2C4,p (γ + 1) p ||λu − Au||∞ . Taking the supremum with respect to x0 ∈ Rd , we conclude the proof of (10.3.1). Step 3. To complete the proof, let us prove that, for every λ ∈ C, with Re λ > λ∞ := inf p∈(d,∞) λp ∨ 1, and every f ∈ L∞ (Rd ; C), the equation λu − Au = f admits a unique 2,p solution u ∈ Cb1+α (Rd ; C) ∩ Wloc (Rd ; C) for every p ∈ [1, ∞) and α ∈ (0, 1), such that ∞ d Au ∈ L (R ; C). For this purpose, we fix λ as above, f ∈ L∞ (Rd ; C) and approximate it

Semigroups of Bounded Operators and Second-Order PDE’s

289

via truncation with a sequence (fn ) ∈ L∞ (Rd ; C) of compactly supported functions such that ||fn ||∞ ≤ ||f ||∞ , for every n ∈ N, and fn converges to f pointwise almost everywhere on Rd . Further, we fix p > d, such that Re λ ≥ λp , and observe that, by Theorem 10.2.16, equation λun − Aun = fn admits a unique solution un ∈ W 2,p (Rd ; C) for every n ∈ N. By 2−d/p the Sobolev embedding theorem 1.3.6, each function un belongs also to Cb (Rd ; C) so that, using (10.3.1) we can estimate p ep ||fn ||∞ ≤ C ep ||f ||∞ . |λ|||un ||∞ + |λ||||∇un |||∞ + sup ||D2 un ||Lp (B(x0 ,|λ|−1/2 );C) ≤ C x0 ∈Rd

(10.3.8) Thus, we infer that (un ) is a bounded sequence in W 2,p (B(0, r); C) for every r > 0 and, up 2,p to a subsequence, we can assume that there exists a function u ∈ Wloc (Rd ; C) such that un 1,p converges to u in W (B(0, r); C) for every r > 0, Dij un converges weakly in Lp (B(0, r); C) to Dij u for every i, j = 1, . . . , d. Moreover, we can also assume that un and ∇un converge, respectively, to u and ∇u pointwise on Rd . Let us fix ϕ ∈ Cc∞ (Rd ; C). Since, for every r > 0, un converges to u in W 1,p (B(0, r); C) and Dij un converges to Dij u weakly on Lp (B(0, r); C) as n tends to ∞, it follows that Aun converges to Au weakly in Lp (B(0, r); C). Therefore, Z Z Z Z f ϕ dx = lim fn ϕ dx = lim (λun − Aun )ϕ dx = (λu − Au)ϕ dx. Rd

n→∞

Rd

n→∞

Rd

Rd

The arbitrariness of ϕ ∈ Cc∞ (Rd ; C) yields that λu − Au = f . 2−d/p To conclude that u belongs to Cb (Rd ; C), it suffices to use (10.3.8), which shows that ||un ||W 2,p (B(x0 ,|λ|−1/2 );C) ≤ C5,p ||f ||∞ for some positive constant C5,p , independent of n, x0 and u. From the Sobolev embedding theorem, it follows that ||un ||C 2−d/p (B(x0 ,|λ|−1/2 );C) ≤ C6,p ||f ||∞ . From the Arzel` a-Ascoli theorem, we infer that u ∈ C 2−d/p (B(x0 , |λ|−1/2 ); C) and ||u||C 2−d/p (B(x0 ,|λ|−1/2 );C) ≤ C6,p ||f ||∞ . Since x0 is arbitrarily fixed in Rd , from this inequality 2−d/p

2−d/p

and Lemma 1.1.3, we conclude that u ∈ Cb (Rd ; C) and its Cb (Rd ; C)-norm can be ∞ bounded from above by a positive constant (which depends on λ) times the Lp (Rd ; C)-norm of f . Moreover, letting n tend to ∞ in the pointwise estimate |λ||un (x)| + |λ||∇un (x)| ≤ p ep ||f ||∞ , we infer that |λ|||u||∞ + |λ||||∇u|||∞ ≤ C ep ||f ||∞ , Hence, u is a solution to the C 2−d/p 2,p d equation λu − Au = f which belongs to Cb (R ; C) ∩ Wloc (Rd ; C). By estimate (10.3.1), this is the unique solution to the previous equation which belongs to such a space. We now fix q ∈ (p, ∞) and observe that, since u ∈ L∞ (Rd ; C), the above arguments show that the equation λq v − Av = (λq − λ)u + f admits a unique solution v ∈ C 2−d/q (Rd ; C) ∩ 2,q 2,q 2,p Wloc (Rd ; C). Clearly, u solves the same equation as v and since Wloc (Rd ; C) ⊂ Wloc (Rd ; C), v belongs to the same space as function u. By uniqueness v ≡ u and, consequently, u ∈ 2,q C 2−d/q (Rd ; C) ∩ Wloc (Rd ; C). By the arbitrariness of q, we conclude that the equation \ 2,q λu − Au = f admits a unique solution, which belongs to Cb1+α (Rd ; C) ∩ Wloc (Rd ; C) q λ∞ , where λ∞ is the same as in Theorem 10.3.1, the equation λu − Au = f admits a unique solution u ∈ 2,p Cb1+α (Rd ; C) ∩ Wloc (Rd ; C), for every p ∈ [1, ∞) and α ∈ (0, 1), such that Au ∈ Cb (Rd ; C). e∞ , which depends only on d and, in a continuous Moreover, there exists a positive constant C way on µ, the sup-norm of the coefficients of the operator A and on the modulus of continuity of the diffusion coefficients, such that p d e∞ ||λu − Au||∞ (10.3.9) |λ|||u||∞ + |λ||||∇u|||∞ + |λ| 2p sup ||D2 u||Lp (B(x0 ,|λ|−1/2 );C) ≤ C x0 ∈Rd

for every λ ∈ C, with Re λ > λ∞ . Moreover, for every α ∈ (0, 1) and λ as above, there exists b∞,λ such that a constant C b∞,λ ||λu − Au||∞ . ||u||C 1+α (Rd ;C) ≤ C

(10.3.10)

b

b∞,λ depends only on α and, in a continuous way on λ, µ, the sup-norm of the The constant C coefficients of the operator A and on the modulus of continuity of the diffusion coefficients. Proof The proof is a straightforward consequence of Theorem 10.3.1. Indeed, since Cb (Rd ; C) is continuously embedded into L∞ (Rd ; C), for every λ ∈ C with Reλ ≥ λ∞ , we can determine a solution u to the equation λu − Au = f , which belongs to T 2,p Cb1+α (Rd ; C) ∩ p 0, then we are done. On the other hand, if c0 < 0, then we observe that the operator Rλ : Cbα (Rd+ ; C) → Cb2+α (Rd+ ; C) ∩ C0 (Rd+ ; C) which, to every f ∈ Cbα (Rd+ ; C) associates the unique solution u ∈ Cb2+α (Rd+ ; C) to problem (11.1.4) with ψ ≡ 0, is well defined for every λ ∈ C with positive real part. Rλ is the resolvent operator of the operator A : Cb2+α (Rd+ ; C) ∩ C0 (Rd+ ; C) → Cbα (Rd+ ; C), defined by Au = Au for every u ∈ Cb2+α (Rd+ ; C) ∩ C0 (Rd+ ; C) and, due to estimate (11.1.1) and Remark A.4.5, it can be extended to every λ ∈ C with real part larger than c0 . This means that for each such λ’s the boundary value problem (11.1.4), with ψ ≡ 0, is solvable in C02+α (Rd+ ; C). The arguments in Step 1 show that such a problem is solvable in Cb2+α (Rd+ ; C) also when 0 6= ψ ∈ Cb2+α (Rd−1 ; C).  Remark 11.1.3 Arguing as in the forthcoming Corollary 14.1.4, the dependence of the constant C in (11.1.1) can be made sharp when ψ ≡ 0. In fact, C depends linearly on |λ|.

11.1.1

Further regularity results

In this subsection, we prove that the more the coefficients of the operator A and the terms f and g are smooth, the more the solution of problem ( λu − Au = f, in Rd+ , (11.1.5) u = g, on ∂Rd+ is itself smooth. For this purpose, we assume the following set of hypotheses. Hypotheses 11.1.4 (i) the coefficients qij = qji , bj (i, j = 1, . . . , d) and c belong to Cbk+α (Rd+ ) for some α ∈ (0, 1) and k ∈ N; (ii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for all x ∈ Rd+ and ξ ∈ Rd . Theorem 11.1.5 Under Hypotheses 11.1.4, fix f ∈ Cbk+α (Rd+ ; C) and g ∈ Cbk+2+α (Rd−1 ; C). Then, for every λ ∈ C with real part greater than c0 , the boundary value problem (11.1.5)

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admits a unique solution which belongs to Cbk+2+α (Rd+ ; C). Moreover, there exists a positive constant C, independent of u, f and g, such that ||u||C k+2+α (Rd ;C) ≤ C(||f ||C k+α (Rd ;C) + ||g||C k+2+α (Rd ;C) ). +

b

+

b

+

b

Proof Fix λ and f as in the statement of the theorem. By Theorem 11.1.2 we already know that problem (11.1.5) admits a unique solution u ∈ Cb2+α (Rd+ ; C). Next, we observe that, replacing u with u − g and f by f − λg + Ag, we can reduce to considering the case where u identically vanishes on ∂Rd+ . Hence, in what follows we assume f ∈ Cbk+α (Rd+ ) and g ≡ 0 on Rd−1 . To prove that u actually belongs to Cbk+2+α (Rd+ ; C), we use the interior Schauder k+2+α estimates in Theorem 10.1.11 to infer that u ∈ Cloc (Rd+ ; C). Now, we fix an index h ∈ {1, . . . , d − 1} and observe that, since u identically vanishes on ∂Rd+ , also Dh u vanishes on ∂Rd+ . Moreover, we can differentiate the equation λu − Au = f and conclude that d X

Dh f = λDh u − Dh Au = λDh u − ADh u −

Dh qij Dij u −

i,j=1

d X

Dh bj Dj u − (Dh c)u.

j=1

Therefore, the function Dh u, which also belongs to Cb (Rd+ ; C), solves the boundary value problem ( λv − Av = fh , in Rd+ , v = 0, on ∂Rd+ , where fh = Dh f +

d X

Dh qij Dij u +

d X

Dh bj Dj u + (Dh c)u.

j=1

i,j=1

Since fh ∈ Cbα (Rd+ ; C), we can apply Theorem 11.1.2 and Theorem 4.2.3 to conclude that Dh u ∈ Cb2+α (Rd+ ; C) and ||Dh u||C 2+α (Rd ;C) ≤ C||fh ||Cbα (Rd+ ;C) ≤ C1 (||f ||C 1+α (Rd ;C) + ||u||C 2+α (Rd ;C) ), b

+

b

+

b

+

where C is the constant in (11.1.1) and C1 is a positive constant, which depends also on the Cb1+α -norm of the coefficients over Rd+ . Using again estimate (11.1.1), we can continue the previous estimate and conclude that ||Dh u||C 2+α (Rd ;C) ≤ C2 ||f ||C 1+α (Rd ;C) , where the b

+

b

+

constant C2 depends on the Cb1+α -norm of the coefficients of the operator A, on d, on λ and on the ellipticity constant of the operator A. To prove that also the derivative Dd u belongs to Cb2+α (Rd+ ; C), we differentiate the equation λu − Au = f thrice with respect to the variable xd , obtaining Dddd u =

 1 λDd u − Dd f − qdd −

d X j=1

X (i,j)6=(d,d)

bi Djd u −

d X

qij Dijd u −

d X

Dd qij Dij u

i,j=1

 Dd bi Dj u − cDd u − (Dd c)u .

j=1

Since the right-hand side of the previous formula belongs to Cbα (Rd+ ; C) and it can be bounded from above in terms of the Cb1+α -norm of f over Rd+ , we conclude that also Dddd u belongs to Cbα (Rd+ ; C), so that the assertion follows when k = 1.

Elliptic Equations in Rd+ with Homogeneous Dirichlet Boundary Conditions

296

We can now proceed by induction on k. Suppose that the assertion is true for some k ∈ N and let us prove it with k + 1 in place of k. We know that u ∈ Cbk+2+α (Rd+ ; C) and f ∈ Cbk+1+α (Rd+ ; C), so that the function fh belongs to Cbk+α (Rd+ ) and ||fh ||C k+α (Rd ;C) ≤ C3 ||f ||C k+1+α (Rd ;C) , b

+

b

+

where the constant C3 depends on the ellipticity constant of the operator A, on d and on the Cbk+1+α (Rd+ ; C)-norm of the coefficients. From the inductive assumption, we infer that Dh u belongs to Cbk+2+α (Rd+ ; C). Therefore, all the derivatives of order k + 1 of u but the ∂ku belong to Cb2+α (Rd+ ; C). To complete the proof, we need to consider the derivative ∂xkd derivative Ddk+1 u. Since Ddd u =

 1 λu − f − qdd

X

qij Dij u −

d X

 bj Dj u − cu

j=1

(i,j)6=(d,d)

we can differentiate the derivative Ddd u (k − 1)-times with respect to the variable xd . The so obtained derivative belongs to Cb2+α (Rd+ ; C) and its H¨older norm can be estimated from above in terms of the Cbk+1+α (Rd+ ; C)-norm of f . This completes the proof. 

11.2

Solutions in Sobolev Spaces

We begin this section with the following useful lemma. Lemma 11.2.1 For every u ∈ W 2,p (Rd+ ; C) ∩ W01,p (Rd+ ; C) it holds that Z Re ∆u u|u|p−2 dx ≤ 0. Rd +

Proof Fix u ∈ W 2,p (Rd+ ; C) ∩ W01,p (Rd+ ; C) and consider its odd extension with respect to the last variable, i.e., the function uo : Rd → C, defined by uo (x) = u(x) if xd > 0 and uo (x) = −u(x0 , −xd ) if xd < 0. By Remark B.4.7, this function belongs to W 2,p (Rd ; C) and the functions ∆uo is odd with respect to the variable xd . Applying Lemma 10.2.14, we deduce that Z Z 0 ≥ Re ∆uo uo |uo |p−2 dx = 2Re ∆u u|u|p−2 dx Rd

whence the assertion follows.

Rd +



Now, we have all the necessary tools to prove the existence-uniqueness result for the Laplace equation. Theorem 11.2.2 For every λ ∈ C with positive real part and f ∈ Lp (Rd+ ; C), there exists a unique function u ∈ W 2,p (Rd+ ; C)∩W01,p (Rd+ ; C) which solves the Laplace equation λu−∆u = f . Moreover, there exists a positive constant Cp , independent of λ, f and u, such that p λ||u||Lp (Rd+ ;C) + |λ||||∇u|||Lp (Rd+ ;C) + ||D2 u||Lp (Rd+ ;C) ≤ Cp ||f ||Lp (Rd+ ;C) . (11.2.1)

Semigroups of Bounded Operators and Second-Order PDE’s

297

Proof Fix λ ∈ C, with positive real part, f ∈ Lp (Rd+ ; C) and denote by fo the odd extension (with respect to the last variable) of f to Rd . Clearly, fo belongs to Lp (Rd ; C) and ||fo ||Lp (Rd ;C) ≤ 21/p ||f ||Lp (Rd ;C) . By Theorem 10.2.13, there exists a unique function v ∈ W 2,p (Rd ; C) which solves the equation λv − ∆v = fo . Moreover, p 1 |λ|||v||Lp (Rd ;C) + |λ||||∇v|||Lp (Rd ;C) + ||D2 v||Lp (Rd ;C) ≤Cp ||fo ||Lp (Rd ;C) ≤ 2 p Cp ||f ||Lp (Rd ) for some positive constant Cp , depending on p but being independent of v, f and λ. Denote by u the restriction of v to Rd+ . Clearly, u solves the Laplace equation λu − ∆u = f and it satisfies the estimate (11.2.1). Let us prove that it belongs also to W01,p (Rd+ ; C). In view of Lemma 1.3.5, it is enough to show that v(x0 , xd ) = −v(x0 , −xd ) for almost every x ∈ Rd . For this purpose, it suffices to observe that also the function w, defined by w(x) = −v(x0 , −xd ) for almost every x ∈ Rd solves the Laplace equation λw − ∆w = fo . By uniqueness, w ≡ v and we are done. To show that u is the unique solution in W 2,p (Rd+ ; C) ∩ W01,p (Rd+ ; C) of the Laplace equation λu − ∆u = f , it suffices to multiply both sides of this equation by u|u|p−2 and take Lemma 11.2.1 into account.  Now, we pass to consider the more general equation λu − Au = f , under the following assumptions on the coefficients of the operator A. Hypotheses 11.2.3 (i) The coefficients qij = qji (i, j = 1, . . . , d) belong to BU C(Rd+ ), whereas the coefficients bj (j = 1, . . . , d) and c belong to L∞ (Rd+ ); (ii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for all x ∈ Rd+ and ξ ∈ Rd . The main result of this subsection is the following theorem. We stress that also in this subsection p is arbitrarily fixed in the interval (1, ∞). Theorem 11.2.4 Under Hypotheses 11.2.3, there exists a positive constant λp such that, for every λ ∈ C with Reλ ≥ λp , and every f ∈ Lp (Rd+ ; C), the equation λu − Au = f admits a unique solution u ∈ W 2,p (Rd+ ; C)∩W01,p (Rd+ ; C). Moreover, there exists a positive constant Cp , which depends only on p, d and, in a continuous way, on the ellipticity constant µ, on M = max{||qij ||∞ , ||bj ||∞ , ||c||∞ : i, j = 1, . . . , d} and on the modulus of continuity of the diffusion coefficients, such that p |λ|||u||Lp (Rd+ ;C) + |λ||||∇u|||Lp (Rd+ ;C) + ||D2 u||Lp (Rd+ ;C) ≤ Cp ||f ||Lp (Rd+ ;C) .

(11.2.2)

(11.2.3)

Proof We sketch the proof, illustrating the main steps. For the missing details, we refer the reader to the proofs of Proposition 10.2.17. Step 1. Here, we prove the assertion in the case when A = Tr(QD2 ) and Q is a constant matrix, with positive eigenvalues, and λ ∈ C with positive real part. If Q is a diagonal matrix, then the assertion follows straightforwardly from Theorem 11.2.2. Indeed, if the function u ∈ W 2,p (Rd+ ; C)∩W01,p (Rd+ ; C) solves the equation λu−Au = f , then the function √ v = u( Q ·) belongs to W 2,p (Rd+ ; C) ∩ W01,p (Rd+ ; C) and solves the equation λv − ∆v = g, where g = f (Q−1/2 ·) belongs to Lp (Rd+ ; C). Moreover, ||g||Lp (Rd ;C) ≤ (detQ)p/2 ||f ||Lp (Rd ;C) . Applying estimate (11.2.1) to function v and, then, writing u in terms of v estimate (11.2.3) follows immediately. If Q is not a diagonal matrix, then, thanks to Lemma 8.1.2, we can determine an

298

Elliptic Equations in Rd+ with Homogeneous Dirichlet Boundary Conditions

invertible matrix P such that P QP T is the identity matrix and P T ed = Ked , where −1/2 λmax ≤ K ≤ µ−1/2 and λmax is the maximum eigenvalue of the matrix Q. If u ∈ W 2,p (Rd+ ; C)∩W01,p (Rd+ ; C) solves the equation λu−Au = f , then the function v = u(P −1 ·) belongs to W 2,p (Rd+ ; C) ∩ W01,p (Rd+ ; C) as well and solves the equation λv − ∆v = f (P −1 ·). Arguing as above, estimate (11.2.3) follows also in this case. Step 2. Here, we address the general case where A = Tr(QD2 ) + hb, ∇i + c and the coefficients of the operator A are not necessarily constant on Rd+ . For this purpose, we fix λ ∈ C, with positive real part, x0 ∈ Rd+ , r > 0 and a smooth function ϑx0 ,r ∈ Cc∞ (Rd ) such that χB+ (x0 ,r/2) ≤ ϑx0 ,r ≤ χB+ (x0 ,r) . Further, we denote by L a positive constant, independent of x0 and r, such that r|||∇ϑx0 ,r |||∞ + r2 ||D2 ϑx0 ,r ||∞ ≤ L for every r > 0. Finally, we fix a function u ∈ W 2,p (Rd+ ; C) ∩ W01,p (Rd+ ; C). Clearly, function v = ϑx0 ,r u belongs to W 2,p (Rd+ ; C) ∩ W01,p (Rd+ ; C) and, since λv − Ax0 v = g, where g =(λu − Ax0 u)ϑx0 ,r − 2hQ∇u, ∇ϑx0 ,r i − uAx0 ϑ =f ϑx0 ,r + ϑTr((Q − Q(x0 ))D2 u) + (hb, ∇ui + cu)ϑx0 ,r − 2hQ∇u, ∇ϑx0 ,r i − uTr(Q(x0 )D2 ϑ) and Ax0 = Tr(Q(x0 )D2 ), it follows ||g||Lp (Rd+ ;C) ≤||λu − Au||Lp (B+ (x0 ,r);C) + dω(r)||D2 u||Lp (B+ (x0 ,r);C) + M |||∇u|||Lp (B+ (x0 ,r);C) + M ||u||Lp (B+ (x0 ,r);C) 2M L ML + |||∇u|||Lp (B+ (x0 ,r);C) + 2 ||u||Lp (B+ (x0 ,r);C) , r r where ω(r) denotes, for every r > 0, the maximum with respect to i, j ∈ {1, . . . , d} of sup|x−y|≤r |qij (x) − qij (y)| and B+ (x0 , r) = B(x0 , r) ∩ Rd+ . Thus, applying estimate (11.2.3), with A = Ax0 and taking into account that ϑx0 ,r ≡ 1 on B+ (x0 , r/2) and ϑx0 ,r ≤ 1 on Rd+ , we deduce that p |λ|||u||Lp (B+ (x0 ,r/2);C) + |λ||||∇u|||Lp (B+ (x0 ,r/2);C) + ||D2 u||Lp (B+ (x0 ,r/2);C) ep ||λu − Au||Lp (B (x ,r);C) + K1 (r, M )||u||Lp (B (x ,r);C) ≤C + 0 + 0
(11.2.4) + K2 (r, M )|||∇u|||Lp (B+ (x0 ,r);C) + ω(r)||D2 u||Lp (B+ (x0 ,r);C) . Applying Lemma 10.2.19, we can determine a sequence (xn ) such that ||w||pLp (Rd ;C) ≤ +

∞ X

||w||pLp (B+ (xn ,r/2);C) ,

n=1

∞ X

||w||pLp (B+ (xn ,r);C) ≤ ξ(d)||w||pLp (Rd ;C)

n=1

+

for every w ∈ Lp (Rd+ ; C). From this inequality, applied to u and its first- and second-order derivatives, and (11.2.4) we deduce that p |λ|||u||Lp (Rd+ ;C) + |λ||||∇u|||Lp (Rd+ ;C) + ||D2 u||Lp (Rd+ ;C) ≤C p [||λu − Au||Lp (Rd+ ;C) + K1 (r, M )||u||Lp (Rd+ ;C) + K2 (r, M )|||∇u|||Lp (Rd+ ;C) + ω(r)||D2 u||Lp (Rd+ ;C) ],

(11.2.5)

where Kj (r, M ) (j = 1, 2) are continuous functions of their entries and blow up as r tends to 0+ , and C p is a positive constant, independent of u and r.

Semigroups of Bounded Operators and Second-Order PDE’s

299

Next, we fix r > 0 sufficiently small such that C p ω(r) ≤ 1/2 to move the term ω(r)||D2 u||Lp (Rd+ ;C) to the left-hand side of (11.2.5) and get |λ|||u||Lp (Rd+ ;C) +

p |λ||||∇u|||Lp (Rd+ ;C) + ||D2 u||Lp (Rd+ ;C)

≤2C p [||λu − Au||Lp (Rd+ ;C) + K1 (r, M )||u||Lp (Rd+ ;C) + K2 (r, M )|||∇u|||Lp (Rd+ ;C) ]. p Finally, we choose the positive constant λp such that |λ| ≥ 4C p K1 (r, M ) and |λ| ≥ 4C p K2 (r, M ) for every λ ∈ C, with Re λ ≥ λp , to move the terms containing the Lp -norm of u and its gradient from the left- to the right-hand side of the previous inequality, and obtain (11.2.3). Step 3. Here, we complete the proof using the continuity method to solve the equation λu − Au = f for every λ ∈ C with Re λ ≥ λp . For this purpose, we introduce the oneparameter family of elliptic operators Aσ (σ ∈ (0, 1)) defined by Aσ = σA + (1 − σ)∆ for every σ ∈ [0, 1] and, for every λ ∈ C, with Re λ ≥ λp , and σ ∈ [0, 1], we consider the operator Tσ = λI − Aσ : W 2,p (Rd+ ; C) ∩ W01,p (Rd+ ; C) → Lp (Rd+ ; C). Such one-parameter family of operators satisfies the assumptions of Theorem 6.1.1, so that the operator T1 is invertible or, equivalently, the equation λu − Au = f admits a unique solution u ∈  W 2,p (Rd+ ; C) ∩ W01,p (Rd+ ; C) for every f ∈ Lp (Rd+ ; C). Remark 11.2.5 From (11.2.3), two a priori estimates follow for functions u which belong ep and C bp , to W 2,p (Rd+ ; C) (p ∈ (1, ∞)). More precisely, there exist two positive constants C which depends on p, d, µ and, in a continuous way, on λ, on the L∞ (Rd+ )-norm of the coefficients of the operator A and the modulus of continuity of the diffusion coefficients qij (i, j = 1, . . . , d), such that p |λ|||u||Lp (Rd+ ;C) + |λ||||∇u|||Lp (Rd+ ;C) + ||D2 u||Lp (Rd+ ;C) p ep ||λu − Au||Lp (Rd ;C) + |λ|||g0 ||Lp (Rd ;C) + |λ||||∇g0 |||Lp (Rd ;C) + |||D2 g0 |||Lp (Rd ;C) ) ≤C + + + + (11.2.6) and ||u||W 2,p (Rd+ ;C) ≤ Cp (||u||Lp (Rd+ ;C) + ||Au||Lp (Rd+ ;C) + ||g0 ||W 2,p (Rd+ ;C) )

(11.2.7)

for every u ∈ W 2,p (Rd+ ; C), λ ∈ C with Re λ ≥ λp (λp being the constant in Theorem 11.2.4) and every function g0 ∈ W 2,p (Rd+ ; C) which coincides with u on ∂Rd+ . Indeed, fix g0 as above. Then, the function v = u − g0 belongs to W 2,p (Rd+ ; C) ∩ 1,p W0 (Rd+ ; C) and solves the equation λv − Av = f , where f = λu − Au − λg0 + Ag0 . From estimate (11.2.3), we infer that p |λ|||v||Lp (Rd+ ;C) + |λ||||∇v|||Lp (Rd+ ;C) + ||D2 v||Lp (Rd+ ;C) ≤Cp (||λu − Au||Lp (Rd+ ;C) + |λ|||g0 ||Lp (Rd+ ;C) + ||Ag0 ||Lp (Rd+ ;C) ). Observing that ||Ag0 ||Lp (Rd+ ;C) ≤ C(||g0 ||Lp (Rd+ ;C) +|||∇g0 |||Lp (Rd+ ;C) +||D2 g0 ||Lp (Rd+ ;C) ) for some positive constant C, which depends only on d, µ and, in a continuous way, on λ, on the L∞ (Rd+ )-norm of the coefficients of the operator A and the modulus of continuity of the diffusion coefficients qij , from the previous estimate and the choice of λp (see the last part of Step 2 in the above proof) (11.2.6) follows at once. Finally, estimate (11.2.7) is a straightforward consequence of (11.2.6). Corollary 11.2.6 Under the assumptions of Theorem 11.2.4, for every p ∈ (1, ∞), f ∈

300

Elliptic Equations in Rd+ with Homogeneous Dirichlet Boundary Conditions

Lp (Rd+ ; C), g ∈ B 2−1/p,p (Rd−1 ; C), λ ∈ C, with real part not less than λp , there exists a unique solution u of the elliptic problem ( λu − Au = f, in Rd+ , (11.2.8) u = g, on ∂Rd+ . Moreover, there exists a positive constant Cp , which depends only on p, d and, in a continuous way, on the ellipticity constant µ, on M (see (11.2.2)), on λ and on the modulus of continuity of the diffusion coefficients, such that ||u||W 2,p (Rd+ ;C) ≤ Cp (||f ||Lp (Rd+ ;C) + ||g||B 2−1/p,p (Rd−1 ;C) ).

(11.2.9)

Proof Fix p ∈ (1, ∞) and let w = Ep g, where Ep is the extension operator defined in Proposition B.4.15. The function w belongs to W 2,p (Rd+ ; C) and ||w||W 2,p (Rd+ ;C) ≤ Cp0 ||g||B 2−1/p,p (Rd−1 ;C) for some positive constant Cp0 , independent of g. By Theorem 11.2.4, there exists a unique function v ∈ W 2,p (Rd+ ; C) ∩ W01,p (Rd+ ; C) which solves the equation λv − Av = f − (λw − Aw). The function u = v + w is the (unique) solution to the Cauchy problem (11.2.8) and satisfies (11.2.9), thanks to estimate (11.2.3).  As in the case of elliptic equations on Rd , a better characterization of λp can be provided in the case when A is in divergence form, i.e., when A = div(Q∇) + hb, ∇i + c, under the following assumptions on its coefficients. Hypotheses 11.2.7 (i) The coefficients qij = qji and bj (i, j = 1, . . . , d) belong to Cb1 (Rd+ ) and W 1,∞ (Rd+ ), respectively, whereas the coefficient c belongs to L∞ (Rd+ ); (ii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for all x ∈ Rd+ and ξ ∈ Rd . Let us set

  1 λp = sup c − div b . p Rd +

(11.2.10)

Theorem 11.2.8 For each λ ∈ C with Re λ > λp and each f ∈ Lp (Rd+ ; C), g ∈ B 2−1/p,p (Rd−1 ; C), there exists a unique u ∈ W 2,p (Rd+ ; C) which solves the boundary value problem ( λu − Au = f, in Rd+ , (11.2.11) u = g, on ∂Rd+ . Moreover, there exists a positive constant Cp , which depends only on p, d and, in a continuous way, on λ, on the ellipticity constant µ, on the Cb1 (Rd )-norm of the coefficients qij and the sup-norm of the coefficients bj and c of the operator A (i, j = 1, . . . , d), such that ||u||W 2,p (Rd+ ;C) ≤ Cp (||f ||Lp (Rd+ ;C) + ||g||B 2−1/p,p (Rd−1 ;C) ). If g ≡ 0, then for every ε > 0 there exists a positive constant Cp,ε , which depends also on the same quantities as the constant Cp , such that p |λ|||u||Lp (Rd+ ;C) + |λ||||∇u|||Lp (Rd+ ;C) + ||D2 u||Lp (Rd+ ;C) ≤ Cp,ε ||f ||Lp (Rd+ ;C) (11.2.12) for every λ ∈ C with Reλ > (λp ∨ 0) + ε.

Semigroups of Bounded Operators and Second-Order PDE’s

301

Proof The proof can be obtained adapting the arguments used in the proof of Lemma 10.2.22 and Theorem 10.2.23. Hence, we refer the reader to the quoted lemma and theorem for additional details. Looking at the problem satisfied by function v, defined by v = u − Eα g, where Ep is the extension operator in Proposition B.4.15, we can reduce to considering problem (11.2.11) with g ≡ 0. Hence, in the rest of the proof, we will deal with solutions u ∈ W 2,p (Rd+ ; C) ∩ W01,p (Rd+ ; C) to the elliptic equation λu − Au = f . The crucial points of the proof are the estimate Z u ∈ W 2,p (Rd+ ; C) ∩ W01,p (Rd+ ; C) (11.2.13) Re div(Q∇u)u|u|p−2 dx ≤ 0, Rd +

and the formula Z Re

hb, ∇uiu|u|p−2 dx = −

Rd +

1 p

Z

(divb)|u|p dx

(11.2.14)

Rd +

for the same functions u. To prove (11.2.13) in the trickier case p ∈ (1, 2), we argue as follows. We approximate u in W 2,p (Rd ; C) by a sequence (ϕn ) ⊂ Cc∞ (Rd ; C) and fix δ > 0. p−2 The function ϕn (|ϕn |2 + δ) 2 belongs to Cc∞ (Rd ; C); hence, as in the proof of Lemma 10.2.22, we can integrate by parts and write Z p−2 div(Q∇ϕn )ϕn (|ϕn |2 + δ) 2 dx Re Rd +

Z p−4 = − (p − 1) hQRe(ϕn ∇ϕn ), Re(ϕn ∇ϕn )i(|ϕn |2 + δ) 2 dx Rd Z p−4 − hQIm(ϕn ∇ϕn ), Im(ϕn ∇ϕn )i(|ϕn |2 + δ) 2 dx d R Z p−4 − δRe hQ∇ϕn , ∇ϕn i(|ϕn |2 + δ) 2 dx Rd

−

d X

Z

qdj (·, 0)Dj ϕn (·, 0)ϕn (·, 0)(|ϕn (·, 0)|2 + δ)

Re

p−2 2

dx0

p−2 2

dx0 .

Rd−1

j=1

Z

hQ∇ϕn , ∇ϕn i(|ϕn |2 + δ)

≤ − δRe

p−4 2

dx

Rd

−

d X

Z

qdj (·, 0)Dj ϕn (·, 0)ϕn (·, 0)(|ϕn (·, 0)|2 + δ)

Re Rd−1

j=1

p−2

As it is easily seen, |ϕn (|ϕn |2 + δ) 2 | ≤ |ϕn |p−1 for every x ∈ Rd and δ > 0. Hence, letting δ tend to 0+ in both sides of the previous inequality, we obtain Z Re div(Q∇ϕn )ϕn |ϕn |p−2 dx Rd +

≤−

d X j=1

Z Re

qdj (·, 0)Dj ϕn (·, 0)ϕn (·, 0)|ϕn (·, 0)|p−2 dx0 .

(11.2.15)

Rd−1

Now, we observe that both sides of the previous inequality converge as n tends to ∞. Clearly, div(Q∇ϕn ) converges to div(Q∇u) in Lp (Rd+ ; C). Moreover, ϕn |ϕn |p−2 converges to u|u|p−2 1 0 in Lp (Rd+ ; C). Indeed, the sequence (vn ), defined by vn = 2 p−1 (|u|p + |ϕn |p ) − |u|u|p−2 −

302

Elliptic Equations in Rd+ with Homogeneous Dirichlet Boundary Conditions 0

ϕn |ϕ|p−2 |p for every n ∈ N, consists of nonnegative functions and it pointwise converges 0 to 2p |u|p as n tends to ∞. Hence, applying Fatou lemma to the sequence (vn ) the claim follows. Applying H¨ older inequality, we can write Z Z p−2 p−2 div(Q∇ϕ )ϕ |ϕ | dx − dx div(Q∇u)u|u| n n n Rd +

Rd +

p−1 ≤||div(Q∇ϕn ) − div(Q∇u)||Lp (Rd+ ;C) ||u||L p (Rd ;C) +

+ ||div(Q∇u)||Lp (Rd+ ;C) ||ϕn |ϕn |p−2 − u|u|p−2 ||Lp0 (Rd ;C) +

and the last side of the previous chain of inequalities vanishes as n tends to ∞. Similarly, we observe that, since the trace operator is bounded from W 1,p (Rd+ ; C) into Lp (Rd−1 , C), we infer that the trace of ϕn converges to the trace u on ∂Rd+ as n tends to ∞. Since u ∈ W01,p (Rd+ ; C), its trace on ∂Rd+ is the trivial function. Thus adapting the computation above, we can easily show that qij (·, 0)Dj ϕn (·, 0)ϕn (·, 0)|ϕn (·, 0)|p−2 converges to 0 in L1 (Rd−1 ; C) as n tends to ∞, for every i, j = 1, . . . , d. Estimate (11.2.13) follows at once. The case p ≥ 2 is easier as it has been claimed. Indeed, in such a situation the function 0 u|u|p−2 belongs to W01,p (Rd+ ; C), where 1/p + 1/p0 = 1, if u ∈ W 2,p (Rd+ ; C) ∩ W01,p (Rd+ ; C) (see Proposition 1.3.9 and Remark 1.3.10 for further details). Therefore, integrating by parts over Rd+ we can write Z Z p−2 div(Q∇u)u|u| dx = − hQ∇u, ∇(u|u|p−2 )i dx. Rd +

Rd +

Computing explicitly ∇(u|u|p−2 ) and using estimate (11.2.15), written with ϕn being replaced by u, (11.2.13) easily follows. To prove (11.2.14) it suffices to observe that the function |w|p belongs to W01,1 (Rd+ ; C) if w ∈ W01,p (Rd+ ; C) for each p ∈ (1, ∞). Therefore, observing that Z Z Z 1 1 hb, ∇(|u|2 )i|u|p−2 dx = hb, ∇(|u|p )i dx hb, ∇uiu|u|p−2 dx = Re d d 2 p R R Rd + + + and, integrating by parts, formula (11.2.14) follows at once. Now, we fix u ∈ W 2,p (Rd+ ; C) ∩ W01,p (Rd+ ; C) and λ ∈ C with real part larger than λp . Then, Z Z p−1 p−2 (λu − Au)u|u| ||λu − Au||Lp (Rd+ ;C) ||u||Lp (Rd ;C) ≥ (λu − Au)u|u|p−2 dx dx ≥ Re +

Rd +

Z ≥ Rd +



Rd +

 1 Re λ + divb − c |u|p dx ≥ (Re λ − λp )||u||pLp (Rd ;C) , + p

so that (Re λ − λp )||u||Lp (Rd ;C) ≤ ||λu − Au||Lp (Rd ;C) . Denote by Ap the realization of the operator A in Lp (Rd+ ; C) with D(A) = W 2,p (Rd+ ; C) ∩ W01,p (Rd ; C). The previous estimate implies that the operator norm of resolvent operator R(λ, Ap ) stays bounded when λ ∈ ρ(Ap )∩{λ ∈ C : Reλ > λp }. This implies that the halfspace {λ ∈ C : Reλ > λp } is contained in ρ(Ap ). Moreover, using the above estimate and writing λu − Au = λ∗ u − Au + (λ − λ∗ )u for every u ∈ W 2,p (Rd+ ; C) ∩ W01,p (Rd ; C), λ ∈ C with Re λ > λp , and λ∗ ∈ R so large that (11.2.3) holds true with λ being replaced by λ∗ , it is not an hard task to prove (11.2.12).  We conclude this subsection with some interesting consequences of Theorems 11.2.4 and 11.2.8, which we prove assuming Hypotheses 11.2.7. The same results can be proved also under Hypotheses 11.2.3. We refer the reader to Corollaries 12.2.5 and 12.2.6 for further details.

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Proposition 11.2.9 Under Hypotheses 11.2.7, the following properties are satisfied. 2,p (i) Let u ∈ Wloc (Rd+ ; C) ∩ W01,p (Rd+ ; C) be a function such that Au and u belong to Lq (Rd+ ; C) for some q ∈ (1, ∞). Then, u ∈ W 2,q (Rd+ ; C) ∩ W01,q (Rd+ ; C).

(ii) Suppose that u ∈ W 2,p (Rd+ ; C)∩W01,p (Rd+ ; C) is such that λu−Au belongs to Lq (Rd+ ; C) for some λ ∈ R and q ∈ (p, ∞). Then, u belongs to W 2,q (Rd+ ; C) ∩ W01,q (Rd+ ; C). Proof (i) Let us set f = λq u − Au, where λq is defined by (11.2.10). By assumptions, f ∈ Lq (Rd+ ; C) and by Theorem 11.2.4, there exists a unique function v ∈ W 2,q (Rd+ ; C) ∩ W01,q (Rd+ ; C) such that λq v − Av = f . To prove that v ≡ u and, thus, conclude the proof, we consider the function w = v − u which clearly belongs to Lq (Rd+ ; C) ∩ W01,r (Rd+ ; C) ∩ 2,r Wloc (Rd+ ; C), where r = p ∧ q. Moreover, λq w − Aw = 0. We multiply both sides of this equation by ϑ ∈ Cc∞ (Rd+ ; C) and integrate over Rd+ getting Z Z Z Z 0 = λq wϑ dx − div(Q∇w)ϑ dx − hb, ∇wiϑ dx − cwϑ dx. (11.2.16) Rd +

Rd +

Rd +

Rd +

Integrating by parts, we can write Z div(Q∇w)ϑ dx = − Rd +

d Z X i,j=1

qij Dj wDi ϑ dx

Rd +

and Z hb, ∇wiϑ dx = − Rd +

d Z X j=1

Z Dj (bj ϑ)w dx = −

Rd +

Z (divb)ϑw dx −

Rd +

hb, ∇ϑiw dx. Rd +

0

By density, we can extend the previous formulas to every ϑ ∈ W01,q (Rd+ ; C), where 1/q + 0 1/q 0 = 1. If in addition, ϑ ∈ W 2,q (Rd+ ; C), we can integrate by parts once more and obtain that Z Z div(Q∇w)ϑ dx = div(Q∇ϑ)w dx. Rd +

Rd +

Replacing these two last formulas in (11.2.16), we conclude that Z 0= (λq ϑ − A∗ ϑ)w dx

(11.2.17)

Rd + 0

0

for every ϑ ∈ W 2,q (Rd+ ; C) ∩ W01,q (Rd+ ; C), where A∗ ϑ = div(Q∇ϑ) − hb, ∇ϑi + (c − divb)ϑ. Note that the coefficients of operator A∗ satisfy Hypotheses 11.2.3 and     1 1 λq0 (A∗ ) = sup c − divb + 0 divb = sup c − divb = λq . q q Rd Rd + + Therefore, we can apply Theorem 11.2.4 which shows that the operator λq − A∗ is surjective 0 0 0 from W 2,q (Rd+ ; C) ∩ W01,q (Rd+ ; C) to Lq (Rd+ ; C). Since the function |u|q−2 u belongs to 0 0 Lq (Rd+ ; C), the equation λq ϑ − Aϑ Z = |w|q−2 w admits a solution ϑ ∈ W 2,q (Rd+ ; C). From |w|q dx = 0, so that w ≡ 0 or, equivalently, u ≡ v.

formula (11.2.17), we conclude that Rd +

Elliptic Equations in Rd+ with Homogeneous Dirichlet Boundary Conditions

304

(ii) The proof can be obtained by a bootstrap argument. By assumptions the function f = λu − Au belongs to Lp (Rd+ ; C) ∩ Lq (Rd+ ; C). Since u ∈ W 2,p (Rd+ ; C), from the Sobolev embedding theorems (see Theorem 1.3.6) it follows that the function u belongs to pd ], if p < d, and r1 is any arbitrary Lr1 (Rd+ ; C), where r1 is any number in the interval [p, d−p number in the interval [p, ∞), otherwise. If r1 ≥ q, then, by difference, we conclude that Au ∈ Lq (Rd+ ; C) and, applying property (i), we deduce that u ∈ W 2,q (Rd+ ; C)∩W01,q (Rd+ ; C). If p < d and r1 < q, then f ∈ Lr1 (Rd+ ; C) and, by difference, Au ∈ Lr1 (Rd+ ; C). Hence, by (i), u ∈ W 2,r1 (Rd ; C) ∩ W01,r1 (Rd+ ; C). Applying the Sobolev embedding theorem, we obtain r1 d r1 d that u ∈ Lr2 (Rd+ ; C) for every r2 ∈ [p, d−r ]. If d−r ≥ q, then we are done. Otherwise, we 1 1 iterate this procedure and in finite number of steps we get to the assertion. 

11.2.1

Further regularity results

In this subsection, we prove that the more the function Au is smooth, the more u itself is smooth. We assume the following set of assumptions. Hypotheses 11.2.10 (i) The coefficients qij = qji , bj (i, j = 1, . . . , d) and c belong to W k,∞ (Rd+ ) for some k ∈ N; (ii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for all x ∈ Rd+ and ξ ∈ Rd . Theorem 11.2.11 Let Hypotheses 11.2.10 be satisfied. Let u ∈ W 2,p (Rd+ ; C) ∩ W01,p (Rd+ ; C) for some p ∈ (1, ∞) be a function such that Au ∈ W k,p (Rd ; C). Then, u ∈ W k+2,p (Rd+ , C) and there exists a positive constant C, depending on p, d, µ and, in a continuous way, on the W k,∞ (Rd+ )-norm of the coefficients of the operator A and the modulus of continuity of the diffusion coefficients qij (i, j = 1, . . . , d), such that ||u||W k+2,p (Rd+ ;C) ≤ C(||u||Lp (Rd+ ;C) + ||Au||W k,p (Rd+ ;C) ). Proof The proof is similar to that of Theorem 10.2.29, so that we sketch it, referring the reader to the quoted theorem for further details. Throughout the proof, we denote by C a positive constant which depends at most on λ, p, d, µ and, in a continuous way, on the W k,∞ (Rd+ )-norm of the coefficients of the operator A and the modulus of continuity of the diffusion coefficients qij (i, j = 1, . . . , d). The constant C may vary from line to line. We fix p ∈ (1, ∞) and argue by induction on k ∈ N. In the case k = 1, for every r > 0 and h ∈ {1, . . . , d − 1}, we introduce the function v (r,h) = r−1 (u(· + reh ) − u), which clearly belongs to W 2,p (Rd+ ; C). It also belongs to W01,p (Rd+ ; C). Indeed, if (ϕn ) is a (r,h) sequence in Cc∞ (Rd+ ; C) converging to u in W 1,p (Rd+ ; C), then each function ϕn belongs ∞ d to Cc (R+ ; C), since its support is translate along the direction eh . Moreover, it converges to v (r,h) in W 1,p (Rd+ ; C). Moreover, as it is easily seen, Au

(r,h)

= (Au)

(r,h)

−

d X

(r,h) qij Dij u(·

+ reh ) −

i,j=1

d X

(r,h)

bj

Dj u(· + reh ) − c(r,h) u(· + reh ),

j=1 (r,h)

(r,h)

where the functions (Au)(r,h) , qij , bj (i, j = 1, . . . , d) and c(r,h) are defined in analogy (r,h) with the definition of the function u . Note that Au(r,h) belongs to Lp (Rd ; C) and ||Au(r,h) ||Lp (Rd+ ;C) ≤ ||Au||W 1,p (Rd+ ;C) + C||u||W 2,p (Rd+ ;C) .

(11.2.18)

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Since u ∈ W 2,p (Rd+ ; C) ∩ W01,p (Rd+ ; C), by Theorem 11.2.4 we know that ||u||W 2,p (Rd+ ;C) ≤ C(||u||Lp (Rd+ ;C) + ||Au||Lp (Rd+ ;C) ). Replacing this inequality in (11.2.18) we conclude that ||Au(r,h) ||Lp (Rd+ ;C) ≤ C(||u||Lp (Rd+ ;C) + ||Au||W 1,p (Rd+ ;C) ). Applying again Theorem 11.2.4 this time to the function u(r,h) , we infer that ||(Dij u)(r,h) ||Lp (Rd+ ;C) = ||Dij u(r,h) ||Lp (Rd+ ;C) ≤ C(||u||Lp (Rd+ ;C) + ||Au||Lp (Rd+ ;C) ). In view of Proposition 1.3.11, the distributional derivative Dijh u belongs to Lp (Rd+ ; C), for every i, j = 1, . . . , d, and its norm can be bounded from above by C(||u||Lp (Rd+ ;C) + ||Au||Lp (Rd+ ;C) ). We have so proved that all the third-order derivatives of u except the derivative Dddd u are in Lp (Rd+ ; C). To show that also this latter derivative belongs to Lp (Rd+ ; C) and satisfies the right estimate, we write Ddd u =

 1 Au − qdd

X

qij Dij u −

d X

 bj Dj u − cu

(11.2.19)

j=1

(i,j)6=(d,d)

and observe that the right-hand side of the previous formula can be weakly differentiate with respect to the variable xd , due to the above results. Hence, we conclude that Dddd u ∈ Lp (Rd+ ; C) and the assertion is proved in the case k = 1. Next we suppose that the assertion holds true for every k 0 ≤ k and prove it with k being replaced by k + 1. By the inductive assumptions, u ∈ W k+2,p (Rd+ ; C). We again fix h ∈ {1, . . . , d − 1} and observe that the function Dh u belongs to W01,p (Rd+ ; C) ∩ W k+1,p (Rd+ ; C). Moreover, ADh u = Dh Au −

d X

Dh qij Dij u −

i,j=1

d X

Dh bj Dj u − (Dh c)u

j=1

which belongs to W k,p (Rd+ , C). Hence, ||ADh u||W k,p (Rd+ ;C) ≤||Dh Au||W k,p (Rd+ ;C) + C||u||W k+2,p (Rd+ ;C) ≤C(||u||Lp (Rd+ ;C) + ||Au||W k+1,p (Rd+ ;C) ). The inductive assumptions allow us to infer that the function Dh u belongs to W k+2,p (Rd+ ; C) and ||Dh u||W k+2,p (Rd+ ;C) ≤C(||Dh u||Lp (Rd+ ;C) + ||ADh u||W k,p (Rd+ ) ) ≤C(||u||Lp (Rd+ ;C) + ||Au||W k+1,p (Rd+ ) ). Using (11.2.19) we can estimate also the missing derivative Ddk+3 u, thus completing the proof.  The following consequences of Theorem 11.2.11 will be used in Chapter 12. Corollary 11.2.12 Let Hypotheses 11.2.10 be satisfied and fix p ∈ (1, ∞) and λ ∈ C with

306

Elliptic Equations in Rd+ with Homogeneous Dirichlet Boundary Conditions

real part greater than λp (see Theorem 11.2.4). Then, for every f ∈ W k,p (Rd+ ; C) and g ∈ B k+2−1/p,p (Rd−1 ; C) the boundary value problem ( λu − Au = f, in Rd+ , (11.2.20) u = g, on ∂Rd+ admits a unique solution u ∈ W k+2,p (Rd+ ; C). Moreover, there exists a positive constant C, depending on p, d, µ and, in a continuous way, on λ on the W k,∞ (Rd+ ; C)-norm of the coefficients of the operator A and the modulus of continuity of the diffusion coefficients qij (i, j = 1, . . . , d), such that ||u||W k+2,p (Rd+ ;C) ≤ C(||f ||W k,p (Rd+ ;C) + ||g||B k+2−1/p,p (Rd−1 ;C) ).

(11.2.21)

Proof Fix p, λ, f and g as in the statement of the corollary, and let u ∈ W 2,p (Rd+ ; C) be the unique solution of problem (11.2.20), see Corollary 11.2.6. Moreover, let Ep be the extension operator in Proposition B.4.15. Function Ep g belongs to W k+2,p (Rd+ ; C) and ||Ep g||W k+2,p (Rd+ ;C) ≤ C1 ||g||B k+2−1/p,p (Rd−1 ;C) for some positive constant C1 , independent of g. It thus follows that the function f − AEp g belongs to W k,p (Rd+ ; C) and ||f − (λ − A)Ep g||W k,p (Rd+ ;C) ≤ ||f ||W k,p (Rd+ ;C) + C2 ||g||W k+2−1/p,p (Rd−1 ;C) for some positive constant C2 , which depends on the W k,∞ -norm of the coefficients of the operator A and on d, but it is independent of f and g. Let us consider the function v = u − Ep g, which solves the equation λv − Av = f − (λ − A)Ep g and vanishes on ∂Rd+ . Clearly, u belongs to W k+2,p (Rd+ ; C) if and only if v does. Hence, in the rest of the proof, we will deal with the function v and prove the assertion by a bootstrap argument. The function Av = λv + (λ − A)Ep g − f belongs to W k∧2,p (Rd+ ; C). Therefore, from Theorem 11.2.11 we deduce that v belongs to W 2+κ∧2,p (Rd+ ; C) and ||v||W 2+k∧2,p (Rd+ ;C) ≤C3 (||v||Lp (Rd+ ;C) + ||Av||W k∧2,p (Rd+ ;C) ) ≤C3 (||v||W 2,p (Rd+ ;C) + ||f − (λ − A)Ep g||W k∧2,p (Rd+ ;C) ) ≤C4 ||f − (λ − A)Ep g||W k∧2,p (Rd+ ;C) ≤C5 (||f ||W k∧2,p (Rd+ ;C) + ||g||B 2+k∧2−1/p,p (Rd−1 ;C) ) for some positive constants C3 , C4 and C5 , which depend on p, d, µ and, in a continuous way, on the W k,∞ (Rd+ ; C)-norm of the coefficients of the operator A and the modulus of continuity of the diffusion coefficients, where in the third estimate we have used (11.2.1). Thus, ||u||W 2+κ∧2,p (Rd+ ;C) ≤ C5 ||f ||W κ∧2,p (Rd+ ;C) + (C5 + 1)||g||B 2+k∧2−1/p,p (Rd−1 ;C) ). If k = 1, 2 then we are done. Otherwise, we can apply the above argument with k ∧ 2 being replaced by k ∧ 4. Iterating this argument, we conclude the proof in a finite number of steps.  Remark 11.2.13 If A is in divergence form, i.e., A = div(Q∇) + hb, ∇i + c and its coefficients satisfy Hypotheses 11.2.10, with qij ∈ W k+1,∞ (Rd+ ; C), then the arguments in the proof of Corollary 11.2.12 can be applied to show that its assertion holds true for every λ in C with Re λ > λp , where λp is given by Theorem 11.2.8. In particular, the constant C in (11.2.21) depends on p, d, µ and, in a continuous way, on λ on the W k+1,∞ (Rd+ ; C)-norm of the coefficients qij and on the W k,∞ (Rd+ ; C)-norm of the coefficients bj and c (i, j = 1, . . . , d).

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Corollary 11.2.14 Assume that Hypotheses 11.2.10 are satisfied and fix p, q ∈ (1, ∞). Further, let u ∈ W 2,p (Rd+ ; C) be a function such that Au ∈ W k,q (Rd+ ; C) and Tru ∈ B k+2−1/q,q (Rd−1 ; C) (and u ∈ Lq (Rd+ ; C), if q < p). Then, u ∈ W k+2,q (Rd+ ; C). Proof Let Eq0 be the extension operator in Proposition B.4.15, set g = Tr u and note that g ∈ B 2−1/p,p (Rd−1 ; C) ∩ B k+2−1/q,q (Rd−1 ; C). The proof of quoted proposition shows that Ep0 g = Eq0 g, so that this function belongs to W 2,p (Rd+ ; C) ∩ W k+2,q (Rd+ ; C). Therefore, the function v = u − Eq0 g belongs to W 2,p (Rd+ ; C) and Av = Au − AEq0 g belongs to W k,q (Rd+ ; C). Moreover, v vanishes on ∂Rd+ , so that Theorem B.3.3 guarantees that u ∈ W01,p (Rd+ ; C). We first assume that q > p. By the Sobolev embedding theorems (see Theorem 1.3.6), W 2,p (Rd+ ; C) is continuously embedded into Lr1 (Rd+ ; C) for some r1 > p. If r1 ≥ q, then u ∈ Lq (Rd+ ; C) and we can apply first Proposition 11.2.9(i), to infer that v ∈ W 2,q (Rd+ ; C) ∩ W01,q (Rd+ ; C), and then Theorem 11.2.11 to conclude that v ∈ W k+2,q (Rd+ ; C). As a byproduct function u itself belongs to this space. If r1 < q, then, since Au belongs to Lp (Rd+ ; C) ∩ Lq (Rd+ ; C), we conclude that Au ∈ Lr1 (Rd+ ; C). Hence, u ∈ W 2,r1 (Rd+ ; C) ∩ W01,r1 (Rd+ ; C) again by Proposition 11.2.9. Applying again the Sobolev embedding theorem, we obtain that u ∈ Lr2 (Rd+ ; C) for some r2 > r. If r2 ≥ q, then we can repeat the above arguments and conclude that u ∈ W k+2,q (Rd+ ; C). Otherwise, u ∈ Lr3 (Rd+ ; C) for some r3 > r2 . Note that in a finite number of iteration, we get an index rn ≥ q. Indeed, if this were not the case, then by the Sobolev embedding theorems it would −1 follow that rn+1 = rn−1 − 2d−1 for every n ∈ N, which, of course, cannot be the case. To complete the case, we consider the case q < p (the case q = p is contained in Corollary 11.2.12). Since v and Av belong to Lq (Rd+ ; C), Proposition 11.2.9 shows that v ∈ W 2,q (Rd+ ; C) ∩ W 1,q (Rd+ ; C). Hence, by Corollary 11.2.12, v belongs to W k+2,q (Rd+ ; C) and the assertion follows.  Corollary 11.2.15 Let f ∈ Lp (Rd+ ; C) be a given function, η a vector of Rd and λ a complex number with positive real part. Further, assume that Q = (qij ) is a symmetric constant matrix with positive eigenvalues. Then, there exists a unique function u ∈ W01,p (Rd+ ; C), which solves the variational equation Z uϕ dx −

λ Rd +

d X i,j=1

Z

Z uDij ϕ dx = −

qij Rd +

f Rd +

∂ϕ dx ∂η

(11.2.22)

for every ϕ ∈ Cc∞ (Rd+ ; C). Moreover, there exists a positive constant Cp , which is independent of u, f and depends linearly on η and, in a continuous way, on the modulus of the coefficients of the operator Q and on the inverse of the minimum eigenvalue of Q, such that p |λ|||u||Lp (Rd+ ;C) + |||∇u|||Lp (Rd+ ;C) ≤ Cp ||f ||Lp (Rd+ ;C) . (11.2.23) ∂f belongs to Lp (Rd+ ; C), then function u belongs to ∂η ∂f . W 2,p (Rd+ ; C) ∩ W01,p (Rd+ ; C) and λu − Tr(QD2 u) = ∂η

In particular, if the weak derivative

Proof We first consider the easiest case when Q = I and we begin by proving that, if a function w ∈ W01,p (Rd+ ; C) solves the variational equation Z Z λ wϕ dx − w∆ϕ dx = 0, ϕ ∈ Cc∞ (Rd+ ; C), Rd +

Rd +

then w ≡ 0. For this purpose, we first extend the function w to Rd , odd with respect to

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Elliptic Equations in Rd+ with Homogeneous Dirichlet Boundary Conditions

the last variable. The so extended function wo belongs to W 1,p (Rd ; C), since w vanishes on ∂Rd+ . To prove that wo vanishes, it suffices to show that Z wo (λϕ − ∆ϕ) dx = 0, ϕ ∈ Cc∞ (Rd ; C). (11.2.24) Rd

0

Indeed, since Cc∞ (Rd ; C) is dense in W 2,p (Rd ; C) (see Theorem 1.3.4), the previous formula 0 0 can be extended to every function in W 2,p (Rd ; C). Hence, denoting by ϕ ∈ W 2,p (Rd ; C) p−2 the unique solution of the equation λϕ − ∆ϕ = wo |wo | , we conclude that wo ≡ 0. Fix ϕ ∈ Cc∞ (Rd ; C) and consider the function ψ : Rd → C, defined by ψ(x) = ϕ(x) − ϕ(x0 , −xd ) for every x ∈ Rd . Clearly ψ belongs to Cc∞ (Rd ; C) and it is odd with respect to the last variable. Moreover, taking into account that also wo is odd with respect to the last variable and integrating by parts, we can write Z Z Z Z 2 wo (λϕ − ∆ϕ) dx = wo (λψ − ∆ψ) dx = λ wo ψ dx + h∇wo , ∇ψi dx Rd Rd Rd Rd   Z Z =2 λ wψ dx + h∇w, ∇ψi dx . (11.2.25) Rd +

Rd + 0

Next we observe that ψ vanishes on ∂Rd+ . Hence, it belongs to W01,p (Rd+ ; C) and we can 0 determine a sequence (ψn ) ⊂ Cc∞ (Rd+ ; C) converging to ψ in W 1,p (Rd+ ; C). Thus,  Z  Z Z Z λ wψ dx + h∇w, ∇ψi dx = lim λ wψn dx + h∇w, ∇ψn i dx Rd +

n→∞

Rd +

Rd +

Rd +

Z w(λψn − ∆ψn ) dx = 0.

= lim

n→∞

(11.2.26)

Rd +

From (11.2.25) and (11.2.26), formula (11.2.24) follows at once. To prove the existence of a solution to the variational equation Z d X ∂ϕ ηj f Dj ϕ dx, dx = − λ wϕ dx − w∆ϕ dx = − f ∂η Rd Rd Rd Rd + + + + j=1 Z

Z

Z

and prove estimate (11.2.23), it suffices to consider the case when η = ej for some j ∈ {1, . . . , d}. If j < d, then we extend the function f to Rd , odd with respect to the last variable. We denote by fj the so obtained function. By Corollary 10.2.32 there exists a unique function wj ∈ W 1,p (Rd ; C) such that Z Z Z λ wj ϕ dx − wj ∆ϕ dx = − fj Dj ϕ dx, ϕ ∈ Cc∞ (Rd ; C). Rd

Rd

Rd

p

This function satisfies the estimate |λ|||wj ||Lp (Rd+ ;C) + |||∇wj |||Lp (Rd+ ;C) ≤ Cp ||f ||Lp (Rd+ ;C) , where the constant Cp is independent of f . Note that the function zj : Rd → C defined by zj (x) = −wj (x0 , −xd ), for every x ∈ Rd , solves the same equation, whence it coincides with wi . This shows that wi is odd with respect to the last variable so that it belongs to W01,p (Rd+ ; C) (see Lemma 1.3.5). Moreover, Z Z Z λ wj ϕ dx − wj ∆ϕ dx = − f Dj ϕ dx, ϕ ∈ Cc∞ (Rd+ ; C). Rd +

Rd +

Rd +

If j = d, then we extend the function f to Rd , even with respect to the last variable and

Semigroups of Bounded Operators and Second-Order PDE’s

309

denote by fd the so obtained function. Arguing as above, we conclude that there exists a (unique solution) wd ∈ W01,p (Rd+ ; C) to the variational equation Z Z Z λ wd ϕ dx − wd ∆ϕ dx = − f Dd ϕ dx, ϕ ∈ Cc∞ (Rd+ ; C). Rd +

Rd +

Rd +

Next, we consider the general case. Let T : Rd → Rd be an invertible linear transformation such that T (Rd+ ) = Rd+ , T (∂Rd+ ) = ∂Rd+ and satisfies the conditions T QT ∗ x = x for −1/2 −1/2 every x ∈ Rd and T ∗ ed = Ked for some positive constant K such that λmax ≤ K ≤ λmin , where λmin and λmax are the minimum and the maximum eigenvalues of Q (see Lemma 8.1.2). Denote by v ∈ W01,p (Rd+ ; C) the unique solution of the variational equation Z Z Z ∂ϕ dx, ϕ ∈ Cc∞ (Rd+ ; C), (11.2.27) λ vϕ dx − v∆ϕ dx = − fT d d d ∂η ∗ R+ R+ R+ where fT = f ◦ T −1 belongs to Lp (Rd+ ; C) and η∗ = T η. Function v satisfies the estimate p |λ|||v||Lp (Rd+ ;C) + |||∇v|||Lp (Rd+ ;C) ≤ Cp ||f ||Lp (Rd+ ;C) (11.2.28) for some constant Cp , independent of v and f . Fix ϕ ∈ Cc∞ (Rd ; C) and consider the function ϕT = ϕ ◦ T −1 . Since T −1 (Rd+ ) = Rd+ and −1 T (∂Rd+ ) = ∂Rd+ , then the function ϕT belongs to Cc∞ (Rd+ ; C). Moreover, ∂ϕT ∂ϕT −1 (T ·), = ∂η∗ ∂η

∆ϕT = Tr(QD2 ϕ)(T −1 ·).

Inserting ϕT it into (11.2.27), via a straightforward change of unknowns we get Z Z Z ∂ϕ λ v(T ·)ϕ dx − v(T ·)Tr(QD2 ϕ) dx = − f dx. d d d R+ R+ R+ ∂η Hence, the function u = v ◦ T , which belongs to W01,p (Rd+ ; C), is the solution to (11.2.22), we are looking for. Moreover, it satisfies (11.2.23), thanks to (11.2.28). By the uniqueness of the solution u ∈ W01,p (Rd+ ; C) to the variational equation (11.2.22), Pd it follows that u = j=1 ηj uj , where η1 , . . . , ηd are the components of the vector η and uj is the solution to (11.2.22) with η = ej , for every j = 1, . . . , d. From this remark, we easily deduce that the constant Cp depends linearly on η. ∂f belongs to To conclude the proof, we observe that in the particular case when ∂η ∂f Lp (Rd+ ; C), then the equation λu − Au = admits a unique solution u ∈ W 2,p (Rd+ ; C) ∩ ∂η W01,p (Rd+ ; C) by Theorem 11.2.8. Clearly, this function satisfies the variational equation (11.2.22). 

11.3

Solutions in L∞ (

R ; C) and in C (R ; C) d +

b

d +

In this section, we study the solvability of the boundary value problem ( λu − Au = f, in Rd+ , u = 0, on ∂Rd+ , when f ∈ L∞ (Rd+ ; C) and f ∈ Cb (Rd+ ; C). In the first case, we prove the following result.

310

Elliptic Equations in Rd+ with Homogeneous Dirichlet Boundary Conditions

Theorem 11.3.1 Assume that Hypotheses 11.2.3 are satisfied. Then, there exists λ∞ > 0 such that, for every f ∈ L∞ (Rd+ ; C) and λ in C, with Re λ > λ∞ , the equation λu − Au = f T 2,p admits a unique solution u ∈ Cb1+α (Rd+ ; C) ∩ p λ∞ , where B+ (x0 , |λ|−1/2 ) = B(x0 , |λ|−1/2 ) ∩ Rd+ . There also bλ , which depends only on d, α and, in a continuous way on µ, the supexists a constant C norm of the coefficients of the operator A and on the modulus of continuity of the diffusion coefficients, such that bλ ||λu − Au||∞ . ||u||C 1+α (Rd ;C) ≤ C

(11.3.2)

b

Proof The proof is similar to that of Theorem 10.3.1. For this reason, we skip some details. We split the proof into three steps. In the first one, we prove that p |λ|||u||Lp (B+ (x0 ,r);C) + |λ||||∇u|||Lp (B+ (x0 ,r);C) + ||D2 u||Lp (B+ (x0 ,r);C)  1 ∗ e ≤Cp ||λu − Au||Lp (B+ (x0 ,(γ+1)r);C) + 2 ||u||Lp (B+ (x0 ,(γ+1)r);C) γr  1 + |||∇u|||Lp (B+ (x0 ,(γ+1)r);C) (11.3.3) γr 2,p for every u ∈ Wloc (Rd+ ; C) ∩ W01,p (Rd+ ; C) (p ∈ (1, ∞)), r ∈ (0, 1], x0 ∈ Rd+ , γ ∈ [1, ∞), λ ∈ C, with Re λ ≥ λp , where λp is the constant appearing in Theorem 11.2.4, and some e ∗ . Using this inequality, in Step 2, we prove estimate (11.3.1). Finally, in positive constant C p Step 3, we complete the proof. Throughout the proof, if not otherwise specified, by Cj,p we denote a positive constant which depends at most only on p, α, d, γ and, in a continuous way on µ, the sup-norm of the coefficients of the operator A and on the modulus of continuity of the diffusion coefficients. The constant Cj,p may vary from line to line. Step 1. Let us fix x0 ∈ Rd+ , p > 1 and r, γ, λ as above, and introduce a smooth function ϑ such that ϑ = 1 on B+ (x0 , r), ϑ = 0 outside B+ (x0 , (γ + 1)r) and ||ϑ||∞ + γr|||∇ϑ|||∞ + γ 2 r2 ||D2 ϑ||∞ ≤ K for some positive constant K, independent of γ and r. 2,p Given a function u ∈ Wloc (Rd+ ; C) ∩ Cb1+α (Rd+ ; C), which vanishes on ∂Rd+ , we apply Theorem 11.2.4 to the function v = uϑ and write p |λ|||v||Lp (Rd+ ;C) + |λ||||∇v|||Lp (Rd+ ;C) + ||D2 v||Lp (Rd+ ;C) ≤ Cp ||g||Lp (Rd+ ;C) , (11.3.4)

where g = ϑ(λu − Au) − u(Aϑ − cϑ) − 2hQ∇u, ∇ϑi. Recalling that ϑ is supported in B+ (x0 , (γ + 1)r) it is easy to estimate ||g||Lp (Rd+ ;C) ≤C1,p ||λu − Au||Lp (B+ (x0 ,(γ+1)r);C) + +

C1,p |||∇u|||Lp (B+ (x0 ,(γ+1)r);C) . γr

C1,p ||u||Lp (B+ (x0 ,(γ+1)r);C) γr2 (11.3.5)

Since v = u on B+ (x0 , r), from (11.3.4) and (11.3.5) estimate (11.3.3) follows at once.

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311

2,p Step 2. We now fix p > d and u ∈ Wloc (Rd+ ; C) ∩ Cb1+α (Rd+ ; C) which vanishes on ∂Rd+ . To simplify the notation, we set p Iλ,x0 =|λ|||u||L∞ (B+ (x0 ,|λ|−1/2 );C) + |λ||||∇u|||L∞ (B+ (x0 ,|λ|−1/2 );C) d

+ |λ| 2p ||D2 u||Lp (B+ (x0 ,|λ|−1/2 );C) . We recall, see the proof of Theorem 10.3.1, that there exists a positive constant C2,p such that d

||w||L∞ (B(x0 ,r);C) ≤ C2,p r− p (||w||Lp (B(x0 ,r);C) + r|||∇w|||Lp (B(x0 ,r);C) )

(11.3.6)

for every w ∈ W 1,p (B(x0 , r); C). We want to prove a similar estimate with B(x0 , r) replaced by B+ (x0 , r), when r is larger than the last component of x0 . Given w ∈ W 1,p (B+ (x0 , r); C), we consider the function we : Λ(x0 , r) → C, defined by we (x) = w(x, |xd |) for x ∈ Λ(x0 , r) = {x ∈ Rd : (x0 , |xd |) ∈ B(x0 , r)} (i.e., we is the even extension of w with respect to the variable xd ). Since we = E1 w, where E1 is the extension operator in the proof of Proposition B.4.5, repeating the same arguments in the quoted proof, it is not difficult to check that the function we belongs to W 1,p (Λ(x0 , r); C) and ||we ||W 1,p (Λ(x0 ,r);C) ≤ 2||w||W 1,p (B+ (x0 ,r);C) . Note that B(x0 , r) ⊂ Λ(x0 , r). Hence, we can apply (11.3.6) to the function we and get d

||w||L∞ (B+ (x0 ,r);C) ≤ 2C2,p r− p (||w||Lp (B+ (x0 ,r);C) + r|||∇w|||Lp (B+ (x0 ,r);C) ).

(11.3.7)

Estimates (11.3.6) and (11.3.7), with w being replaced by u and its first-order derivatives, together with (11.3.3), allow us to deduce that p d Iλ,x0 ≤C3,p |λ| 2p (|λ|||u||Lp (B+ (x0 ,|λ|−1/2 );C) + |λ||||∇u|||Lp (B+ (x0 ,|λ|−1/2 );C) + ||D2 u||Lp (B+ (x0 ,|λ|−1/2 );C) ) d

≤C4,p |λ| 2p [||λu − Au||Lp (B+ (x0 ,(γ+1)|λ|−1/2 );C) + γ −1 |λ|||u||Lp (B+ (x0 ,(γ+1)|λ|−1/2 );C) p + γ −1 |λ||||∇u|||Lp (B+ (x0 ,(γ+1)|λ|−1/2 );C) ] for every λ ∈ C, with Reλ ≥ λp ∨ 1. Next, observing that d

1

||v||Lp (B+ (x0 ,(γ+1)|λ|−1/2 );C) ≤ [(γ + 1)d d−1 ωd |λ|− 2 ] p ||v||∞ for every v ∈ L∞ (Rd+ ; C), we can continue the previous chain of inequalities and get d

Iλ,x0 ≤ C5,p (γ + 1) p [||λu − Au||∞ + γ −1 |λ|||u||∞ + γ −1

p

|λ||||∇u|||∞ ].

(11.3.8)

Taking the supremum with respect to x0 ∈ Rd+ of both sides of (11.3.8), we obtain that |λ|||u||∞ +

p d |λ||||∇u|||∞ + |λ| 2p sup ||D2 u||Lp (B+ (x0 ,|λ|−1/2 );C) x0 ∈Rd +

d p

≤3C5,p (γ + 1) [||λu − Au||∞ + γ −1 |λ|||u||∞ + γ −1

p |λ||||∇u|||∞ ].

(11.3.9)

Since d/p < 1, we can fix γ large enough such that 3C5,p γ −1 (γ + 1)d/p < 1/2. With this choice of γ, we can move the terms containing the L∞ -norm of u and its gradient from the right- to the left-hand side of (11.3.9) and (11.3.1) follows. Step 3. To complete the proof, let us prove that, for every f ∈ L∞ (Rd+ ; C) and λ ∈ C, with Re λ > λ∞ := inf p∈(d,∞) λp ∨ 1, the equation λu − Au = f admits a unique solution 2,p u ∈ Cb1 (Rd+ ; C) ∩ Wloc (Rd+ ; C) for every p ∈ [1, ∞), which vanishes on ∂Rd+ . The uniqueness

312

Elliptic Equations in Rd+ with Homogeneous Dirichlet Boundary Conditions

follows from (11.3.1). To prove the existence part, we fix λ and f as above and denote by p any real number greater than d such that Reλ > λp . Further, we approximate f via truncation with a sequence (fn ) ∈ L∞ (Rd+ ; C) of compactly supported functions, such that ||fn ||∞ ≤ ||f ||∞ for every n ∈ N and fn converges to f pointwise almost everywhere on Rd+ . Further, we observe that, by Theorem 11.2.4, for every n ∈ N there exists a unique function un ∈ W 2,p (Rd+ ; C) ∩ W01,p (Rd+ ; C) such that λun − Aun = fn . By the Sobolev embedding 2−d/p theorem 1.3.6, each function un belongs also to Cb (Rd+ ; C) so that p d ep ||f ||∞ . (11.3.10) |λ|||un ||∞ + |λ||||∇un |||∞ + |λ| 2p sup ||D2 un ||Lp (B+ (x0 ,|λ|−1/2 );C) ≤ C x0 ∈Rd +

2,p From this estimate, we infer that there exists a function u ∈ Wloc (Rd+ ; C) such that, up 1,p to a subsequence, un converges to u in W (B+ (0, r); C) for every r > 0, Dij un converges weakly in Lp (B+ (0, r); C) to Dij u for every i, j = 1, . . . , d. Since Aun converges to Au weakly in Lp (B+ (0, r); C), we easily obtain that λu − Au = f . 2−d/p Next, we prove that u ∈ Cb (Rd+ ; C) and vanishes on ∂Rd+ . For this purpose, we observe that (11.3.10) implies that the sequence (un ) is bounded in W 2,p (B+ (x0 , |λ|1/2 ); C) by a constant times the sup-norm of f and the constant is independent of x0 . By the Sobolev embedding theorems, (un ) is bounded in C 2−d/p (B+ (x0 , |λ|1/2 ); C). Hence, by Arzel`a-Ascoli theorem we infer that un converges locally uniformly on Rd+ to a function which belongs to C 2−d/p (B+ (x0 , |λ|1/2 ); C). Clearly, this function is u. The arbitrariness of x0 ∈ Rd+ and 2−p/d Lemma 1.1.3 imply that u ∈ Cb (Rd+ ; C) and ||u||C 2−p/d (Rd+ ;C) ≤ C6,p ||f ||∞ , where the

constant C6,p depends also, in a continuous way, on λ. Since un vanishes on ∂Rd+ for every n ∈ N, it follows that u ≡ 0 on ∂Rd+ . T 2,q Finally, to prove that u ∈ q p allows us to conclude that u ∈ qd λp ∨ 1, where λp is defined by (11.2.10). The constants ep and C bλ depend, in a continuous way, also on the C 1 (Rd+ ; C)-norm of the coefficients qij C b and bj (i, j = 1, . . . , d) of the operator A. 2,p 2,p (Rd+ ; C) ∩ Remark 11.3.3 Suppose that u ∈ Wloc (Rd+ ; C) ∩ Cb1 (Rd+ ; C) and let g ∈ Wloc 1 d d Cb (R+ ; C) be a function such that u = g on ∂R+ . Then, applying (11.3.3) to the function u − g, which belongs to the same function space as u, we deduce that p |λ|||u||Lp (B+ (x0 ,|λ|−1/2 );C) + |λ||||∇u|||Lp (B+ (x0 ,|λ|−1/2 );C) + ||D2 u||Lp (B+ (x0 ,|λ|−1/2 );C)  |λ| e ≤Cp ||λu − Au||Lp (B+ (x0 ,(γ+1)|λ|−1/2 );C) + ||u||Lp (B+ (x0 ,(γ+1)|λ|−1/2 );C) γ p |λ| |||∇u|||Lp (B+ (x0 ,(γ+1)|λ|−1/2 );C) + |λ|||g||Lp (B+ (x0 ,|λ|−1/2 );C) + γ  p + |λ||||∇g|||Lp (B+ (x0 ,|λ|−1/2 );C) + ||D2 g||Lp (B+ (x0 ,|λ|−1/2 );C) (11.3.11)

Semigroups of Bounded Operators and Second-Order PDE’s

313

for every λ ∈ C, with Reλ ≥ λp , where λp is the same constant in the proof of Theorem 11.3.1. Repeating the arguments in the proof of Theorem 11.3.1, from (11.3.11), we can show that p d |λ|||u||∞ + |λ||||∇u|||∞ + |λ| 2p sup ||D2 u||Lp (B+ (x0 ,|λ|−1/2 );C) x0 ∈Rd +

d



ep |λ| 2p ≤C

sup ||λu − Au||Lp (B+ (x0 ,(γ+1)|λ|−1/2 );C) + |λ| sup ||g||Lp (B+ (x0 ,|λ|−1/2 );C)

x0 ∈Rd +

+ |λ|

1 2

x0 ∈Rd +



2

sup |||∇g|||Lp (B+ (x0 ,|λ|−1/2 );C) + sup ||D g||Lp (B+ (x0 ,|λ|−1/2 );C)

x0 ∈Rd +

x0 ∈Rd +

(11.3.12) for every λ ∈ C with Reλ ≥ λp ∨ 1. Now, we observe that there exists a positive constant K, independent of λ, such that sup ||λu − Au||Lp (B+ (x0 ,(γ+1)|λ|−1/2 );C) ≤ K sup ||λu − Au||Lp (B+ (x0 ,|λ|−1/2 );C) . (11.3.13)

x0 ∈Rd +

x0 ∈Rd +

To prove this estimate, it suffices to observe that there exist N ∈ N, independent of |λ| and, for every x0 ∈ Rd+ , N points x1 , . . . , xN such that B(x0 , (γ + 1)|λ|−1/2 ) is contained in the union of the balls B(xj , |λ|−1/2 ) (j = 1, . . . , N ). Indeed, the ball B(0, γ +1) is contained into the union of all the balls B(y, 1) centered at the points y ∈ B(0, γ + 1). By compactness, we can extract a subcovering {B(y1 , 1), . . . , B(yN , 1)} of B(0, γ + 1). It thus follows that B(x0 , (γ + 1)|λ|−1/2 ) ⊂

N [

B(xj , |λ|−1/2 ),

j=1

where xj = x0 + |λ|−1/2 yj for every j = 1, . . . , N . Based on this remark, we can estimate Z

|λu − Au|p dx ≤

B(x0 ,(γ+1)|λ|−1/2 )

N Z X j=1

|λu − Au|p dx,

B(xj ,|λ|−1/2 )

and (11.3.13) follows with K = N 1/p . Replacing (11.3.13) in (11.3.12), we conclude that p d |λ|||u||∞ + |λ||||∇u|||∞ + |λ| 2p sup ||D2 u||Lp (B+ (x0 ,|λ|−1/2 );C) x0 ∈Rd +

d

ep |λ| 2p ≤C



d

sup ||λu − Au||Lp (B+ (x0 ,|λ|−1/2 );C) + |λ|1+ 2p sup ||g||Lp (B+ (x0 ,|λ|−1/2 );C)

x0 ∈Rd +

x0 ∈Rd +

1

d

+ |λ| 2 + 2p sup |||∇g|||Lp (B+ (x0 ,|λ|−1/2 );C) + sup ||D2 g||Lp (B+ (x0 ,|λ|−1/2 );C) x0 ∈Rd +



x0 ∈Rd +

2,p (Rd+ ; C) and every λ ∈ C, with Re λ > λp ∨ 1, where for every u ∈ Cb1 (Rd+ ; C) ∩ Wloc 2,p 1 g ∈ Cb (Rd+ ; C) ∩ Wloc (Rd+ ; C) is an arbitrary function, which coincides with u on ∂Rd+ .

We can now consider the case when the datum f belongs to Cb (Rd+ ; C) assuming the following set of assumptions. Hypotheses 11.3.4 (i) The coefficients qij = qji (i, j = 1, . . . , d) belong to BU C(Rd+ ), whereas the coefficients bj (j = 1, . . . , d) and c belong to Cb (Rd+ );

Elliptic Equations in Rd+ with Homogeneous Dirichlet Boundary Conditions

314

(ii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for all x ∈ Rd+ and ξ ∈ Rd . Theorem 11.3.5 For every f ∈ Cb (Rd+ ; C) and λ ∈ C, with Re λ > λ∞ , the equation 2,p λu − Au = f admits a unique solution u ∈ Cb1+α (Rd+ ; C) ∩ Wloc (Rd+ ; C), for every p ∈ [1, ∞) d e∞ , which and α ∈ (0, 1), which vanishes on ∂R+ . Moreover, there exists a positive constant C depends only on d and, in a continuous way on µ, the sup-norm of the coefficients of the operator A and on the modulus of continuity of the diffusion coefficients, such that p d e∞ ||λu − Au||∞ |λ|||u||∞ + |λ||||∇u|||∞ + |λ| 2p sup ||D2 u||Lp (B+ (x0 ,|λ|−1/2 );C) ≤ C x0 ∈Rd +

(11.3.14) for every λ ∈ C, with Reλ > λ∞ . Here, λ∞ is the same as in Theorem 11.3.1. Moreover, e∞,λ , which depends only only on d, α and, in a continuous there exists a positive constant C way on µ, the sup-norm of the coefficients of the operator A and on the modulus of continuity of the diffusion coefficients, such that ||u||C 1+α (Rd ;C) ≤ C∞,λ ||λu − Au||∞ . b

+

Proof The proof is a straightforward consequence of Theorem 11.3.1.



Remark 11.3.6 If A is in divergence form and Hypotheses 11.3.4 are satisfied, with the coefficients qij and bj (i, j = 1, . . . , d) in Cb1 (Rd+ ; C) and W 1,∞ (Rd+ ; C), respectively, then Theorem 11.3.5 holds true with λ∞ = inf p>d λp ∨ 1, where λp is defined by (11.2.10). The e∞ and C b∞,λ depend, in a continuous way, also on the C 1 (Rd+ ; C)-norm of the constants C b coefficients qij and the W 1,∞ -norm of the coefficients bj (i, j = 1, . . . , d) of the operator A.

11.4

Exercises

1. Add the missing details of the proof of Theorem 11.2.4. 2. Add the missing details of the proof of Theorem 11.2.8. 3. Add the missing details of the proof of Theorem 11.2.11. 4. Add the missing details of the proof of Theorem 11.3.1.

Chapter 12

R

Elliptic Equations in Boundary Conditions

d +

with General

In this chapter, we extend the results of Chapter 11 to the case when the Dirichlet boundary conditions are replaced with more general first-order boundary conditions. More precisely, we consider the operator A = Tr(Q∇) + hb, ∇i + c, assuming that hQ(x)ξ, ξi ≥ µ|ξ|2 for every x ∈ Rd+ , ξ ∈ Rd and some positive constant µ, and the boundary value problem ( λu − Au = f, in Rd+ , (12.0.1) Bu = g, on ∂Rd+ , ∂ is a first-order nontangential boundary operator. As in the previous ∂η chapters, we first analyze problem (12.0.1) in the context of the space of H¨older continuous functions. The starting point is the proof of the a priori estimate

where B = aI +

||u||C 2+α (Rd ;C) ≤ C(||λu − Au||Cbα (Rd+ ;C) + ||Bu||C 1+α (Rd−1 ;C) ) b

+

b

for λ sufficiently large. To prove such an estimate, we take advantage of the Schauder estimates proved in Chapter 8 for solutions to Cauchy problems associated with the pair (A, B). Then, by a reflection argument, we prove that, if A = ∆ and B is the trace on ∂Rd+ of the normal derivative, then, for every f ∈ Cbα (Rd+ ; C) and λ as above, the Cauchy problem (12.0.1) admits a (unique) solution which belongs to Cb2+α (Rd+ ; C). The method of continuity then allows us to cover the general case. Then, we move to the Lp -setting for p ∈ (1, ∞) and, finally, based on these results we consider the L∞ - and Cb -setting. The presence of the boundary operator B causes new technical difficulties to prove the a priori Lp -estimate p |λ|||u||Lp (Rd+ ;C) + |λ||||∇u|||Lp (Rd+ ;C) + ||D2 u||Lp (Rd+ ;C) p ≤Cp (||λu − Au||Lp (Rd+ ;C) + |λ|||g0 ||Lp (Rd+ ;C) + |||∇g0 |||Lp (Rd+ ;C) ) (12.0.2) satisfied by every function u ∈ W 2,p (Rd+ ; C), where g0 is any extension in W 1,p (Rd+ ; C) of the function Bu (which belongs to the Besov space B 1−1/p,p (Rd+ ; C)). This estimate is the key-tool to solve the boundary value problem (12.0.1) in Lp (Rd ; C). Things are easy if A = ∆ and B is the trace on ∂Rd+ of the normal derivative since in this case, up to using ∂u = 0, we can extend u by reflection with an extension operator to reduce to the case ∂ν respect to the axis xd = 0 and obtain a function which belongs to W 2,p (Rd ; C) to which the a priori estimate in Chapter 10 can be applied. Already the case when the normal vector ν is replaced by a nontangential vector η is much trickier and demands some efforts to be addressed. Estimate (12.0.2) and the solvability of the Neumann-boundary value problem associated with operator A (which can be easily proved) allow to make the method of continuity work, which yields the solvability of (12.0.1) for more general operator A and B. 315

Elliptic Equations in Rd+ with General Boundary Conditions

316

12.1

The C α -Theory

In this section, we assume the following set of hypotheses. Hypotheses 12.1.1 (i) The coefficients qij = qji , bj (i, j = 1, . . . , d) and c are bounded and α-H¨ older continuous in Rd+ for some α ∈ (0, 1); (ii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for all x ∈ Rd+ and ξ ∈ Rd ; (iii) the functions a and η belong to Cb1+α (Rd−1 ) and Cb1+α (Rd−1 , Rd ), respectively. Moreover, a(y) ≥ 0, |η(y)| = 1 for each y ∈ Rd−1 and η0 := supRd−1 ηd < 0, where, as usually, ηd denotes the last component of the function η. To simplify the notation, in what follows we will find convenient to denote by B the ∂ . differential boundary operator aI + ∂η To begin with, we introduce the constant b c0 that will play a crucial role in our analysis. It is defined as follows:  sup c, if inf a > 0,   Rd−1  Rd+    (12.1.1) b c0 = 2 1  qdd Ddd ψ + bd Dd ψ − qdd (Dd ψ)2 , if inf a = 0, c−   sup d−1 ψ ψ d R R+

where ψ(x) = 1 − (xd + 2)−1 for every x ∈ Rd+ . Proposition 12.1.2 There exists a positive constant C, which depends in a continuous way on λ, the ellipticity constant of the operator A, the Cbα (Rd+ ; C)-norms of the coefficients of the operator A, on the Cb1+α (Rd−1 )- and Cb1+α (Rd−1 ; Rd )-norms of the coefficients of the operator B and on the supremum over Rd−1 of the function η, such that ||u||C 2+α (Rd ;C) ≤ C(||λu − Au||Cbα (Rd+ ;C) + ||Bu||C 1+α (Rd−1 ;C) ) b

+

(12.1.2)

b

c0 , where b c0 is defined in (12.1.1). for every u ∈ Cb2+α (Rd+ ; C) and λ > b Proof We fix u and λ as in the statement. Moreover, we introduce a function ϑ ∈ C ∞ ([0, ∞)) such that ϑ(t) = 0 for every t ∈ [0, 1/2] and ϑ(t) = 1 for every t ≥ 1. Finally, we consider the function v : [0, ∞) × Rd+ → C, defined by v(t, x) = ϑ(t)eλt u(x) for every 1+α/2,2+α (t, x) ∈ [0, ∞) × Rd+ . This function belongs to Cb ((0, 2) × Rd+ ; C) and solves the Cauchy problem  d   Dt v(t, x) = Av(t, x) + f (t, x), t ∈ [0, 2], x ∈ R+ , Bv(t, x) = h(t, x), t ∈ [0, 2], x ∈ Rd−1 ,   v(0, x) = 0, x ∈ Rd+ , where f (t, x) = ϑ0 (t)eλt u(x) + eλt ϑ(t)(λu(x) − Au(x)) for every (t, x) ∈ [0, 2] × Rd+ and α/2,α h(t, x) = ϑ(t)eλt Bu(x) for every (t, x) ∈ [0, 2] × Rd−1 . Clearly, f ∈ Cb ((0, 2) × Rd+ ; C) and h ∈ C (1+α)/2,1+α ((0, 2) × Rd−1 ; C). Moreover, ||f ||C α/2,α ((0,2)×Rd ;C) ≤ Cλ (||λu − Au||Cbα (Rd+ ;C) + ||u||Cbα (Rd+ ;C) ) b

+

Semigroups of Bounded Operators and Second-Order PDE’s

317

||h||C (1+α)/2,1+α ((0,2)×Rd−1 ;C) ≤ Cλ ||Bu||C 1+α (Rd−1 ;C) b

b

for some positive constant Cλ , independent of u. Thus, Theorem 8.0.2 shows that ||v||C 1+α/2,2+α ((0,2)×Rd ;C) ≤ C(||λu − Au||Cbα (Rd+ ;C) + ||u||Cbα (Rd+ ;C) + ||Bu||C 1+α (Rd−1 ;C) ), b

+

b

(12.1.3) where the constant C depends in a continuous way on λ, the ellipticity constant of the operator A, on η0 , on the Cbα (Rd+ )-norm of the coefficients of the operator A and on the Cb1+α (Rd−1 )-norms of the coefficients of the operator B. Taking Corollary 1.1.5 into account and using Young’s inequality, we can estimate 2

α

2+α ||u||Cbα (Rd+ ;C) ≤ C∗ ||u||∞ ≤ ε||u||C 2+α (Rd ;C) + Cε ||u||∞ ||u||C2+α 2+α (Rd ;C) b

+

b

+

for every ε > 0 and some positive constants C∗ and Cε , this latter blowing up as ε tends to 0. Choosing ε such that Cε ≤ eλ /2, we can move the term ||u||C 2+α (Rd ;C) to the left-hand +

b

side of (12.1.3) and, observing that ||v||C 1+α/2,2+α ((0,2)×Rd ;C) ≥ eλ ||u||C 2+α (Rd ;C) , we conclude b

b

+

+

that ||u||C 2+α (Rd ;C) ≤ 2Ce−λ (||λu − Au||Cbα (Rd+ ;C) + ||u||∞ + ||Bu||C 1+α (Rd−1 ;C) ). b

+

(12.1.4)

b

Finally, using Proposition 4.2.10 we can estimate ||u||∞ ≤

1 1 ||Bu||∞ , ||λu − Au||∞ + λ−b c0 inf Rd−1 a

if inf Rd−1 a > 0 and ||u||∞ ≤

2 4 ||λu − Au||∞ + ||Bu||∞ , λ−b c0 |η0 |

if inf Rd−1 a = 0. Replacing these inequalities into the right-hand side of (12.1.4), estimate (12.1.2) follows immediately.  We can now prove the following existence and uniqueness result for the solution of the boundary value problem ( λu(x) − Au(x) = f (x), x ∈ Rd+ , (12.1.5) Bu(x) = g(x), x ∈ Rd−1 . Theorem 12.1.3 Let b c0 be the constant in (12.1.1). Then, for every λ > b c0 , f ∈ Cbα (Rd+ ; C) and g ∈ Cb1+α (Rd−1 ; C) there exists a unique solution u ∈ Cb2+α (Rd+ ; C) of the boundary value problem (12.1.5). Moreover, there exists a positive constant C, which depends in a continuous way on λ, the ellipticity constant of the operator A, the Cbα (Rd+ )-norms of the coefficients of the operator A, on the Cb1+α (Rd−1 )- and Cb1+α (Rd−1 ; Rd )-norms of the coefficients of the operator B and on the supremum over Rd−1 of the function η, but independent of u, f and g, such that ||u||C 2+α (Rd ;C) ≤ C(||f ||Cbα (Rd+ ;C) + ||g||C 1+α (Rd−1 ;C) ). b

+

(12.1.6)

b

Proof The uniqueness of the solution to problem (12.1.5) as well as estimate (12.1.6) follow straightforwardly from Proposition 12.1.2. To prove the existence part, we use the

318

Elliptic Equations in Rd+ with General Boundary Conditions

method of continuity. For this purpose, for every t ∈ [0, 1] we introduce the operator St : Cb2+α (Rd+ ; C) → Cbα (Rd+ ; C) × Cb1+α (Rd−1 ; C), defined by   ∂u , u ∈ Cb2+α (Rd+ ; C), St u = λu − tAu − (1 − t)∆u, tBu + (1 − t) ∂xd Cbα (Rd+ ; C) × Cb1+α (Rd−1 ; C) being endowed with the norm ||(f, g)||C α (Rd ;C)×C 1+α (Rd−1 ;C) = ||f ||Cbα (Rd+ ;C) + ||g||C 1+α (Rd−1 ;C) b

+

b

b

for every (f, g) ∈ Cbα (Rd+ ; C) × Cb1+α (Rd−1 ; C). Clearly, Cbα (Rd+ ; C) × Cb1+α (Rd−1 ; C) is a Banach space when endowed with this norm. From Proposition 12.1.2 it follows that ||St u||C α (Rd ;C)×C 1+α (Rd−1 ;C) ≥ C∗ ||u||C 2+α (Rd ;C) b

+

b

b

+

for some positive constant C∗ , independent of t and every λ > b c0 (t), where b c0 (t) is defined as in (12.1.1), with qdd being replaced by tqdd +1−t. It is easy to check that c∗ = supt∈[0,1] b c0 (t) is finite. So, we now assume that λ > c∗ . To apply the method of continuity, we prove that the operator S0 is invertible. We fix f ∈ Cbα (Rd+ ; C), g ∈ Cb1+α (Rd−1 ; C) and observe that u ∈ Cb2+α (Rd+ ; C) satisfies the equation S0 u = (f, g) if and only if the function v = u − Eg solves the boundary value problem ( λv(x) − ∆v(x) = fe(x), x ∈ Rd+ , (12.1.7) Dd v(x) = 0, x ∈ Rd−1 , where fe = f − λEg + AEg and E is the bounded operator defined in Proposition B.4.12(i), which satisfies the condition Dd Eg = g on ∂Rd+ . Function fe belongs to Cbα (Rd+ ; C). We now extend the function fe to Rd , even with respect to the variable xd , i.e., ( fe(x), if xd ≥ 0, b f (x) = 0 e f (x , −xd ), if xd < 0. Clearly, fb belongs to Cbα (Rd ; C). Theorem 10.1.1 shows that there exist a unique function w ∈ Cb2+α (Rd ; C) which solves the equation λw − ∆w = fb. Note that the function x 7→ w(x0 , −xd ) belongs to Cb2+α (Rd ; C) and solves the same equation. Therefore, w(x0 , xd ) = w(x0 , −xd ) for every x ∈ Rd+ . From this formula it follows immediately that Dd w(·, 0) = 0 on Rd−1 . The restriction v of the function w to Rd+ is the solution to problem (12.1.7) we were looking for. We have so proved that the operator S0 is invertible and, consequently, also the operator S1 is invertible, i.e., the boundary value problem (12.1.5) is solvable for every λ > c∗ and every f ∈ Cbα (Rd+ ; C), g ∈ Cb1+α (Rd−1 ; C). If c∗ = b c0 , then we are done. So, let us suppose that b c0 < c∗ and prove that problem (12.1.5) is solvable for every λ ∈ (b c0 , c∗ ]. First of all, we observe that it suffices to solve such a problem when g identically vanishes on Rd−1 . Indeed, if g does not identically vanish, then we consider the function u0 ∈ Cb2+α (Rd+ ; C) solution to the equation 2c∗ u0 − Au0 = 0 on Rd+ and satisfying the condition Bu0 = g on Rd−1 . Clearly, u ∈ Cb2+α (Rd+ ; C) solves problem (12.1.5) if and only if the function v = u − u0 solves the boundary value problem ( λv(x) − Av(x) = fb(x), x ∈ Rd+ , (12.1.8) Bv(x) = 0, x ∈ Rd−1 ,

Semigroups of Bounded Operators and Second-Order PDE’s

319

where fb = f + (2c∗ − λ)u0 . We denote by Aα the realization of the operator A in Cbα (Rd+ ; C) with D(Aα ) = {u ∈ 2+α Cb (Rd+ ; C) : Bu = 0 on ∂Rd+ }. Then the resolvent operator R(λ, Aα ) is defined for every λ > c∗ . We denote by ρ(Aα ) the resolvent set of operator Aα . By Proposition 12.1.2, the constant C in (12.1.2) stays bounded when λ varies in compact subsets of (b c0 , ∞). In particular, ||u||Cbα (Rd+ ;C) ≤ C||λu − Au||Cbα (Rd+ ;C) ,

λ>b c0 ,

for every u ∈ Cb2+α (Rd+ ; C) or, equivalently, ||R(λ, Aα )||L(Cbα (Rd+ ;C)) ≤ C for every λ ∈ ρ(Aα ) ∩ (b c0 , ∞). In view of Remark A.4.5, we conclude that the interval (b c0 , ∞) is contained in ρ(Aα ), i.e., the boundary value problem (12.1.8) admits a solution in Cb2+α (Rd+ ; C) for every λ > b c0 and every f ∈ Cbα (Rd+ ; C). This completes the proof. 

12.1.1

Further regularity

In this subsection, we prove that the solution u of the boundary value problem (12.1.5) is more regular, if the data f and g are themselves more regular. Hypotheses 12.1.4 (i) the coefficients qij = qji , bj (i, j = 1, . . . , d) and c belong to Cbk+α (Rd+ ) for some α ∈ (0, 1) and k ∈ N; (ii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for all x ∈ Rd+ and ξ ∈ Rd . (iii) the functions a and η belong to Cbk+1+α (Rd−1 ) and Cbk+1+α (Rd−1 , Rd ), respectively. Moreover, a(y) ≥ 0, |η(y)| = 1 for each y ∈ Rd−1 and η0 := supRd−1 ηd < 0, where, as usually, ηd denotes the last component of the function η. Theorem 12.1.5 Under Hypotheses 12.1.4, let f and g be two functions which bec0 (see long to Cbk+α (Rd+ ; C) and Cbk+1+α (Rd−1 ; C), respectively. Then, for every λ > b (12.1.1)) the boundary value problem (12.1.5) admits a unique solution u which belongs to Cbk+2+α (Rd+ ; C). Moreover, there exists a positive constant Ck , independent of u, f and g, such that ||u||C k+2+α (Rd ;C) ≤ Ck (||f ||C k+α (Rd ;C) + ||g||C k+1+α (Rd−1 ;C) ). +

b

+

b

b

Proof Fix λ, f and g as in the statement of the theorem. Theorem 12.1.3 implies that problem (12.1.5) admits a unique solution u ∈ Cb2+α (Rd+ ; C). The interior Schauder k+2+α estimates in Theorem 10.1.11 apply and show that u ∈ Cloc (Rd+ ; C). To prove that u k+2+α d actually belongs to C (R+ ; C), we argue as follows. We observe that Dh g = Dh Bu = BDh u + (Dh a)u + h∇u, Dh ηi, so that BDh u = Dh g − (Dh a)u − h∇u, Dh ηi =: gh on ∂Rd+ . Function gh belongs to Cb1+α (Rd−1 ; C) and ||gh ||C 1+α (Rd−1 ;C) ≤||Dh g||C 1+α (Rd−1 ;C) + ||Dh a||C 1+α (Rd−1 ;C) ||u||C 1+α (∂Rd ;C) b

b

b

b

+

+ ||∇u||C 1+α (∂Rd ;Cd ) ||Dh η||C 1+α (Rd−1 ;Cd ) b

+

b

≤||g||C 2+α (Rd−1 ;C) + C1 ||u||C 2+α (Rd ;C) , b

b

+

where the constant C1 depends on the C 2+α -norms of the coefficients a and η over Rd−1 . Using estimate (12.1.6), we conclude that ||gh ||C 1+α (Rd−1 ;C) ≤ C2 (||f ||C 2+α (Rd ;C) + ||g||C 2+α (Rd−1 ;C) ), b

b

+

b

(12.1.9)

320

Elliptic Equations in Rd+ with General Boundary Conditions

where the constant C2 is independent of f and g. Next, we differentiate the equation λu − Au = f and obtain that the function Dh u ∈ 1 Cb (Rd+ ; C) ∩ C 2 (Rd+ ; C) solves the boundary value problem ( λv − Av = fh , in Rd+ , (12.1.10) Bv = gh , on ∂Rd+ , where fh := Dh f +

d X

Dh qij Dij u +

i,j=1

d X

Dh bj Dj u + (Dh c)u.

j=1

Function fh belongs to Cbα (Rd+ ; C) and ||fh ||Cbα (Rd+ ;C) ≤ ||f ||C 1+α (Rd ;C) + C3 ||u||C 2+α (Rd ;C) , +

b

+

b

where the constant C3 depends of the Cb1+α (Rd+ ; C)-norms of the coefficients of the operator A. Taking (12.1.6) into account, we can estimate ||fh ||Cbα (Rd+ ;C) ≤ C4 (||f ||C 1+α (Rd ;C) + ||g||C 1+α (Rd−1 ;C) ), b

+

b

where the constant C4 is independent of f and g. We can thus apply Theorem 12.1.3 to infer that there exists a unique solution v ∈ Cb2+α (Rd+ ; C) of problem (12.1.10). Moreover, ||v||C 2+α (Rd ;C) ≤ C5 (||f ||C 1+α (Rd ;C) + ||g||C 1+α (Rd−1 ;C) ), +

b

b

+

b

the constant C5 being independent of v, f and g. By Proposition 4.2.10, there exists a unique solution to problem (12.1.10) which belongs to Cb1 (Rd+ ; C) ∩ C 2 (Rd+ ; C), so that v = Dh u for h ∈ {1, . . . , d − 1}. To prove that also the derivative Dd belongs to Cb2+α (Rd+ ; C), we observe that  1 Ddd u = λu − f − qdd

X

qij Dij u −

d X

 bi Dj u − cu .

(12.1.11)

j=1

(i,j)6=(d,d)

Since the right-hand side of the previous formula belongs to Cb1+α (Rd+ ; C) and it can be bounded from above in terms of the Cb1+α -norm of f over Rd+ and the Cb2+α -norm of g over Rd−1 , we conclude that also Dddd u belongs to Cbα (Rd+ ; C), so that the assertion follows when k = 1. We can now proceed by induction on k: we suppose that the assertion is true for some k ∈ N and prove it with k + 1 in place of k. We know that u ∈ Cbk+2+α (Rd+ ; C) and f ∈ Cbk+1+α (Rd+ ; C), so that the function fh belongs to Cbk+α (Rd+ ; C) and ||fh ||C k+α (Rd ;C) ≤ C6 (||f ||C k+1+α (Rd ;C) + ||g||C k+2+α (Rd−1 ;C) ), b

+

+

b

b

where the constant C6 depends on the ellipticity constant of the operator A, on d and on the Cbk+1+α (Rd+ )-norm of the coefficients of the operator A and on the Cbk+2+α (Rd−1 )and Cbk+2+α (Rd−1 ; Rd )-norms of the coefficients of the operator B. Similarly, the function g belongs to Cbk+2+α (Rd−1 ; C) and ||gh ||C k+1+α (Rd ;C) ≤ C7 (||f ||C k+1+α (Rd ;C) + ||g||C k+2+α (Rd−1 ;C) ), b

+

b

+

b

Semigroups of Bounded Operators and Second-Order PDE’s

321

where C7 depends on the same quantities as the constant C6 . From the inductive assumption, we infer that Dh u belongs to Cbk+2+α (Rd+ ; C). Therefore, all the derivatives of order k + 1, but the derivative Ddk+1 u, belong to Cb2+α (Rd+ ; C). To complete the proof, we use (12.1.11) to infer that also that the function Ddd u can be differentiated (k − 1)-times with respect to the variable xd in Rd+ . The so obtained derivative belongs to Cb2+α (Rd+ ; C) and its H¨ older norm can be estimated from above in terms of the Cbk+1+α (Rd+ ; C)-norm of f and the Cbk+2+α (Rd−1 ; C)-norm of g. This completes the proof. 

12.2

R ; C)

Elliptic Equations in Lp (

d +

Throughout this section, we assume the following set of hypotheses. Hypotheses 12.2.1 (i) The coefficients qij = qji (i, j = 1, . . . , d) belong to BU C(Rd+ ), whereas bj (j = 1, . . . , d) and c belong to L∞ (Rd+ ); (ii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for all x ∈ Rd+ and ξ ∈ Rd ; (iii) the functions a and η belong to Cb1 (Rd−1 ) and Cb1 (Rd−1 ; Rd ), respectively. Moreover, |η(y)| = 1 for each y ∈ Rd−1 and η0 := supRd−1 ηd < 0, where, as usually, ηd denotes the last component of the function η. To begin with, we prove a priori Lp -estimates for the solution of the boundary value problem ( λu(x) − Au(x) = f (x), x ∈ Rd+ , (12.2.1) Bu(x) = g(x), x ∈ Rd−1 . Proposition 12.2.2 There exist λp > 0 and a positive constant Cp such that p |λ|||u||Lp (Rd+ ;C) + |λ||||∇u|||Lp (Rd+ ;C) + ||D2 u||Lp (Rd+ ;C) p ≤Cp (||λu − Au||Lp (Rd+ ;C) + |λ|||g0 ||Lp (Rd+ ;C) + |||∇g0 |||Lp (Rd+ ;C) )

(12.2.2)

for every u ∈ W 2,p (Rd+ ; C) and λ ∈ C with Re λ ≥ λp , where g0 ∈ W 1,p (Rd+ ; C) is any function which agrees with Bu on ∂Rd+ . The constants Cp and λp depend on p, d and, in a continuous way, on the ellipticity constant of the operator A, on the modulus of continuity of its diffusion coefficients, on the sup-norms of the coefficients of the operator A, on the Cb1 (Rd−1 )- and Cb1 (Rd−1 ; Rd )-norms of the coefficients of the operator B and on the supremum over Rd−1 of the function η. Proof The proof is very long and articulated in several steps. To make it easier to read, we summarize the strategy that we follow. The main step is the proof of estimate (12.2.2) when A = Tr(QD2 ), Q is a constant matrix and B has constant coefficients. This is the content of Steps 2 to 4. Based on these results and freezing the coefficients, we can address in Step 5 the case when A = Tr(QD2 ), Q being a matrix-valued function, and B has C 1 smooth coefficients. Even if rather long, this step is not particularly tricky. Passing to the case when A is a general second-order elliptic operator is then an easy task. Coming back to the content of Steps 2 to 4, we first consider the easiest case when A = ∆ and B is (the opposite of) the normal derivative on ∂Rd+ . We use Step 1, where we prove

322

Elliptic Equations in Rd+ with General Boundary Conditions

that, for every vector η = (η 0 , ηd ), with ηd 6= 0, every λ ∈ C, with positive real part and ∂ζ every g0 ∈ W 1,p (Rd+ ; C) there exists a function ζ ∈ W 2,p (Rd+ ; C) such that = g0 on ∂Rd+ ∂η and its W 1,p (Rd+ ; C)-norm decays in a good way, as |λ| tends to ∞. Function ζ allows us to deal with functions u ∈ W 2,p (Rd+ ; C) whose normal derivative vanishes on ∂Rd+ . Next step is the case when the normal derivative is replaced by a nontangential derivative and A = ∆. ∂u vanishes on ∂Rd+ . Here, It suffices to consider the case when η = ηd−1 ed−1 + ηd ed and ∂η the main idea is to apply the a priori estimate in Theorem 10.2.13 to a “variant” of the function u(·, xd ) which belongs to W 2,p (Rd−1 ; C) for almost every xd ∈ (0, ∞). Raising this inequality to the power p and then integrating on (0, ∞) with respect to xd , we get (12.2.2) also in this case. Finally, when the Laplacian is replaced by the operator A = Tr(QD2 ), we use a linear change of variables to transform A into the Laplacian. This transformation ∂ changing η to a new vector η ∗ , but this does modifies the boundary operator Bu = au + ∂η not cause particular problems and, applying the results in Step 3, (12.2.2) follows. bp a positive constant, which depends at most on Throughout the proof, we denote by C p, d and, in a continuous way, on the ellipticity constant of the operator A, on the modulus of continuity of its diffusion coefficients, on the sup-norms of the coefficients of the operator A, on the Cb1 (Rd−1 )- and Cb1 (Rd−1 ; Rd )-norms of the coefficients of the operator B and on bp may vary from line the supremum over Rd−1 of the function η. In particular, constant C to line. Step 1. Here, we prove that for every function g ∈ W 1,p (Rd+ ; C), every vector η ∈ Rd , with ηd 6= 0, and every λ ∈ C with positive real part, there exists a function ζ ∈ W 2,p (Rd+ ; C) ∂ζ = g on ∂Rd+ and such that ∂η   p |η 0 | b |λ|||ζ||Lp (Rd+ ;C) + |||∇ζ|||Lp (Rd+ ;C) ≤ 1 + 2 C , (12.2.3) p ||g||Lp (Rd + ;C) ηd   |η 0 |2 b 2 (12.2.4) ||D ζ||Lp (Rd+ ;C) ≤ 1 + 4 Cp |||∇g|||Lp (Rd+ ;C) , ηd where η 0 = (η1 , . . . , ηd−1 ). For this purpose, we observe that we can reduce ourselves to considering the easiest case when η = ed . Indeed, let P : Rd → Rd be the linear, operator defined by P x = (x0 + ηd−1 η 0 xd , xd ) for every x ∈ Rd , which acts as the identity operator on ∂Rd+ and transforms Rd+ into itself. Moreover, for every w ∈ W 2,p (Rd+ ; C), the function ∂u v = w ◦ P belongs to W 2,p (Rd+ ; C). Moreover, Dd v = ηd−1 ◦ P. ∂η Set ge = ηd−1 g ◦ P . By Corollary 11.2.12, there exists a unique w ∈ W 3,p (Rd+ ; C) ∩ 1,p W0 (Rd+ ; C) such that λw − ∆w = −e g . The function ζe = Dd w belongs to W 2,p (Rd+ ; C) and solves the equation λζ − ∆ζ = −Dd ge. Moreover, Dd ζe = λw −

d−1 X

Djj w + ge

(12.2.5)

j=1

in Rd+ . Since w vanishes on ∂Rd+ , also the tangential derivatives Djj w (j = 1, . . . , d − 1) have null traces on ∂Rd+ . Thus, taking the trace of both sides of (12.2.5), we deduce that the trace of Dd ζe on ∂Rd+ equals the trace of ge on ∂Rd+ . Moreover, by Theorem 11.2.2 and Corollary 11.2.15, we can estimate p e p d e bp ||e ≤C g ||Lp (Rd+ ;C) , |λ|||ζ|| L (R+ ;C) + |||∇ζ|||Lp (Rd + ;C)

Semigroups of Bounded Operators and Second-Order PDE’s

323

e p d b ||D2 ζ|| e||Lp (Rd+ ;C) . L (R+ ;C) ≤ Cp ||Dd g ∂ζ It is easy to check that the function ζ = ζe ◦ P −1 belongs to W 2,p (Rd+ ; C), = g on ∂Rd+ ∂η and estimates (12.2.3) and (12.2.4) hold true. Indeed, the Lp -norm of ζ coincides with the e whereas norm of the function ζ, e ◦ P −1 | + |(Dd ζ) e ◦ P −1 |)2 |∇ζ|2 =|(∇x0 ζ) ◦ P −1 |2 + (ηd−2 |η 0 ||(∇x0 ζ) e ◦ P −1 |2 , ≤2(η −4 |η 0 |2 + 1)|(∇ζ) d

where ∇x0 ζe is the vector of partial derivatives of ζe with respect to the first d−1 variables, √the −2 e p d . Similarly, so that |||∇ζ|||Lp (Rd+ ;C) ≤ 2(ηd |η 0 | + 1)|||∇ζ||| L (R+ ;C) e ◦ P −1 |2 |D2 ζ|2 =(1 + 2ηd−4 |η 0 |2 + 4ηd−8 |η 0 |4 )|(Dx20 ζ) + 2(1 + 8ηd−4 |η 0 |2 )

d−1 X

e ◦ P −1 |2 + 4|(Ddd ζ) e ◦ P −1 |2 |(Did ζ)

i=1

≤4(1 +

4ηd−4 |η 0 |2

+

e 2, ηd−8 |η 0 |4 )|D2 ζ|

e p d . Estimates (12.2.3) and (12.2.4) follow. so that ||D2 ζ||Lp (Rd+ ;C) ≤ 2(1+ηd−2 |η 0 |)2 ||D2 ζ|| L (R+ ;C) Step 2. Fix u ∈ W 2,p (Rd+ ; C), g0 ∈ W 1,p (Rd+ ; C), such that g0 = Dd u on ∂Rd+ , and λ ∈ C with positive real part. Further, let ζ ∈ W 2,p (Rd ; C) be the function in Step 1 such that Dd ζ = g0 on ∂Rd+ . The function v = u − ζ belongs to W 2,p (Rd+ ; C) and its normal derivative (η = ed ) on ∂Rd+ vanishes. Therefore, the even extension, with respect to xd , of the function v to Rd (say ve ) belongs to W 2,p (Rd ; C) (see Remark B.4.8) and ∆ve is the even extension with respect to the last variable of the function ∆v. Applying Theorem 10.2.13, we infer that, for every λ ∈ C with positive real part, p bp ||λve − ∆ve ||Lp (Rd ;C) . (12.2.6) |λ|||ve ||Lp (Rd ;C) + |λ||||∇ve |||Lp (Rd ;C) + ||D2 ve ||Lp (Rd ;C) ≤ C Since ||λve − ∆ve ||Lp (Rd ;C) ≤ 21/p ||λv − ∆v||Lp (Rd+ ;C) , from (12.2.6) we deduce that |λ|||v||Lp (Rd+ ;C) +

p

bp ||λv − ∆v||Lp (Rd ;C) . |λ||||∇v|||Lp (Rd+ ;C) + ||D2 v||Lp (Rd+ ;C) ≤ 21/p C +

Recalling that u = v + ζ and taking (12.2.3) and (12.2.4) into account, we can estimate p |λ|||u||Lp (Rd+ ;C) + |λ||||∇u|||Lp (Rd+ ;C) + ||D2 u||Lp (Rd+ ;C) p ≤|λ|||v||Lp (Rd+ ;C) + |λ||||∇v|||Lp (Rd+ ;C) + ||D2 v||Lp (Rd+ ;C) + |λ|||ζ||Lp (Rd+ ;C) p + |λ||||∇ζ|||Lp (Rd+ ;C) + ||D2 ζ||Lp (Rd+ ;C) p bp (||λu − ∆u||Lp (Rd ;C) + ||λζ − ∆ζ||Lp (Rd ;C) + |λ|||g0 ||Lp (Rd ;C) + |||∇g0 |||Lp (Rd ;C) ) ≤C + + + + p b ≤Cp (||λu − ∆u||Lp (Rd+ ;C) + |λ|||g0 ||Lp (Rd+ ;C) + |||∇g0 |||Lp (Rd+ ;C) ), which yields the assertion in this case. Step 3. Here, we consider the case when η is a general vector with ηd 6= 0. We fix ∂u u ∈ W 2,p (Rd+ ; C) and g0 ∈ W 1,p (Rd+ ; C) such that = g0 on ∂Rd+ . ∂η We begin by assuming that d ≥ 3 and introduce a rotation Θ0 on Rd−1 which transforms the vector η 0 into a new vector which has all the first d − 2 components equal to zero. We

Elliptic Equations in Rd+ with General Boundary Conditions

324

denote by Θ : Rd → Rd the rotation defined by Θx = (Θ0 x0 , xd ) for every x ∈ Rd . Clearly, Θ transforms Rd+ and ∂Rd into themselves. We set v = u ◦ Θ and η ∗ = (Θ0 η 0 , ηd ). As it is ∂u ∂v = ◦ Θ and ∆v = (∆u) ◦ Θ. easily seen, ∂η ∗ ∂η 2,p d Let ζ ∈ W (R+ ; C) be the function provided by Step 1 which satisfies the condition ∂ζ = g0 ◦ Θ and estimates (12.2.3) and (12.2.4). The function ve = v − ζ belongs to ∂η ∗ 2,p W (Rd+ ; C). Moreover, since λζ − ∆ζ = ηd−2

d−1 X

2 Dij ζηi0 ηj + 2

i,j=1

d−1 X

Did ζηi0 ηd−1 − (Dd ge0 ) ◦ P −1 ,

i=1

it follows that λe v − ∆e v = fe := (λu − ∆u − Dd g0 ) ◦ Θ + G(η, D2 ζ), where G depends linearly on the second-order derivatives of ζ and, in a continuous way, on the entries of the vector ∂e v ∂e v vanishes on ∂Rd+ . Thus, w = belongs to W01,p (Rd+ ; C) and satisfies η. Moreover, ∂η ∗ ∂η ∗ the variational equation Z Z Z ∂ϕ λ wϕ dx + h∇w, ∇ϕi dx = − fe ∗ dx, ϕ ∈ Cc∞ (Rd+ ; C). d d d ∂η R+ R+ R+ By Corollary 11.2.15, p bp (||λu − ∆u||Lp (Rd ;C) + |||∇g0 |||Lp (Rd ;C) ). |λ|||w||Lp (Rd+ ;C) + |||∇w|||Lp (Rd+ ;C) ≤ C + +

(12.2.7)

A straightforward computation shows that Ddd ve = =

  ∗ ηd−1 1 1 ∗ ∗ 2 (D w − η D D v e ) = D w − (D w − η D v e ) d d d−1 d−1 d−1 d d−1 d−1 ηd∗ ηd ηd (η ∗ )2 2 η∗ 1 Dd−1 w + d−1 Dd−1 ve, Dd w − d−1 2 ηd ηd ηd2

so that λe v−

d−2 X

Dj2 ve −



j=1

∗ 2  η η∗ 1 2 1 + d−1 Dd−1 ve = fe + Dd w − d−1 Dd−1 w =: fb. ηd ηd ηd2

(12.2.8)

We denote by A the subset of (0, ∞) consisting of all the points xd > 0 such that the function ve(·, xd ) belongs to W 2,p (Rd−1 ; C). The set (0, ∞) \ A is negligible; moreover, we can assume that for each xd ∈ A and each multi-index α ∈ (N ∪ 0)d−1 , Dxβ0 ve(·, xd ) = (Dxβ0 ve)(·, xd ), and Z Z Z |e v |p dx = dxd |v(x0 , xd )|p dx0 , (12.2.9) Rd +

Z Rd +

Rd−1

A

|Dxβ0 ve|p dx =

Z

Z dxd

A

Rd−1

|Dxβ0 v(x0 , xd )|p dx0 .

(12.2.10)

Up to replacing A with a smaller set with null complement, if needed, from (12.2.8) it follows that for each x ∈ A the function ve(·, xd ) solves the equation λe v (·, xd ) −

d−2 X j=1

Dj2 ve(·, xd )

∗ 2  η 2 ve(·, xd ) = fb(·, xd ) − 1 + d−1 Dd−1 ηd 

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∗ on Rd−1 . If we set a = 1+|ηd−1 ηd−1 |2 , then we easily see that the function χ : Rd−1 → C, de√ fined by χ(y) = ve(y1 , . . . , yd−2 , ayd−1 ) for y ∈ Rd−1 , solves the equation λχ(y) − ∆χ(y) = √ fb(y1 , . . . , yd−2 , ayd−1 , xd ) for y ∈ Rd−1 . Applying Theorem 10.2.13 to this function and then changing the variable, we infer that

p

Z

0

|λ|

0

p

|e v (x , xd )| dx + |λ| Rd−1

+

d−1 Z X i,j=1

Rd−1

p 2

d−1 Z X i=1

Rd−1

bpp |Dij ve(x0 , xd )|p dx0 ≤ C

|Di ve(x0 , xd )|p dx0 ,

Z

|fb(x0 , xd )|p dx0 .

(12.2.11)

Rd−1

Integrating both sides of (12.2.11) with respect to xd in A, raising both sides of the so obtained inequality to the power 1/p and taking (12.2.7), (12.2.9), (12.2.10) and the definition of the function fb into account, we obtain |λ|||e v ||Lp (Rd+ ;C) +

p

|λ|

d−1 X

||Dj ve||Lp (Rd+ ;C) +

j=1

d−1 X

||Dij ve||Lp (Rd+ ;C)

i,j=1

bp (||fe||Lp (Rd ;C) + |ηd |−1 ||Dd w||Lp (Rd ;C) + |η 0 |η −2 ||Dd−1 w||Lp (Rd ;C) ) ≤C d + + + bp (1 + |η|2 )(1 + η −2 )(||λu − ∆u||Lp (Rd ;C) + |||∇g0 |||Lp (Rd ;C) ). ≤C d + +

(12.2.12)

Pd−1 Next, we observe that Ddd ve = λe v − fe − j=1 Djj ve, so that, from (12.2.12) we deduce that ||Ddd ve||Lp (Rd+ ;C) ≤|λ|||e v ||Lp (Rd+ ) +

d−1 X

||Djj ve||Lp (Rd+ ;C) + ||fe||Lp (Rd+ ;C)

j=1

bp (1 + |η|2 )(1 + η −2 )(||λu − ∆u||Lp (Rd ;C) + |||∇g0 |||Lp (Rd ;C) ). (12.2.13) ≤C d + + To estimate the derivatives Djd ve when j = 1, . . . , d − 1, we use the formula Djd ve = ∗ ηd−1 [Dj w − ηd−1 Dj Dd−1 ve] together with (12.2.13), to get bp (1 + |η|3 )(1 + |ηd |−3 )(||λu − ∆u||Lp (Rd ;C) + |||∇g0 |||Lp (Rd ;C) ). (12.2.14) ||Djd ve||Lp (Rd+ ;C) ≤C + + From the interpolation estimate in Proposition 1.3.8 and estimates (12.2.12)–(12.2.14), we deduce that 1

1

bp ||e ||Dd ve||Lp (Rd+ ;C) ≤C v ||L2 p (Rd ;C) ||D2 ve||L2 p (Rd ;C) +

1

bp |λ|− 2 (1 + |η|3 )(1 + |ηd |−3 )(||λu − ∆u||Lp (Rd ;C) + |||∇g0 |||Lp (Rd ;C) ). ≤C + + Summing up, so far we have proved that p |λ|||e v ||Lp (Rd+ ;C) + |λ||||∇e v |||Lp (Rd+ ;C) + ||D2 ve||Lp (Rd+ ;C) bp (1 + |η|3 )(1 + |ηd |−3 )(||λu − ∆u||Lp (Rd ;C) + |||∇g0 |||Lp (Rd ;C) ). ≤C + +

(12.2.15)

Now, we are almost done. Since v = ve + ζ, taking (12.2.3), (12.2.4) and (12.2.15) into account we can estimate p |λ|||v||Lp (Rd+ ;C) + |λ||||∇v|||Lp (Rd+ ;C) + ||D2 v||Lp (Rd+ ;C)

326

Elliptic Equations in Rd+ with General Boundary Conditions p bp (1 + |η|3 )(1 + |ηd |−3 )(||λu − ∆u||Lp (Rd ;C) + |λ|||g0 ||Lp (Rd ;C) + |||∇g0 |||Lp (Rd ;C) ). ≤C + + +

Finally, recalling that u = v ◦ Θ−1 , estimate (12.2.2) follows at once. If d = 2, then things are easier, since we do not need to introduce the rotation Θ. ∂ Step 4. Here, A = Tr(QD2 ) and B = a + , Q and η being a constant matrix and ∂η a vector of Rd with ηd 6= 0, respectively. Taking advantage of Lemma 8.1.2, we introduce a linear operator T : Rd → Rd such that T (Rd+ ) = Rd+ , T (∂Rd ) = ∂Rd , T QT ∗ x = x for −1/2 −1/2 every x ∈ Rd and T ∗ ed = Ked , where λmax ≤ K ≤ λmin . Fix u ∈ W 2,p (Rd+ ; C) and ∂u = g0 on ∂Rd+ . The function v = u ◦ T −1 belongs to g0 ∈ W 1,p (Rd+ ; C) such that au + ∂η ∂v W 2,p (Rd+ ; C). Moreover, av+ ∗ = g0 ◦T −1 on ∂Rd+ , where η∗ = T η. It thus follows that the ∂η ∂v −1 on ∂Rd+ . function ge0 = (g0 − au) ◦ T belongs to W 1,p (Rd+ ; C) and extends the trace of ∂η ∗ 1/2 Note that ηd∗ = hT η, ed i = hη, T ∗ ed i = Kηd so that |ηd∗ |−1 ≤ |η0 |−1 λmax . Therefore, the proof the quoted lemma shows that the Jacobian matrix of T is the product P2 DP1 , where P1 and P2 are orthogonal matrices and D is a diagonal matrix, whose entries are the inverse 1/2 of the roots of the eigenvalues of Q. Therefore, ||T ||L(Rd ) ≤ µ−1/2 and ||T −1 ||L(Rd ) ≤ λmax . Applying Step 3 to the function v, we can estimate p |λ|||v||Lp (Rd+ ;C) + |λ||||∇v|||Lp (Rd+ ;C) + ||D2 v||Lp (Rd+ ;C) p bp (1 + |η|3 )(1 + |η0 |−3 )λ3max (||λv − ∆v||Lp (Rd ;C) + |λ|||e ≤C g0 ||Lp (Rd+ ;C) + |||∇e g0 |||Lp (Rd+ ;C) ) + (12.2.16) for every λ ∈ C with positive real part. Since λv − ∆v = (λu − Tr(QD2 u)) ◦ T −1 , it follows that ||λv − ∆v||Lp (Rd+ ;C) ≤ µ−d/(2p) ||λu − Au||Lp (Rd+ ;C) . Similarly, d

||e g0 ||Lp (Rd+ ;C) ≤ µ− 2p (||g0 ||Lp (Rd+ ;C) + |a|||u||Lp (Rd+ ;C) ), p d |||∇e g0 |||Lp (Rd+ ;C) ≤ λmax µ− 2p (|||∇g0 |||Lp (Rd+ ;C) + |a||||∇u|||Lp (Rd+ ;C) ). Finally, since p

|λ||||∇u|||Lp (Rd+ ;C) + ||D2 u||Lp (Rd+ ;C) d p 2p ≤λmax (|λ|||v||Lp (Rd+ ;C) + |λ|dµ−1/2 |||∇v|||Lp (Rd+ ;C) + d2 µ−1 ||D2 v||Lp (Rd+ ;C) ) |λ|||u||Lp (Rd+ ;C) +

from (12.2.16), it follows that p |λ|||u||Lp (Rd+ ;C) + |λ||||∇u|||Lp (Rd+ ;C) + ||D2 u||Lp (Rd+ ;C) p ep (||λu − Au||Lp (Rd ;C) + |λ|||g0 ||Lp (Rd ;C) + |||∇g0 |||Lp (Rd ;C) + ||u||Lp (Rd ;C) ), ≤C + + + +

(12.2.17)

where d

d 2p +3 − 2p ep = λmax C µ (1 + µ−1 )(1 +

p

bp . λmax )(1 + |a|)(1 + |η|3 )(1 + |η0 |−3 )C

e p = 2C ep , then we can move the Lp -norm of u from the right- to the left-hand Choosing λ ep and Cp = 2C ep . side of (12.2.17) and obtain (12.2.2) with λp = λ Step 5. Here, A = Tr(QD2 u), Q being a matrix-valued function, and the coefficients

Semigroups of Bounded Operators and Second-Order PDE’s

327

of the operator B are C 1 -smooth functions. We fix x0 ∈ Rd+ , r > 0, u ∈ W 2,p (Rd+ ; C) and a smooth function ϑx0 ,r ∈ Cc∞ (Rd ) such that χB+ (x0 ,r/2) ≤ ϑx0 ,r ≤ χB+ (x0 ,r) , where B+ (x0 , σ) = B(x0 , σ)∩Rd+ for every σ > 0. Moreover, we denote respectively by Ax0 and Bx0 ∂ . the elliptic operator Tr(Q(x0 )D2 ) and the boundary differential operator a(x0 ) + ∂η(x0 ) Note that the function ge0 = g0 +(Bx0 u−Bu) belongs to W 1,p (Rd+ ; C), due to the assumptions on the coefficients of the operator B, and agrees with the function Bx0 u on ∂Rd+ . Thus, taking into account that ϑx0 ,r ≡ 1 on B+ (x0 , r/2) and ϑx0 ,r ≤ 1 on Rd+ , from Step 4, applied to the pair (Ax0 , Bx0 ) and the function v = ϑx0 ,r u, which belongs to W 2,p (Rd+ ; C), we get p |λ|||u||Lp (B+ (x0 ,r/2);C) + |λ||||∇u|||Lp (B+ (x0 ,r/2);C) + ||D2 u||Lp (B+ (x0 ,r/2);C) p ≤|λ|||v||Lp (Rd+ ;C) + |λ||||∇v|||Lp (Rd+ ;C) + ||D2 v||Lp (Rd+ ;C) p ep (||λv − Ax v||Lp (Rd ;C) + |λ|||b ≤C g0 ||Lp (Rd+ ;C) + |||∇b g0 |||Lp (Rd+ ;C) (12.2.18) 0 ep , where gb0 is any function in W 1,p (Rd ; C), which agrees for every λ ∈ C with Re λ ≥ C + with the function Bx0 v on ∂Rd+ and d

d 2p +3 − 2p ep = 2||Q||∞ bp . (12.2.19) C µ (1 + µ−1 )(1 + ||Q||∞ )(1 + ||a||∞ )(1 + |||η|||3∞ )(1 + |η0 |−3 )C

To estimate the last side of (12.2.18), we denote by L a positive constant, independent of x0 and r, such that r|||∇ϑx0 ,r |||∞ + r2 ||D2 ϑx0 ,r ||∞ ≤ L for every r > 0. Thus, we can write ||λv − Ax0 v||Lp (Rd+ ;C) ≤||λu − Au||Lp (B+ (x0 ,r);C) + dω(r)||D2 u||Lp (B+ (x0 ,r);C) +

ML 2M L ||u||Lp (B+ (x0 ,r);C) + |||∇u|||Lp (B+ (x0 ,r);C) , 2 r r

(12.2.20)

where ω(r) denotes the maximum of sup|x−y|≤r |qij (x) − qij (y)| when i, j ∈ {1, . . . , d} and M = max{||qij ||∞ , ||bj ||∞ , ||c||∞ : i, j = 1, . . . , d}. Next, we observe that the function ϑx0 ,r ge0 + h∇ϑx0 ,r , η(x0 )iu belongs to W 1,p (Rd+ ; C) and agrees with the function Bx0 v on ∂Rd+ . Hence, we can take it as function gb0 . Note that ||e g0 ||Lp (B+ (x0 ,r);C) ≤ ||g0 ||Lp (B+ (x0 ,r);C) +r|||∇a|||∞ ||u||Lp (B+ (x0 ,r);C) + Kr|||∇u|||Lp (B+ (x0 ,r);C) , where K =

Pd

j=1

|||∇ηj |||∞ , and

|||∇e g0 |||Lp (B+ (x0 ,r);C) ≤|||∇g0 |||Lp (B+ (x0 ,r);C) + |||∇a|||∞ (||u||Lp (B+ (x0 ,r);C) + r|||∇u|||Lp (B+ (x0 ,r);C) ) + K(|||∇u|||Lp (B+ (x0 ,r);C) + r||D2 u||Lp (Rd ;C) ). Therefore, ||b g0 ||Lp (Rd+ ;C) ≤ ||g0 ||Lp (B+ (x0 ,r);C) + C1 (a, L, η, r)||u||Lp (B+ (x0 ,r);C) + Kr|||∇u|||Lp (B+ (x0 ,r);C) (12.2.21) and |||∇b g0 |||Lp (Rd+ ;C) ≤Lr−1 ||g0 ||Lp (B+ (x0 ,r);C) + |||∇g0 |||Lp (B+ (x0 ,r);C) + C2 (a, K, L, η, r)||u||Lp (B+ (x0 ,r);C) + Kr||D2 u||Lp (B+ (x0 ,r);C)

Elliptic Equations in Rd+ with General Boundary Conditions

328

+ C3 (a, K, L, η, r)|||∇u|||Lp (B+ (x0 ,r);C) ,

(12.2.22)

where C1 (a, L, η, r) = r|||∇a|||∞ + Lr−1 ||η||∞ , C2 (a, K, L, η, r) = (L + 1)|||∇a|||∞ + KLr−1 + KLr−2 + Lr−1 |||η|||∞ , C3 (a, K, L, η, r) = KL + r|||∇a|||∞ + K + Lr−1 |||η|||∞ . Replacing (12.2.20)–(12.2.22) in the right-hand side of (12.2.18) yields p |λ|||u||Lp (B+ (x0 ,r/2);C) + |λ||||∇u|||Lp (B+ (x0 ,r/2);C) + ||D2 u||Lp (B+ (x0 ,r/2);C) p  ep∗ ||λu − Au||Lp (B (x ,r);C) + ( |λ| + Lr−1 )||g0 ||Lp (B (x ,r);C) ≤C + 0 + 0 p −2 + (M Lr + |λ|C1 (a, L, η, r) + C2 (a, K, L, η, r))||u||Lp (B+ (x0 ,r);C) p + (2M Lr−1 + |λ|Kr + C3 (a, K, L, η, r))|||∇u|||Lp (B+ (x0 ,r);C)  (12.2.23) + (dω(r) + Kr)||D2 u||Lp (B(x0 ,r);C) + |||∇g0 |||Lp (B+ (x0 ,r);C) , ep∗ is defined as in (12.2.19), with a possible different choice of the where the constant C bp . By Lemma 10.2.19 and Remark 10.2.20, we can determine a sequence (xn ) constant C such that ||w||pLp (Rd ;C) ≤ +

∞ X n=1

||w||pLp (B+ (xn ,r/2);C) ,

∞ X n=1

||w||pLp (B+ (xn ,r);C) ≤ ξ(d)||w||pLp (Rd ;C) +

for every w ∈ Lp (Rd+ ; C). Applying these inequalities to u and its first- and second-order ep ∨ 1, from (12.2.23) we deduce that derivatives and assuming that Reλ ≥ C p |λ|||u||Lp (Rd+ ;C) + |λ||||∇u|||Lp (Rd+ ;C) + ||D2 u||Lp (Rd+ ;C) p  ep∗∗ ||λu − Au||Lp (Rd ;C) + |λ|(1 + Lr−1 )||g0 ||Lp (Rd ;C) + |||∇g0 |||Lp (Rd ;C) ≤C + + + p 2 + |λ|C4 (a, K, L, η, r)||u||Lp (Rd+ ;C) + (dω(r) + Kr)||D u||Lp (Rd+ ;C) p  (12.2.24) + ( |λ|Kr + C5 (a, K, L, η, r))|||∇u|||Lp (Rd+ ;C) , where C4 (a, K, L, η, r) = r−2 M L + C1 (a, L, η, r) + C2 (a, K, L, η, r), C5 (a, K, L, η, r) = 2M (Lr)−1 + C3 (a, K, L, η, r) and Cp∗∗ is defined as in (12.2.19) with a different choice of bp . the constant C ep (dω(r) + Kr) ≤ 1/2 to move the terms C p Now, we fix r sufficiently small such that 2 |λ|Kr|||∇u|||Lp (Rd+ ;C) and (dω(r) + Kr)||D u||Lp (Rd+ ;C) to the left-hand side of (12.2.24) and get p |λ|||u||Lp (Rd+ ;C) + |λ||||∇u|||Lp (Rd+ ;C) + ||D2 u||Lp (Rd+ ;C) p  ep∗∗ ||λu − Au||Lp (Rd ;C) + |λ|(1 + Lr−1 )||g0 ||Lp (Rd ;C) + |||∇g0 |||Lp (Rd ;C) ≤2C + + + p  + |λ|C4 (a, K, L, η, r)||u||Lp (Rd+ ;C) + C5 (a, K, L, η, r)|||∇u|||Lp (Rd+ ;C) . (12.2.25) p ep ∨ 1, such that 4C ep∗∗ C4 (a, K, L, η, r) ≤ Finally, we fix λp ≥ C λp , and p ep∗∗ C5 (a, K, L, η, r) ≤ λp , and λ ∈ C with Reλ ≥ λp , to move the terms containing 4C the Lp -norms of u and |∇u| to the left-hand side of (12.2.25). Estimate (12.2.2) follows.

Semigroups of Bounded Operators and Second-Order PDE’s

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Step 6. For a complete operator A = Tr(QD2 ) + hb, ∇x i + c we apply the estimate obtained in Step 5 to the pair A0 = Tr(QD2 ) and B to write p |λ|||u||Lp (Rd+ ;C) + |λ||||∇u|||Lp (Rd+ ;C) + ||D2 u||Lp (Rd+ ;C) p ≤Cp (||λu − A0 u||Lp (Rd+ ;C) + |λ|||g0 ||Lp (Rd+ ;C) + |||∇g0 |||Lp (Rd+ ;C) ). (12.2.26) Note that ||λu − A0 u||Lp (Rd+ ;C) ≤ ||λu − A0 u||Lp (Rd+ ;C) + |||b|||∞ |||∇u|||Lp (Rd+ ;C) + ||c||∞ ||u||Lp (Rd+ ;C) . Inserting this inequality in the right-hand side of (12.2.26) and taking λp larger, if needed, we can move the terms containing the Lp -norms of u and its gradient to the left-hand side and get (12.2.2). The proof is complete.  Corollary 12.2.3 There exists a positive constant Cp , which depends on p and, in a continuous way, on the ellipticity constant of the operator A, on the modulus of continuity of its diffusion coefficients, on the sup-norms of the coefficients of the operator A, on the Cb1 (Rd−1 )- and Cb1 (Rd−1 ; Rd )-norms of the coefficients of the operator B and on the supremum over Rd−1 of the function η, such that ||u||W 2,p (Rd+ ;C) ≤ Cp (||Au||Lp (Rd+ ;C) + ||u||Lp (Rd+ ;C) + ||g0 ||W 1,p (Rd+ ;C) )

(12.2.27)

for every u ∈ W 2,p (Rd+ ; C), where g0 ∈ W 1,p (Rd+ ; C) is any function which agrees with Bu on ∂Rd+ . Proof It suffices to apply estimate (12.2.2) taking λ = λp .



We can now prove the existence-uniqueness result for problem (12.2.1). Theorem 12.2.4 Let λp > 0 be as in Proposition 12.2.2. Then, for every p ∈ (1, ∞), f ∈ Lp (Rd+ ; C), g ∈ B 1−1/p,p (Rd−1 ; C) and λ ∈ C with Reλ ≥ λp , there exists a unique function u ∈ W 2,p (Rd+ ; C) which solves problem (12.2.1). Moreover, there exists a positive constant Cp , which, as λp , depends in a continuous way on λ, on the ellipticity constant of the operator A, on the modulus of continuity of its diffusion coefficients, on the supnorms of the coefficients of the operator A, on the Cb1 (Rd−1 )- and Cb1 (Rd−1 ; Rd )-norms of the coefficients of the operator B and on the supremum over Rd−1 of the function η, such that p |λ|||u||Lp (Rd+ ;C) + |λ||||∇u|||Lp (Rd+ ;C) + ||D2 u||Lp (Rd+ ;C) p ≤Cp (||f ||Lp (Rd+ ;C) + |λ|||g0 ||Lp (Rd+ ;C) + |||∇g0 |||Lp (Rd+ ;C) ), (12.2.28) where g0 ∈ W 1,p (Rd+ ; C) is any function which agrees with g on ∂Rd+ . Proof Fix p ∈ (1, ∞). The uniqueness of a solution u ∈ W 2,p (Rd+ ; C) to the boundary value problem (12.2.1) and estimate (12.2.28) follow from (12.2.2). The existence of a solution to that problem, can be proved using the method of continuity. To make such a method work, we prove that the boundary value problem ( λu(x) − ∆u(x) = f (x), x ∈ Rd+ , (12.2.29) Dd u(x) = g(x), x ∈ Rd−1 . admits a unique solution u, which belongs to W 2,p (Rd+ ; C), for every f ∈ Lp (Rd+ ; C), and

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Elliptic Equations in Rd+ with General Boundary Conditions

g ∈ B 1−1/p,p (Rd−1 ; C). This can be obtained adapting the arguments in the first two steps of the proof of Proposition 12.2.2. Indeed, fix λ ∈ C, with positive real part and let g0 = Ep g, where Ep is the operator defined in Proposition B.4.15. Function g0 belongs to W 1,p (Rd+ ; C) and g is its trace on ∂Rd+ . Further, let w ∈ W 3,p (Rd+ , C)∩W01,p (Rd+ ; C) be the unique solution to the equation λw −∆w = −g0 . Clearly, the function ζ = Dd w belongs to W 2,p (Rd+ ; C) and Pd−1 solves the equation λζ − ∆ζ = −Dd g0 . Moreover, Dd ζ = Ddd w = λw − j=1 Djj w + g0 . Taking the trace at ∂Rd+ , we easily see that Dd ζ = g. Clearly, u solves (12.2.29) if and only if the function v = u − ζ solves the equation λv − ∆v = fe := f + Dd g0 and has null normal derivative on ∂Rd+ . Extend fe to Rd , even with respect to the last variable and denote by fee the so obtained function. By Theorem 10.2.13, there exists a unique function vb ∈ W 2,p (Rd ; C) which solves the equation λw b − ∆w b = fbe . As it is easily seen, also the 0 2,p d function x 7→ vb(x , −xd ) belongs to W (R ; C) solves the same equation. By uniqueness, it follows that it coincides with vb. Thus, vb is even with respect to the last variable. This is enough to infer that Dd vb vanishes on ∂Rd+ . As a byproduct, we infer that the function u = vb + ζ belongs to W 2,p (Rd+ ; C) and solves the boundary value problem (12.2.29). We can now make the method of continuity works. Without loss of generality, we can assume that ηd > 0 on ∂Rd+ . For every σ ∈ [0, 1], we consider the operators Aσ and Bσ ∂ defined by Aσ = (1−σ)A+σ∆ = Tr(Qσ D2 )+hb, ∇i+c and Bσ = (1−σ)B+σDd = a+ σ , ∂η where Qσ = (1 − σ)Q + σI and ησ = (1 − σ)η + σed for every σ ∈ [0, 1]. As it is easily seen, hQσ (x)ξ, ξi ≥ (µ ∧ 1)|ξ|2 for every x ∈ Rd+ , ξ ∈ Rd , and ηdσ ≥ η0 ∧ 1 on Rd−1 for every σ ∈ [0, 1]. Moreover, ||Qσ ||∞ ≤ ||Q||∞ ∨ 1 and |||η σ |||∞ ≤ |||η|||∞ ∨ 1 for every σ ∈ [0, 1]. Therefore, from Proposition 12.2.2 we deduce that there exist two positive constants λp and Cp = Cp (λ), independent of σ ∈ [0, 1], such that ||u||W 2,p (Rd+ ;C) ≤ Cp (||λu − Aσ u||Lp (Rd+ ;C) + ||g0 ||W 1,p (Rd+ ;C) ), where g0 is any extension in W 1,p (Rd+ ; C) of the function Bσ u defined on ∂Rd+ . We take g0 = Ep Bσ u, which satisfies the estimate ||g0 ||W 1,p (Rd+ ;C) ≤ C∗ ||Bσ u||W 1,p (Rd−1 ;C) for some positive constant C∗ independent of σ. Therefore, ep (||λu − Aσ u||Lp (Rd ;C) + ||Bσ u||W 1,p (Rd−1 ;C) ) ||u||W 2,p (Rd+ ;C) ≤ C + ep , depending on λ but being independent of σ. for some positive constant C For every σ ∈ [0, 1], denote by Tσ : W 2,p (Rd ; C) → Lp (Rd+ ; C) × W 1,p (Rd−1 ; C) (where the latter space is endowed with the product norm) the operator defined by Tσ u = (λu − Aσ u, Bσ u) for every u ∈ W 2,p (Rd+ ; C) and σ ∈ [0, 1]. The above estimate ep−1 ||u||W 2,p (Rd ;C) . Since the operator T0 is invertshows that ||Tσ u||Lp (Rd+ ;C)×W 1,p (Rd−1 ;C) ≥ C + ible, all the operators Tσ are invertible. In particular, T1 is invertible and this means that, for every f ∈ Lp (Rd+ ; C) and g ∈ W 1,p (Rd−1 ; C), there exists a unique u ∈ W 2,p (Rd+ ; C) such that λu − Au = f on Rd+ and Bu = g on ∂Rd+ . This completes the proof.  As a consequence of Theorem 12.2.4, we can prove the following result. Corollary 12.2.5 Suppose that f belongs to Lp (Rd+ ; C) ∩ Lq (Rd+ ; C) and g belongs to B 1−1/p,p (Rd+ ; C) ∩ B 1−1/q,q (Rd+ ; C) for some p, q ∈ (1, ∞). Then, for every λ ∈ C, with Re λ ≥ λp (where the constant λp is defined in Proposition 12.2.2) the solution u ∈ W 2,p (Rd+ ; C) to problem (12.2.1) belongs to W 2,q (Rd+ ; C). Proof Fix p and q as in the statement of the corollary and let u ∈ W 2,p (Rd+ ; C) be the unique solution of the equation λu−∆u = f , which satisfies the boundary condition Dd u = g

Semigroups of Bounded Operators and Second-Order PDE’s

331

on ∂Rd+ . We first assume that Re λ ≥ λp ∨ λq . Going through the proof of Proposition B.4.15, it can be easily checked that the extension operator Ep0 maps B 1−1/p,p (Rd−1 ; C) ∩ B 1−1/q,q (Rd−1 ; C) into W 1,p (Rd+ ; C) ∩ W 1,q (Rd+ ; C). Hence, the function Ep0 g belongs both to W 1,p (Rd+ ; C) and to W 1,q (Rd+ ; C). As in the proof of Theorem 12.2.4, we consider the function ζ = Dd w, where w is the unique solution in W 3,p (Rd+ ; C) ∩ W01,p (Rd+ ; C) of the equation λw − ∆w = −Ep g. The trace of Dd ζ on ∂Rd+ coincides with the function g. Moreover, since Ep g belongs to W 1,p (Rd+ ; C) ∩ W 1,q (Rd+ ; C), applying Corollary 11.2.14, we infer that ζ belongs to W 2,p (Rd+ ; C) ∩ W 2,q (Rd+ ; C). Therefore, the function u − ζ solves the boundary value problem ( λv − ∆v = f + Dd Ep g, in Rd+ Dd v = 0, on ∂Rd+ . Denote by fee the even extension to Rd of the function f + Dd Ep g. By the proof of Theorem 12.2.4, v is the restriction to Rd+ of the unique solution in Rd of the equation λw − ∆w = fee , which belongs to W 2,p (Rd ; C). Clearly, fee belongs also to Lq (Rd ; C) so that, by Proposition 10.2.25, w belongs also W 2,q (Rd ; C). As a byproduct, the function v is an element of W 2,p (Rd+ ; C) ∩ W 2,q (Rd+ ; C). In other terms, we have shown that, for every f ∈ Lp (Rd+ ; C) ∩ Lq (Rd+ ; C) and every g ∈ B 1−1/p,p (Rd+ ; C) ∩ B 1−1/q,q (Rd+ ; C), the equation λu − ∆u = f admits a unique solution u ∈ W 2,p (Rd+ ; C) ∩ W 2,q (Rd+ ; C) such that Dd u = g. We now apply the method of continuity to address the general case, as in the proof of Theorem 12.2.4. On X p,q = Lp (Rd+ ; C) ∩ Lq (Rd+ ; C) we consider the norm || · ||X p,q = || · ||Lp (Rd+ ;C) + || · ||Lq (Rd+ ;C) , which makes Xp,q a Banach space. Similarly, on Y p,q = B 1−1/p,p (Rd−1 ; C) ∩ B 1−1/q,q (Rd−1 ; C) and on Z p,q = W 2,p (Rd+ ; C) ∩ W 2,q (Rd+ ; C) we consider the norms || · ||Y p,q = || · ||B 1−1/p,p (Rd−1 ;C) + || · ||B 1−1/q,q (Rd−1 ;C) and || · ||Z p,q = || · ||W 2,p (Rd+ ;C) + || · ||W 2,q (Rd+ ;C) , respectively. These norms make Y p,q and Z p,q Banach spaces. Next, for every σ ∈ [0, 1] we consider the bounded linear operator Tσ : Y p,q → X p,q × Z p,q (where the latter space is endowed with the product norm) the operator defined by Tσ u = (λu − σAu − (1 − σ)∆u, (1 − σ)Dd u + σBu) for every u ∈ X p,q and σ ∈ [0, 1]. The ep ∨ C eq )−1 ||u||X p,q . Since the estimate of Proposition 12.2.2 shows that ||Tσ u||Y p,q ×Z p,q ≥ (C operator T0 is invertible, then all the operators Tσ are invertible. In particular, T1 is invertible and this means that, for every f ∈ Lp (Rd+ ; C) ∩ Lq (Rd+ ; C) and g ∈ B 1−1/p,p (Rd−1 ; C) ∩ B 1−1/q,q (Rd−1 ; C), there exists a unique function u ∈ W 2,p (Rd+ ; C) ∩ W 2,q (Rd+ ; C) such that λu − Au = f on Rd+ and Bu = g on ∂Rd+ . If λp < λq and Reλ < λq , then we apply the Sobolev embedding theorem, to infer that u ∈ Lr (Rd ; C). If r ≥ q, then, since λq u − Au = f + (λq − λ)u and the function f + (λ − λq )u belongs to Lq (Rd+ ; C), we can apply the arguments in the first part of the proof. If r < q, then λq u − Au = f + (λq − λ)u ∈ Lr (Rd+ ; C) and the first part of the proof shows that u ∈ W 2,r (Rd+ ; C). Iterating this procedure, in a finite number of steps we conclude that u ∈ W 2,q (Rd+ ; C).  Corollary 12.2.6 Suppose that u ∈ W 2,p (Rd+ ; C) for some p ∈ (1, ∞) is a function such that Au ∈ Lq (Rd+ ; C) and Bu ∈ B 1−1/p,p (Rd−1 ; C) ∩ B 1−1/q,q (Rd−1 ; C) for some q > p. Then, u belongs to W 2,q (Rd ; C). Proof Applying the Sobolev embedding theorems, we deduce that u ∈ Lr (Rd+ ; C) for some r > p. If r ≥ q, then the function λp u − Au belongs to Lq (Rd+ ; C). We can thus apply Corollary 12.2.5 and conclude that u ∈ W 2,q (Rd+ ; C). If r < q, then the function λp u − Au belongs to Lr (Rd+ ; C). Since B 1−1/p,p (Rd−1 ; C) ∩ B 1−1/q,q (Rd−1 ; C) is contained

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Elliptic Equations in Rd+ with General Boundary Conditions

in B 1−1/r,r (Rd−1 ; C), we can apply Corollary 12.2.5 to infer that u ∈ W 2,r (Rd+ ; C). Iterating this procedure, in a finite number of steps we deduce that u ∈ W 2,q (Rd+ ; C). 

12.2.1

Further regularity results

In this subsection, we prove that the more the function Au is smooth, the more u itself is smooth. We assume the following set of assumptions. Hypotheses 12.2.7 (i) The coefficients qij = qji , bj (i, j = 1, . . . , d) and c belong to W k,∞ (Rd+ ) for some k ∈ N; (ii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for all x ∈ Rd+ and ξ ∈ Rd ; (iii) the functions a and η belong to Cbk+1 (Rd−1 ) and Cbk+1 (Rd−1 ; Rd ), respectively. Moreover, |η(y)| = 1 for each y ∈ Rd−1 and η0 := supRd−1 ηd < 0, where, as usually, ηd denotes the last component of the function η. Theorem 12.2.8 Let Hypotheses 12.2.7 be satisfied. Let u ∈ W 2,p (Rd+ ; C) for some p ∈ (1, ∞) be a function such that Au ∈ W k,p (Rd ; C) and Bu ∈ B k+1−1/p,p (Rd−1 ; C). Then, u ∈ W k+2,p (Rd+ , C) and there exists a positive constant C, depending on p, d and, in a continuous way, on µ, the W k,∞ (Rd+ ; C)-norm of the coefficients of the operator A, the modulus of continuity of the diffusion coefficients qij (i, j = 1, . . . , d), the Cbk+1 (Rd−1 )- and Cbk+1 (Rd−1 ; Rd )-norms of the coefficients of the operator B, such that ||u||W k+2,p (Rd+ ;C) ≤ C(||u||Lp (Rd+ ;C) + ||Au||W k,p (Rd+ ;C) + ||Bu||B k+1−1/p,p (Rd−1 ;C) ).

(12.2.30)

Proof Throughout the proof, we denote by C a positive constant, independent of the functions u and Au that we will consider as well as of the positive parameter r, which may vary from line to line. Even if we do not make it explicit, the constants in the proof may depend on p and, in a continuous way, on µ, the W k,∞ (Rd+ ; C)-norms of the coefficients of the operator A and the Cbk+1 (Rd−1 )- and Cbk+1 (Rd−1 ; Rd )-norms of the coefficients of the operator B We fix p ∈ (1, ∞) and argue by induction on k ∈ N. Suppose that k = 1. For every r > 0 and h ∈ {1, . . . , d − 1}, we introduce the function u(r,h) = r−1 (u(· + reh ) − u), which clearly belongs to W 2,p (Rd+ ; C). As it is easily seen, Au(r,h) = (Au)(r,h) −

d X

(r,h)

qij

i,j=1 (r,h)

Dij u(· + reh ) −

d X

(r,h)

bj

Dj u(· + reh ) − c(r,h) u(· + reh ),

j=1 (r,h)

where the functions qij , bj (i, j = 1, . . . , d) and c(r,h) are defined in analogy with the definition of the function u(r,h) . Note that Au(r,h) belongs to Lp (Rd ; C) and ||Au(r,h) ||Lp (Rd+ ;C) ≤ C(||Au||W 1,p (Rd+ ;C) + ||u||W 2,p (Rd+ ;C) ).

(12.2.31)

Moreover, Bu(r,h) = (Bu)(r,h) +a(r,h) u(·+reh , 0)+h∇u(·+reh , 0), η (r,j) i, so that the function g0 = Ep0 (Bu)(r,h) + a(r,h) u(· + reh ) + h∇u(· + reh ), η (r,j) i, where Ep0 is the extension operator in Proposition B.4.15, belongs to W 1,p (Rd+ ; C) and coincides with the function Bu(r,h) on ∂Rd+ . Moreover, taking Proposition B.4.15 into account, we can show that ||g0 ||W 1,p (Rd+ ;C) ≤ C(||Bu||B 2−1/p,p (Rd−1 ;C) + ||u||W 2,p (Rd+ ;C) ).

(12.2.32)

Semigroups of Bounded Operators and Second-Order PDE’s

333

Applying repeatedly estimate (12.2.27) and using also estimates (12.2.31), (12.2.32) and Proposition 1.3.11(i), to infer that ||u(r,h) ||Lp (Rd+ ;C) ≤ |||∇u|||Lp (Rd+ ;Ω) , we deduce that ||u(r,h) ||W 2,p (Rd+ ;C) ≤C(||u(r,h) ||Lp (Rd+ ;C) + ||Au(r,h) ||Lp (Rd+ ;C) + ||g0 ||W 1,p (Rd+ ;C) ) ≤C(||u||W 2,p (Rd+ ;C) + ||Au||W 1,p (Rd+ ;C) + ||Bu||B 2−1/p,p (Rd−1 ;C) ) ≤C(||u||Lp (Rd+ ;C) + ||Au||W 1,p (Rd+ ;C) + ||Bu||B 2−1/p,p (Rd−1 ;C) ). In particular, ||(Dij u)(r,h) ||Lp (Rd+ ;C) =||Dij u(r,h) ||Lp (Rd+ ;C) ≤C(||u||Lp (Rd+ ;C) + ||Au||W 1,p (Rd+ ;C) + ||Bu||B 2−1/p,p (Rd−1 ;C) ). Taking advantage of Proposition 1.3.11(ii) and the arbitrariness of r and h, we infer that every third-order distributional derivative Dijh u of u belongs to Lp (Rd+ ; C) and its norm can be bounded from above by C(||u||Lp (Rd+ ;C) + ||Au||W 1,p (Rd+ ;C) + ||u||B 2−1/p,p (Rd−1 ;C) ). To show that also the distributional derivative Dddd u belongs to Lp (Rd+ ; C), it suffices to write   d X X 1 Ddd u = Au − qij Dij u − bj Dj u − cu (12.2.33) qdd j=1 (i,j)6=(d,d)

and observe that the right-hand side of the previous formula can be weakly differentiate with respect to the variable xd , due to the above results. Hence, we conclude that Dddd u ∈ Lp (Rd+ ; C) and satisfies the same estimate as the other third-order derivatives of u. The assertion is proved in the case k = 1. Next, we suppose that the assertion holds true for every k 0 ≤ k and prove that with k being replaced by k + 1. By the inductive assumptions, u ∈ W k+2,p (Rd+ ; C) and ||u||W k+2,p (Rd+ ;C) ≤ C(||u||Lp (Rd+ ;C) + ||Au||W k,p (Rd+ ;C) + ||Bu||B k+1−1/p,p (Rd−1 ;C) ).

(12.2.34)

We again fix h as above and observe that the function Dh u belongs to W k+1,p (Rd+ ; C) and ADh u = Dh Au −

d X i,j=1

Dh qij Dij u −

d X

Dh bj Dj u − (Dh c)u

j=1

which belongs to W k,p (Rd+ , C). Moreover, due to (12.2.34) we can estimate ||ADh u||W k,p (Rd+ ;C) ≤||Dh Au||W k,p (Rd+ ;C) + C||u||W k+2,p (Rd+ ;C) ≤C(||u||Lp (Rd+ ;C) + ||Au||W k+1,p (Rd+ ;C) + ||Bu||B k+1−1/p,p (Rd−1 ;C) ). (12.2.35) Similarly, BDh u = Dh Bu − (Dh a)u(·, 0) − h∇u(·, 0), Dh ηi, applying Theorem B.3.1, one deduces that BDh u belongs to B k+1−1/p,p (Rd−1 ; C) and ||BDh u||B k+1−1/p,p (Rd−1 ;C) ≤C(||Bu||B k+2−1/p,p (Rd−1 ;C) + ||u||W k+2,p (Rd+ ) ) ≤C(||u||Lp (Rd+ ;C) + ||Au||W k,p (Rd+ ;C) + ||Bu||B k+2−1/p,p (Rd−1 ;C) ). (12.2.36) Applying the inductive assumption to the function Dh u and taking (12.2.35) and (12.2.36) into account, we obtain that all the distributional derivatives of u of order k + 3, but the derivative Ddk+3 u, belong to Lp (Rd+ ; C) and their Lp -norm is bounded from above by C(||u||Lp (Rd+ ;C) + ||Au||W k,p (Rd+ ;C) + ||Bu||B k+2−1/p,p (Rd−1 ;C) ). Formula (12.2.33) allows to complete the proof. 

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Elliptic Equations in Rd+ with General Boundary Conditions

Corollary 12.2.9 Under Hypotheses 12.2.7, let u ∈ W 2,p (Rd+ ; C) be a function which satisfies the conditions Au ∈ W k,p (Rd+ ; C) ∩ W k,q (Rd+ ; C) and Bu ∈ B k−1/p,p (Rd−1 ; C) ∩ B k−1/q,q (Rd−1 ; C) for some p, q ∈ (1, ∞), with p < q. Then, u belongs to W k+2,p (Rd+ ; C) ∩ W k+2,q (Rd+ ; C). Proof Fix p, q ∈ (1, ∞) and let u be as in the statement. By Theorem 12.2.8, u belongs to W k+2,p (Rd+ ; C). Hence, by the Sobolev embedding theorems (see Theorem 1.3.6), u belongs to W 2,r (Rd+ ; C) for some r > p. If r ≥ q, then u ∈ W 2,q (Rd+ ; C) and, by assumptions, Au ∈ W k,q (Rd+ ; C) and Bu ∈ W k−1/q,q (Rd−1 ; C). Hence, by Theorem 12.2.8, u ∈ W k+2,q (Rd+ ; C) and we are done. On the other hand, if r < q, then observing that W k,p (Rd+ ; C) ∩ W k,q (Rd+ ; C) ⊂ W k,r (Rd+ ; C) and Bu ∈ B k−1/p,p (Rd−1 ; C) ∩ B k−1/q,q (Rd−1 ; C) ⊂ Bu ∈ B k−1/r,r (Rd−1 ; C), we conclude, by Theorem 12.2.8, that u ∈ W k+2,r (Rd+ ; C). Iterating this procedure, in a finite number of steps we obtain that u ∈ W k+2,q (Rd+ ; C). 

12.3

Solutions in L∞ (

R ; C) and in C (R ; C) d +

b

d +

In this section, we study the solvability of the boundary value problem ( λu − Au = f, in Rd+ , Bu = 0, on ∂Rd+ , when f ∈ L∞ (Rd+ ; C) and f ∈ Cb (Rd+ ; C) respectively. In the first case we prove the following result. We recall that B+ (x0 , r) = B(x0 , r) ∩ Rd+ for every r > 0. Theorem 12.3.1 Assume that Hypotheses 12.2.1 are satisfied. Then, there exists λ∞ ∈ R such that, for every f ∈ L∞ (Rd+ ; C) and λ in C, with Re λ > λ∞ , the equation λu − Au = f T 2,p admits a unique solution u, which belongs to Cb1+α (Rd+ ; C) ∩ p λ∞ . Proof The proof is similar to that of Theorem 11.3.1. For this reason, we skip some details. The key tool is the proof of the a priori estimate (12.3.1), which is obtained as a consequence of the estimate p |λ|||u||Lp (B+ (x0 ,r);C) + |λ||||∇u|||Lp (B+ (x0 ,r);C) + ||D2 u||Lp (B+ (x0 ,r);C)

Semigroups of Bounded Operators and Second-Order PDE’s 335 p  ≤C1,p ||λu − Au||Lp (B+ (x0 ,(γ+1)r);C) + γ −1 r−2 (1 + r |λ|)||u||Lp (B+ (x0 ,(γ+1)r);C)  + γ −1 r−1 |||∇u|||Lp (B+ (x0 ,(γ+1)r);C) (12.3.3) 2,p for every p > 1, u ∈ Wloc (Rd+ ; C) ∩ Cb1 (Rd ; C), with Bu = 0, r ∈ (0, 1], x0 ∈ Rd+ , γ ∈ [1, ∞), ep . Here, λp is the constant given λ ∈ C, with Re λ ≥ λp , and some positive constant C by Proposition 12.2.2. Throughout the proof, if not otherwise specified, by Cj,p we denote a positive constant, which depends at most only on p, d and, in a continuous way, on the ellipticity constant of the operator A, on the modulus of continuity of its diffusion coefficients, on the sup-norms of the coefficients of the operator A and on the Cb1 (Rd−1 )and Cb1 (Rd−1 ; Rd )-norms of the coefficients of the operator B. To prove (12.3.3), let us fix x0 ∈ Rd+ , p, r, γ and λ as above, and introduce a smooth function ϑ such that ϑ = 1 on B+ (x0 , r), ϑ = 0 outside B+ (x0 , (γ + 1)r) and ||ϑ||∞ + γr|||∇ϑ|||∞ + γ 2 r2 ||D2 ϑ||∞ ≤ K for some positive constant K, independent of γ and r. 2,p Given a function u ∈ Wloc (Rd+ ; C), we apply Theorem 12.2.4 to the function v = uϑ. For this purpose, we observe that the function g0 = uhη, ∇ϑi belongs to W 1,p (Rd+ ; C) and agrees C ||u||Lp (B+ (x0 ,(γ+1)r)) and with the function Bv on ∂Rd+ . Clearly, ||g0 ||Lp (Rd+ ;C) ≤ γr

|||∇g0 |||Lp (Rd+ ;C) ≤

C (||u||Lp (B+ (x0 ,(γ+1)r)) + r|||∇u|||Lp (B+ (x0 ,(γ+1)r)) ). γr2

Moreover, λv − Av = f , where f = ϑ(λu − Au) − u(Aϑ − cϑ) − 2hQ∇u, ∇ϑi, so that ||f ||Lp (Rd ;C) ≤C2,p ||λu − Au||Lp (B+ (x0 ,(γ+1)r);C) + +

C2,p ||u||Lp (B+ (x0 ,(γ+1)r);C) γr2

C2,p |||∇u|||Lp (B+ (x0 ,(γ+1)r);C) . γr

Therefore, applying estimate (12.2.2) and recalling that v coincides with function u on B+ (x0 , r), estimate (12.3.3) follows at once. Now, using (12.3.3), the steps needed to prove (12.3.1) are the following. (i) The inequality ||w||L∞ (B+ (x0 ,|λ|−1/2 );C) d

1

≤C3,p |λ| p (||w||Lp (B+ (x0 ,|λ|−1/2 );C) + |λ|− 2 |||∇w|||Lp (B+ (x0 ,|λ|−1/2 );C) ), which holds true for every w ∈ W 1,p (B+ (x0 , |λ|−1/2 ); C), every λ ∈ C \ {0} and some positive constant C3,p , is used to estimate the left-hand side of (12.3.1) by a positive constant C4,p times |λ|d/(2p) times the left-hand side of (12.3.3) for every λ ∈ C with Reλ ≥ λp ∨ 1. d

1

(ii) The inequality ||w||Lp (B+ (x0 ,(γ+1)|λ|−1/2 );C) ≤ C5,p [(γ + 1)d |λ|− 2 ] p ||w||∞ , which holds true for every w ∈ L∞ (Rd+ ; C) and it can be straightforwardly proved applying H¨older’s inequality, is used to estimate the right-hand side of (12.3.3) in terms of the sup-norm over Rd+ of u and its gradient. (iii) Taking the supremum over all x0 ∈ Rd+ , in the estimate obtained combining points (i) and (ii), it follows that p d |λ|||u||∞ + |λ||||∇u|||∞ + |λ| 2p sup ||D2 u||Lp (B+ (x0 ,|λ|−1/2 );C) x0 ∈Rd +

d p

≤C6,p (γ + 1) [||λu − Au||∞ + γ −1 |λ|||u||∞ + γ −1

p

|λ||||∇u|||∞ ].

(12.3.4)

Elliptic Equations in Rd+ with General Boundary Conditions

336

(iv) Fixing γ sufficiently large, the terms containing the L∞ -norm of u and its gradient can be moved from the right- to the left-hand side of (12.3.4) to obtain estimate (12.3.1). To complete the proof, let us prove that the equation λu − Au = f admits a unique 2,p solution u, which belongs to Cb1+α (Rd+ ; C) ∩ Wloc (Rd+ ; C), for every p ∈ [1, ∞) and α ∈ d (0, 1), and satisfies the condition Bu = 0 on ∂R+ , for every f ∈ L∞ (Rd+ ; C) and λ ∈ C, with Reλ > λ∞ := inf p∈(d,∞) λp ∨ 1. The uniqueness follows from (12.3.1). To prove the existence part, we fix λ, f and p > d such that Re λ ≥ λp . Further, we introduce a sequence (fn ) ∈ L∞ (Rd+ ; C) of compactly supported functions, such that ||fn ||∞ ≤ ||f ||∞ for every n ∈ N and fn converges to f pointwise almost everywhere on Rd+ , as n tends to ∞. By Theorem 12.2.4, for every n ∈ N there exists a unique function un ∈ W 2,p (Rd+ ; C) such that λun − Aun = fn and Bun = 0 on ∂Rd+ . By the Sobolev embedding theorem (see Theorem 1.3.6), each function un belongs also to Cb1 (Rd+ ; C) so that we can estimate |λ|||un ||∞ +

p ep ||f ||∞ . |λ||||∇un |||∞ + sup ||D2 un ||Lp (B+ (x0 ,|λ|−1/2 );C) ≤ C x0 ∈Rd +

Therefore, the sequence (un ) is bounded in W 2,p (B+ (x0 , |λ|−1/2 ); C) for every x0 ∈ Rd+ , by a 2−d/p 2,p (Rd+ , C) constant independent of x0 , so that there exists a function u ∈ Wloc (Rd+ ; C)∩Cb 2−d/p such that, up to a subsequence, un converges to u in C (B+ (0, r); C) for every r > 0, Dij un converges weakly to Dij u in Lp (B+ (0, r); C) for every i, j = 1, . . . , d. This is enough to infer that λu − Au = f on Rd+ and Bu = 0 on ∂Rd+ . Moreover, estimate (12.3.2) holds true with α = 1 − d/p. T 2,q Finally, to prove that u ∈ q p allows us to conclude T 2,q that u ∈ q d there exist two positive constants λ |λ|||u||∞ +

p d |λ||||∇u|||∞ + |λ| 2p sup ||D2 u||Lp (B+ (x0 ,|λ|−1/2 );C) x0 ∈Rd +

d



ep |λ| 2p ≤C

sup ||λu − Au||Lp (B+ (x0 ,|λ|−1/2 );C) +

x0 ∈Rd +

p |λ| sup ||g||Lp (B+ (x0 ,|λ|−1/2 );C) x0 ∈Rd +

 + sup |||∇g|||Lp (B+ (x0 ,|λ|−1/2 );C)

(12.3.5)

x0 ∈Rd +

2,p bp , where g ∈ W 1,p (Rd ; C) for every u ∈ Wloc (Rd+ ; C) ∩ Cb1 (Rd+ ; C) and λ ∈ C with Reλ ≥ λ + d is any function which coincides with Bu on ∂R+ . The constant C depends only on d, p and, in a continuous way, on the ellipticity constant of the operator A, on the modulus of continuity of its diffusion coefficients, on the sup-norms of the coefficients of the operator A and on the Cb1 (Rd−1 )- and Cb1 (Rd−1 ; Rd )-norms of the coefficients of the operator B. In particular, if u ∈ Cb2 (Rd+ ; C), then

|λ|||u||∞ +

p p b |λ||||∇u|||∞ + ||D2 u||∞ ≤ C(||λu − Au||∞ + |λ|||Bu||∞ + |||∇Bu|||∞ ), (12.3.6)

Semigroups of Bounded Operators and Second-Order PDE’s

337

b depends only on d and, in a continuous way, on the ellipticity constant where the constant C of the operator A, on the modulus of continuity of its diffusion coefficients, on the sup-norms of the coefficients of the operator A and on the Cb1 (Rd−1 )- and Cb1 (Rd−1 ; Rd )-norms of the coefficients of the operator B. Proof Throughout the proof, we denote by C a positive constant, which may vary from line to line and depends at most on d, p and, in a continuous way, on the ellipticity constant of the operator A, on the modulus of continuity of its diffusion coefficients, on the sup-norms of the coefficients of the operator A and on the Cb1 (Rd−1 )- and Cb1 (Rd−1 ; Rd )-norms of the coefficients of the operator B. Using the arguments in points (i) and (ii) of the proof of Theorem 12.3.1, it can be proved that p d |λ|||u||∞ + |λ||||∇u|||∞ + |λ| 2p sup ||D2 u||Lp (B+ (x0 ,|λ|−1/2 );C) x0 ∈Rd +

 d d 1 1p p ≤Cp (γ + 1) |λ| 2p sup ||λu − Au||Lp (B+ (x0 ,(1+γ)|λ|−1/2 ;C)) + |λ|||u||∞ + |λ||||∇u|||∞ γ γ x0 ∈Rd + 1

d

+ |λ| 2 + 2p sup ||g||Lp (B+ (x0 ,(1+γ)|λ|−1/2 );C) x0 ∈Rd +

 + sup |||∇g|||Lp (B+ (x0 ,(1+γ)|λ|−1/2 );C)

(12.3.7)

x0 ∈Rd +

2,p for every u ∈ Wloc (Rd+ ; C) ∩ Cb1 (Rd+ ; C), every g ∈ W 1,p (Rd+ ; C) which coincides with the d function Bu on ∂R+ , and every λ ∈ C with Re λ > λp ∨ 1. Up to taking γ large enough and replacing λp ∨ 1 with a larger value if needed, we can move the L∞ -norm of u and its gradient from the right- to the left-hand side of (12.3.7) and conclude that p d |λ|||u||∞ + |λ||||∇u|||∞ + |λ| 2p sup ||D2 u||Lp (B+ (x0 ,|λ|−1/2 );C) x0 ∈Rd +

 d d p ≤Cp (γ + 1) |λ| 2p sup ||λu − Au||Lp (B+ (x0 ,(1+γ)|λ|−1/2 );C) x0 ∈Rd + 1

d

+ |λ| 2 + 2p sup ||g||Lp (B+ (x0 ,(1+γ)|λ|−1/2 );C) x0 ∈Rd +

+ |λ|

d 2p

 sup |||∇g|||Lp (B+ (x0 ,(1+γ)|λ|−1/2 );C) .

(12.3.8)

x0 ∈Rd +

Finally, we observe that there exist N ∈ N, independent of |λ| and, for every x0 ∈ Rd+ , N points x1 , . . . , xN such that B(x0 , (γ + 1)|λ|−1/2 ) is contained in the union of the balls B(xj , |λ|−1/2 ) (j = 1, . . . , N ). Indeed, the ball B(0, γ + 1) is contained into the union of all the balls B(y, 1) centered at the points y ∈ B(0, γ + 1). By compactness, we can extract a subcovering {B(y1 , 1), . . . , B(yN , N )} of B(0, γ + 1). It thus follows that B(x0 , (γ + 1)|λ|−1/2 ) ⊂

N [

B(xj , |λ|−1/2 ),

j=1

where xj = x0 + |λ|−1/2 yj for every j = 1, . . . , N . Based on this remark, we can estimate Z N Z X |λu − Au|p dx ≤ |λu − Au|p dx, B(x0 ,(γ+1)|λ|−1/2 )

j=1

B(xj ,|λ|−1/2 )

Elliptic Equations in Rd+ with General Boundary Conditions

338 so that

1

||λu − Au||Lp (B(x0 ,(γ+1)|λ|−1/2 );C) ≤ N p sup ||λu − Au||Lp (B(x,|λ|−1/2 );C) . x∈Rd +

Hence, replacing this inequality in the right-hand side of (12.3.8), estimate (12.3.5) follows. Finally, estimate (12.3.6) follows from (12.3.5), using H¨older inequality to estimate the Lp -norms of the terms in (12.3.5) with their L∞ -norms.  We can now consider the case when the datum f belongs to Cb (Rd+ ; C) assuming the following set of assumptions. Hypotheses 12.3.3 (i) The coefficients qij = qji (i, j = 1, . . . , d) belong to BU C(Rd+ ), whereas the coefficients bj (j = 1, . . . , d) and c belong to Cb (Rd+ ; C); (ii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for all x ∈ Rd+ and ξ ∈ Rd . Theorem 12.3.4 For every f ∈ Cb (Rd+ ; C) and λ ∈ C, with Re λ > λ∞ , the equation 2,p (Rd+ ; C), for every p ∈ [1, ∞) λu − Au = f admits a unique solution u ∈ Cb1+α (Rd+ ; C) ∩ Wloc d e∞ and α ∈ (0, 1), such that Bu = 0 on ∂R+ . Moreover, there exist two positive constant C b b and C∞,λ , which depend only on d (the constant C∞,λ depends also on α and λ) and, in a continuous way, on the ellipticity constant of the operator A, on the modulus of continuity of its diffusion coefficients, on the sup-norms of the coefficients of the operator A and on the Cb1 (Rd−1 )- and Cb1 (Rd−1 ; Rd )-norms of the coefficients of the operator B, such that p e∞ ||λu − Au||∞ |λ|||u||∞ + |λ||||∇u|||∞ ≤ C and b∞,λ ||λu − Au||∞ ||u||C 1+α (Rd ;C) ≤ C b

+

for every λ ∈ C, with Re λ > λ∞ . Here, λ∞ is the same as in Theorem 12.3.1. Proof The proof is a straightforward consequence of Theorem 12.3.1.

12.4

Exercises

1. Add the missing details in the proof of Theorem 12.3.1.



Chapter 13 Elliptic Equations on Smooth Domains Ω

In this chapter, we analyze the boundary value problems ( λu − Au = f, in Ω, u = g, on ∂Ω and

(

λu − Au = f, in Ω, Bu = g, on ∂Ω,

(13.0.1)

(13.0.2)

when Ω is a bounded open set of class C k , with k = 2, k = 2 + α for some α ∈ (0, 1) or with k even larger, depending on the function spaces that the will consider, where A = Pd Pd ∂ is a first-order nontangential boundary i,j=1 qij Dij + j=1 bj Dj + c and B = aI + ∂η operator. To solve these problems we will take advantage of the results in Chapters 10, 11 and 12. Indeed, through local charts and suitable partitions of the unity, we will transform both problems (13.0.1) and (13.0.2) into different problems: one “far away” from the boundary of Ω and the other one “close” to the boundary of Ω. The first problem is then reduced to a problem in the whole Rd to which the results in Chapter 10 can be applied. The second problem is transformed into a problem on Rd+ to which the results in Chapters 11 and 12 apply. The chapter is organized as follows. In Section 13.1, we study problems (13.0.1) and (13.0.2) in spaces of H¨ older continuous functions. In Theorems 13.1.4 and 13.1.6, we prove optimal Schauder estimates for the two above problems. Then, in Subsection 13.1.1, assuming that Ω, the coefficients of the operators A and B and the data f and g, the solution to problems (13.0.1) and (13.0.2) are themselves smoother. In Section 13.2, we set our analysis in the context of Lp -spaces, for p ∈ (1, ∞). The main result of this section are Theorem 13.2.3 and Corollary 13.2.4, which provide an existence and uniqueness result of a solution u ∈ W 2,p (Ω; C) to problem (13.0.1) (resp. (13.0.2)) and an estimate of the W 2,p -norm of the solution in terms of the data of the problem. Also in this context, we prove some further regularity results, which are the natural counterpart of the results in Subsection 13.1.1. Finally, in Section 12.3, we study problems (13.0.1) and (13.0.2) when g ≡ 0 and f belongs to L∞ (Ω; C), to C(Ω; C) and to Cb (Ω; C), respectively. These results are the key-tool to prove the generation results contained in Chapter 14.

13.1

Elliptic Equations in C α (Ω;

C)

In this section, we assume the following conditions on Ω and the coefficients of the operators A and B. 339

340 Hypotheses 13.1.1

Elliptic Equations on Smooth Domains Ω (i) Ω is a bounded domain of class C 2+α for some α ∈ (0, 1);

(ii) the coefficients of the operator A belong to C α (Ω). Moreover, qij = qji for every i, j = 1, . . . , d; (iii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for all x ∈ Ω and ξ ∈ Rd ; (iv) the functions a and η belong to C 1+α (∂Ω) and C 1+α (∂Ω, Rd ), respectively. Moreover, a(x) ≥ 0, |η(x)| = 1 and hη(x), ν(x)i > 0 for each x ∈ ∂Ω, where ν(x) is the unit outward normal vector to ∂Ω at x ∈ ∂Ω. Throughout the section, we set c0 = max c(x)

(13.1.1)

x∈Ω

and consider the following lemmas. Lemma 13.1.2 Let Ω be an open set of class C k for some k > 0. Then, there exists a finite covering {Uj : j = 1, . . . , N − 1} of ∂Ω such that each Uj is an open subset of Rd . Moreover, for every j = 1, . . . , N − 1 there exists a bijective function ψj : Ui → B(0, r) of class C k , with inverse of class C k , such that ψj (Uj ∩ Ω) = B+ (0, r) := B(0, r) ∩ Rd+ and ψj (Uj ∩ ∂Ω) = B(0, r) ∩ ∂Rd+ . Finally, the following properties are satisfied. (i) There exist an open set UN , compactly contained in Ω, such that {Uj : j = 1, . . . , N } is a covering of Ω, and a family of functions {ϑi : i = 1, . . . , N } such that supp(ϑi ) ⊂ Uj PN for every j = 1, . . . , N and j=1 ϑ2j = 1 on Ω. 0 0 } is a , compactly contained in Ω, such that {U1 , . . . UN −1 , UN (ii) There exist an open set UN covering of Ω, and two families of functions {φi : i = 1, . . . , N } and {ζi : i = 1, . . . , N } such that supp(φj ) and supp(ζj ) are contained in Uj for every j = 1, . . . , N − 1, PN ∂ζj 0 , j=1 φj ζj = 1 on Ω and supp(φN ) and supp(ζN ) are contained in UN = 0 on ∂η ∂Ω.

Proof See Corollary B.1.3, Remark B.2.4 and Proposition B.2.13



Lemma 13.1.3 Under Hypotheses 13.1.1(i)–(iii), the following properties are satisfied. (i) Let Ω0 and Ω00 be two open subsets of Ω such that Ω0 b Ω00 b Ω. Then, there exists e with coefficients in C α (Rd ), such that the diffusion a uniformly elliptic operator A, b e is the trivial extension to Rd of the matrix is symmetric for every x ∈ Rd and Au function Au for every function u ∈ C 2 (Rd ; C), with support contained in Ω0 . (ii) Let Uk and ψk (k = 1, . . . , N − 1) be as in Lemma 13.1.2. For every function ζ : Ω ∩ Uk → R, denote by ζ ] the function defined by ζ ] = ζ ◦ ψk−1 : B+ (0, r) → R. Then, for every smooth enough function ϑ : Uk → R with compact support in Uk , there exists a uniformly elliptic operator Abk , with coefficients in Cbα (Rd+ ), such that the diffusion matrix is symmetric for every x ∈ Rd+ and, if u : Uk ∩ Ω → R is a smooth enough function and v is the trivial extension to Rd+ of the function (uϑ)] , then Abk v is the trivial extension outside B+ (0, r) of the function (A(uϑ))] , defined on B+ (0, r). (iii) Under the same notation as in (ii), there exists a first-order boundary differential ∂ operator Bbk = b a(k) I + (k) with coefficients in Cb1+α (Rd−1 ) and in Cb1+α (Rd−1 ; Rd ), ∂b η η (k) , ed i < 0 and Bbk v is the trivial extension outside respectively, such that inf Rd−1 hb d B(0, r) ∩ ∂R+ of the function (B(uϑ))] .

Semigroups of Bounded Operators and Second-Order PDE’s

341

If Ω is of class C k+2+α and the coefficients of the operator A belong to C k+α (Ω) for some k ∈ N, then the coefficients of operator Ae (resp. Abk ) belong to Cbk+α (Rd ) (resp. to Cbk+α (Rd+ )). Similarly, if the coefficients of the operator B belong to C k+1+α (∂Ω) and C k+1+α (∂Ω; Rd ), respectively, then the coefficients of the operator Bbk belong to Cbk+1+α (Rd−1 ) and Cbk+1+α (Rd−1 ; Rd ), respectively. Proof Throughout the proof, given a function ϑ defined on a subset of Rd , we denote by ϑ its trivial extension to Rd . Moreover, we denote by ψk,j (j = 1, . . . , d) the components of the function ψk . (i) We fix a function % ∈ Cc∞ (Rd ) such that χΩ0 ≤ % ≤ χΩ00 and define the coefficients qeij , bej (i, j = 1, . . . , d) and e c of the operator Ae as follows: qeij = %qij + (1 − %)δij , ebj = %bj and e c = %c for every i, j = 1, . . . , d. It is easy to check that the so defined functions belong to Cbα (Rd ) and d X

qeij (x)ξi ξj =%(x)

i,j=1

d X

qij (x)ξi ξj + (1 − %(x))|ξ|2

i,j=1

≥[%(x)µ + 1 − %(x)]|ξ|2 ≥ (1 ∧ µ)|ξ|2 e is the trivial for every x, ξ ∈ Rd , so that the operator Ae is uniformly elliptic. Moreover, Au extension to Rd of the function Au, if u is compactly supported in Ω0 . Finally, we note that, if the coefficients of the operator A belong to C k+α (Ω) for some k ∈ N, then the coefficients of the operator Ae belong to Cbk+α (Rd ). (ii) Let ϑ and u be as in the statement. A straightforward computation shows that A(uϑ) = (Ak (uϑ)] ) ◦ ψk , where Ak = Tr(Q(k) D2 ) + hb(k) , ∇i + c(k) and its coefficients are defined as follows: (k)

qij = [(Jac ψk )Q(Jac ψk )∗ ]]ij ,

(k)

bj

= [Tr(QD2 ψk,j ) + hb, ∇ψk,j i]] ,

c(k) = c]

for every i, j = 1, . . . , d. It is easy to check that the coefficients of the operator Ak belong to C α (B+ (0, r)) if the coefficients of the operator A belong to C α (Ω). Further, we note that hQ(k) (y)ξ, ξi =hQ(ψk−1 (y))(Jac ψk (ψk−1 (y)))∗ ξ, (Jac ψk (ψk−1 (y)))∗ ξi ≥µ|(Jac ψk (ψk−1 (y)))∗ ξ|2 for every y ∈ B+ (0, r) and ξ ∈ Rd . Since the matrix Jac ψk (x) is invertible at every point x ∈ Uxk , it follows that det(Jac ψk (x)) > 0 for every x ∈ Uxk . Thus, the function y 7→ ||(Jac ψk (ψk−1 y))−1 || is bounded by a positive constant Mk in B(0, r0 ), where r0 ∈ (0, r) is chosen in such a way that ψk (supp(ϑ)) ⊂ B(0, r0 ) . As a byproduct, we deduce that |(Jac ψk (ψk−1 (y)))∗ ξ| ≥ Mk−1 |ξ| for every y ∈ B(0, r0 ), ξ ∈ Rd , and this implies that there exists a positive constant µk such that hQ(k) (y)ξ, ξi ≥ µk |ξ|2 for every ξ ∈ Rd and y ∈ B+ (0, r0 ). Now, the operator Abk is obtained by extending the coefficients of the operator Ak to Rd+ arguing as in (i). The only difference is the definition of the function % ∈ Cc∞ (Rd ), which now is chosen in such a way that χB(0,r0 ) ≤ % ≤ χB(0,r00 ) , where r00 is arbitrarily fixed in the interval (r0 , r). Finally, it is easy to check that, if Ω is of class C k+2+α and the coefficients of the operator A belong to C k+α (Ω) for some k ∈ N, then the coefficients of the operator Abk belong to Cbk+α (Rd+ ). ∂(uϑ)] ∂(uϑ) (iii) A straightforward computation shows that = ◦ ψk on ∂Ω, where ∂η ∂η (k)

342

Elliptic Equations on Smooth Domains Ω

η (k) ◦ ψk = (Jacψk )η on Uk ∩ ∂Ω. We recall that ν(x) = −|∇ψk,d (x)|−1 ∇ψk,d (x) for every x ∈ Uk ∩ ∂Ω. Hence, ]  ∂ψk,d (k) ] = −|(∇ψk,d )] |hν ] , η ] i hη , ed i = h((Jacψk )η) , ed i = ∂η and, observing that |(∇ψk,d )] | nowhere vanishes on B(0, r) ∩ ∂Rd+ , we conclude that ∂(uϑ)] hη (k) , ed i < 0 on B(0, r) ∩ ∂Rd+ . It thus follows that (B(uϑ))] = a] (uϑ)] + on ∂η (k) B(0, r) ∩ ∂Rd+ . Clearly, a] and η (k) belong to C 1+α (B 0 (0, r)) and to C 1+α (B 0 (0, r); Rd ) where B 0 (0, r) denotes the ball of Rd−1 centered at zero and with radius r. We now extend these coefficients to Rd−1 with functions b a(k) and ηb(k) which belong to Cb1+α (Rd−1 ) 1+α d−1 d and Cb (R ; R ), respectively, in such a way that the condition inf Rd−1 hb η (k) , ed i < 0 is satisfied. For this purpose, we set b a(k) = ϕa ◦ ψk−1 (·, 0) + 1 − ϕ,

ηb(k) = ϕ(Jacψk )η ◦ ψk−1 (·, 0) + ϕ − 1,

where ϕ : Rd−1 → R is a smooth function compactly supported in B 0 (0, r) which is equal to one in B 0 (0, re), where re is chosen in such a way that ψk−1 (·, 0)(B 0 (0, re)) ⊂ supp ϑ. It is easy to check that, if Ω is of class C k+2+α for some k ∈ N, then the functions a(k) and µ b(k) belong to C k+1+α (Rd−1 ) and to C k+1+α (Rd−1 ; Rd ), respectively.  The main result of this section are the following two theorems. Theorem 13.1.4 Under Hypotheses 13.1.1(i)–(iii), there exists a sequence (λ0n ) ⊂ C \ [c0 , ∞) such that, for every λ ∈ C, with λ 6= λ0n for each n ∈ N, every f ∈ C α (Ω; C) and g ∈ C 2+α (∂Ω; C), the boundary value problem ( λu − Au = f, in Ω, (13.1.2) u = g, on ∂Ω, admits a unique solution u which belongs to C 2+α (Ω; C). Moreover, there exists a positive constant C, which depends on d, Ω and, in a continuous way, on λ, µ and the C α (Ω)-norm of the coefficients of the operator A, such that ||u||C 2+α (Ω;C) ≤ C(||f ||C α (Ω;C) + inf{||G||C 2+α (Ω;C) : G|∂Ω = g}),

(13.1.3)

where G is any function in C 2+α (Ω; C) which coincides with the function g on ∂Ω. Remark 13.1.5 Note that Proposition B.4.9 guarantees the existence of a function G with the properties in the statement of Theorem 13.1.4. Theorem 13.1.6 Under Hypotheses 13.1.1, there exists a sequence (λ00n ) ⊂ C \ (c0 , ∞) such that for every λ ∈ C, with λ 6= λ00n for each n ∈ N, every f ∈ C α (Ω; C) and every g ∈ C 1+α (∂Ω; C), the boundary value problem ( λu − Au = f, in Ω, (13.1.4) Bu = g, on ∂Ω, admits a unique solution u which belongs to C 2+α (Ω; C). Moreover, there exists a positive constant C, which depends on d, Ω and, in a continuous way, on λ, µ, the C α (Ω)-norm of the coefficients of the operator A and the C 1+α (∂Ω)- and C 1+α (∂Ω; Rd )-norm of the coefficients of the operator B, such that ||u||C 2+α (Ω;C) ≤ C(||f ||C α (Ω;C) + inf{||G||C 1+α (Ω;C) : G|∂Ω = g}),

(13.1.5)

Semigroups of Bounded Operators and Second-Order PDE’s

343

where G is any extension of the function g which belongs to C 1+α (Ω; C). Remark 13.1.7 Note that Proposition B.4.14 guarantees the existence of a function G with the properties in the statement of Theorem 13.1.6. Arguing as in the forthcoming Corollary 14.1.4, the dependence of the constant C in (13.1.3) and (13.1.4) can be made sharp when g ≡ 0. In fact, C depends linearly on |λ|. Proof of Theorem 13.1.4 In the first part of the proof, we will study problem (13.1.2) for λ ∈ R sufficiently large. Based on this result, we will complete the proof using a compactness argument. The uniqueness of the solution u ∈ C 2+α (Ω; C) to problem (13.1.2), for λ ≥ c0 , follows immediately from Corollary 4.2.5, applied to the real and the imaginary parts of u. Let us now prove the existence part. Throughout the rest of the proof, C denotes a positive constant, which may vary from line to line and depends (if not otherwise specified) at most on d, Ω and, in a continuous way, on the ellipticity constant of the operator A and the C α (Ω)-norm of its coefficients. Finally, given a function h, defined in a subset of Rd , we denote by h its trivial extension to Rd . Let us fix f ∈ C α (Ω; C), λ > c0 and G as in the statement of the theorem. Then, u ∈ C 2+α (Ω; C) solves the boundary value problem (13.1.2) if and only if the function w = u−G, which belongs to C 2+α (Ω; C) and vanishes on ∂Ω, solves the equation λw−Aw = fe := f − λG + AG on Ω. Hence, in the rest of the proof, we assume that g identically vanishes on ∂Ω. We will determine λ∗ > 0 and, for every λ > λ∗ a bounded operator S(λ) mapping C α (Ω; C) into C 2+α (Ω; C) such that S(λ)g identically vanishes on ∂Ω and (λ − A)S(λ)φ = φ + T (λ)φ for every φ ∈ C α (Ω; C), where T (λ) belongs to L(C α (Ω; C)) and its operator norm is less than 1/2. We then deduce that the operator I + T (λ) is invertible in C α (Ω; C), so that S(λ)(I + T )−1 f is the solution to problem (13.1.2) (with g = 0) for every f ∈ C α (Ω; C). In the rest of the proof, we consider the covering {Uk : k = 1, . . . , N } of Ω and the partition of the unity {ϑ2k : k = 1, . . . , N } in Lemma 13.1.2(i). We will define the operator PN S(λ), for λ large enough, as the sum S(λ) = k=1 Sk (λ), where the operators Sk (λ) (k = 1, . . . , N ) are defined in Steps 1 and 2. Step 1 (the operator SN (λ)). We fix φ ∈ C α (Ω; C) and set φN = ϑN φ. Clearly, φN belongs to Cbα (Rd ; C) and its norm can be bounded from above in terms of the α-H¨older norm of φ. Next, we introduce the uniformly elliptic operator AeN in Lemma 13.1.3(i) such that AeN v is the trivial extension of the function Av for every smooth enough function v with support contained in UN . From Theorem 10.1.4 it follows that there exists a positive constant λN ≥ c0 , which depends only on d, Ω and, in a continuous way on the ellipticity constant µ and the C α -norm of the coefficients of the operator A, such that the equation λv−AeN v = φN admits a unique solution uN ∈ Cb2+α (Rd ; C) for every λ ∈ C with Reλ ≥ λN . Moreover, ||uN ||C 2+α (Rd ;C) ≤ C||φN ||Cbα (Rd ;C) ≤ C||φ||C α (Ω;C) ,

(13.1.6)

b

where the constant C depends in a continuous way also on λ. For simplicity we set RN (λ)φ := uN . Due to the previous estimate, RN (λ) is a bounded operator mapping C α (Ω; C) into Cb2+α (Rd ; C). Moreover, from Corollary 4.2.2(i), it follows that ||RN (λ)||L(C(Ω),Cb (Rd ;C)) ≤ C(λ − λN )−1 ,

λ > λN .

(13.1.7)

Now, for every λ > λN we introduce the operator SN (λ) = ϑN RN (λ). Clearly, SN (λ) is bounded from C α (Ω; C) into Cc2+α (UN ; C). Moreover, (λI − A)SN (λ)φ =ϑN (λ − AeN )RN (λ)φ + hb, ∇ϑN iRN (λ)φ

344

Elliptic Equations on Smooth Domains Ω − 2hQ∇ϑN , ∇RN (λ)φi + Tr(QD2 ϑN )RN (λ)φ = : ϑ2N φ + TN φ.

Step 2 (the operator Sk (λ)). We fix k < N and extend the function φ ∈ C α (Ω; C) to Rd with the function Eα φ, where Eα is the extension operator in Proposition B.4.1. Next, we introduce the function φk = (ϑk Eα φ) ◦ ψk−1 defined and compactly supported in B(0, r). The function φk belongs to Cbα (Rd ; C) and is α-H¨older norm over Rd can be estimated from above by a positive constant, independent of φ, times the α-H¨older norm of φ. We consider the operator Abk in Lemma 13.1.3(ii), where Abk (wϑk ) ◦ ψk−1 is the trivial extension of the function (A(ϑk w)) ◦ ψk−1 for every function w ∈ C 2 (Ω; C), and apply Theorem 11.2.4, which guarantees that there exists a positive constant λk , which depends only on d, Ω and, in a continuous way, on the ellipticity constant µ and the C α (Ω; C)-norm of the coefficients of the operator A, such that the equation λv − Abk v = φk admits a unique solution vk ∈ Cb2+α (Rd+ ; C), which vanishes on ∂Rd+ , for every λ > λk . Moreover, ||vk ||C 2+α (Rd ;C) ≤ C||φ||C α (Ω;C) ,

(13.1.8)

+

b

where the constant C also depends, in a continuous way, on λ, and, arguing as in the first part of the proof, we infer that ||vk ||∞ ≤ C(λ − λk )−1 ||φ||∞ .

(13.1.9)

We set Rk (λ)φ := vk for λ > λk and introduce the operator Sk (λ) defined on C α (Ω; C) by Sk (λ)φ = ϑk (Rk (λ)φ) ◦ ψk for every φ ∈ C α (Ω; C). Clearly, Sk (λ) is bounded from C α (Ω; C) into C 2+α (Ω; C). Moreover, (λI − A)Sk (λ)φ = ϑ2k φ + Tk φ, where Tk f = −([A, ϑk ◦ ψk−1 ]Rk (λ)f ) ◦ ψk , and Sk (λ)φ identically vanishes on ∂Ω, since (Rk (λ)φ) ◦ ψk vanishes on Uk ∩ ∂Ω. e := maxk=1,...,N λk , we can define the operator S(λ) Step 3 (Conclusion). For every λ > λ as illustrated above. Clearly, S(λ) ∈ L(C α (Ω; C), C 2+α (Ω; C)) and, from the computations in Steps 1 and 2, it follows that (λ − A)S(λ)φ =

N X

ϑ2k φ +

k=1

N X

Tk (λ)φ := φ + T (λ)φ

k=1

for every φ ∈ C α (Ω; C) as it has been claimed. Applying (1.1.2) and (1.1.6), with (θ, β, α) = (0, 1 + α, 2 + α) and taking (13.1.6), (13.1.7), (13.1.8) and (13.1.9) into account, it can be shown that 1

e − 2+α ||Rk (λ)||L(C α (Ω;C),C 1+α (Rd ;C)) ≤ C(λ − λ) b

+

1 e so that ||Tk (λ)f ||L(C α (Ω;C)) ≤ C(λ−λ) e − 2+α e and k = 1, . . . , N . for every λ > λ for every λ > λ e Now, we can fix λ∗ > λ such that ||T (λ)||L(C α (Ω;C)) ≤ 1/2 for every λ ≥ λ∗ , and the solvability of (13.1.2) follows for such values of λ. To complete the proof, we show that problem (13.1.2) is actually solvable up to sequence (λn ) of complex numbers and that the half-line [c0 , ∞) is contained in the set of λ’s for which problem (13.1.2) admits a solution in C 2+α (Ω; C). Again we assume that g identically vanishes. Suppose that λ∗ > c0 (otherwise there is nothing to prove). Then, from (13.1.3) it follows that

||u||C 2+α (Ω;C) ≤C||λ∗ u − Au||C α (Ω;C)

Semigroups of Bounded Operators and Second-Order PDE’s

345

≤C(|λ∗ − λ|||u||C α (Ω;C) + ||λu − Au||C α (Ω;C) ) ≤C((λ∗ − c0 )||u||C α (Ω;C) + ||λu − Au||C α (Ω;C) ) ≤C(||u||C α (Ω;C) + ||λu − Au||C α (Ω;C) )

(13.1.10)

for every u ∈ C 2+α (Ω; C) which vanishes on ∂Ω and every λ ∈ [c0 , λ∗ ]. Note that the constant C is independent of λ. Using Corollary 1.1.5 and applying Young’s inequality, we can estimate ||u||C α (Ω;C) ≤ ε||u||C 2+α (Ω;C) + Cε ||u||∞ for every ε > 0 and some positive constant Cε , independent of u and blowing up as ε tends to 0+ . Replacing this inequality in (13.1.10) and properly choosing ε, we can show that ||u||C 2+α (Ω;C) ≤ C(||u||C(Ω;C) + ||λu − Au||C α (Ω;C) ). Using Corollary 4.2.5 we conclude that ||u||C 2+α (Ω;C) ≤ C||λu − Au||C α (Ω;C) .

(13.1.11)

Denote by A : {u ∈ C 2+α (Ω; C) : u|∂Ω ≡ 0} → C α (Ω; C) the operator defined by Au = Au for every u ∈ C 2+α (Ω; C) which vanishes on ∂Ω. Estimate (13.1.11) shows that ||R(λ, A)||L(C α (Ω;C)) ≤ C for every λ ∈ ρ(A). Since the norm of the resolvent operator should blow up as λ tends to a point in σ(A) (see Remark A.4.5), the interval [c0 , ∞) is contained in the resolvent set of the operator A, i.e., the boundary value problem (13.1.2) is solvable for every λ ≥ c0 . Now, we observe that, since C 2+α (Ω; C) is compactly embedded into C α (Ω; C), due to Arzel` a-Ascoli theorem, the operator R(c0 , A) is compact from C α (Ω; C) into itself. Consequently its spectrum is a sequence of eigenvalues. As a byproduct, the spectrum of the operator A consists of a sequence (λn ) of eigenvalues only. This implies that the boundary value problem (13.1.2) is solvable for every f ∈ C α (Ω; C) and g ∈ C 2+α (∂Ω; C) for every λ ∈ C such that λ 6= λn for every n ∈ N.  Proof of Theorem 13.1.6 Let E be the extension operator in Proposition B.4.14(i). Since BEg ≡ g, it follows easily that u solves problem (13.1.4) if and only if function v = u−Eg solves the same problem with g ≡ 0 on ∂Ω. Hence, in the rest of the proof we solve this latter problem. The strategy that we will adopt is the same used to prove Theorem 13.1.4, so that we skip some details. The main step consists in proving the solvability of problem (13.1.4), with g ≡ 0, for λ ∈ R sufficiently large and f ∈ C α (Ω). Then, the compactness of the resolvent operator associated to the operator A : D(A) := {u ∈ C 2+α (Ω) : Bu ≡ 0 on ∂Ω} → C α (Ω), defined by Au = Au for every u ∈ D(A), will allow to infer that the spectrum of A consists of a sequence of eigenvalues. The estimate ||u||C 2+α (Ω;C) ≤ C||λu − Au||C α (Ω;C) for λ > c0 will also show that (c0 , ∞) ⊂ ρ(A). Here and throughout the proof, C denotes a positive constant, which may vary from line to line, and depends at most on d, Ω and, in a continuous way, on the ellipticity constant of the operator A, the C α (Ω)norm of its coefficients and the C 1+α (∂Ω)- and C 1+α (∂Ω; Rd )-norm of the coefficients of the operator B. Moreover, given a function ϑ defined in a subset of Rd , we denote by ϑ its trivial extension by zero to Rd . The uniqueness part of the proof follows, for λ > c0 , from Corollary 4.2.9(ii). So, let us deal with the existence part. In the rest of the proof, we consider the covering 0 {U1 , . . . UN −1 , UN } and the functions φ1 , . . . , φN , ζ1 , . . . , ζN in Lemma 13.1.2(ii). We will determine λ∗ > 0 and, for every λ > λ∗ , a bounded operator S 0 (λ) mapping C α (Ω; C) into C 2+α (Ω; C), such that the normal derivative of S 0 (λ)Φ identically vanishes on ∂Ω and (λ − A)S 0 (λ)Φ = Φ + T 0 (λ)Φ for every Φ ∈ C α (Ω; C), where T 0 (λ) belongs to L(C α (Ω; C)) and its operator norm is less than 1/2. We then deduce that the operator I + T 0 (λ) is invertible in C α (Ω; C), so that S 0 (λ)(I + T 0 (λ))−1 f is the solution to problem (13.1.4) (with

346

Elliptic Equations on Smooth Domains Ω

g ≡ 0) for every f ∈ C α (Ω; C). We will define the operator S 0 (λ), for λ large enough, as the PN sum S(λ) = k=1 Sk0 (λ), where the operators Sk (λ) (k = 1, . . . , N ) are defined in Steps 1 and 2. Once this result is achieved, we will be almost done. Indeed, using the compactness of the resolvent operator R(2λ∗ , A), it follows easily that the spectrum of the operator A consists of a sequence (λ00n ) of eigenvalues. This implies that the boundary value problem (13.1.4) admits a unique solution in C 2+α (Ω; C) for every λ ∈ C \ {λ00n : n ∈ N}. Finally, λ00n ∈ / (c0 , ∞) for every n ∈ N. Indeed, we claim that the function ||R(·, A)||L(C α (Ω;C)) stays bounded in ρ(A) ∩ [c1 , ∞) for every c1 > c0 , and Remark A.4.5 allows us to conclude that the interval [c1 , ∞) is contained in ρ(A). To prove the claim, we assume that c1 < λ∗ , otherwise there is nothing to prove, and observe that, for λ ∈ (c1 , λ∗ ], we have ||u||C 2+α (Ω;C) ≤C||2λ∗ u − Au||C α (Ω;C) ≤C(|2λ∗ − λ|||u||C α (Ω;C) + ||λu − Au||C α (Ω;C) ) ≤C(2λ∗ − c0 |||u||C α (Ω;C) + ||λu − Au||C α (Ω;C) ) ≤C(||u||C α (Ω;C) + ||λu − Au||C α (Ω;C) )

(13.1.12)

using the interpolative estimate (1.1.6) and Corollary 4.2.9(ii) to estimate ||u||C α (Ω;C) ≤ ε||u||C 2+α (Ω;C) + Kε ||u||∞ ≤ ε||u||C 2+α (Ω;C) + Kε (λ − c0 )−1 ||λu − Au||∞ . Replacing this estimate in (13.1.12) and choosing ε sufficiently small, we conclude that the function ||R(·, A)||L(C α (Ω;C)) is bounded in (c1 , ∞) for every c1 > c0 . 0 0 0 0 (λ) (λ) = ζN RN (λ) by setting SN (λ)). We define the operator SN Step 1 (the operator SN α 0 for λ > λN , where RN (λ) is the operator which to any Φ ∈ C (Ω; C) associates the unique e = φN Φ. Here, Ae is an elliptic operator solution in Cb2+α (Rd ; C) of the equation λu − Au α d e with coefficients in Cb (R ) such that A(vφN ) = A(vφN ) on Ω for every v ∈ C 2+α (Ω; C). It 0 0 (λ) (λ)Φ = ζN φN Φ+TN0 (λ)Φ in Ω, where TN0 = −[A, ζn ]RN is easy to check that (λI −A)SN and [A, ζn ] is the commutator between the operators A and ζn I. Moreover, 0 ||RN (λ)||L(C α (Ω;C),C 2+α (Rd ;C)) ≤ C

(13.1.13)

b

0 ||RN (λ)||L(C(Ω),Cb (Rd ;C)) ≤ C(λ − λN )−1 ,

λ > λN .

(13.1.14)

Step 2 (the operator Sk0 (λ)). We fix k < N and extend the function Φ ∈ C α (Ω; C) to Rd with the function Eα Φ. Next, we introduce the function Φk = (φk Eα Φ) ◦ ψk−1 , defined and compactly supported in B(0, r). Therefore, the function Φk belongs to Cbα (Rd ; C) and is α-H¨ older norm over Rd can be estimated from above by a positive constant, independent of Φ, times the α-H¨ older norm of Φ. Using Lemma 13.1.3(ii), we introduce a uniformly elliptic b operator Ak with coefficients in Cbα (Rd+ ) such that Abk vk = A(φk w) ◦ ψk−1 on B+ (0, r) for every function w ∈ C 2 (Ω ∩ Uk ), where vk is the trivial extension outside B+ (0, r) of the function (φk w) ◦ ψk−1 . bk and Bbk the elliptic operator and the boundary operator in Lemma 13.1.3. TheLet A orem 12.1.3 guarantees that there exists a positive constant λk , which depends only on d, Ω and, in a continuous way, on the ellipticity constant µ, the C α -norm of the coefficients of the operator A and the C 1+α -norm of the coefficients of the operator B, such that the boundary value problem ( λv − Abk v = Φk , in Rd+ , Bbk v = 0, on ∂Rd+ , admits a unique solution vk ∈ Cb2+α (Rd+ ; C) for every λ > λk and vk satisfies the estimate ||vk ||C 2+α (Rd ;C) ≤ C||Φ||C α (Ω;C) . b

+

(13.1.15)

Semigroups of Bounded Operators and Second-Order PDE’s

347

Moreover, Proposition 4.2.10(ii) shows that ||vk ||∞ ≤ C(λ − λk )−1 ||Φ||∞ .

(13.1.16)

We set vk =: Rk0 (λ)Φ for λ > λk and introduce the operator Sk0 (λ) defined on C α (Ω; C) by Sk (λ)Φ = ζk (Rk0 (λ)Φ) ◦ ψk for every Φ ∈ C α (Ω; C) and λ > λk . As it is easily seen, Sk0 (λ) ∈ L(C α (Ω; C), C 2+α (Ω)) for every λ > λk . Moreover, (λI − A)Sk0 (λ)Φ = φk ζk Φ + Tk0 (λ)Φ, where Tk0 (λ)Φ = −([A, ζk ◦ ψk−1 ]Rk0 (λ)Φ) ◦ ψk . Finally, BSk0 (λ)Φ = ζk Bbk Rk0 (λ)Φ +

∂ζk 0 (Rk (λ)Φ) ◦ ψk = 0 ∂η

on Uk ∩ ∂Ω and, hence, on the whole ∂Ω. e := maxk=1,...,N λk , we can define the operator S 0 (λ) Step 3 (Conclusion). For every λ > λ 0 as illustrated above. Clearly, S (λ) ∈ L(C α (Ω; C), C 2+α (Ω; C)) and, from the computations in Steps 1 and 2, it follows that (λ − A)S 0 (λ)Φ =

N X k=1

ζk φk Φ +

N X

Tk0 (λ)Φ := Φ + T (λ)Φ

k=1

for every Φ ∈ C α (Ω; C). Moreover, BS(λ)Φ = 0 on ∂Ω. Applying (1.1.2) and (1.1.6), with (θ, β, α) = (0, 1 + α, 2 + α) and taking (13.1.13), (13.1.14), (13.1.15) and (13.1.16) into account, it can be shown that 1

0 e − 2+α ||Rk0 (λ)||L(C α (Ω;C),C 1+α (Rd ;C)) + ||RN (λ)||L(C α (Ω;C),C 1+α (Rd ;C)) ≤ C(λ − λ) b

+

b

1 e and k = 1, . . . , N − 1, so that ||Tk (λ)f ||L(C α (Ω;C)) ≤ C(λ − λ) e − 2+α for every λ > λ for every e e λ > λ and k = 1, . . . , N . Now, we can fix λ∗ > λ such that ||T (λ)||L(C α (Ω;C)) ≤ 1/2 for every λ ≥ λ∗ , and the solvability of (13.1.4) follows for such values of λ. 

Corollary 13.1.8 The following properties are satisfied. (i) Under Hypotheses 13.1.1(i)–(iii), there exists a positive constant C1 , which depends on d, Ω and, in a continuous way, on µ and the C α (Ω)-norms of the coefficients of the operator A, such that ||u||C 2+α (Ω;C) ≤ C1 (||u||∞ + ||Au||Cbα (Ω;C) + ||G||C 2+α (Ω;C) )

(13.1.17)

for every u ∈ Cb2+α (Ω; C), where G is any function in C 2+α (Ω) which coincides with u on ∂Ω. (ii) Under Hypotheses 13.1.1, there exists a positive constant C2 , which depends on d, Ω and, in a continuous way, on µ, the C α (Ω)-norms of the coefficients of the operator A and on the C 1+α (∂Ω)- and C 1+α (∂Ω; C)-norms of the coefficients of the operator B, such that ||u||C 2+α (Ω;C) ≤ C2 (||u||∞ + ||Au||Cbα (Ω;C) + ||G||C 1+α (Ω;C) )

(13.1.18)

for every u ∈ Cb2+α (Ω; C), where G is any function in C 1+α (Ω) which coincides with Bu on ∂Ω.

348

Elliptic Equations on Smooth Domains Ω

Proof (i) Fix u ∈ C 2+α (Ω; C) and G ∈ C 2+α (Ω; C) which coincides with u on ∂Ω. Applying estimate (13.1.3) with λ = 2c0 and f = λu − Au, we can write e1 (||u||C α (Ω;C) + ||Au||C α (Ω;C) + ||G||C 2+α (Ω;C) ) ||u||C 2+α (Ω;C) ≤ C

(13.1.19)

e1 , which depends only on d, Ω and, in a continuous way, on for some positive constant C α µ and the C (Ω)-norms of the coefficients of the operator A. Using (1.1.6) and Young’s inequality, we estimate ||u||C α (Ω;C) ≤ ε||u||C 2+α (Ω;C) + Kε ||u||∞ for every ε > 0 and some positive constant Kε , independent of u and blowing up as ε tends to 0+ . Replacing this estimate in (13.1.19) and choosing ε sufficiently small we can get rid of the C 2+α (Ω; C)-norm of u from the right-hand side of (13.1.19), and estimate (13.1.17) follows. (ii) We consider the function EBu, where E is the extension operator in Proposition B.4.14, which is bounded from C 1+α (∂Ω; C) into C 2+α (Ω; C), and apply the same arguments as in the proof of property (i) to the function u − EBu to infer that e2 (||u||C α (Ω;C) + ||Au||C α (Ω;C) + ||Bu||C 1+α (∂Ω;C) ) ||u||C 2+α (Ω;C) ≤ C

(13.1.20)

e2 which depends only on d, Ω and, in a continuous way, on µ, the for some positive constant C α C (Ω)-norms of the coefficients of the operator A and on the Cb1+α (∂Ω)- and Cb1+α (∂Ω; C)norms of the coefficients of the operator B. Clearly, if G belongs to C 1+α (Ω; C) and coincides with the function Bu on ∂Ω, then ||G||C 1+α (Ω;C) ≥ ||Bu||C 1+α (∂Ω;C) . Replacing this estimate in the last side of (13.1.20), estimate (13.1.18) follows immediately.  Weakening the assumptions on g we can prove the following result. Theorem 13.1.9 Let Hypotheses 13.1.1(i)–(iii) be satisfied and let λ ≥ c0 , where c0 is defined in (13.1.1). Then, for every f ∈ C α (Ω; C) and g ∈ C(∂Ω; C) there exists a unique 2+α solution u ∈ Cloc (Ω; C) ∩ C(Ω; C) of problem (13.1.2). Proof In view of Theorem 13.1.4, it suffices to prove that for every g ∈ C(∂Ω; C) the 2+α boundary value problem (13.1.2) with f ≡ 0 admits a unique solution u ∈ Cloc (Ω; C) ∩ C(Ω; C). The uniqueness follows from Theorem 4.2.4 as in the proof of Theorem 13.1.4. To prove the existence part, we fix g ∈ C(∂Ω; C). Using Proposition B.4.9, we extend g with a function ψ ∈ C(Ω; C). Then, we consider a sequence {gn } of functions in C 2+α (Ω; C) which converges to ψ uniformly on Ω. In view of Theorem 13.1.4, for every λ ≥ c0 and n ∈ N there exists a unique function un ∈ C 2+α (Ω; C) which solves the elliptic equation λun − Aun = 0 and satisfies the condition un ≡ gn on ∂Ω. From Corollary 4.2.5 it follows that ||un − um ||∞ ≤ C||gn − gm ||∞ for every m, n ∈ N and some positive constant C, independent of m and n. This estimate can be used to infer that un converges uniformly on Ω to a continuous function, which coincides with g on ∂Ω. Next, we fix an open set Ω∗ b Ω. Using the interior Schauder estimates in Theorem e n − um ||∞ for some positive constant C, e 10.1.9, we obtain that ||un − um ||C 2+α (Ω∗ ;C) ≤ C||u independent of m and n, so that un converges to u in C 2+α (Ω∗ ; C). We thus deduce that 2+α u ∈ Cloc (Ω; C). Finally, letting n tend to ∞ in the equation λun − Aun = 0 we conclude that λu − Au = 0 in Ω. 

13.1.1

Further regularity results

In this subsection, assuming Ω more regular, we prove that the more the data f and g and the coefficients of the operators A, B are smooth the more the solutions of the boundary value problems (13.1.2) and (13.1.4) are smooth.

Semigroups of Bounded Operators and Second-Order PDE’s Hypotheses 13.1.10 and k ∈ N;

349

(i) Ω is a bounded domain of class C k+2+α for some α ∈ (0, 1)

(ii) the coefficients of the operator A belong to C k+α (Ω). Moreover, qij = qji on Ω for every i, j = 1, . . . , d; (iii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for all x ∈ Ω and ξ ∈ Rd ; (iv) the functions a and η belong to C k+1+α (∂Ω) and C k+1+α (∂Ω, Rd ), respectively. Moreover, a(x) ≥ 0, |η(x)| = 1, and hη(x), ν(x)i > 0 for each x ∈ ∂Ω. Theorem 13.1.11 Let f and g1 , g2 be three functions which belong to C k+α (Ω; C), C k+2+α (∂Ω; C) and C k+1+α (∂Ω; C), respectively. Then the following properties are satisfied. (i) Under Hypotheses 13.1.10(i)–(iii), suppose that u ∈ C 2+α (Ω; C) solves the boundary value problem ( λu − Au = f, in Ω, (13.1.21) u = g1 , on ∂Ω, for some λ ∈ C \ {λ0n : n ∈ N}, where {λ0n } is the sequence in Theorem 13.1.4. Then, u belongs to C k+2+α (Ω; C). Moreover, there exists a positive constant C, which depends on d, Ω and, in a continuous way, on µ and the C α (Ω)-norm of the coefficients of the operator A, such that ||u||C k+2+α (Ω;C) ≤ C||f ||C k+α (Ω;C) + inf{||G||C k+2+α (Ω;C) : G = g1 on ∂Ω}.

(13.1.22)

(ii) Under Hypotheses 13.1.10, let u ∈ C 2+α (Ω; C) solve the boundary value problem ( λu − Au = f, in Ω, (13.1.23) Bu = g2 , on ∂Ω, for some λ ∈ C \ {λ00n : n ∈ N}, where {λ00n } is the sequence in Theorem 13.1.6. Then, u belongs to C k+2+α (Ω; C). Moreover, there exists a positive constant C, which depends on d, Ω and, in a continuous way, on µ, the C α (Ω)-norm of the coefficients of the operator A and the C 1+α (∂Ω)- and C 1+α (∂Ω; Rd )-norms of the operator B, such that ||u||C k+2+α (Ω;C) ≤ C||f ||C k+α (Ω;C) + inf{||G||C k+1+α (Ω;C) : G = g2 on ∂Ω}.

(13.1.24)

Proof Fix λ u, f, g1 and g2 as in the statement of the theorem. Note that, as in the proof of Corollary 13.1.8, it suffices to consider the case when g1 (resp. g2 ) identically vanishes on ∂Ω. Indeed, for a general g1 (resp. g2 ), the function u − G1 (resp. u − G2 ), where G1 is any function in C k+2+α (Ω; C) such that G1 ≡ g1 (resp. G2 ∈ C k+1+α (Ω; C) such that G2 ≡ g2 ) on ∂Ω belongs to C 2+α (Ω; C) and u ≡ 0 (resp. Bu = 0) on ∂Ω. Note that Proposition B.4.9 (resp. Proposition B.4.12) guarantees the existence of a function G ∈ C k+2+α (Ω; C) (resp. C k+1+α (Ω; C)) which coincides with g1 (resp. g2 ) on ∂Ω. So, in the rest of the proof, we assume that g1 ≡ 0 (resp. g2 ≡ 0) on ∂Ω. Moreover, throughout the proof, we denote by C∗ a positive constant, which depends at most on d, Ω and, in a continuous way on λ, µ, the Cbk+α (Ω)-norm of the coefficients of the operator A and the C 1 -norm of the coefficients of the operator B, and may vary from line to line. Moreover, given a function ζ, defined on a subset of Rd , we denote by ζ its trivial extension to Rd . To prove that u belongs to C k+2+α (Ω; C), we use the usual procedure based on local charts. For this purpose, we consider the open coverings {Uh : h = 1, . . . , N } of Ω, defined

350

Elliptic Equations on Smooth Domains Ω

in Lemma 13.1.2. We also consider a partition of the unity {ϑi : i = 1, . . . , N } subordinated to such a covering. PN We split the solution u to problem (13.1.21) (resp. (13.1.23)) into the sum u = h=1 uh and prove that each function uh = uϑh belongs to C k+2+α (Ω; C). We begin by considering the case k = 1. The function uN . Using Lemma 13.1.3(i), we introduce the uniformly elliptic operator e e is the trivial extension of the function AN , whose coefficients belong to Cbk+α (Rd ) and Av Av for every sufficiently smooth function v, compactly supported in UN . Observe that the function uN solves the elliptic equation λuN − AeN uN = fN on Rd , where fN = f ϑN − 2hQ∇u, ∇ϑN i − u(AϑN − cϑN ). Since it is compactly supported in Ω, function fN belongs to Cb1+α (Rd ). Thus, we can apply Theorem 10.1.7 to show that uN belongs to Cb3+α (Rd ) and (13.1.25) ||uN ||C 3+α (Rd ;C) ≤ C∗ (||fN ||C 1+α (Rd ;C) + ||u||C 1+α (Ω;C) ). b

b

(If the real part of λ is not larger than the supremum over Rd of the potential term of the operator AeN (say e c0 ), it suffices to write the equation λuN − AeN uN = fN in the form c0 .) λ∗ uN − AeN uN = fN + (λ∗ − λ)uN , where λ∗ ∈ C has real part greater than e Note that ||fN ||C 1+α (Rd ;C) ≤ C∗ (||f ||C 1+α (Ω;C) + ||u||C 2+α (Ω;C) ). Hence, taking Theorem b b 13.1.4 into account, from (13.1.25) we conclude that ||uN ||C 3+α (Rd ;C) ≤ C∗ ||f ||C 1+α (Ω;C) .

(13.1.26)

b

The function uh (h < N ). We fix h < N and introduce the uniformly elliptic operator b Ah in Lemma 13.1.3(ii) with coefficients in Cbk+α (Rd+ ) and such that (A(ϑh u))◦ψh−1 = Abh vh on B+ (0, r), where vh is the trivial extension outside B+ (0, r) of the function (ϑh u) ◦ ψh−1 . Function vh belongs to Cb2+α (Rd+ ; C) and solves the equation λvh − Abh vh = fh , where fh = (f ϑh )] − 2hQ(h) ∇u] , ∇ϑ]h i − u] (Abh ϑ]h − c(h) ϑ]h )u] . It follows easily that fh belongs to Cb1+α (Rd+ ; C) and, taking (13.1.3) (resp. (13.1.5)) into account, we can estimate ||fh ||C 1+α (Rd ;C) ≤ C∗ ||f ||C 1+α (Ω;C) . + b Further, we note that ˆ vh vanishes on ∂Rd+ if u vanishes on ∂Ω;

 ∂ϑh u ◦ ψk−1 , if Bu identically ∂η vanishes on ∂Ω, where Bbh is the operator in Lemma 13.1.3(iii).

ˆ Bbh vh is the trivial extension to Rd+ of the function



Therefore, we can apply Theorem 11.1.5 (resp. Theorem 12.1.5) with the operator Abh (resp. with the operator Abh and Bbh ) to deduce that vh belongs to Cb3+α (Rd+ ; C) and ||vh ||C 3+α (Rd ;C) ≤ C∗ ||fh ||C 1+α (Rd ;C) . Coming back to the function uh , we conclude that b

+

b

+

it belongs to C 3+α (Ω; C) and ||uh ||C 3+α (Ω;C) = ||uh ||C 3+α (Ω∩Uh ;C) ≤ C∗ ||f ||C 1+α (Ω;C) .

(13.1.27)

if u vanishes on ∂Ω and ||uh ||C 3+α (Ω;C) = ||uh ||C 3+α (Ω∩Uh ;C) ≤ C∗ (||f ||C 1+α (Ω;C) + ||u||C 2+α (Ω,C) ),

(13.1.28)

otherwise. Combining (13.1.26) and (13.1.27), we get the assertion of (i) in the case k = 1.

Semigroups of Bounded Operators and Second-Order PDE’s

351

Similarly, combining (13.1.26), (13.1.28) and using also estimate (13.1.3), the assertion of (ii) follows in the case k = 1. In a recursively way, we can complete the proof of the theorem, if k ≥ 2. Indeed, since u ∈ C 3+α (Ω; C), the functions fN and fh belong, respectively, to Cb2+α (Rd ; C) and Cb2+α (Rd+ ; C), so that Theorems 10.1.7, 11.1.5 and 12.1.5 guarantee that u ∈ C 4+α (Ω; C) and satisfies estimate (13.1.22) (resp. (13.1.24)) with k = 2. In a finite number of iterations, we obtain the assertion. 

13.2

Elliptic Equations in Lp (Ω;

C)

In this section, we assume the following conditions on Ω and the coefficients of the operators A, B. Hypotheses 13.2.1

(i) Ω is a bounded domain of class C 2 ;

(ii) the coefficients of the operator A are bounded and measurable on Ω. Moreover, the diffusion coefficients qij = qji (i, j = 1, . . . , d) are continuous over Ω; (iii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for all x ∈ Ω and ξ ∈ Rd . (iv) the functions a and η belong to C 1 (∂Ω) and C 1 (∂Ω, Rd ), respectively. Moreover, |η(x)| = 1 for each x ∈ ∂Ω and hη(x), ν(x)i > 0. The idea to solve the boundary value problems ( λu − Au = f, in Ω, u = g, on ∂Ω, and

(

λu − Au = f, in Ω, Bu = g, on ∂Ω,

(13.2.1)

(13.2.2)

is the same used in Section 13.1: through local charts problems (13.2.1) and (13.2.2) are transformed into problems on Rd and Rd+ , to which we can apply the results of Chapters 10, 11 and 12. The main tool is the a priori estimates contained in the following Theorem 13.2.3. Before stating and proving it, we consider the following lemma, which is the counterpart of Lemma 13.1.3. Lemma 13.2.2 Under Hypotheses 13.2.1(i)–(iii), the following properties are satisfied. (i) Let Ω0 and Ω00 be two open subsets of Ω such that Ω0 b Ω00 b Ω. Then, there exists e with diffusion coefficients in BU C(Rd ) and the other a uniformly elliptic operator A, ∞ d coefficients in L (R ), such that the diffusion matrix is symmetric for every x ∈ Rd e is the trivial extension to Rd of the function Au for every u ∈ W 2,p (Rd ; C) and Au (p ∈ (1, ∞)), with support contained in Ω0 . (ii) Let Uk and ψk (k = 1, . . . , N −1) be as in Lemma 13.1.2. For every k and every function ζ : Ω ∩ Uk → R, denote by ζ ] the function defined by ζ ] = ζ ◦ ψk−1 : B+ (0, r) → R. Then, for every smooth enough function ϑ : Uk → R with compact support in Uk , there exists a uniformly elliptic operator Abk , with diffusion coefficients in BU C(Rd+ )

352

Elliptic Equations on Smooth Domains Ω and the other coefficients in L∞ (Rd+ ), such that the diffusion matrix is symmetric for every x ∈ Rd+ and Abk v is the trivial extension to Rd+ of the function (A(ϑu))] for every function u ∈ W 2,p (Uk ∩ Ω; C) (p ∈ (1, ∞)), where v denotes the trivial extension to Rd+ of the function (ϑu)] .

(iii) Under the same notation as in (ii), there exists a first-order boundary differential ∂ operator Bbk = a(k) I + (k) with coefficients in Cb1 (Rd−1 ) and in Cb1 (Rd−1 ; Rd ), re∂b η spectively, such that inf Rd−1 hb η (k) , ed i < 0 and Bbk v is the trivial extension outside d B(0, r) ∩ ∂R+ of the function (B(uϑ))] . If Ω is of class C k+2 and the coefficients of the operator A belong to W k,∞ (Ω) for some k ∈ N, then the coefficients of operator Ae (resp. Abk ) belong to W k,∞ (Rd ) (resp. to W k,∞ (Rd+ )). Similarly, if the coefficients of the operator B belong to C k+1 (∂Ω) and to C k+1 (∂Ω; Rd ), respectively, then the coefficients of the operator Bbk belong to C k+1 (Rd+ ) and to C k+1 (Rd+ ; Rd ), respectively. Proof The proof can be obtained arguing as in the proof Lemma 13.1.3.



bp,j (j = Theorem 13.2.3 For every p ∈ (1, ∞), there exist positive constants λp,j and C 1, 2), depending on d, p, Ω and in continuous way on µ, the modulus of continuity of the diffusion coefficients of the operator A and on the sup-norm of all its coefficients, such that p |λ|||u||Lp (Ω;C) + |λ||||∇u|||Lp (Ω;C) + ||D2 u||Lp (Ω;C) p bp,1 (||λu − Au||Lp (Ω;C) + |λ|||G1 ||Lp (Ω;C) + |λ||||∇G1 |||Lp (Ω;C) + ||D2 G1 ||Lp (Ω;C) ) ≤C (13.2.3) for every λ ∈ C with Re λ ≥ λp,1 and u ∈ W 2,p (Ω; C), where G1 ∈ W 2,p (Ω; C) is any function which coincides with u on ∂Ω, and p |λ|||u||Lp (Ω;C) + |λ||||∇u|||Lp (Ω;C) + ||D2 u||Lp (Ω;C) p bp,2 (||λu − Au||Lp (Ω;C) + |λ|||G2 ||Lp (Ω;C) + |||∇G2 |||Lp (Ω;C) ) ≤C (13.2.4) for every λ ∈ C with Re λ ≥ λp,2 and u ∈ W 2,p (Ω; C), where G2 ∈ W 1,p (Ω; C) is any function which coincides with Bu on ∂Ω. Proof We fix u ∈ W 2,p (Ω; C) and consider the covering {Uk : k = 1, . . . , N } defined in Lemma 13.1.2. Moreover, we introduce a partition of the unity {ϑk : k = 1, . . . , N } PN subordinated to the above covering. Finally, we split u = k=1 uk , where uk = uϑk for every k = 1, . . . , N . Throughout the proof, given a function h defined on a subset of Rd , we denote by h its trivial extension to Rd . Since UN is compactly supported on Ω, the function uN belongs to W 2,p (Rd ; C). Moreover, Lemma 13.1.3(i) shows that there exists an elliptic operator AeN , whose diffusion coefficients belong to BU C(Rd ) and the other ones belong to L∞ (Rd ), such that AeN uN is the trivial extension to Rd of the function AuN . Applying Theorem 10.2.16 and observing that AuN = ϑN Au + u(A − c)ϑN + 2hQ∇u, ∇ϑN i, we deduce that p |λ|||uN ||Lp (Ω;C) + |λ||||∇uN |||Lp (Ω;C) + ||D2 uN ||Lp (Ω;C) p ≤|λ|||uN ||Lp (Rd ;C) + |λ||||∇uN |||Lp (Rd ;C) + ||D2 uN ||Lp (Rd ;C) ≤C||λuN − AeN uN ||Lp (Rd ;C)

Semigroups of Bounded Operators and Second-Order PDE’s

353

=C||λuN − AuN ||Lp (Ω;C) ≤C(||λu − Au||Lp (Ω;C) + |||∇u|||Lp (Ω;C) + ||u||Lp (Ω;C) )

(13.2.5)

eN , where the positive constants λ eN and C, depend on p, d, for every λ ∈ C with Re λ ≥ λ Ω and, in a continuous way, on the sup-norm of the coefficients of the operator A, on the modulus of continuity of its diffusion coefficients and on the ellipticity constant. In the rest of the proof, we denote by C all the constants that will appear. They depend at most on p, d, Ω and, in a continuous way, on the sup-norm of the coefficients of the operator A, on the modulus of continuity of its diffusion coefficients and on the ellipticity constant. Next, we consider the function uk when k < N . We set vk = uk ◦ ψk−1 and observe that, by Lemma 13.2.2(ii), there exists an elliptic operator Abk whose diffusion coefficients belong to BU C(Rd+ ) and the other ones belong to L∞ (Rd+ ), such that (Auk ) ◦ ψk−1 coincides with Abk vk on B+ (0, r). Function vk belong to W 2,p (Rd+ ; C). Moreover, ˆ the function G1,k = (ϑk G1 ) ◦ ψk−1 belongs to W 2,p (Rd+ ; C) and coincides with the function vk on ∂Rd+ , if uG1 on ∂Ω;   ∂ϑk −1 ˆ the function G2,k = (ϑk G2 ) ◦ ψk + u ◦ ψk−1 belongs to W 1,p (Rd+ ; C) and ∂η coincides with the function Bbk vk on ∂Rd+ if Bu ≡ G2 on ∂Ω, where Bbk is the boundary operator in Lemma 13.2.2(iii).

If u ≡ G1 on ∂Ω, then we can apply (11.2.6) to conclude that p |λ|||vk ||Lp (B+ (0,r);C) + |λ||||∇vk |||Lp (B+ (0,r);C) + ||D2 vk ||Lp (B+ (0,r);C) p ≤|λ|||vk ||Lp (Rd+ ;C) + |λ||||∇vk |||Lp (Rd+ ;C) + ||D2 vk ||Lp (Rd+ ;C) ≤C(||λvk − Abk vk ||Lp (B+ (0,r);C) + |λ|||G1,k ||Lp (Rd+ ;C) p + |λ||||∇G1,k |||Lp (Rd+ ;C) + ||D2 G1,k ||Lp (Rd+ ;C) ) ≤C(||λuk − Auk ||Lp (Ω∩Uk ;C) + |λ|||G1 ||Lp (Ω∩Uk ;C) p |λ| + |||∇G1 |||Lp (Ω∩Uk ;C) + ||D2 G1 ||Lp (Ω∩Uk ;C) ) ≤C(||λu − Au||Lp (Ω;C) + ||u||Lp (Ω;C) + |||∇u|||Lp (Ω;C) + |λ|||G1 ||Lp (Ω;C) p + |λ||||∇G1 |||Lp (Ω;C) + ||D2 G1 ||Lp (Ω;C) )

(13.2.6)

ek,1 and some real constant λ ek,1 , which we can assume to be for every λ ∈ C with Reλ ≥ λ larger than one, and depends on p, d, Ω and, in a continuous way, on the sup-norm of the coefficients of the operator A, on the modulus of continuity of its diffusion coefficients and on the ellipticity constant. On the other hand, if Bu ≡ G2 on ∂Ω, then from (12.2.2) it follows that p |λ|||vk ||Lp (B+ (0,r);C) + |λ||||∇vk |||Lp (B+ (0,r);C) + ||D2 vk ||Lp (B+ (0,r);C) p ≤C(||λu − Au||Lp (Ω;C) + |λ|||u||Lp (Ω;C) + |||∇u|||Lp (Ω;C) p (13.2.7) + |λ|||G2 ||Lp (Ω;C) + |||∇G2 |||Lp (Ω;C) ). ek,2 and some real constant λ ek,2 , which we can assume to be for every λ ∈ C with Reλ ≥ λ larger than one, and depends on p, d, Ω and, in a continuous way, on the sup-norm of the coefficients of the operator A, on the modulus of continuity of its diffusion coefficients, on

354

Elliptic Equations on Smooth Domains Ω

the ellipticity constant and on the C 1 (∂Ω)- and C 1 (∂Ω; Rd )-norm of the coefficients of the operator B. Since ||D2 uk ||Lp (Ω∩Uk ;C) ≤ C(|||∇vk |||Lp (B+ (0,r);C) + ||D2 vk ||Lp (B+ (0,r);C) ), from (13.2.6) and (13.2.7) we deduce that p |λ|||uk ||Lp (Ω∩Uk ;C) + |λ||||∇uk |||Lp (Ω∩Uk ;C) + ||D2 uk ||Lp (Ω∩Uk ;C) p ≤C(|λ|||vk ||Lp (B+ (0,r);C) + |λ||||∇vk |||Lp (B+ (0,r);C) + ||D2 vk ||Lp (B+ (0,r);C) ) ≤C(||λu − Au||Lp (Ω;C) + ||u||Lp (Ω;C) + |||∇u|||Lp (Ω;C) + |λ|||G1 ||Lp (Ω;C) p + |λ||||∇G1 |||Lp (Ω;C) + ||D2 G1 ||Lp (Ω;C) ),

(13.2.8)

if u ≡ G1 on ∂Ω and p

|λ||||∇uk |||Lp (Ω∩Uk ;C) + ||D2 uk ||Lp (Ω∩Uk ;C) p ≤C(||λu − Au||Lp (Ω;C) + |λ|||u||Lp (Ω;C) + |||∇u|||Lp (Ω;C) p + |λ|||G2 ||Lp (Ω;C) + |||∇G2 |||Lp (Ω;C) ). |λ|||uk ||Lp (Ω∩Uk ;C) +

(13.2.9)

if Bu ≡ G2 on ∂Ω. PN From (13.2.5), (13.2.8), (13.2.9) and recalling that u = k=1 uk , we infer that p |λ|||u||Lp (Ω;C) + |λ||||∇u|||Lp (Ω;C) + ||D2 u||Lp (Ω;C) ≤C(||λu − Au||Lp (Ω;C) + ||u||Lp (Ω;C) + |λ|||G1 ||Lp (Ω;C) + |||∇u|||Lp (Ω;C) p + |λ||||∇G1 |||Lp (Ω;C) + ||D2 G1 ||Lp (Ω;C) ) ep,1 = max{λ ej,1 : j = 1, . . . , N } if u ≡ G1 on ∂Ω and for every λ ∈ C with Re λ ≥ λ p |λ|||u||Lp (Ω;C) + |λ||||∇u|||Lp (Ω;C) + ||D2 u||Lp (Ω;C) p ≤C(||λu − Au||Lp (Ω;C) + |λ|||u||Lp (Ω;C) + |||∇u|||Lp (Ω;C) p + |λ|||G2 ||Lp (Ω;C) + |||∇G2 |||Lp (Ω;C) ) ep,2 = max{λ ej,2 : j = 1, . . . , N }, if Bu ≡ G2 on ∂Ω. Here, for every λ ∈ C with Reλ ≥ λ e e e e ep,2 with a larger value if needed, we can λN,1 = λN,2 = λN . Up to replacing λp,1 and λ ep,1 ≥ 2C and λ ep,j ≥ 4C 2 (j = 1, 2), so that we can move the Lp -norm of assume that λ u and its gradient from the right- to the left-hand side of the previous two inequalities. Estimates (13.2.3) and (13.2.4) follow.  The following result is a straightforward consequence of Theorem 13.2.3. Hence, the proof is omitted. ep , which depend on p, on d Corollary 13.2.4 There exist two positive constants Cp and C and, in a continuous way, on the L∞ -norm of the coefficients of the operator A, on µ and ep on the modulus of continuity of the diffusion coefficients of the operator A (constant C depends also, in a continuous way, on the C 1 (∂Ω)- and C 1 (∂Ω; Rd )-norm of the coefficients of the operator B), such that ||u||W 2,p (Ω;C) ≤ Cp (||u||Lp (Ω;C) + ||Au||Lp (Ω;C) + ||G1 ||W 2,p (Ω;C) ) for every function u ∈ W 2,p (Ω; C), where G1 is any function in W 2,p (Ω; C) which coincides with u on ∂Ω, and ep (||u||Lp (Ω;C) + ||Au||Lp (Ω;C) + ||G2 ||W 1,p (Ω;C) ) ||u||W 2,p (Ω;C) ≤ C for every function u ∈ W 2,p (Ω; C), where G2 is any function in W 1,p (Ω; C) which coincides with Bu on ∂Ω.

Semigroups of Bounded Operators and Second-Order PDE’s We have all the necessary tools to solve the boundary value problems ( λu − Au = f, in Ω, u = g1 , on ∂Ω, and

(

λu − Au = f, Bu = g2 ,

in Ω, on ∂Ω.

355

(13.2.10)

(13.2.11)

Theorem 13.2.5 For every p ∈ (1, ∞), there exist two sequences (λ∗n ) and (λ∗∗ n ) of complex numbers such that, for every f ∈ Lp (Ω, C), g1 ∈ B 2−1/p,p (∂Ω; C) and g2 ∈ B 1−1/p,p (Ω; C), the following properties are satisfied. (i) For every λ ∈ C such that λ 6= λ∗n for each n ∈ N, the boundary value problem (13.2.10) admits a unique solution u ∈ W 2,p (Ω; C). Moreover, there exists a positive constant Cp , depending on p, d, Ω and, in continuous way, on µ, on the modulus of continuity of the diffusion coefficients of the operator A and the sup-norm of all its coefficients, such that p |λ|||u||Lp (Ω;C) + |λ||||∇u|||Lp (Ω;C) + ||D2 u||Lp (Ω;C)   p ≤Cp ||λu − Au||Lp (Ω;C) + |λ|||G1 ||Lp (Ω;C) + |λ||||∇G1 |||Lp (Ω;C) + ||D2 G1 ||Lp (Ω;C) (13.2.12) for every λ ∈ C with Reλ ≥ λp,1 , where λp,1 is the positive constant in Theorem 13.2.3 and G1 is any function in W 2,p (Ω; C), whose trace on ∂Ω coincides with function g1 . (ii) For every λ ∈ C such that λ 6= λ∗∗ n for each n ∈ N, the boundary value problem (13.2.11) admits a unique solution u ∈ W 2,p (Ω; C). Moreover, there exists a positive constant Cp0 , depending on p, d, Ω and, in continuous way, on µ, on the modulus of continuity of the diffusion coefficients of the operator A, the sup-norm of all its coefficients and the C 1 (∂Ω)- and C 1 (∂Ω; Rd )-norm of the coefficients of the operator B, such that p |λ|||u||Lp (Ω;C) + |λ||||∇u|||Lp (Ω;C) + ||D2 u||Lp (Ω;C) p (13.2.13) ≤Cp0 (||λu − Au||Lp (Ω;C) + |λ|||G2 ||Lp (Ω;C) + |||∇G2 |||Lp (Ω;C) ) for every λ ∈ C with Reλ ≥ λp,2 , where λp,2 is the positive constant in Theorem 13.2.3 and G2 is any function in W 1,p (Ω; C) such that BG2 = g2 on ∂Ω. Proof As usually in this chapter, given a function h, defined in a subset of Rd , we denote by h its trivial extension to Rd . (i) Fix p ∈ (1, ∞), f ∈ Lp (Ω; C), g1 ∈ B 2−1/p,p (Ω; C) and let λp,1 be as in Theorem 13.2.3. The uniqueness of the solution to the boundary value problem (13.2.10) for λ ∈ C, with Re λ ≥ λp,1 , as well as estimate (13.2.12) follow immediately from (13.2.3). So in the rest of the proof, we prove that problem (13.2.10) actually admits a solution u ∈ W 2,p (Ω; C). We can limit ourselves to considering the case g1 ≡ 0 on ∂Ω. Indeed, u ∈ W 2,p (Ω; C) solves problem (13.2.10) if and only if the function v = u−Ep0 g1 , where Ep0 is the extension operator in Proposition B.4.15, solves the equation λu − Au = f and vanishes on ∂Ω. So, in the rest of the proof, we assume that g1 ≡ 0 on ∂Ω. Let {Uk : k = 1, . . . , N } be the covering of Ω and {ϑ2k : k = 1, . . . , N } be the associated partition of the unity in Lemma 13.1.2. As in the proof of Theorem 13.1.4, we will define an

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Elliptic Equations on Smooth Domains Ω PN operator S(λ), for λ large enough, as the sum S(λ) = k=1 Sk (λ), where the operators Sk (λ) (k = 1, . . . , N ) are defined in Steps 1 and 2. We will see that this operator maps Lp (Ω; C) into W 2,p (Ω; C) ∩ W01,p (Ω; C) and λS(λ)φ − AS(λ)φ = φ + T (λ)φ for every ψ ∈ Lp (Ω; C), where T (λ) is a bounded operator from Lp (Ω; C) in itself, whose norm is less than 1/2 if λ ∈ C has a sufficiently large norm. This will imply that, for such values of λ, the operator I + T (λ) is invertible in Lp (Ω; C). Therefore, the function S(λ)(I + T (λ))−1 f will be the solution to problem (13.2.10) that we are looking for. Then, a compactness argument, will allow to complete the proof. Since the strategy of the proof is similar to that used to prove Theorem 13.1.4, we omit some details. Step 1 (the operator SN (λ)). Since the function ϑN f belongs to Lp (Rd ; C) and is compactly supported in Ω, Theorem 10.2.16, shows that there exists a positive constant λp,N , which depends only on p, d, Ω and, in a continuous way on the ellipticity constant µ, the modulus of continuity of the diffusion coefficients of the operator A and on the sup-norm of all its coefficients, such that the equation λw − AeN w = ϑN f admits a unique solution uN ∈ W 2,p (Rd ; C) for every λ ∈ C with Re λ ≥ λp,N . Here, AeN is a uniformly elliptic operator, whose coefficients are bounded on Rd with diffusion coefficients which are uniformly continuous over Rd , such that AeN uN = AuN . Moreover, p |λ|||uN ||Lp (Rd ;C) + |λ||||∇uN |||Lp (Rd ;C) + ||D2 uN ||Lp (Rd ;C) ≤ C||f ||Lp (Ω;C) (13.2.14) for every λ ∈ C with Re λ ≥ λp,N , where the positive constant C, which as all the forthcoming constants that appear in the proof of property (i) (still denoted by C), depend (at most) on p, d and, in a continuous way on the ellipticity constant µ, the modulus of continuity of the diffusion coefficients of the operator A and on the sup-norm of all its coefficients. We set uN := RN (λ)f and define the operator SN (λ) on Lp (Ω; C) by setting SN (λ)φ = ϑN RN (λ)φ for every φ ∈ Lp (Ω; C). Clearly, SN is bounded from Lp (Ω; C) into W 2,p (Rd ; C) and SN (λ)φ has compact support in ΩN for every φ ∈ Lp (Ω; C). Moreover, (λI −A)SN (λ)φ = ϑ2N φ+TN (λ)φ for every φ ∈ Lp (Ω; C), where TN = −[A, ϑN ] and [A, ϑN ] denotes the commutator between A and ϑN . Using (13.2.14), it can be easily shown that 1

||TN (λ)f ||Lp (Rd ;C) ≤ C|λ|− 2 ||f ||Lp (Ω;C) ,

Re λ ≥ λp,N .

(13.2.15)

Step 2 (the operator Sk (λ)). We fix k and consider Abk the operator defined in Lemma 13.1.3, whose coefficients belong to L∞ (Rd+ ), with the diffusion coefficients uniformly continuous in Rd+ , and Abk vk is the trivial extension of the function (A(ϑu)) ◦ ψk−1 , where vk = (ϑu) ◦ ψk−1 . Theorem 11.2.4, applied to the operator Abk guarantees that there exists a positive constant λp.k , which depends only on p, d, Ω and, in a continuous way on the ellipticity constant µ, the modulus of continuity of the diffusion coefficients of the operator A and on the sup-norm of all its coefficients, such that the equation λv − Abk v = ϑk f admits a unique solution vk ∈ W 2,p (Rd+ ; C) ∩ W01,p (Rd+ ; C) for every λ ∈ C with Re λ ≥ λp,k . Moreover, p |λ|||vk ||Lp (Rd+ ;C) + |λ||||∇vk |||Lp (Rd+ ;C) + ||D2 vk ||Lp (Rd+ ;C) ≤ Cp ||f ||Lp (Ω;C) . (13.2.16) We set Rk (λ)f := vk and introduce the operator Sk (λ) defined on Lp (Ω; C) by Sk (λ)g = [(ϑk ◦ ψk−1 )Rk (λ)g] ◦ ψk for every g ∈ Lp (Ω; C). This operator is bounded from Lp (Ω; C) into W 2,p (Ω; C) ∩ W01,p (Ω; C) and (λI − A)Sk (λ)g = ϑ2k g + Tk (λ)g for every g ∈ Lp (Ω; C), where Tk (λ)f = −([A, ϑk ◦ ψk−1 ]Rk (λ)g) ◦ ψk . Using estimate (13.2.16) it can be proved that 1

||Tk (λ)f ||Lp (Ω;C) ≤ Cp |λ|− 2 ||f ||Lp (Ω;C) ,

Re λ ≥ λp,k .

(13.2.17)

Step 3 (Conclusion). We now introduce the operator S(λ) on Lp (Ω; C) as explained

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at the beginning of the proof. Clearly, S(λ) maps Lp (Ω; C) into W 2,p (Ω; C) ∩ W01,p (Ω; C). Moreover, (λI − A)S(λ)φ = φ + T (λ)φ for every φ ∈ Lp (Ω; C) and λ ∈ C with Re λ ≥ bp := max{λp,1 , . . . , λp,N }, where T (λ) = PN Tk (λ) Using (13.2.15) and (13.2.17) we can λ k=1 1 bp . Hence, replacing λ bp estimate ||T (λ)||L(Lp (Ω;C)) ≤ Cp |λ|− 2 for every λ ∈ C with Re λ ≥ λ with a larger value if needed, we can assume that the norm of T (λ) is less than 1/2 for every bp . We have proved that the boundary problem (13.2.10) is solvable, λ ∈ C with Re λ ≥ λ bp . with a solution in W 2,p (Ω; C) ∩ W01,p (Ω; C), for every λ ∈ C with Re λ ≥ λ 2,p To complete the proof of (i), we introduce the operator Ap : W (Ω; C) ∩ W01,p (Ω; C) → p L (Ω; C) defined by Ap u = Au for every u ∈ W 2,p (Ω; C) ∩ W01,p (Ω; C). The above results bp , Ap ) is compact from and the Rellich-Kondrachov theorem show that the operator R(2λ p L (Ω; C) into itself. Hence, its spectrum consists of a sequence of eigenvalues accumulating at zero. Consequently, the spectrum of operator Ap consists of a sequence (λn ) of eigenvalues as well. This implies that for λ 6= λn for every n ∈ N, the boundary value problem (13.2.10) admits a (unique) solution in W 2,p (Ω; C). (ii) The proof is analogous to that of property (i). Fix p ∈ (1, ∞), f ∈ Lp (Ω; C), g2 ∈ B 1−1/p,p (Ω; C) and let λp,2 be as in Theorem 13.2.3. The uniqueness of the solution to the boundary value problem (13.2.11) as well as estimate (13.2.13), for λ ∈ C with Re λ ≥ λp,2 , follow immediately from (13.2.4). So in the rest of the proof, we prove that problem (13.2.11) actually admits a solution u ∈ W 2,p (Ω; C) for every λ ∈ C \ {λ0n : n ∈ N} for a suitable sequence (λ0n ). Also in this situation we can limit ourselves to proving the solvability of problem (13.2.11) for λ ∈ R sufficiently large. Indeed, then a compactness argument, similar to that used in Step 3, will allow to complete the proof. 0 Let {U1 , . . . , UN −1 , UN } be the covering of Ω and φk , ζk (k = 1, . . . , N ) be the functions in Lemma 13.1.2(ii). We will define an operator S 0 (λ), for λ large enough, as the sum S 0 (λ) = PN 0 0 k=1 Sk (λ), where the operators Sk (λ) (k = 1, . . . , N ) are defined in Steps 4 and 5. We will p see that this operator maps L (Ω; C) into W 2,p (Ω; C), λS 0 (λ)φ − AS 0 (λ)φ = φ + T 0 (λ)φ and BS 0 (λ)φ = g2 for every φ ∈ Lp (Ω; C), where T 0 (λ) is a bounded operator from Lp (Ω; C) in itself, whose norm is less than 1/2, if λ ∈ C has a sufficiently large norm. 0 (λ)). This operator can be defined as the operator SN (λ) in Step 4 (the operator SN Step 2. The only difference is that now ϑN is replaced by the function φN . This operator is bounded from Lp (Ω; C) into W 2,p (Ω; C) for every λ ∈ C such that Re λ ≥ λ∗p,N , where the positive constant λ∗p,N depends on p, d, Ω and, in a continuous way on the ellipticity constant µ, the modulus of continuity of the diffusion coefficients of the operator A, on the sup-norm of all its coefficients and the C 1 -norm of the coefficients of the operator B. Step 5 (the operator Sk0 (λ)). By analogy with Step 2, we set Sk0 (λ)Φ = [(ζk ◦ −1 ψk )R0 (λ)(Φ, g)] ◦ ψk for every Φ ∈ Lp (Ω; C), where R0 (λ)(Φ, g) is the unique solution in W 2,p (Rd+ ; C) of the boundary value problem ( λv − Aek v = Φφk , in Rd+ , Bbk v = gbk , on ∂Rd+ , and Abk and Bbk are the elliptic operator and the boundary operator defined in Lemma 13.2.2. Moreover, gbk = (g2 ϑk ) ◦ ψk−1 . Function gbk belongs to B 1−1/p,p (∂Ω; C). Theorem 12.2.4 guarantees that the previous problem is solvable in W 2,p (Rd+ ; C) for every λ ∈ C, with Reλ ≥ λ∗k,p , and a suitable positive constant λ∗p,k which depends on p, d, Ω and, in a continuous way on the ellipticity constant µ, the modulus of continuity of the diffusion coefficients of the operator A, on the sup-norm of all its coefficients and the C 1 -norm of the coefficients of the operator B. Step 6 (Conclusion). It is easy to check that the operator S 0 (λ) is well defined for every

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b∗ = max{λ∗ : k = 1, . . . , N }. Moreover, (λS 0 (λ)h − A0 S(λ)h) = λ ∈ C, such that Re λ ≥ λ p p,k PN 0 0 φ ζ h + T (λ)h = h + T (λ)h and k k k=1 T 0 (λ)h = −

N −1 X

([Abk , ζN ◦ ψk−1 ]Rk0 (λ)(h, g)) ◦ ψk − [A, ζN ]RN (λ)h

(13.2.18)

k=1

for h ∈ Lp (Ω; C) and such values of λ. Finally, BS 0 (λ)h ≡ g2 on ∂Ω. Suppose first that g2 ≡ 0 on ∂Ω. Then, arguing as in Steps 1 and 2 and taking also estimates (10.2.23) and (12.2.2) into account, it is easy to check that all the terms in (13.2.18) are bounded operators from Lp (Ω; C) into itself and their operator norms can be estimated b∗ . Hence, up from above by a positive constant times |λ|−1/2 for every λ ∈ C with Reλ ≥ λ p b∗ with a larger value if needed, we can assume that ||T 0 (λ)||L(Lp (Ω)) ≤ 1/2 for to replacing λ p b∗ . The assertion follows in this case. Finally, if g2 6≡ 0 then the every λ ∈ C with Reλ ≥ λ p first part of this step shows that there exists a function v ∈ W 2,p (Ω; C) such that Bv ≡ g2 on ∂Rd+ . Hence, if g2 6≡ 0, then the solution to problem (13.2.11) can be obtained as the sum of function v and the solution to the boundary value problem ( λw − Aw = f − λv + Av, in Ω, Bw = 0, on ∂Ω. This completes the proof.



Corollary 13.2.6 Let p, q ∈ (1, ∞), with p 6= q. Then, the following properties are satisfied. (i) Let u ∈ W 2,p (Ω; C) ∩ W01,p (Ω; C) be such that u and Au belong to Lq (Ω; C). Then u ∈ W 2,q (Ω; C) ∩ W01,q (Ω; C). (ii) Denote by Ar the realization of the operator A in Lr (Ω; C) with domain D(Ar ) = W 2,r (Ω; C) ∩ W01,r (Ω; C). Then, σ(Ap ) = σ(Aq ) and R(λ, Ap )f = R(λ, Aq )f for all f ∈ Lp (Ω; C) ∩ Lq (Ω; C) and λ ∈ ρ(Ap ). (iii) Let u ∈ W 2,p (Ω; C) be such that u and Au belong to Lq (Ω; C) and Bu ≡ 0 on ∂Ω. Then u ∈ W 2,q (Ω; C) ∩ W01,q (Ω; C). r B (iv) Denote by AB r the realization of the operator A in L (Ω; C) with domain D(Ar ) = B B B {W 2,r (Ω; C) : Bu = 0}. Then, σ(AB ) = σ(A ) and R(λ, A )f = R(λ, A )f for all p q p q p q B f ∈ L (Ω; C) ∩ L (Ω; C) and λ ∈ ρ(Ap ).

Proof (i) Clearly, we only need to consider the case p < q, since, if p > q, then W 2,p (Ω; C) ∩ W01,p (Ω; C) ,→ W 2,q (Ω; C) ∩ W01,q (Ω; C). We fix λ ≥ (λp ∨ λq ), where λp and λq are defined in the statement of Theorem 13.2.5. The function f := λu − Au belongs to Lq (Ω; C), so that the quoted theorem guarantees that there exists a unique function v ∈ W 2,q (Ω; C) ∩ W01,q (Ω; C) which solves the equation λv − Av = f . Since W 2,q (Ω; C) ∩ W01,q (Ω; C) ,→ W 2,p (Ω; C) ∩ W01,p (Ω; C), the function v belongs to W 2,p (Ω; C) ∩ W01,p (Ω; C). By uniqueness, u ≡ v and this concludes the proof. (ii) To fix the ideas, we assume that p < q. Clearly, σ(Aq ) ⊂ σ(Ap ). Indeed, if λ ∈ σ(Aq ), then there exists a nontrivial function u ∈ W 2,q (Ω; C) ∩ W01,q (Ω; C) such that λu − Au = 0. Since u belongs also to W 2,p (Ω; C) ∩ W01,p (Ω; C), we conclude that λ ∈ σ(Ap ). Let us now suppose that λ ∈ σ(Ap ). Then, there exists a nontrivial function v ∈ W 2,p (Ω; C) ∩ W01,p (Ω; C) such that λu − Au = 0. By the Sobolev embedding theorems (see Theorem 1.3.6), W 2,p (Ω; C) ,→ Lr1 (Ω; C), where 1/r1 = 1/p − 2/d, if p < d, r1 is any arbitrarily

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number in the range (1, ∞), if p = d, and r1 = ∞, if p > d. By difference, Au ∈ Lr (Ω; C) as well. By property (i), we deduce that u ∈ W 2,r (Ω; C) ∩ W01,r (Ω; C). If r ≥ q, then, we are done. So, we suppose that r1 < q, repeat the previous argument, with p being replaced by r1 and denote by r2 the exponent provided by the Sobolev embedding theorems. If r2 ≥ q then we are done. Otherwise we iterate the procedure once more. Note that this procedure concludes in a finite number of steps. Indeed, supposing that this is not the case, then rn < d for every n ∈ N and the sequence (rn ) is recursively defined by  1 2 1   = − , n ∈ N,  rn+1 rn d 1 1   = .  r1 p 1 1 2n = − for every n ∈ N, which is definitively negative: a contradiction. rn p d To prove that the functions R(λ, Ap )f and R(λ, Aq )f coincide when f ∈ Lp (Ω; C) ∩ q L (Ω; C), it suffices to observe that they both solve the equation λu − Au = f and W 2,q (Ω; C) ∩ W01,q (Ω; C) ,→ W 2,p (Ω; C) ∩ W01,p (Ω; C). (iii) & (iv): the proofs can be obtained arguing as in (i) and (ii). 

Hence,

When the operator A is in divergence form, we can give more precise information on the solvability of the boundary value problem ( λu − Au = f, in Ω, (13.2.19) u = g, on ∂Ω. We assume the following set of assumptions Hypotheses 13.2.7

(i) Ω is a bounded domain of class C 2 ;

(ii) the coefficients qij = qji and bj of the operator A belong to W 1,∞ (Ω; C) (i, j = 1, . . . , d) and c ∈ L∞ (Ω); (iii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for all x ∈ Ω and ξ ∈ Rd . For every p ∈ (1, ∞), we set   1 λp = max c − div b . p Ω Theorem 13.2.8 Under Hypotheses 13.2.7, for every λ ∈ C, with Re λ > λp , every f ∈ Lp (Ω; C) and g ∈ B 2−1/p,p (∂Ω; C), the boundary value problem (13.2.19) admits a unique solution u ∈ W 2,p (Ω; C). Moreover, estimate (13.2.3) holds true for every λ ∈ C with Reλ > λ+ p. Proof The arguments used here are similar to those in the proof of Theorem 11.2.8. In particular we can reduce to considering problem (13.2.19) with g ≡ 0. Hence, we skip some technical details. Fix f ∈ Lp (Ω; C), λ ∈ C, with Re λ > λp , and suppose that u ∈ W 2,p (Ω; C)∩W01,p (Rd ; C) solves the equation λu − Au = f . Multiplying both sides of this equation by u|u|p−2 and observing that the real part of the integral over Ω of the function div(Q∇u)u|u|p−2 is nonpositive, we get Z Z Z (13.2.20) Re f u|u|p−2 dx ≥ (Re λ − c)|u|p dx − Re hb, ∇uiu|u|p−2 dx. Ω

Ω

Ω

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Elliptic Equations on Smooth Domains Ω

Since ∇(|u|p ) = 2−1 p(u∇u + u∇u)|u|p−2 , we can write Z Z 1 p−2 Re hb, ∇uiu|u| dx = Re [hb, ∇uiu + hb, ∇uiu]|u|p−2 dx 2 Ω Z Ω Z 1 1 = hb, ∇(|u|p )i dx = − (div b)|u|p dx. p Ω p Ω This formula replaced in the right-hand side of (13.2.20) gives  Z  Z 1 p−1 p Reλ − c + div b |u| dx ≤ Re f u|u|p−2 dx ≤ ||f ||Lp (Ω;C) ||u||L p (Ω;C) . p Ω Ω In particular, (Reλ − λp )||u||Lp (Ω;C) ≤ ||f ||Lp (Ω;C) . Now, we can complete the proof arguing as in Step 3 of the proof of Theorem 13.1.4. We introduce the operator Ap : W 2,p (Ω; C) ∩ W01,p (Ω; C) → Lp (Ω; C) defined by Ap u = Au for every u ∈ W 2,p (Ω; C) ∩ W01,p (Ω; C) and observe that the previous estimate shows that the norm of R(λ, Ap ) stays bounded if λ ∈ ρ(A) ∩ {λ ∈ C : Reλ ≥ λp }. Hence, Remark A.4.5 implies that the set {λ ∈ C : Reλ ≥ λp } is contained in ρ(Ap ) and the assertion follows. The last details of the proof are left to the reader (see Exercise 13.4.1).  Now, we discuss the connections with the solutions of problems (13.1.2) (resp. (13.1.4)) provided by Theorem 13.1.4 (resp. Theorem 13.1.6) and Theorem 13.2.5. Theorem 13.2.9 Let Hypotheses 13.1.1(i)–(iii) (resp. Hypotheses 13.1.1) be satisfied and let u ∈ W 2,p (Ω; C) be such that u and Au both belong to C α (Ω; C) and u (resp. Bu) vanishes on ∂Ω. Then, u belongs to C 2+α (Ω; C). Proof The function f = 2c0 u − Au belongs to C α (Ω; C) so that, by Theorem 13.1.4 (resp. Theorem 13.1.6), there exists a unique function v ∈ C 2+α (Ω; C) which solves the equation 2c0 v − Av = f and vanishes (resp. Bv vanishes) on ∂Ω. Clearly, v belongs to W 2,p (Ω; C) so that, by the uniqueness part of Theorem 13.2.5, v ≡ u on Ω.  Corollary 13.2.10 Under Hypotheses 13.1.1, for every p ∈ (1, ∞) the interval [c0 , ∞) belongs to the resolvent set ρ(Ap ), i.e., for every f ∈ Lp (Ω; C) and λ ≥ c0 there exists a unique u ∈ W 2,p (Ω; C) ∩ W01,p (Ω; C) which solves the equation λu − Au = f , where c0 is the maximum of the function c over Ω. Similarly, the interval (c0 , ∞) belongs to the resolvent set of the operator AB p. Proof Since the spectrum of Ap (resp. AB p ) is independent of p (see Corollary 13.2.6), we can suppose that p = (2 − 2α)−1 d, so that W 2,p (Ω; C) is continuously embedded into C α (Ω; C). The quoted corollary also shows that the spectrum of Ap (resp. AB p ) consists of eigenvalues only. So, the assertion will follow immediately if we prove that the operator λI − Ap (resp. λI − Ap ) is injective for every λ ≥ c0 (resp. λ > c0 ). Fix λ ≥ c0 (resp. λ > c0 ) and u ∈ W 2,p (Ω; C) such that λu − Au = 0 and u (resp. Bu) vanishes on ∂Ω. Since both u and Au belong to C α (Ω; C), Theorem 13.2.9 guarantees that u ∈ C 2+α (Ω; C). Hence, u = 0 on Ω, by Theorem 13.1.4 (resp. Theorem 13.1.6). 

13.2.1

Further regularity results

In this subsection, we prove that the more the functions Au (and Bu) are smooth, the more u itself is smooth. For this purpose, we assume the following set of assumptions. Hypotheses 13.2.11

(i) Ω is a bounded domain of class C 2+k for some k ∈ N;

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(ii) the coefficients qij = qji belong to Cbk (Ω) (i, j = 1, . . . , d) , whereas bj (j = 1, . . . , d) and c belong to W k,∞ (Ω); (iii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for all x ∈ Ω and ξ ∈ Rd ; (iv) the coefficients of the operator B belong to C k+1 (∂Ω) and C k+1 (∂Ω; Rd ), respectively. Moreover, hη(x), ν(x)i > 0 for each x ∈ ∂Ω. Theorem 13.2.12 Let Hypotheses 13.2.11(i)–(iii) (resp. Hypotheses 13.2.11) be satisfied. Let u ∈ W 2,p (Ω; C), for some p ∈ (1, ∞), be a function such that Au ∈ W k,p (Ω; C) and u (resp. Bu) vanishes on ∂Ω. Then, u ∈ W k+2,p (Ω, C) and there exists a positive constant C, depending on p, d and, in a continuous way on µ and the W k,∞ (Ω)-norm of the coefficients of the operator A (resp. also, in a continuous way, on the C k+1 (∂Ω)- and C k+1 (∂Ω; Rd )norm of the coefficients of the operator B), such that ||u||W k+2,p (Ω;C) ≤ C(||u||Lp (Ω;C) + ||Au||W k,p (Ω;C) ).

(13.2.21)

Proof Throughout the proof, we denote by C a positive constant, which depends at most on p, d, Ω and, in a continuous way on µ, the W k,∞ (Ω)-norm of the coefficients of the operator A and the C k+1 (∂Ω)- and C k+1 (∂Ω; Rd )-norm of the coefficients of the operator B, which may vary from line to line. Moreover, given a function ζ defined in a subset of Rd , we denote by ζ its trivial extension to Rd . We first consider the case k = 1. We consider the usual covering {U1 , . . . , UN } of Ω in Lemma 13.1.2 and a partition of the unity {ϑi : i = 1, . . . , N } associated to such a PN covering. Moreover, we fix u ∈ W 2,p (Ω; C) and split u = h=1 uh , where uh = uϑh for every h = 1, . . . , N . Since ϑN is compactly supported in Ω, the function uN is compactly supported in Ω as well. Therefore, the function uN belongs to W 2,p (Rd ; C). Thanks to Lemma 13.1.3(i) we can determine a uniformly elliptic operator AeN , with coefficients in BU C 1 (Rd ) (see the proof of the quoted lemma) such that AeN uN is the trivial extension of the function AuN to Rd . Since this latter function is compactly supported in Ω and AuN = ϑN Au+2hQ∇u, ∇ϑN i+u(A−c)ϑN belongs to W 1,p (Ω; C), it follows that AeN uN ∈ W 1,p (Rd ; C) and ||AeN uN ||W 1,p (Rd ;C) = ||AuN ||W 1,p (Ω;C) . We can thus apply Theorem 10.2.29 to the operator AeN and infer that ||uN ||W 3,p (Ω;C) ≤C(||uN ||Lp (Rd ;C) + ||AeN uN ||W 1,p (Rd ;C) ) ≤C(||uN ||Lp (Ω;C) + ||AuN ||W 1,p (Ω;C) ) ≤C(||u||W 2,p (Ω;C) + ||Au||W 1,p (Ω;C) ). From Theorem 13.2.5, we know that ||u||W 2,p (Ω;C) ≤ C||2c0 u − Au||Lp (Ω;C) ≤ C(||u||Lp (Ω;C) + ||Au||Lp (Ω;C) ),

(13.2.22)

where c0 is the maximum over Ω of the function c. Therefore, ||uN ||W 3,p (Ω;C) ≤ C(||u||Lp (Ω;C) + ||Au||W 1,p (Ω;C) ).

(13.2.23)

Next, we consider the case h < N . Set vh = uh ◦ ψh−1 , where ψh is the diffeomorphism of class C 2+α introduced in Lemma 13.1.2. As it is easily seen, the function vh belongs to W 2,p (Rd+ ; C). Moreover, ˆ vh vanishes on ∂Rd+ if u vanishes on ∂Ω;

362

Elliptic Equations on Smooth Domains Ω   ∂ϑh ˆ Bbh vh is the trivial extension to Rd+ of the function u ◦ ψh−1 , if Bu vanishes on ∂η ∂Ω, where Bbh is the boundary operator in Lemma 13.2.2(iii).

By properties (ii) and (iii) in Lemma 13.2.2, we can determine a uniformly elliptic bh , with coefficients in W k,∞ (Rd+ ) such that Abh v h coincides with the funcoperator A tion (A(ϑh u)) ◦ ψh−1 on B+ (0, r) and vanishes outside B+ (0, r). Hence, Abh v h belongs to W 1,p (Rd+ ; C). Therefore, Theorems 11.2.11 and 12.2.8 show that the function vh belongs to W 3,p (Rd+ ; C) and ||vh ||W 3,p (B+ (0,r);C) =||vh ||W 3,p (Rd+ ;C) ≤ C(||vh ||Lp (Rd+ ;C) + ||Abh vh ||W 1,p (Rd+ ;C) ) ≤C(||vh ||Lp (B+ (0,r);C) + ||Auh ||W 1,p (Ω;C) ),

(13.2.24)

if u vanishes on ∂Ω, whereas ||vh ||W 3,p (B+ (0,r);C) ≤C(||vh ||Lp (B+ (0,r);C) + ||Auh ||W 1,p (Ω;C) + ||u||W 1,p (Ω;C) ),

(13.2.25)

otherwise. Therefore, the function uh belongs to W 3,p (Ω ∩ Uh ; C) and, from (13.2.24), (13.2.25), we get ||uh ||W 3,p (Ω∩Uh ;C) ≤C(||u||W 2,p (Ω;C) + ||Au||W 1,p (Ω;C) ).

(13.2.26)

Since the function ϑh is compactly supported in Uh , uh belongs to W 3,p (Ω; C) and ||uh ||W 3,p (Ω;C) = ||uh ||W 3,p (Ω∩Uh ;C) . Taking (13.2.22) into account, from (13.2.26) we deduce that ||uh ||W 3,p (Ω;C) ≤ C(||u||Lp (Ω;C) + ||Au||W 1,p (Ω;C) ), (13.2.27) in PNboth cases. We obtain the same estimate, if Bu vanishes on ∂Ω. Recalling that u = h=1 uh and combining (13.2.23) and (13.2.27), estimate (13.2.21), with k = 1, follows. Let us suppose that the assertion is true for every k 0 = 1, . . . , k and some k ≥ 2, and let us prove (13.2.21), with k replaced by k + 1. So, let us assume that u ∈ W 2,p (Ω; C) is such that Au ∈ W k+1,p (Ω; C) and u (resp. Bu) vanishes on ∂Ω. Then, Au ∈ W k,p (Ω; C) so e N that u ∈ W k+2,p (Ω; C). The strategy is the same used in the case k = 1. The function Au k+1,p d belongs to W (R ; C), so that Theorem 10.2.29 and the inductive assumption show that uN belongs to W k+3,p (Rd ; C) and ||uN ||W k+3,p (Ω;C) ≤ C(||u||Lp (Ω;C) + ||Au||W k+1,p (Ω;C) ).

(13.2.28)

b h Similarly, if h < N , then the function Abh vh belongs to W k+1,p (Rd+ ; C). Moreover, Bv belongs to B k+2−1/p,p (∂Ω; C) if Bu vanishes on ∂Ω. Thus, by Theorems 11.2.11 and 12.2.8, the function vh belongs to W k+3,p (Rd+ ; C) and ||vh ||W k+3,p (Rd+ ;C) ≤ C(||vh ||Lp (B+ (0,r);C) + ||Auh ||W k+1,p (Ω;C) ), if u ≡ 0 on ∂Ω, whereas ||vh ||W k+3,p (Rd+ ;C) ≤ C(||vh ||Lp (B+ (0,r);C) + ||Auh ||W k+1,p (Ω;C) + ||u||W k+2,p (Ω;C) ), otherwise. Coming back to the function uh , we infer that ||uh ||W k+3,p (Ω;C) ≤ C(||u||W k+2,p (Ω;C) + ||Au||W k+1,p (Ω;C) ).

(13.2.29)

Using the inductive assumption, we can replace the W k+2,p -norm of u by the sum of the Lp -norm of u and the W k,p -norm of Au. From (13.2.28) and (13.2.29) the assertion follows also in this case.  The following result now follows from Theorem 13.2.12.

Semigroups of Bounded Operators and Second-Order PDE’s

363

Corollary 13.2.13 The following properties are satisfied. (i) Let Hypotheses 13.2.11(i)-(iii) be satisfied. Further, assume that u ∈ W 2,p (Ω; C) solves the boundary value problem ( λu − Au = f, in Ω, (13.2.30) u = g, on ∂Ω, for some λ ∈ C, f ∈ W k,p (Ω; C) and g ∈ B k+2−1/p,p (∂Ω; C). Then, u ∈ W k+2,p (Ω; C) and there exists a positive constant C, depending on p, d, λ and, in a continuous way, on µ and the W k,∞ (Ω; C)-norm of the coefficients of the operator A, such that ||u||W k+2,p (Ω;C) ≤ C(||f ||W k,p (Ω;C) + ||g||B k+2−1/p,p (∂Ω;C) ).

(13.2.31)

(ii) Let Hypotheses 13.2.11 be satisfied. Further, assume that u ∈ W 2,p (Ω; C) solves the boundary value problem ( λu − Au = f, in Ω, (13.2.32) Bu = g, on ∂Ω, for some λ ∈ C, f ∈ W k,p (Ω; C) and g ∈ B k+1−1/p,p (∂Ω; C). Then, u ∈ W k+2,p (Ω; C) and there exists a positive constant C, depending on p, d, λ and, in a continuous way, on µ and the W k,∞ (Ω; C)-norm of the coefficients of the operator A and the C k+1 -norm of the coefficients of the operator B, such that ||u||W k+2,p (Ω;C) ≤ C(||f ||W k,p (Ω;C) + ||g||B k+1−1/p,p (∂Ω;C) ).

(13.2.33)

Proof As one can easily see, the proofs of (i) and (ii) are similar. For this reason we give here only the proof of (i). The other one is left to the reader. Replacing the function u with the function u − Ep0 g, where Ep0 is the extension operator in Proposition B.4.15, we can assume that g ≡ 0 on ∂Ω and, hence u ∈ W 2,p (Ω; C) ∩ W01,p (Ω; C). Since Au = λu − f , it follows that Au ∈ W k∧2,p (Ω; C) and Theorem 13.2.12 shows that u ∈ W 2+k∧2,p (Ω; C). Hence, if k ≤ 2, then u ∈ W k+2,p (Ω; C). If k > 2, then we infer that u ∈ W 4,p (Ω; C). Hence, Au ∈ W 4∧k,p (Ω; C) and Theorem 13.2.12 implies that u ∈ W 2+k∧4,p (Ω; C). If k ≤ 4, then we are done. Otherwise we iterate this procedure and, in a finite number of steps, we obtain that u ∈ W k+2,p (Ω; C). Finally, estimate (13.2.31) follows from (13.2.21). Indeed, iterating estimate (1.3.9) in Proposition 1.3.8, it can be shown that ||u||W k,p (Ω;C) ≤ ε||u||W k+2,p (Ω;C) + Cε ||u||Lp (Ω;C) for every ε > 0 and some positive constant Cε , independent of u and blowing up as ε tends to 0. Starting from (13.2.21) and using this estimate we get ||u||W k+2,p (Ω;C) ≤C1 (||u||Lp (Ω;C) + ||Au||W k,p (Ω;C) ) ≤C2 (||u||W k,p (Ω;C) + ||f ||W k,p (Ω;C) ) ≤Kε (||u||Lp (Ω;C) + ε||u||W k+2,p (Ω;C) + ||f ||W k,p (Ω;C) ), where the constants C1 , C2 and Kε depend at most on p, d, λ and, in a continuous way on µ and the W k,∞ (Ω; C)-norm of the coefficients of the operator A. Constant Kε depends also on ε and blows up as ε tends to 0. Taking ε sufficiently small, we can get rid of the W k+2,p (Ω; C)-norm of u from the last side of the previous chain of inequalities. Finally, estimate (13.2.12) allows us to estimate the Lp -norm of u in terms of the Lp -norm of f and the B 2−1/p,p -norm of g. Estimate (13.2.31) follows. 

364

Elliptic Equations on Smooth Domains Ω

13.3

Solutions in L∞ (Ω;

C), in C(Ω; C) and in C (Ω; C) b

In this section, we study the solvability of the boundary value problems ( ( λu − Au = f, in Ω, λu − Au = f, in Ω, u = 0, on ∂Ω, Bu = 0, on ∂Ω,

(13.3.1)

when f ∈ L∞ (Ω; C), f ∈ C(Ω; C) and f ∈ Cb (Ω; C). In the first case we prove the following result. Theorem 13.3.1 Assume that Hypotheses 13.2.1 are satisfied. Then, there exists λ∞ ∈ R (resp. λ∗ ∈ R) such that, for every f ∈ L∞ (Rd+ ; C) and λ in C, with Re λ > λ∞ (resp. Reλ > λ∗∞ ), T the equation λu − Au = f admits a unique solution u, which belongs to 2,p Cb1+α (Ω; C) ∩ p λ∞ (resp. Re λ > λ∗∞ ). Proof Throughout the proof, if not otherwise specified, by Cj,p we denote a positive constant, which depends at most only on p, d and, in a continuous way, on the ellipticity constant of the operator A, on the modulus of continuity of its diffusion coefficients, on the sup-norms of the coefficients of the operator A and on the C 1 -norms of the coefficients of the operator B. First, we prove the a priori estimate (13.3.2). For this purpose, we begin by observing that p |λ|||u||Lp (Ω∩B(x0 ,r);C) + |λ||||∇u|||Lp (Ω∩B(x0 ,r);C) + ||D2 u||Lp (Ω∩B(x0 ,r);C)
≤C1,p ||λu − Au||Lp (B(x0 ,(γ+1)r)∩Ω;C) + γ −1 r−2 ||u||Lp (B(x0 ,(γ+1)r)∩Ω;C) (13.3.4) 2,p for every p > 1, u ∈ Wloc (Ω; C) ∩ C 1 (Ω; C), with u ≡ 0 on ∂Ω, r ∈ (0, 1], x0 ∈ Ω, γ ∈ [1, ∞), λ ∈ C, with Reλ ≥ λp,1 , and some positive constant C1,p . Here, λp,1 is the constant in Theorem 13.2.3. Fixed u and the other quantities described above, estimate (13.3.4) follows applying (13.2.3) to the function v = uϑ, where ϑ is a smooth function such that χB(x0 ,r) ≤ ϑ ≤ χB(x0 ,(γ+1)r) and ||ϑ||∞ + γr|||∇ϑ|||∞ + γ 2 r2 ||D2 ϑ||∞ ≤ K for some positive constant K, independent of γ and r. Using the same technique, it can be easily checked that p |λ|||u||Lp (Ω∩B(x0 ,r);C) + |λ||||∇u|||Lp (Ω∩B(x0 ,r);C) + ||D2 u||Lp (Ω∩B(x0 ,r);C)

≤C2,p

Semigroups of Bounded Operators and Second-Order PDE’s 365 p ||λu − Au||Lp (B(x0 ,(γ+1)r)∩Ω;C) + (γ −1 r−2 + |λ| + r−1 )||u||Lp (B(x0 ,(γ+1)r)∩Ω;C)
+ |||∇u|||Lp (B(x0 ,(γ+1)r)∩Ω;C) , (13.3.5)

if Bu ≡ 0 on ∂Ω. Indeed, the function u

∂ϑ belongs to W 1,p (Ω; C) and coincides with the ∂η

function Bv on ∂Ω. Now, we consider the covering {Uj : j = 1, . . . , N } of Ω in Lemma 13.1.2 and an associated partition of the unity {ϑj : j = 1 . . . , N }. We split u ∈ W 1,p (Ω; C) into the sum PN u = j=1 uj , where uj = ϑj u for every j = 1, . . . , N . Next step consists in proving that d

||uj ||L∞ (B(x0 ,r)∩Ω;C) ≤ C3,p r− p (||u||Lp (B(x0 ,r)∩Ω;C) + r|||∇u|||Lp (B(x0 ,r)∩Ω;C) )

(13.3.6)

for p > d, every j = 1 . . . , N , every r ∈ (0, 1] and some positive constant C3,p . To prove this estimate, arguing as in the proof of Proposition B.4.5, we extend all the functions uj (j = 1, . . . , N − 1) to Rd and denote by u bj the extended function, which satisfies the estimates ||b uj ||Lp (B(x0 ,r);C) ≤ C4,j ||uj ||Lp (B(x0 ,r)∩Ω;C) and ||b uj ||W 1,p (B(x0 ,r);C) ≤ C4,j ||uj ||W 1,p (B(x0 ,r)∩Ω;C) . Moreover, since the function uN is compactly supported in Ω ∩ B(x0 , r), its trivial extension uN to Rd belongs to W 1,p (B(x0 , r); C). Now, we use d the estimate ||w||∞ ≤ C5,p r− p (||w||Lp (B(x0 ,r);C) + r||w||W 1,p (B(x0 ,r);C) ), which holds true for every w ∈ W 1,p (B(x0 , r); C), see the proof of Theorem 10.3.1. This estimate implies that d

||uN ||L∞ (Ω∩B(x0 ,r);C) ≤C6,j r− p (||uN ||Lp (B(x0 ,r)∩Ω;C) + r||uN ||W 1,p (B(x0 ,r)∩Ω;C) ) d

≤C7,j r− p (||u||Lp (B(x0 ,r)∩Ω;C) + r||u||W 1,p (B(x0 ,r)∩Ω;C) ).

(13.3.7)

Similarly, ||uj ||L∞ (B(x0 ,r)∩Ω;C) d

≤C8,j r− p (||b uj ||Lp (B(x0 ,r);C) + r|||∇b uj |||Lp (B(x0 ,r);C) ) d

≤C9,j r− p (||uj ||Lp (B(x0 ,r)∩ω;C) + r||uj ||Lp (B(x0 ,r)∩Ω;C) + r|||∇uj |||Lp (B(x0 ,r)∩Ω;C) ) d

≤C10,j r− p ((1 + r)||u||Lp (B(x0 ,r)∩Ω;C) + r|||∇u|||Lp (B(x0 ,r)∩Ω;C) )

(13.3.8)

for every j = 1, . . . , N − 1. From (13.3.7) and (13.3.8), estimate (13.3.6) follows at once. Using (13.3.6), with r = |λ|−1/2 and Re λ ≥ λp,1 ∨ 1 (resp. Re λ ≥ λp,2 ∨ 1), and taking also (13.3.4) and (13.3.5) into account, we can estimate p |λ|||u||L∞ (B(x0 ,|λ|−1/2 );C) + |λ||||∇u|||L∞ (B(x0 ,|λ|−1/2 );C) + ||D2 u||Lp (B(x0 ,|λ|−1/2 );C) d

≤C11,p |λ| 2p ||λu − Au||Lp (B(x0 ,(γ+1)|λ|−1/2 )∩Ω;C) + |||∇u|||Lp (B(x0 ,(γ+1)|λ|−1/2 )∩Ω;C) p
+ (γ −1 |λ| + |λ|)||u||Lp (B(x0 ,(γ+1)|λ|−1/2 )∩Ω;C) . To proceed further, we use the inequality ||w||Lp (B(x0 ,(γ+1)|λ|−1/2 )∩Ω;C) ≤ C12,p [(γ + d

1

1) |λ|− 2 ] p ||w||∞ , applied to u and its gradient and in the end we get p |λ|||u||L∞ (B(x0 ,|λ|−1/2 )∩Ω;C) + |λ||||∇u|||L∞ (B(x0 ,|λ|−1/2 )∩Ω;C) d

+ ||D2 u||Lp (B(x0 ,|λ|−1/2 );C) ≤C13,p ||λu − Au||∞ + (γ −1 |λ| +

p


|λ|)||u||∞ + |||∇u|||∞ .

Taking first the supremum with respect to x0 ∈ Ω and then taking γ sufficiently large,

366

Elliptic Equations on Smooth Domains Ω

and also replacing λp with a larger value if needed, to get rid of the L∞ -norm of u and its gradient from the right-hand side of the previous inequality, estimate (13.3.2) follows at once. To complete the proof, we can argue as in the proof of Theorem 11.3.1. Hence, we skip some details. We fix f ∈ L∞ (Ω; C) and λ ∈ C, with Re λ > λ∞ := inf p∈(d,∞) λp ∨ 1. Moreover, we approximate f by a sequence (fn ) of compactly supported functions such that ||fn ||∞ ≤ ||f ||∞ for every n ∈ N and fn converges to f pointwise almost everywhere on Ω. Finally, we fix p > d such that λ > λp and, for every n ∈ N, denote by un ∈ W 2,p (Ω; C) the unique solution of the equation λu − Au = fn such that un ≡ 0 (resp. Bun ≡ 0) on ∂Ω (see Theorem 13.2.5). Applying estimates (13.2.12), (13.2.13) and (13.3.2) to each function un , using also the Sobolev embedding theorems and a compactness argument, we conclude that the sequence (un ) is bounded in C 2−d/p (Ω; C) and in W 2,p (Ω ∩ B(x0 , r); C) for every r > 0. Hence, up to a subsequence, it converges to a function u which belongs to 2,p C 2−d/p (Ω; C) ∩ Wloc (Ω; C). Moreover, u (resp. Bu) vanishes on ∂Ω and solves the equation T 2,q λu − Au = f . To prove that u ∈ q p allows us to conclude that u ∈ q d there exist two positive constants λ p d |λ|||u||∞ + |λ||||∇u|||∞ + |λ| 2p sup ||D2 u||Lp (Ω∩B(x0 ,|λ|−1/2 );C) x0 ∈Ω  p d ep |λ| 2p sup ||λu − Au||Lp (Ω∩B(x ,|λ|−1/2 );C) + |λ| sup ||g||Lp (Ω∩B(x ,|λ|−1/2 );C) ≤C 0 0 x0 ∈Ω x0 ∈Ω  + sup |||∇g|||Lp (Ω∩B(x0 ,|λ|−1/2 );C) (13.3.9) x0 ∈Ω

2,p bp , where g ∈ W 2,p (Ω; C) is for every u ∈ Wloc (Ω; C) ∩ Cb1 (Ω; C) and λ ∈ C with Reλ ≥ λ any function which coincides with Bu on ∂Ω. The constant C depends only on d, p and, in a continuous way, on the ellipticity constant of the operator A, on the modulus of continuity of its diffusion coefficients, on the sup-norms of the coefficients of the operator A and on the Cb1 (∂Ω)- and Cb1 (∂Ω; Rd )-norms of the coefficients of the operator B. In particular, if u ∈ Cb2 (Ω; C), then p p b |λ|||u||∞ + |λ||||∇u|||∞ + ||D2 u||∞ ≤ C(||λu − Au||∞ + |λ|||Bu||∞ + |||∇Bu|||∞ ), (13.3.10)

b depends only on d and, in a continuous way, on the ellipticity constant where the constant C of the operator A, on the modulus of continuity of its diffusion coefficients, on the supnorms of the coefficients of the operator A and on the Cb1 (∂Ω)- and Cb1 (∂Ω; Rd )-norms of the coefficients of the operator B. Proof The proof is similar to that of Corollary 12.3.2. Hence, we skip some details. Throughout the proof, we denote by C a positive constant, which may vary from line to line and depends at most on d, p and, in a continuous way, on the ellipticity constant of the operator A, on the modulus of continuity of its diffusion coefficients, on the sup-norms of the coefficients of the operator A and on the Cb1 (∂Ω)- and Cb1 (∂Ω; Rd )-norm of the coefficients

Semigroups of Bounded Operators and Second-Order PDE’s

367

of the operator B. Note that (13.3.10) is a straightforward consequence of (13.3.9). So, we limit ourselves to proving estimate (13.3.9). Using estimate (13.2.4) and arguing as in the proof of Theorem 13.3.1, it can be proved that p d |λ|||u||∞ + |λ||||∇u|||∞ + |λ| 2p sup ||D2 u||Lp (Ω∩B(x0 ,|λ|−1/2 );C) x0 ∈Ω  d 1 1p ≤Cγ |λ| 2p sup ||λu − Au||Lp (Ω∩B(x0 ,(1+γ)|λ|−1/2 ;C)) + |λ|||u||∞ + |λ||||∇u|||∞ γ γ x0 ∈Ω  d 1 + |λ| 2 + 2p sup ||g||Lp (Ω∩B(x0 ,(1+γ)|λ|−1/2 );C) + sup |||∇g|||Lp (Ω∩B(x0 ,(1+γ)|λ|−1/2 );C) x0 ∈Ω

x0 ∈Ω

(13.3.11) 2,p for every u ∈ Wloc (Ω; C)∩Cb1 (Ω; C), every g ∈ W 1,p (Ω; C) which coincides with the function Bu on ∂Ω, and every λ ∈ C with Reλ > λp ∨1. Here, the constant Cγ depends in a continuous way on γ. Taking γ large enough and replacing λp ∨ 1 with a larger value if needed, we can move the L∞ -norm of u and its gradient from the right- to the left-hand side of (13.3.11) and conclude that p d |λ|||u||∞ + |λ||||∇u|||∞ + |λ| 2p sup ||D2 u||Lp (Ω∩B(x0 ,|λ|−1/2 );C) x0 ∈Ω  d ep |λ| 2p sup ||λu − Au||Lp (Ω∩B(x ,(1+γ)|λ|−1/2 ;C) + |λ| 12 sup ||g||Lp (Ω∩B(x ,(1+γ)|λ|−1/2 );C) ≤C 0 0 x0 ∈Ω x0 ∈Ω  + sup |||∇g|||Lp (Ω∩B(x0 ,(1+γ)|λ|−1/2 );C) . (13.3.12) x0 ∈Ω

Finally, we observe that there exist N ∈ N, independent of |λ| and, for every x0 ∈ Ω, N [ N points x1 , . . . , xN such that B(x0 , (γ + 1)|λ|−1/2 ) ⊂ B(xj , |λ|−1/2 ). Thus, we can j=1

estimate ||λu − Au||Lp (Ω∩B(x0 ,(γ+1)|λ|−1/2 );C) ≤ N 1/p supx∈Ω ||λu − Au||Lp (Ω∩B(x,|λ|−1/2 );C) . Replacing this inequality in the right-hand side of (13.3.12), estimate (13.3.9) follows.  We can now consider the case when the datum f belongs to C(Ω; C) assuming the following set of assumptions. Hypotheses 13.3.3

(i) Ω is a bounded domain of class C 2 ;

(ii) The coefficients qij = qji (i, j = 1, . . . , d) belong to C(Ω), whereas the coefficients bj (j = 1, . . . , d) and c belong to Cb (Ω); (iii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for all x ∈ Ω and ξ ∈ Rd ; (iv) the functions a and η belong to C 1 (∂Ω) and C 1 (∂Ω, Rd ), respectively. Moreover, |η(x)| = 1 for each x ∈ ∂Ω and hη(x), ν(x)i > 0. Theorem 13.3.4 Under Hypotheses 13.3.3(i)-(iii) (resp. Hypotheses 13.3.3), for every f ∈ C(Ω; C) and λ ∈ C, with Re λ > λ∞ (resp. with Re λ > λ∗∞ ), the equation λu − Au = f 2,p admits a unique solution u ∈ C 1+α (Ω; C) ∩ Wloc (Ω; C), for every p ∈ [1, ∞) and α ∈ (0, 1), such that u ≡ 0 (resp. Bu ≡ 0) on ∂Ω. Moreover, there exist two positive constant e∞ and C b∞,λ , which depend only on d (the constant C b∞,λ depends also on λ) and, in a C continuous way, on the ellipticity constant of the operator A, on the modulus of continuity

368

Elliptic Equations on Smooth Domains Ω

of its diffusion coefficients and on the sup-norms of the coefficients of the operator A (and also on the C 1 -norms of the coefficients of the operator B, if the second problem in (13.3.1) is considered) such that p e∞ ||λu − Au||∞ |λ|||u||∞ + |λ||||∇u|||∞ ≤ C and b∞,λ ||λu − Au||∞ ||u||C 1+α (Ω;C) ≤ C for every λ ∈ C, with Re λ > λ∞ (resp. with Re λ > λ∗∞ ). Here, λ∞ and λ∗∞ are the same constants as in Theorem 13.3.1. Proof The proof is a straightforward consequence of Theorem 13.3.1.



Remark 13.3.5 The assertion of Theorem 13.3.4 is true also if we take f ∈ Cb (Ω; C), with no differences in the proof.

13.4

Exercises

1. Complete the proof of Theorem 13.2.8. [Hint: argue as in the proof of Theorem 11.2.8.] 2. Prove property (ii) in Corollary 13.2.13.

Chapter 14 Elliptic Operators and Analytic Semigroups

In this chapter, taking advantage of the results proved in all the previous chapters, we show that the semigroups considered in Chapters 6 to 9 are analytic and we characterize the interpolations spaces of order α and 1 + α for every α ∈ (0, 1) \ {1/2}. Moreover, using tools from semigroup theory we provide the proof of the regularity results in Section 7.4. This to further enlighten the connections between parabolic equations and analytic semigroups of bounded operators.

14.1

R ; C)

The Semigroup {T (t)} in Cb (

d

Here, we consider the operator A, defined on smooth functions ψ : Rd → R by Aψ(x) =

d X i,j=1

qij Dij ψ(x) +

d X

bj (x)Dj ψ(x) + c(x)ψ(x),

x ∈ Rd ,

j=1

under the following assumptions on its coefficients. Hypotheses 14.1.1 (i) The coefficients qij = qji , bj (i, j = 1, . . . , d) and c are bounded and α-H¨ older continuous in Rd for some α ∈ (0, 1); (ii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for all x, ξ ∈ Rd . In Section 6.4, we have introduced a semigroup {T (t)} of bounded operators in Cb (Rd ), which is associated to the operator A. For every f ∈ Cb (Rd ) and every t > 0, T (t)f is the 1+α/2,α value at t of the unique solution u ∈ C([0, ∞) × Rd ) ∩ Cloc (0, ∞) × Rd ) to the Cauchy problem ( Dt u(t, x) = Au(t, x), t ∈ (0, ∞), x ∈ Rd , (14.1.1) u(0, x) = f (x), x ∈ Rd , which is bounded in [0, T ] × Rd for every T > 0. Clearly, we can extend each operator T (t) to Cb (Rd ; C) by setting T (t)f = T (t)Ref + iT (t)Imf for every f ∈ Cb (Rd ; C). Since the coefficients of the operator A are real-valued, it follows that, for every f ∈ Cb (Rd ; C), the function T (·)f is the unique solution to problem (14.1.1) in C([0, ∞) × Rd ; C) ∩ 1+α/2,α Cloc (0, ∞) × Rd ; C), which is bounded in the strip [0, T ] × Rd for every T > 0. In particular, if f ∈ Cb2+α (Rd ; C), then T (t)f belongs to Cb2+α (Rd ; C) for every t > 0. Moreover, the function t 7→ T (t)f is continuous in [0, ∞) with values in Cb2 (Rd ; C) (see Theorem 6.2.6). For further use, we introduce the sets   \ 2,p d d d D = u ∈ Cb (R ; C) ∩ Wloc (R ; C) : Au ∈ Cb (R ; C) , p 0. This property, together with Theorem 6.4.2, allows to conclude that T (t)f = S(t)f for every t > 0. So, we fix f and (fn ) as above. As a first step, we prove that R(λ, A)fn converges to R(λ, A)f pointwise in Rd , for every λ ∈ C with Reλ > λ∞ , where λ∞ is the constant in Theorem 10.3.1. By Theorem 10.3.4, for every n ∈ N the function un = R(λ, A)fn belongs to D, solves the equation λun −Aun = fn and satisfies the estimate ||un ||C 1+β (Rd ;C) + sup ||D2 un ||Lp (B(x0 ,|λ|−1/2 );C) ≤ C1 ||fn ||∞ ≤ C2 b

x0 ∈Rd

for every n ∈ N, β ∈ (0, 1), p ∈ [1, ∞) and some positive constants C1 and C2 , which are independent of n. From the previous inequality it follows that the sequence (un ) is bounded in Cb1+β (Rd ; C) and in W 2,p (B(0, R); C), for every β ∈ (0, 1), p ∈ [1, ∞) and R > 0. A 2,p compactness argument shows that there exists a function u ∈ Cb1+β (Rd ; C) ∩ Wloc (Rd ; C), 1+β 2 for every β ∈ (0, 1) and p ∈ [1, ∞), such that un tends to u in C (B(0, R); C) and Dij un 2 converges to Dij u weakly in Lp (B(0, R); C), as n tends to ∞, for every β ∈ (0, 1), p ∈ [1, ∞), R > 0 and i, j = 1, . . . , d. Therefore, for every ϕ ∈ Cc∞ (Rd ; C) it holds that Z Z Z Z f ϕ dx = lim f ϕ dx = lim (λun − Aun )ϕ dx = (λu − Au)ϕ dx, Rd

n→∞

Rd

n→∞

Rd

Rd

so that u solves the equation λu − Au = f . Since u ∈ D, we conclude that u = R(λ, A)f . From the above results, it follows that un converges to u actually locally uniformly on Rd . By recurrence, we can show that R(λ, A)k fn converges to R(λ, A)k f locally uniformly in Rd for every k ∈ N.

Semigroups of Bounded Operators and Second-Order PDE’s

371

Next, we prove that R(λ, A)fn converges pointwise in Rd to R(λ, A)f for λ in a sector of the complex plane. For this purpose, we argue as in the proof of Proposition 3.2.8. This proposition shows how to determine a sector contained in the resolvent set of the operator A. We set ω = λ∞ +1 and denote by M a positive constant such that ||λR(λ, A)||L(Cb (Rd ;C)) ≤ M for every λ ∈ C with Reλ ≥ λ∞ + 1. Then, by Proposition 3.2.8, ρ(A) contains the sector Σω,θ2M = {λ ∈ C : |arg(λ − ω)| ≤ θ2M }, where θ2M = π − arctan(2M ). Moreover, if λ ∈ Σω,θ2MP and Re λ < ω, then there exists r ∈ R such that λ ∈ B(ω + ir, |ω + ir|/M ) and ∞ R(λ, A) = k=0 (−1)k (λ − ω − ir)(R(ω + ir, A))k+1 . Therefore, (R(λ, A)fn )(x) =

∞ X

(−1)k (λ − ω − ir)k ((R(ω + ir, A))k+1 fn )(x),

x ∈ Rd , n ∈ N.

k=0

From this formula and the dominated convergence theorem, it follows that R(λ, A)fn converges to R(λ, A)f pointwise on Rd . We are almost done. Indeed, Z eωt ∞ ρ cos(η)t iρ sin(η)t e e (R(ω + ρeiη , A)fn )(x) dρ (S(t)fn )(x) = 2πi r Z eωt ∞ ρ cos(η)t −iρ sin(η)t e e (R(ω + ρe−iη , A)fn )(x) dρ − 2πi r Z eωt r η (r cos(θ)+ir sin(θ))t e (R(ω + reiθ , A)fn )(x)eiθ dθ, + 2πi −η for every t > 0 and x ∈ Rd , where r > 0 and η ∈ (π/2, θ2M ) are arbitrarily fixed (see Section 3.2). Applying the dominated convergence theorem, we can show that, for every t > 0, S(t)fn converges to S(t)f , pointwise on Rd , as n tends to ∞. (ii) Here, we prove that {T (t)} extends to L∞ (Rd ; C) with an analytic semigroup. For this purpose, we observe that Proposition 3.2.8 and Theorem 10.3.1 guarantee that the operator A∞ generates an analytic semigroup {T∞ (t)} on L∞ (Rd ; C). The resolvent set of A∞ contains the sector Σω,θ2M and R(λ, A)f = R(λ, A∞ )f for every f ∈ Cb (Rd ; C) and λ ∈ Σω,θ2M . Since Z eωt ∞ ρ cos(η)t iρ sin(η)t T∞ (t)f = e e R(ω + ρeiη , A∞ )f dρ 2πi r Z eωt ∞ ρ cos(η)t −iρ sin(η)t − e e R(ω + ρe−iη , A∞ )f dρ 2πi r Z eωt r η (r cos(θ)+ir sin(θ))t + e R(ω + reiθ , A∞ )f eiθ dθ 2πi −η for every f ∈ L∞ (Rd ; C) and t > 0, it follows that T∞ (t)f = T (t)f for every f ∈ Cb (Rd ; C) and t > 0. (iii) To prove that D = D∞ = BU C(Rd ; C), we will show that BU C(Rd ; C) ⊂ D and D∞ ⊂ BU C(Rd ; C). Since every function f ∈ BU C(Rd ; C) is the uniform limit of a sequence (fn ) ⊂ Cb2 (Rd ; C) and Cb2 (Rd ; C) is contained in D, we conclude that BU C(Rd ; C) ⊂ D. Suppose now that f ∈ D∞ . Then, the function T∞ (t)f converges to f uniformly in Rd . Note that T∞ (t)f ∈ D∞ ⊂ Cb1 (Rd ; C) for every t > 0. The last inclusion follows from Theorem 10.3.1. Hence, T∞ (t)f ∈ BU C(Rd ; C) for every t > 0 and, as a byproduct, f ∈ BU C(Rd ; C).  We now characterize the interpolation spaces DA (β, ∞) and DA (1 + β, ∞).

372

Elliptic Operators and Analytic Semigroups

Theorem 14.1.3 The following properties are satisfied. (i) For every β ∈ (0, 1) \ {1/2} it holds that DA (β, ∞) = Cb2β (Rd ; C), with equivalence of the corresponding norms. (ii) For every β ∈ (0, α/2], it holds that DA (1 + β, ∞) = Cb2+2β (Rd ; C), with equivalence of the corresponding norms. (iii) If β ∈ (1/2, 1) and the coefficients of the operator A belong to Cb2β (Rd ), then DA (1 + β, ∞) = Cb2+2β (Rd ; C), with equivalence of the corresponding norms. Proof (i) The proof is similar to that of Theorem 5.3.6 and is based on the estimates of the semigroup {T (t)} in Theorem 6.4.3. Hence, we refer to Theorem 5.3.6 for the missing details. We first assume that the supremum c0 of the function c over Rd is negative. Throughout the proof, we denote by C a positive constant, independent of f and t, which may vary from line to line. Fix β ∈ (0, 1) \ {1/2}. The embedding Cb2β (Rd ; C) ,→ DA (β, ∞) follows easily from Theorem 6.4.3 which shows that ||AT (t)f ||∞ ≤ Ctβ−1 ||f ||C 2β (Rd ;C) for every t ∈ (0, 1], b

f ∈ Cbβ (Rd ; C) and β ∈ (0, 1). To check the other embedding, we fix f ∈ DA (β, ∞) and prove some preliminary results. First of all, we use Remark 3.3.5 to write Z s (T (s)f )(x) = (T (t)f )(x) + (AT (r)f )(x) dr, 0 < t < s, x ∈ Rd . (14.1.3) t

Next we observe that, since ||T (t)||L(Cb (Rd ;C)) ≤ ec0 t for every t > 0 (see (6.4.2)), using Theorem 6.4.3 we can estimate ||Dγ T (t)||L(Cb (Rd ;C)) ≤||Dγ T (1)T (t − 1)||L(Cb (Rd ;C)) ≤||Dγ T (1)||L(Cb (Rd ;C)) ||T (t − 1)||L(Cb (Rd ;C)) ≤Cec0 (t−1) ≤ Ct−

|γ| 2

(14.1.4)

for t > 1, every multi-index γ ∈ (N ∪ {0})d with length equal to one or two. From estimate (14.1.4) it follows that ||Dγ AT (t)f ||∞ = ||Dγ T (t/2)AT (t/2)f ||∞ ≤ Ctβ−1−

|γ| 2

||f ||DA (β,∞)

(14.1.5)

for every γ as above. We can thus differentiate formula (14.1.3) with respect to the spatial variables and write Z s γ γ (D T (s)f )(x) = (D T (t)f )(x) + (Dγ AT (r)f )(x) dr, 0 < t < s, x ∈ Rd , |γ| = 1, 2. t

(14.1.6) If |γ| = 2 (resp. |γ| = 1 and β < 1/2), then, thanks to (14.1.5), we can let s tend to ∞ in (14.1.6) and then take the sup-norms of both the sides of the so obtained formula to infer that |γ| ||Dγ T (t)f ||∞ ≤ Ctβ− 2 ||f ||DA (β,∞) , t > 0, |γ| = 1, 2. (14.1.7) If β > 1/2, then, from (14.1.5) and (14.1.6) we deduce that the function Dγ T (·)f can be extended to [0, ∞) with a continuous function with values in Cb (Rd ; C) for every multiindex γ with length equal to one. In particular, f belongs to Cb1 (Rd ; C) and ||f ||Cb1 (Rd ;C) ≤ C||f ||DA (β,∞) . So, from (14.1.6), with (s, t) = (t, 0), we obtain that 1

||Dγ T (t)f − Dγ f ||∞ ≤ Ctβ− 2 ||f ||DA (β,∞) ,

t > 0, |γ| = 1.

(14.1.8)

Semigroups of Bounded Operators and Second-Order PDE’s

373

Now, we can complete the proof. If β < 1/2, we fix x, y ∈ Rd , with x 6= y, and estimate |f (x) − f (y)| ≤|(T (t)f )(x) − f (x)| + |(T (t)f )(x) − (T (t)f )(y)| + |(T (t)f )(y) − f (y)| ≤C(||f ||DA (β,∞) tβ + |||∇x T (t)f |||∞ |x − y|). Using (14.1.7) with t = |x − y|2 , the embedding DA (β, ∞) ,→ Cb2β (Rd ; C) follows. If β > 1/2, then we split |Dγ f (x) − Dγ f (y)| ≤|(Dγ T (t)f )(x) − Dγ f (x)| + |(Dγ T (t)f )(x) − (Dγ T (t)f )(y)| + |(Dγ T (t)f )(y) − Dγ f (y)| 1

≤C(||f ||DA (β,∞) tβ− 2 + |||∇x Dγ T (t)f |||∞ |x − y|) for every x, y ∈ Rd , with x 6= y, and every multi-index γ with length equal to one. Taking t = |x − y|2 and using (14.1.7) and (14.1.8), we conclude that [Dγ f ]C 2β−1 (Rd ;C) ≤ C||f ||DA (β,∞) , b

and the embedding DA (β, ∞) ,→ Cb2β (Rd ; C) follows also in this case. To remove the condition on c0 , it suffices to observe that, if c0 ≥ 0, then, for every c1 > c0 , the semigroup {Tc1 (t)}, defined by Tc1 (t) = e−c1 t T (t) for every t > 0, is analytic, ||Tc1 (t)||L(Cb (Rd ;C)) ≤ e−(c1 −c0 )t for every t > 0. The semigroup {Tc1 (t)} is associated with the sectorial operator A0 = A − c1 , having D as domain. Moreover, the graph norm of A and A0 are equivalent. Indeed ||u||Cb (Rd ;C) + ||Au||Cb (Rd ;C) ≤ (1 + c1 )||u||Cb (Rd ;C) + ||A0 u||Cb (Rd ;C) ≤ (1 + c1 )||u||D(A0 ) , ||u||Cb (Rd ;C) + ||A0 u||Cb (Rd ;C) ≤ (1 + c1 )||u||Cb (Rd ;C) + ||Au||Cb (Rd ;C) ≤ (1 + c1 )||u||D(A) . Since DA (β, ∞) = (Cb (Rd ; C); D)β,∞ (see Proposition 3.3.7) and the norms of the spaces D and D(A0 ) are equivalent, it follows that DA (β, ∞) = (Cb (Rd ; C); D)β,∞ = (Cb (Rd ; C); D(A0 ))β,∞ = Cb2β (Rd ; C), with equivalence of the corresponding norms. (ii) Fix β ∈ (0, 1/2). To begin with, we prove that Cb2+2β (Rd ; C) is continuously embedded into DA (1 + β, ∞). For this purpose, we observe that Theorem 14.1.2 shows that Cb2+2β (Rd ; C) ,→ D. Moreover, Af belongs to Cb2β (Rd ; C) for f ∈ Cb2+2β (Rd ; C), so that, by property (i), Af belongs to DA (β, ∞) and ||Af ||DA (β,∞) ≤ ||f ||C 2+2β (Rd ;C) . The embedding b

Cb2+2β (Rd ; C) ,→ DA (1 + β, ∞) follows. Vice versa, let us suppose that f ∈ DA (1 + β, ∞). Then, u ∈ D and Au ∈ DA (β, ∞) = 2β Cb (Rd ; C). Since D(A) ,→ DA (β, ∞) (see Remark 3.3.2), u and Au belong to Cb2β (Rd ; C). Set f = λ0 u − Au, where λ0 = 1 + (c0 ∨ λ∞ ). Theorem 10.1.4 guarantees that there exists a unique function v ∈ Cb2+2β (Rd ; C) such that λ0 v − Av = f . Since v ∈ D, by uniqueness it follows that v ≡ u. Hence, u ∈ Cb2+2β (Rd ; C). Moreover, estimate (10.1.6) shows that ||u||C 2+α (Rd ;C) ≤ C||f ||Cbα (Rd ;C) ≤ C(|λ0 |||u||C 2β (Rd ;C) + ||Au||C 2β (Rd ;C) ) ≤ C||u||DA (1+β,∞) , b

b

b

β, ∞) ,→ Cb2+2β (Rd ; C) follows. ∈ Cb2+2+β (Rd ; C). Then, u ∈ D(A)

The embedding DA (1 + (iii) Suppose that f and the function Au belongs to Cb2β (Rd ; C). Hence, by property (i), Au ∈ DA (β, ∞), so that u ∈ DA (1 + β, ∞). The embedding C 2+2β (Rd ; C) ,→ DA (1 + β, ∞) follows. To prove the other embedding, let us assume that u ∈ DA (1 + β, ∞). Then, for λ ∈ ρ(A), the function f = λu − Au belongs to Cb2β (Rd ). By Theorem 10.1.7, the equation λv − Av = f admits a unique solution v ∈ Cb2+2β (Rd ; C). Moreover, ||v||C 2+2β (Rd ;C) ≤ C||f ||C 2β (Rd ;C) ≤ C||u||DA (1+β,∞) , where the b b constant C is independent of u. Since v ∈ D(A), by uniqueness, it follows that v ≡ u and the embedding DA (1 + β, ∞) ,→ Cb2+2β (Rd ; C). follows.  We conclude this section with the following corollary, which improves Theorem 10.1.4.

374

Elliptic Operators and Analytic Semigroups

Corollary 14.1.4 For every λ ∈ C with Re λ > λ∞ and every f ∈ Cbα (Rd ; C), the elliptic equation λu − Au = f admits a unique solution u ∈ Cb2+α (Rd ; C). Moreover, there exists a positive constant C, independent of u, λ and f , such that |λ|||u||C 2+α (Rd ;C) ≤ C||f ||Cbα (Rd ;C) .

(14.1.9)

b

Proof The proof follows immediately from Theorems 3.3.13 and 14.1.3, which show that the part of operator A in Cbα (Rd ; C) generates an analytic semigroup and its spectrum is contained in σ(A). Hence, for every λ ∈ C, with Reλ > λ∞ and every f ∈ Cbα (Rd ; C), the elliptic equation λu − Au = f admits a unique solution u ∈ Cb2+α (Rd ; C). Moreover, the proof of Theorem 3.3.13 shows that ||u||C 2+α (Rd ;C) ≤ ||R(λ, A)||L(Cb (Rd ;C)) ||f ||Cbα (Rd ;C) . Hence, b estimate (10.3.9) yields (14.1.9). 

14.2

R ; C)

The Semigroups in Cb (

d +

In this section, we consider the semigroups associated with the elliptic operator A, defined on smooth enough functions ψ : Rd+ → R by Aψ(x) =

d X i,j=1

qij (x)Dij ψ(x) +

d X

bj (x)Dj ψ(x) + c(x)ψ(x)

j=1

for every x ∈ Rd+ , under the following assumptions on its coefficients. Hypotheses 14.2.1 (i) The coefficients qij = qji , bj (i, j = 1, . . . , d) and c are bounded and α-H¨ older continuous in Rd+ for some α ∈ (0, 1); (ii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for every x ∈ Rd+ and ξ ∈ Rd . Associated to these operators, in Chapters 7 and 8 we have considered the CauchyDirichlet problem  d   Dt u(t, x) = Au(t, x), t ∈ (0, ∞), x ∈ R+ , u(t, x) = 0, t ∈ (0, ∞), x ∈ ∂Rd+ , (14.2.1)   d u(0, x) = f (x), x ∈ R+ and the Cauchy problem with first-order boundary conditions  d   Dt u(t, x) = Au(t, x), t ∈ (0, ∞), x ∈ R+ , Bu(t, x) = 0, t ∈ (0, ∞), x ∈ ∂Rd+ ,   u(0, x) = f (x), x ∈ Rd+ , where B = aI +

(14.2.2)

∂ , under the following conditions on the coefficients of the operator B. ∂η

Hypotheses 14.2.2 respectively;

(i) Functions a and η belong to Cb1+α (Rd−1 ) and Cb1+α (Rd−1 , Rd ),

Semigroups of Bounded Operators and Second-Order PDE’s

375

(ii) |η(y)| = 1 for each y ∈ Rd−1 and η0 := supRd−1 ηd < 0, where ηd denotes the last component of the function η. In Sections 7.5 and 8.5, we have shown that the solutions to the previous Cauchy problems are governed by semigroups of bounded operators. More precisely, there exist two semigroups {T1 (t)} and {T2 (t)} on Cb (Rd+ ; C), such that, for every f ∈ Cb (Rd+ ), T1 (·)f is the unique solution in C([0, ∞) × Rd+ ) ∩ C((0, ∞) × Rd+ ) ∩ C 1,2 ((0, ∞) × Rd+ ) to problem (14.2.1), which is bounded in each strip (0, T ) × Rd+ . Similarly, T2 (·)f is the unique solution in C([0, ∞) × Rd+ ) ∩ C 0,1 ((0, ∞) × Rd+ ) ∩ C 1,2 ((0, ∞) × Rd+ ) to problem (14.2.2), which is bounded in each strip (0, T ) × Rd+ . In Chapters 7 and 8 we have considered the semigroups {T1 (t)} and {T2 (t)} in Cb (Rd+ ), by setting Tj (t)f = Tj (t)Ref + iTj (t)Imf for every t > 0 and f ∈ Cb (Rd+ ; C), we can deal with complex-valued functions. As in Section 14.1, we can start proving that the semigroups {T1 (t)} and {T2 (t)} are analytic. For this purpose, we introduce the following sets.   \ 2,p D0 = u ∈ Cb (Rd+ ; C) ∩ Wloc (Rd+ ; C) : Au ∈ Cb (Rd+ ; C), u|∂Rd+ ≡ 0 , p 0. A 2,p compactness argument shows that there exists a function u ∈ Cb1+β (Rd+ ; C) ∩ Wloc (Rd+ ; C) 1+β for every β ∈ (0, 1) and p ∈ [1, ∞), such that un tends to u in C (B+ (0, R); C) and 2 2 Dij un converges to Dij u weakly in Lp (B+ (0, R); C), as n tends to ∞, for every β ∈ (0, 1), p ∈ [1, ∞), R > 0 and i, j = 1, . . . , d. Moreover, u solves the equation λu − Au = f and vanishes on ∂Rd+ . Hence, u = R(λ, A0 )f . We have so proved, that R(λ, A0 )fn converges to R(λ, A0 )f locally uniformly in Rd+ . By recurrence, we can show that R(λ, A0 )k fn converges to R(λ, A0 )k f locally uniformly in Rd for every k ∈ N. Arguing as in the proof of Theorem 14.1.2 with no differences, it can be shown that R(λ, A)fn converges pointwise in Rd+ to R(λ, A)f for λ in a sector of the complex plane. This result and the representation formula of the semigroup {S(t)} through the Dunford integral (see Section 3.2), allow us to conclude that S(t)fn converges to S(t)f pointwise on Rd+ . (ii) This property can be obtained arguing as in the proof of Theorem 14.1.2, taking Theorem 11.3.1 into account. (iii) The proof is similar to that of property (i). It suffices to repeat the same arguments, taking Theorems 8.5.3 and 12.3.4 into account. (iv) This property can be obtained arguing as in the proof of Theorem 14.1.2, taking Theorem 12.3.1 into account. (v) To prove that D0 = D0,∞ = {f ∈ BU C(Rd+ ; C) : f|∂Rd+ ≡ 0}, we will show that {f ∈ BU C(Rd+ ; C) : f|∂Rd+ ≡ 0} ⊂ D0,∞ and D0,∞ ⊂ {f ∈ BU C(Rd+ ; C) : f|∂Rd+ ≡ 0}. Fix f ∈ BU C(Rd+ ; C), which vanishes on ∂Rd+ , and extend it odd with respect to the last variable. Then, regularize this function by convolution with a sequence of mollifiers, which are even functions with respect to the last variable. Denote by (fn ) the so obtained sequence. Each function fn belongs to Cb2 (Rd+ ; C) and vanishes on ∂Rd+ . Hence, it belongs to D0 . Moreover, fn converges to f uniformly on Rd+ . Hence, {f ∈ BU C(Rd+ ; C) : f|∂Rd+ ≡ 0} is contained in D0 . Vice versa, suppose that f ∈ D0,∞ . Then, the function T0,∞ (t)f converges

Semigroups of Bounded Operators and Second-Order PDE’s

377

to f uniformly in Rd+ . Since T0,∞ (t)f ∈ D0,∞ ⊂ Cb1 (Rd+ ; C), T0,∞ (t)f ∈ BU C(Rd+ ; C) for every t > 0 and, of course, it vanishes on ∂Rd+ . Hence, f belongs to BU C(Rd+ ; C) and vanishes on Rd+ . The inclusion BU C(Rd+ ; C) ⊂ DB follows from Lemma 8.5.1. The inclusion DB , ∞ ⊂  BU C(Rd+ ; C) can be proved as the embedding D0 ⊂ {f ∈ BU C(Rd+ ; C) : f|∂Rd+ ≡ 0}. We now characterize the interpolation spaces DA0 (β, ∞), DA0 (1 + β, ∞), DAB (β, ∞), DAB (1 + β, ∞). We begin by considering the first two spaces. Theorem 14.2.4 Let Hypotheses 14.2.1 be satisfied. Then, the following properties hold true. (i) For every β ∈ (0, 1) \ {1/2}, it holds that DA0 (β, ∞) = {u ∈ Cb2β (Rd+ ; C) : u|∂Rd+ ≡ 0}, with equivalence of the corresponding norms. (ii) For every β ∈ (0, α/2], it holds that DA0 (1 + β, ∞) = {u ∈ Cb2+2β (Rd ; C) : u|∂Rd+ ≡ (Au)|∂Rd+ ≡ 0}, with equivalence of the corresponding norms. (iii) If β ∈ (1/2, 1) and the coefficients of the operator A belong to C 2β (Rd+ ), then DA0 (1 + β, ∞) = {u ∈ Cb2+2β (Rd ; C) : u|∂Rd+ ≡ (Au)|∂Rd+ ≡ 0}, with equivalence of the corresponding norms. Proof (i) To begin with, we prove that, for every θ ∈ [0, 1), there exists a positive constant C1 such that 1+θ

1−θ

||f ||C 1+θ (Rd ;K) ≤ C1 ||f ||∞2 ||f ||D20 , b

f ∈ D0 .

(14.2.3)

+

Let λ∞ be the constant in Theorem 11.3.1. Applying estimate (11.3.14), we get |||∇R(λ, A0 − λ∞ )g|||∞ = |||∇R(λ + λ∞ , A0 )g|||∞ ≤ √

C2 ||g||∞ λ + λ∞

for every g ∈ Cb (Rd+ ; C) and λ > 0 or, even, ||∇f ||∞ ≤ √

C2 C2 (λ||f ||∞ + ||(A0 − λ∞ )f ||∞ ) ≤ √ (λ||f ||∞ + ||(A0 − λ∞ )f ||∞ ) λ + λ∞ λ

for every λ > 0 and f ∈ D0 , where the constant C2 is independent of λ, g and f . Minimizing with respect to λ ∈ (0, ∞), estimate (14.2.3) with θ = 0 follows. To prove (14.2.3) with θ > 0, we fix j ∈ {1, . . . , d}. From (11.3.14), with p = (1 − θ)−1 d, and arguing as above, we can show that d

sup ||Dj f ||W 1,p (B+ (x0 ,|λ|−1/2 );C) ≤ C3 λ− 2p ||λf − (A0 f − λ∞ f )||∞

x0 ∈Rd +

for every λ > 0 and some positive constant C3 , which is independent of λ and f , as all the other forthcoming constants. Then, the Sobolev embedding theorem shows that Dj f ∈ C θ (B+ (x0 , |λ|−1/2 ); C) and d

sup ||Dj f ||C θ (B+ (x0 ,|λ|−1/2 );C) ≤ C4 λ− 2p ||λf − (A0 f − λ∞ f )||∞ .

x0 ∈Rd +

Fix x, y ∈ Rd+ , with |x − y| ≤ |λ|−1/2 . The previous estimate shows that d |Dj f (x) − Dj f (y)| ≤ C5 λ− 2p ||λf − (A0 f − λ∞ f )||∞ . θ |x − y|

(14.2.4)

378

Elliptic Operators and Analytic Semigroups

On the other hand, if |x − y| > |λ|−1/2 , then using (11.3.14) we can estimate θ−1 θ |Dj f (x) − Dj f (y)| ≤2|λ| 2 ||∇f ||∞ ≤ C6 |λ| 2 ||λf − (A0 f − λ∞ f )||∞ θ |x − y| d

=C6 |λ|− 2p ||λf − (A0 f − λ∞ f )||∞ . We have so proved that [Dj f ]Cbθ (Rd+ ;C) ≤ C7 |λ|−d/(2p) ||λf − (A0 f − λ∞ f )||∞ for every λ > 1−θ

1+θ

0. Minimizing with respect to λ > 0, we obtain [Dj f ]Cbθ (Rd+ ;C) ≤ C8 ||f ||∞2 ||f ||D20 . Since ||f ||∞ ≤ ||f ||D0 we can also estimate 1

1−θ

1

1+θ

2 2 ||Dj f ||∞ ≤ C9 ||f ||∞ ||f ||D ≤ C9 ||f ||∞2 ||f ||D20 0

Combining these two estimate, (14.2.3) follows. Now, for notational convenience, for every γ > 0, we set C∗γ (Rd+ ; C) = Cbγ (Rd+ ; C) ∩ C0 (Rd+ ; C), endowed with the norm of Cbβ (Rd+ ; C). Since C∗2 (Rd + ; C) is continuously embedded in D0 and DA0 (β, ∞) = (Cb (Rd+ ; C), D0 )β,∞ (see Proposition 3.3.7), we can write C∗2β (Rd+ ; C) = (Cb (Rd+ ; C), C∗2 (Rd+ ; C))β,∞ ,→ (Cb (Rd+ ; C), D0 )β,∞ = DA0 (β, ∞), where the first equality, with equivalence of the corresponding norms, follows from Example A.5.11. Let us prove the other embedding. Fix θ ∈ (0, 1) such that 2β < 1 + θ. From the second part of the proof of Theorem A.5.12 and estimate (14.2.3), it follows that (Cb (Rd+ ; C), D0 )β,∞ ,→ (Cb (Rd+ ; C), C∗1+θ (Rd+ ; C))γ,∞ , where γ = 2(1 + θ)−1 β. Finally, we use again Example A.5.11 and the reiteration theorem (see Theorem A.5.12) to infer that (Cb (Rd+ ; C), C∗1+θ (Rd+ ; C))γ,∞ =(Cb (Rd+ ; C), (Cb (Rd+ ; C), C∗2 (Rd+ ; C)) 1+θ ,∞ ) 2

2β 1+θ ,∞

=(Cb (Rd+ ; C), C∗2 (Rd+ ; C))β,∞ = C∗2β (Rd+ ; C), with equivalence of the corresponding norms. The embedding DA0 (β, ∞) ,→ C∗2β (Rd+ ; C) follows. (ii) Suppose that u ∈ Cb2+2β (Rd+ ; C) vanishes on ∂Rd+ together with the function Au. Clearly, u belongs to D0 and ||u||D0 ≤ C1 ||u||C 2+2β (Rd ;C) for some positive constant C1 , inb

+

dependent of u. Moreover, Au belongs to Cb2β (Rd+ ; C) and, since it vanishes on ∂Rd+ , it belongs to DA0 (1 + β, ∞). From (i), we also deduce that ||A0 u||DA0 (β,∞) ≤ C2 ||Au||C 2β (Rd ;C) ≤ b

+

C3 ||u||C 2+2β (Rd ;C) , where the constants C2 and C3 are independent of u. We have so proved +

b

that {u ∈ Cb2+2β (Rd+ ; C) : u|∂Rd+ ≡ (Au)|∂Rd+ ≡ 0} ,→ DA0 (1 + β, ∞). Vice versa, let us assume that u ∈ DA0 (1 + β, ∞). Then, u and Au belong to Cb2β (Rd+ ; C). We set f = c1 u − Au, where c1 > c0 ∨ λ∞ and λ∞ is the constant in Theorem 11.3.1. By Theorem 11.1.2, there exists a unique function v ∈ Cb2+2β (Rd+ ; C) such that c1 v − Av = f on Rd+ and v ≡ Av ≡ 0 on ∂Rd+ . Clearly, v ∈ D0 . Hence, estimate (11.3.14) in Theorem 11.3.5, which can be applied to functions in D0 , implies that u ≡ v. This theorem also shows that ||v||C 2+2β (Rd ;C) ≤ C||f ||C 2β (Rd ;C) ≤ C||u||DA0 (1+β,∞) . b

+

b

+

(iii) Suppose that u ∈ Cb2+2β (Rd ; C) is such that u and Au identically vanish on ∂Rd+ . Then, Au belongs to DA0 (β, ∞), so that u ∈ DA0 (1 + β, ∞). The embedding {u ∈ Cb2+2β (Rd ; C) : u|∂Rd+ ≡ (Au)|∂Rd+ ≡ 0} ,→ DA0 (1 + β, ∞) follows. To prove the other embedding, we fix u ∈ DA0 (1 + β, ∞) and λ ∈ ρ(A0 ). Since the function f = λu − Au

Semigroups of Bounded Operators and Second-Order PDE’s

379

belongs to Cb2β (Rd+ ; C) and vanishes on ∂Rd+ (see by property (i)), by Theorem 11.1.5, there exists a unique solution v ∈ Cb2+2β (Rd+ ; C) to the elliptic equation λv − Av = f , which vanishes on ∂Rd+ . Moreover, ||v||C 2+2β (Rd ;C) ≤ C||f ||C 2β (Rd ) ≤ C||u||DA0 (1+β,∞) for some positive b

+

b

+

constant C, independent of u. Since v ∈ D(A0 ), by uniqueness it follows that u ≡ v and the embedding DA0 (1 + β, ∞) ,→ Cb2+2β (Rd+ ; C) follows.  To characterize the interpolation spaces DAB (β, ∞), DAB (1 + β, ∞), we shall take advantage of the following result. Proposition 14.2.5 Let Hypotheses 14.2.1 and 14.2.2 be satisfied. Then, Cb1 (Rd+ ; C) is continuously embedded into DAB (1/2, ∞). Proof In view of Proposition 3.3.6, to prove that Cb1 (Rd+ ; C) is continuously embedded in DAB (1/2, ∞), it suffices to show that lim supλ→∞ λ1/2 ||AR(λ, AB )f ||∞ ≤ C1 ||f ||C 1 (Rd ;C) b

+

for every f ∈ Cb1 (Rd+ ; C) and some positive constant C1 , independent of f . For this purpose, we fix f ∈ Cb1 (Rd+ ; C) and extend it with a function Eα f ∈ Cb1 (Rd ; C), using Proposition B.4.1. Then, we regularize this function by convolution with a sequence of mollifiers. More precisely, we set Z ft (x) = %(y)(Eα f )(x − ty) dy, t > 0, x ∈ Rd , Rd

where % ∈ Cc∞ (Rd ) has L1 (Rd ) norm equal to one. It is easy to check that ||ft − f ||∞ ≤ C2 t||f ||C 1 (Rd ;C) , b

+

+

||ft ||Cbk (Rd ;C) ≤ C2 t−(k−1) ||f ||C 1 (Rd ;C)

(14.2.5)

+

b

for every t > 0, k = 0, 1, 2 and some positive constant C2 , independent of f and t. Now, we split f = fλ−1/2 + (f − fλ−1/2 ) for every λ > 0 and, consequently, AR(λ, AB )f = AR(λ, AB )fλ−1/2 + AR(λ, AB )(f − fλ−1/2 ).

(14.2.6)

Using the first estimate in (14.2.5), we get ||AR(λ, AB )(f − fλ−1/2 )|| ≤ λ−1/2 ||AR(λ, AB )||L(Cb (Rd+ ;C)) ||f ||C 1 (Rd ;C) . b

(14.2.7)

+

As far as the first term in the right-hand side of (14.2.6) is concerned, we introduce the function uλ = λR(λ, AB )fλ−1/2 , which belongs to DB . As a byproduct, the function T 2,p vλ = uλ − fλ−1/2 belongs to Cb1 (Rd+ ; C) ∩ p 1/2. Case β < 1/2. From Proposition 14.2.5, it follows that Cb1 (Rd+ ; C) is continuously embedded into (Cb (Rd+ ; C); DB )1/2,∞ . Therefore, applying the reiteration theorem, we deduce that DAB (β, ∞) = (Cb (Rd+ ; C); Cb1 (Rd+ ; C))2β,∞ = Cb2β (Rd+ ; C), with equivalence of the corresponding norms, where the last equality follows from Example A.5.14 (see formula (A.5.12)). Case β > 1/2. The proof here demands more efforts. Since CB1 (Rd+ ; C) ,→ Cb1 (Rd+ ; C), from Proposition 14.2.5 it follows that CB1 (Rd+ ; C) ,→ (Cb (Rd+ ); DB )1/2,∞ . From this property, estimate (14.2.10), with θ = 0, and the reiteration theorem, we obtain that (Cb (Rd+ ; C), DB )(1+γ)/2,∞ = (CB1 (Rd+ ; C), DB )γ,∞ for every γ ∈ (0, 1). Since CB2 (Rd+ ; C) is continuously embedded into DB , from Example A.5.14, see formula (A.5.13), it follows that CB1+γ (Rd+ ; C) = (CB1 (Rd+ ; C), CB2 (Rd+ ; C))γ,∞ ,→ (CB1 (Rd+ ; C), DB )γ,∞ . Hence, CB1+γ (Rd+ ; C) is continuously embedded into (Cb (Rd+ ; C), DB )(1+γ)/2,∞ . Summing up, we have proved that 1

1

2 2 ||f ||CB1 (Rd+ ;C) ≤ C||f ||∞ ||f ||D , B

Cb1 (Rd+ ; C) ,→ (Cb (Rd+ ); DB )1/2,∞ ,

(14.2.11)

Semigroups of Bounded Operators and Second-Order PDE’s 1+θ

1−θ

||f ||C 1+θ (Rd ;C) ≤ C||f ||∞2 ||f ||D2B , B

+

381

CB1+θ (Rd+ ; C) ,→ (Cb (Rd+ ; C), DB )(1+θ)/2,∞ . (14.2.12)

Applying the reiteration theorem, from (14.2.11) and (14.2.12), with θ > β, we infer that (CB1 (Rd+ ; C), CB1+θ (Rd+ ; C))γ,∞ = (Cb (Rd+ ; C), DB )ω,∞ for every γ ∈ (0, 1), where ω = (γθ + 1)/2, Taking γ = (2β − 1)/θ, we conclude that DAB (β, ∞) = (CB1 (Rd+ ; C), CB1+θ (Rd+ ; C)) 2β−1 ,∞ , with equivalence of the corresponding norms. Finally, usθ ing again the reiteration theorem and Example A.5.14, we obtain that (CB1 (Rd+ ; C), CB1+θ (Rd+ ; C)) 2β−1 ,∞ =(CB1 (Rd+ ; C), (CB1 (Rd+ ; C), CB2 (Rd+ ; C))θ,∞ ) 2β−1 ,∞ θ

θ

=(CB1 (Rd+ ; C), CB2 (Rd+ ; C))2β−1,∞

=

CB2β (Rd+ ; C),

with equivalence of the corresponding norms. This completes the proof. (ii) It suffices to repeat the arguments in the proof of property (ii) of Theorem 14.2.4, taking Theorem 12.1.3 and property (i) into account. (iii) It suffices to adapt the arguments in the proof of property (iii) of Theorem 14.2.4, taking Theorem 12.1.5 into account.  To show some interesting consequences of Theorem 14.2.4, we need first to prove this abstract result. Proposition 14.2.7 Let A be a sectorial operator and {T (t)} be the associated semigroup. Then, for every T > 0, θ, β ∈ [0, 1), with θ < β, and γ ∈ (0, 1), there exists a positive constant C such that ||T (t)||L(DA (θ,∞),DA (β,∞)) ≤ Ctθ−β ,

(14.2.13)

θ−γ−1

,

(14.2.14)

θ−β

,

(14.2.15)

||T (t)||L(DA (θ,∞),DA (1+γ,∞)) ≤ Ct

||T (t)||L(DA (1+θ,∞),DA (1+β,∞)) ≤ Ct where DA (0, ∞) = X and DA (1, ∞) = D(A).

Proof We fix T > 0 and first prove (14.2.13) and (14.2.14), with θ = 0. Throughout the proof, we denote by C a positive constant, independent of t ∈ (0, T ], which may vary from line to line. By Theorem 3.2.2, ||T (t)||L(X) ≤ C and ||T (t)||L(X,D(Ak )) ≤ Ct−k , for every t ∈ (0, T ] and k = 1, 2. Moreover, by Proposition 3.3.7 it follows that DA (β, ∞) = (X, D(A))β,∞ , with equivalence of the corresponding norms. Hence, Remark A.5.7 shows that β ||y||DA (β,∞) ≤ C||y||1−β X ||y||D(A) ,

y ∈ D(A).

(14.2.16)

Here, C is independent of y. Writing (14.2.16) with y = T (t)x, estimate (14.2.13) follows in this case. Similarly, writing (14.2.16) with β = γ, y = AT (t)x and x ∈ X, we conclude that ||AT (t)||L(X,DA (γ,∞)) ≤ Ct−1−γ for every t ∈ (0, T ], and from this estimate (14.2.14) follows immediately. We now assume that θ ∈ (0, 1). The reiteration theorem (see Theorem A.5.12) shows that DA (β, ∞) = (DA (θ, ∞), D(A))ω,∞ , where ω = (1 − θ)−1 (β − θ). Since ||T (t)||L(DA (θ,∞)) ≤ C and ||T (t)||L(DA (θ,∞);D(A)) ≤ Ctθ−1 (this last inequality follows immediately from the definition of the interpolation space DA (θ, ∞)), using Proposition A.5.6, the first estimate in (14.2.13) follows easily. To prove estimate (14.2.14), it suffices to argue as above estimating γ ||AT (t)x||DA (γ,∞) ≤C||AT (t)x||1−γ X ||AT (t)x||D(A)

382

Elliptic Operators and Analytic Semigroups γ(θ−2) ≤Ct(1−γ)(θ−1) ||x||1−γ ||x||γDA (θ,∞) DA (θ,∞) t

for every t ∈ (0, T ]. Finally, we prove (14.2.15). We fix x ∈ DA (1 + θ, ∞). Since AT (t)x = T (t)Ax, for every t > 0 and Ax ∈ DA (θ, ∞), from estimate (14.2.13) we infer that ||AT (t)x||DA (β,∞) ≤ Ctθ−β ||Ax||DA (θ,∞) ≤ Ctθ−β ||x||DA (1+θ,∞) for every t ∈ (0, T ], and (14.2.15) follows easily.



Corollary 14.2.8 Let Hypotheses 14.2.1 be satisfied. Fix β, θ ∈ [0, 2 + α], with θ ≤ β and T > 0. Then, the following properties hold true. (i) There exists a positive constant C such that ||T1 (t)f ||C β (Rd ;C) ≤ Ct b

+

θ−β 2

||f ||Cbθ (Rd+ ;C) ,

t ∈ (0, T ],

(14.2.17)

for every f ∈ Cbθ (Rd+ ; C) such that f ≡ 0 on ∂Rd+ , if θ ≤ 2, and f ≡ Af ≡ 0 on ∂Rd+ , otherwise. e such (ii) Let also Hypotheses 14.2.2 be satisfied. Then, there exists a positive constant C that e θ−β 2 ||f || θ ||T2 (t)f ||C β (Rd ;C) ≤ Ct , t ∈ (0, T ], (14.2.18) Cb (Rd + ;C) b

for every f ∈ otherwise.

+

Cbθ (Rd+ ; C),

if θ ≤ 1, and f ∈ Cbθ (Rd+ ; C) such that Bf ≡ 0 on ∂Rd+ ,

Proof (i) If β = 0 then estimate (14.2.17) is trivial. If θ ∈ [0, 2 + α) \ {1, 2} and β ∈ (0, 2 + α) \ {1, 2}, then (14.2.17) is a straightforward consequence of Theorem 14.2.4 and Proposition 14.2.7. We now observe that {f ∈ Cb1 (Rd+ ; C) : f|∂Rd+ = 0} ,→ DA0 (1/2, ∞). Indeed, fix f ∈ Cb1 (Rd+ ; C) which vanishes on ∂Rd+ and extend it to Rd , odd with respect to the last variable. Denote by fe ∈ Cb1 (Rd ; C) the so obtained function. For every t ∈ (0, 1], set Z ft (x) = %(y)fe (x − ty) dy, x ∈ Rd , Rd

where % ∈ Cc∞ (B(0, 1)) is even with respect to the last variable and such that ||%t ||L1 (Rd ) = 1. Each function ft is smooth and vanishes on ∂Rd+ . Moreover, ||ft ||Cb2 (Rd ;C) ≤ C1 t−1 ||f ||C 1 (Rd ;C) b

+

and ||ft − f ||∞ ≤ C1 t||f ||C 1 (Rd ;C) for every t ∈ (0, 1]. Therefore, ||ft ||D0 ≤ C2 t−1 ||f ||C 1 (Rd ;C) . +

b

b

+

Here, C1 , C2 , as the other constants appearing in the proof, denoted by Cj , are independent of t and f . We can thus estimate √ K(t, f, Cb (Rd+ ; C); D0 ) ≤ ||f − ft1/2 ||∞ + t||ft1/2 ||D0 ≤ C3 t||f ||C 1 (Rd ;C) , t ∈ (0, 1]. b

+

Similarly, {u ∈ Cb2 (Rd+ ; C) : u|∂Rd+ ≡ 0} is continuously embedded into D0 . Hence, Theorem 14.2.4, Proposition 14.2.7 yield (14.2.17) for every θ ∈ [0, 2 + α) and β ∈ (0, 2 + α) \ {1, 2} such that θ ≤ β. If β = 1 and θ ∈ [0, 1), then we use estimate (1.1.6) to write α

||T (t)f ||C 1 (Rd ;C) ≤C4 ||T (t)f || 2+α−θ θ d +

Cb (R+ ;C)

2−θ

||T (t)f ||C2+α−θ 1+α (Rd ;C) b

+

Semigroups of Bounded Operators and Second-Order PDE’s

383

2−θ

α

2+α−θ 2+α−θ ||T (t)f ||D ≤C5 ||T (t)f ||D A (θ/2,∞) A ((1+α)/2,∞) 0

0

for every f ∈ DA (θ/2, ∞) (where DA (0, ∞) = Cb (Rd+ ; C)). Again Theorem 14.2.4, Proposition 14.2.7 and the embedding {f ∈ Cb1 (Rd+ ; C) : f∂Rd+ = 0} ,→ DA0 (1/2, ∞) yield the assertion. In a completely similar way, we can address the case β = 2. (ii) The proof of this property is completely similar to that of property (i), if we take Proposition 14.2.5 into account. Hence, the details are omitted.  Remark 14.2.9 We stress that property (i) of Corollary 14.2.8 does not hold if function f does nor satisfy the conditions on the boundary of Rd+ . Indeed, suppose that f ∈ Cbθ (Rd+ ) does not vanish on ∂Rd+ . Then, by Corollary 14.2.8 we know that the function T1 (t)f belongs to Cbβ (Rd+ ) as well for every t > 0. Moreover, there exists a positive constant C, independent of f , such that β

β

||T1 (t)f ||C β (Rd ) ≤ t− 2 ||f ||∞ ≤ Ct− 2 ||f ||C β (Rd ) , b

+

b

t ∈ (0, 1].

(14.2.19)

+

The previous estimate is sharp. Indeed, let us consider the particular case when A is the one-dimensional Laplacian and f ≡ 1l. It is easy to check that  Z ∞ Z √x t (y−x)2 (y+x)2 z2 1 1 (T1 (t)1l)(x) = √ e− 4 dz =: gt (x) e− 4t − e− 4t dy = √ π 0 4πt 0 for every t > 0 and x ∈ R+ . Note that Z 1 √ √ z2 1 gt ( t) − gt (0) = gt ( t) = √ e− 4 dz, π 0

t > 0.

√ β β Thus, if β ∈ (0, 1], then [T1 (t)1l]C β (R+ ;C) ≥ t− 2 |gt ( t) − gt (0)| = Ct− 2 and this shows the b sharpness of estimate (14.2.19) in this case. If β ∈ (1, 2], then, √ √ 1 1 |gd0 (2 t) − gd0 ( t)| = √ (e− 4 − e−1 ), t > 0, πt √ √ β−1 β so that [DT (t)1l]C β−1 (Rd ;C) ≥ t− 2 |gt0 (2 t) − gt ( t)| = Ct− 2 and again the sharpness of b

+

estimate (14.2.19) follows. The case β ∈ (2, 3] can be addressed in the same way. Similarly, estimate (14.2.18) does not hold with θ > 1 if we remove the condition Bf ≡ 0 on ∂Rd+ . Suppose, by contradiction, that ||T2 (t)f ||Cbθ (Rd+ ) ≤ C||f ||Cbθ (Rd+ ) for every t ∈ (0, 1] and Bf does not identically vanish on ∂Rd+ . Then, splitting T2 (t)f = T2 (t/2)T2 (t/2)f for t ∈ (0, 1] and observing that T2 (t/2)f ∈ CBθ (Rd+ ), we can estimate θ

||T2 (t)f ||Cb2 (Rd+ ) ≤ ||T2 (t/2)||L(CBθ (Rd+ ),Cb2 (Rd+ )) ||T2 (t/2)f ||Cbθ (Rd+ ) ≤ Ct−1+ 2 ||f ||Cbθ (Rd+ ) and, consequently, ||AT2 (t)f ||∞ ≤ C 0 t−1+θ/2 ||f ||Cbθ (Rd+ ) for every t ∈ (0, 1], where C and C 0 are positive constants, independent of f and t. Hence, f ∈ DAB (θ, ∞) and the contradiction follows since, by Theorem 14.2.6, DAB (θ, ∞) consists of functions which belong to the kernel of operator B.

384

Elliptic Operators and Analytic Semigroups

14.2.1

Proof of Theorems 7.4.1 and 7.4.3

In this section, taking advance of the results so far proved, we provide the proofs of the regularity results, stated in Section 7.4, for solutions to the Cauchy Dirichlet problem  d   Dt u(t, x) = Au(t, x) + g(t, x), t ∈ (0, T ], x ∈ R+ , u(t, x) = ψ(t, x), t ∈ (0, T ], x ∈ ∂Rd+ , (14.2.20)   u(0, x) = f (x), x ∈ Rd+ Proof of Theorem 7.4.1 Let us assume that f belongs to Cb2+α−2θ (Rd+ ) satisfies, on ∂Rd+ , the conditions f ≡ 0 and Af ≡ 0 on ∂Rd+ (this latter condition if 2 + α − 2θ > 1) and g ∈ C((0, T ] × Rd+ ) is such that supt∈(0,T ] tθ ||g(t, ·)||Cbα (Rd+ ) < ∞ for some θ ∈ (0, 1) and g(t, ·) = 0 on ∂Rd+ for every t ∈ (0, T ]. The uniqueness of the classical solution to the Cauchy problem (14.2.20) follows from Corollary 4.1.5. Hence, we just need to prove the existence part. We will show that the function u : [0, T ] × Rd+ → R, defined by Z

t

(t, x) ∈ [0, T ] × Rd+ ,

(T1 (t − s)g(s, ·))(s) ds,

u(t, x) = (T1 (t)f )(x) +

(14.2.21)

0

is a solution to problem (14.2.20) with the properties listed in the statement of the theorem. In view of Corollary 14.2.8, we just need to deal with the integral term in the right-hand side of (14.2.21), which we denote by v. We follow the arguments in the proof of Theorem 5.4.5 and we refer the reader to the proof of that theorem for the missing details. As a first step, we observe that estimate (14.2.17) shows that α

||Dij T1 (t − s)g(s, ·)||∞ ≤ Cs−θ (t − s) 2 −1 sup rθ ||g(r, ·)||Cbα (Rd+ ) r∈(0,T ]

for every s ∈ (0, t) and t ∈ (0, T ]. This estimate implies that the function v is twice continuously differentiable on (0, T ] × Rd+ with respect to the spatial variables and α

||v(t, ·)||Cb2 (Rd ) ≤C1 t 2 −θ sup rθ ||g(r, ·)||Cbα (Rd ) ,

t ∈ (0, T ],

(14.2.22)

r∈(0,T ]

where C1 is a positive constant, independent of t, g and v. To prove that the function v(t, ·) belongs to Cb2+α (Rd ) for every t ∈ (0, T ], it suffices to use the characterization of the interpolation space DA0 (α/2, ∞) in Theorem 14.2.4 and show that the function τ 7→ α τ 1− 2 ||AT1 (τ )Dij v(t, ·)||∞ is bounded on (0, 1] for every i, j = 1, . . . , d and t ∈ (0, T ]. Since α

||ADij T1 (t + τ − s)g(s, ·)||∞ ≤ C2 s−θ (t + τ − s) 2 −2 sup rθ ||g(r, ·)||Cbα (Rd+ ) , r∈(0,T ]

where C2 is a positive constant, independent of s, t and g, we deduce that Z t α 1− α 1− α θ 2 2 τ ||AT1 (τ )Dij v(t, ·)||∞ ≤C2 τ sup r ||g(r, ·)||Cbα (Rd+ ) s−θ (t + τ − s) 2 −2 ds r∈(0,T ]

≤C3 t

−θ

0

θ

sup r ||g(r, ·)||Cbα (Rd+ )

r∈(0,T ]

for every τ ∈ (0, 1] and some positive constant C3 , which is independent of t and g as all the other forthcoming constants. Hence, Dij v(t, ·) belongs to DA0 (α/2, ∞) and ||Dij v(t, ·)||Cbα (Rd+ ) ≤ C4 t−θ sup sθ ||g(s, ·)||Cbα (Rd+ ) s∈(0,T ]

(14.2.23)

Semigroups of Bounded Operators and Second-Order PDE’s

385

for every t ∈ (0, T ] and i, j = 1, . . . , d. From (14.2.22) and (14.2.23) it follows that v(t, ·) ∈ Cb2+α (Rd+ ) and v(t, ·)|∂Rd+ = 0 for every t ∈ (0, T ]. Moreover, ||v(t, ·)||C 2+α (Rd ) ≤ C5 t−θ sup rθ ||g(r, ·)||Cbα (Rd+ ) , b

+

t ∈ (0, T ].

(14.2.24)

r∈(0,T ]

Combining (14.2.17) and (14.2.24), estimate   sup tθ (||u(t, ·)||C 2+α (Rd ) +||Dt u(t, ·)||Cbα (Rd+ ) ) ≤ C6 ||f ||C 2+α−2θ (Rd ) + sup tθ ||g(t, ·)||Cbα (Rd+ ) b

t∈(0,T ]

+

+

b

t∈(0,T ]

(14.2.25) follows at once. To prove that the first- and second-order spatial derivatives of v are continuous in [0, T ] × Rd+ , we fix t0 > 0 and apply estimate (1.1.6) to get 2

α

2+α ||v(t, ·) − v(t0 , ·)||Cb2 (Rd+ ) ≤C7 ||v(t, ·) − v(t0 , ·)||C2+α ||v(t, ·) − v(t0 , ·)||∞ 2+α (Rd ) +

b

≤C8 t−θ 0

α

θ

2+α sup s ||g(s, ·)||Cbα (Rd ) ||v(t, ·) − v(t0 , ·)||∞

s∈(0,T ]

for every t ∈ [t0 /2, T ]. Since the function v is continuous in (0, T ] with values in Cb (Rd+ ), from the previous chain of inequalities the continuity in (0, T ]×Rd+ of the spatial derivatives of v follows. To prove that the function v is continuously differentiable with respect to time on (0, T ]× d R+ , we introduce, for each ε ∈ (0, 1), the function vε : [0, T ] × Rd+ → R, defined by Z εt vε (t, ·) = (T1 (t − s)g(s, ·))(x) ds, (t, x) ∈ [0, T ] × Rd+ , 0

which converges to v as ε → 1− , uniformly in [0, T ] × Rd+ , is differentiable with respect to the time variable on (0, T ] × Rd+ and Z εt Dt vε (t, x) = ε(T1 ((1 − ε)t)g(εt, ·))(x) + (AT1 (t − s)g(s, ·))(x) ds 0

for every (t, x) ∈ (0, T ] × Rd+ . Letting ε tend to 1− , we conclude that Dt vε (t, x) converges to g(t, x)+Av(t, x), for every (t, x) ∈ (0, T ]×Rd+ . This is enough to infer that v is differentiable with respect to t in (0, T ] × Rd+ and therein Dt v = Av + g. Next, to prove that v belongs to Cb0,2+α−2θ ([0, T ] × Rd+ ), it suffices to argue as in the proof of (14.2.22) and estimate Z t θ ||v(t, ·)||C 2+α−2θ (Rd ) ≤C9 sup s ||g(s, ·)||Cbα (Rd+ ) s−θ (t − s)θ−1 ds b

+

s∈(0,T ]

=C9 sup sθ ||g(s, ·)||Cbα (Rd+ ) s∈(0,T ]

0

Z

1

s−θ (1 − s)θ−1 ds

(14.2.26)

0

for every t ∈ (0, T ]. Summing up, we have proved that u is bounded on [0, T ] with values in Cb2+α−2θ (Rd+ ). Moreover, from (14.2.25) and (14.2.26), estimate (7.4.1) follows at once. Now, taking Corollaries 7.5.4 and 14.2.8, it is easy to check that the function u is a classical solution to the Cauchy problem (14.2.20).

386

Elliptic Operators and Analytic Semigroups

Finally, the last part of the statement and estimate (7.4.2) follow immediately form Theorems 3.4.8(iv) and 14.2.4.  Proof of Theorem 7.4.3 We prove the assertion by induction on k. As a first step, we observe that it suffices to consider the case ψ ≡ 0. Indeed, if v is the solution in (k+2+α)/2,k+2+α Cb ((0, T ) × Rd+ ) to problem (14.2.1), which satisfies estimate (7.4.5), with ψ ≡ 0, then the function v + ψ solves the Cauchy problem (14.2.1) and satisfies estimate (7.4.5). Throughout the proof, C denotes a positive constant, independent of the functions that we consider, that may vary from line to line. Step 1. Here, we suppose that k = 1. Since f vanishes on ∂Rd , due to the assumption ψ ≡ 0, it follows that f ∈ D(A0 ). Moreover, the function Af + g(0, ·) belongs to Cb3+α (Rd+ ) and, by the compatibility condition in (7.4.4), with ψ ≡ 0, it follows that Af + g(0, ·) vanishes on ∂Rd+ . Hence, Theorem 14.2.6 implies that Af + g(0, ·) belongs to DA0 ((1 + α)/2, ∞). Moreover, ||Af + g(0, ·)||DA0 ((3+α)/2,∞) ≤ C1 ||f ||C 3+α (Rd ) for some pos+ b itive constant C1 , independent of f . Applying property (iii) in Theorem 3.4.8, we deduce that the classical solution u to problem (14.2.20) is such that the time derivative belongs to C (1+α)/2 ([0, T ]; Cb (Rd+ )) and is bounded in [0, T ] with values in DA0 ((1 + α)/2, ∞). Hence, (1+α)/2,1+α Dt u ∈ Cb ((0, T ) × Rd+ ) and ||Dt u||C (1+α)/2,1+α ((0,T )×Rd ) ≤ C(||f ||C 3+α (Rd ) + ||g||C (1+α)/2,1+α ((0,T )×Rd+ ) .

(14.2.27)

+

b

+

b

Since Au = Dt u − g, then Au is bounded in [0, T ] with values in Cb1+α (Rd+ ). Fix t ∈ [0, T ], λ ∈ ρ(A0 ) and set ft = λu(t) − Au(t, ·). Then, by Theorem 12.1.5 there exists a unique v ∈ Cb3+α (Rd+ ), which vanishes on ∂Rd+ and solve the equation λv − Av = ft and ||v||C 3+α (Rd ) ≤ C||ft ||C 1+α (Rd ) . By uniqueness, v ≡ u(t). By the arbitrariness of t ∈ [0, T ], +

b

+

b

we conclude that u(t, ·) ∈ Cb3+α (Rd+ ) for every t ∈ [0, T ] and sup ||u(t, ·)||C 3+α (Rd ) ≤C(||u||C 0,1+α (Rd ) + ||g||C 0,1+α ([0,T ]×Rd ) ) +

b

t∈[0,T ]

+

b

+

b

≤C(||f ||C 2+α (Rd ) + ||g||C (1+α)/2,1+α ((0,T )×Rd ) ). +

b

(3+α)/2,3+α

To conclude that u ∈ Cb

((0, T ) × Rd+ ), we first show that

||u(t, ·) − u(s, ·)||C 1+α (Rd ) ≤ C|t − s|α ||Dt u||C (1+α)/2,1+α ((0,T )×Rd ) . +

b

(14.2.28)

+

b

(14.2.29)

+

b

Z

t

Dt u(r, x) dr for every (s, x), (t, x) ∈

For this purpose, it suffices to write u(t, x)−u(s, x) = s

[0, T ] × Rd+ , with s < t, so that t

Z ||u(t, ·) − u(s, ·)||C 1 (Rd ) ≤ b

+

s

||Dt u(r, ·)||C 1 (Rd ) dr ≤ ||Dt u(r, ·)||C (1+α)/2,1+α ((0,T )×Rd ) |t − s| b

+

b

+

(14.2.30) Similarly, Z Dj u(t, x) − Dj u(s, x) − Dj u(t, y) + Dj u(s, y) =

t

(Dj Dt u(r, x) − Dj Dt u(r, y)) dr s

for s, t ∈ [0, T ], such that s ≤ t, and j = 1, . . . , d, so that ||Dj u(t, ·) − Dj u(s, ·)||C 1+α (Rd ) ≤ ||Dt u(r, ·)||C (1+α)/2,1+α ((0,T )×Rd ) |t − s|. b

+

b

+

(14.2.31)

Semigroups of Bounded Operators and Second-Order PDE’s

387

From (14.2.30) and (14.2.31), estimate (14.2.29) follows at once. Now, we observe that h−1−α

3+α−h

2 2 ||u(t, ·) − u(s, ·)||C 3+α ||u(t, ·) − u(s, ·)||C h (Rd ) ≤C||u(t, ·) − u(s, ·)||C 1+α (Rd ) (Rd ) b

+

+

b

≤C|t − s|

3+α−h 2

+

b

3+α−h 2 (1+α)/2,1+α ((0,T )×Rd Cb +)

||Dt u||

h−1−α 2 sup ||u(t, ·)||C 3+α (Rd ) b

t∈[0,T ]

+

for h = 2, 3. From this estimate and Exercise 1.5.5, which shows that also the derivatives (3+α)/2,3+α Dt Dj u exist for every j = 1, . . . , d, it follows at once that u ∈ Cb ((0, T ) × Rd+ ). Estimate (7.4.5) follows from (14.2.27) and (14.2.28). Step 2. Let us now suppose that k = 2 and consider the Cauchy problem  d   Dt v(t, x) = Av(t, x) + Dt g(t, x), t ∈ [0, T ], x ∈ R+ , v(t, x0 , 0) = 0, t ∈ [0, T ], x0 ∈ Rd−1 , (14.2.32)   v(0, x) = Af (x) + g(0, x), x ∈ Rd+ . Since Dt g ∈ C α/2,α ((0, T ) × Rd+ ), Af + g(0, ·) ∈ Cb2+α (Rd+ ) and the compatibility conditions Af + g(0, ·) ≡ 0, A(Af + g(0, ·)) + gt (0, ·) ≡ 0 on ∂Rd+ are satisfied, by Theorem 7.0.2, there 1+α/2,2+α exists a unique solution v ∈ Cb ((0, T ) × Rd+ ) to the above problem. Moreover, ||v||C 1+α/2,2+α ((0,T )×Rd ) ≤ C(||f ||C 4+α (Rd ) + ||g||C 1+α/2,2+α ((0,T )×Rd ) ). +

b

+

b

+

b

Let us prove that Z u(t, x) =

t

(t, x) ∈ [0, T ] × Rd+ .

v(s, x) ds + f (x),

(14.2.33)

0

For this purpose, we will show that the function in the right-hand side of (14.2.33) (which 1+α/2,2+α we denote by w) belongs to Cb ((0, T ) × Rd+ ) and solves the same problem as u 1+α/2,2+α does. Showing that w ∈ Cb ((0, T ) × Rd+ ) is an easy task. Moreover, w(0, ·) = f and w(t, x0 , 0) = f (x0 , 0) = 0 for every t ∈ [0, T ] and x0 ∈ Rd−1 . Finally, integrating the differential equation in (14.2.32) gives  Z t  Z t v(t, x) − v(0, x) = Dt v(s, x) ds = A v(s, ·) ds (x) + g(t, x) − g(0, x) 0

0

or, equivalently, Dt w(t, x) = Aw(t, x) − Af (x) + v(0, x) + g(t, x) − g(0, x) = Aw(t, x) + g(t, x) for every (t, x) ∈ [0, T ] × Rd+ . Thus, we can conclude that w ≡ u. Hence, Dt u ∈ 1+α/2,2+α Cb ((0, T ) × Rd+ ) and ||Dt u||C 1+α/2,2+α ((0,T )×Rd ) ≤ C(||f ||C 4+α (Rd ) + ||g||C 1+α/2,2+α ((0,T )×Rd ) ). b

+

b

+

b

+

Now, we prove that u is four time continuously differentiable in [0, T ] × Rd+ with respect to the spatial variables. For this purpose, we fix h ∈ {1, . . . , d − 1}, r > 0 and introduce the 1+α/2,2+α function u(r,h) ∈ Cb ((0, T )×Rd+ ), defined by u(r,h) (t, x) = r−1 [u(t, x+reh )−u(t, x)] for every (t, x) ∈ [0, T ] × Rd+ . As it is easily seen, this function solves the Cauchy problem  (r,h) (t, x) = Au(r,h) (t, x) + gr,h (t, x), t ∈ [0, T ], x ∈ Rd+ ,   Dt u v(t, x0 , 0) = 0, t ∈ [0, T ], x0 ∈ Rd−1 , (14.2.34)   (r,h) d v(0, x) = f (x), x ∈ R+ ,

388

Elliptic Operators and Analytic Semigroups

where f (r,h) is defined as the function u, whereas gr,h = g (r,h) +

d X

(r,h)

qij

Dij u(·, · + reh ) +

i,j=1

d X

(r,h)

bj

Dj u(·, · + reh ) + c(r,h) u(·, · + reh )

j=1

in [0, T ] × Rd+ . Since function g belongs to C 1+α/2,2+α ((0, T ) × Rd+ ), we can esti(r,h) mate ||g (r,h) ||C (1+α)/2,1+α ((0,T )×Rd ) ≤ C||g||C 1+α/2,2+α ((0,T )×Rd ) . Similarly, ||qij ||C 1+α (Rd ) ≤ +

b

(r,h)

C||qij ||C 2+α (Rd ) , ||bj +

b

(r,h)

b

+

+

b

+

b

+

b

||C 1+α (Rd ) ≤ C||bj ||C 2+α (Rd ) and ||cij

||C 1+α (Rd ) ≤ C||c||C 2+α (Rd ) . b

+

b

+

Summing up, function gr,h belongs to C (1+α)/2,1+α ((0, T ) × Rd+ ) and ||gr,h ||C (1+α)/2,1+α ((0,T )×Rd+ ) ≤ C(||u||C 1+α/2,2+α ((0,T )×Rd ) + ||g||C 1+α/2,2+α ((0,T )×Rd+ ) ). +

b

Similarly, ||f (r,h) ||C 3+α (Rd ) ≤ C||f ||C 4+α (Rd ) . Hence, from Step 1 we infer that u(r,h) ∈ b

(3+α)/2,3+α

Cb

+

b

+

((0, T ) × Rd+ ) and

||u(r,h) ||C (3+α)/2,3+α ((0,T )×Rd ) ≤C(||f (r,h) ||C 3+α (Rd ) + ||gr,h ||C (1+α)/2,1+α ((0,T )×Rd ) +

b

+

b

+

b

≤C(||f ||C 4+α (Rd ) + ||g||C 1+α/2,2+α ((0,T )×Rd ) +

b

+

b

Since u(r,h) converges to Dh u as r tends to 0 pointwise in Rd+ , a compactness argument 1+α/2,2+α based on Arzel` a-Ascoli theorem shows that Dh u ∈ Cb ((0, T ) × Rd+ ) and ||Dh u||C (3+α)/2,3+α ((0,T )×Rd ) ≤ C(||f ||C 4+α (Rd ) + ||g||C 1+α/2,2+α ((0,T )×Rd ) ) b

+

b

+

b

+

Finally, writing Ddd u =

−1 qdd

 Dt u −

X (i,j)6=(d,d)

qij Dij −

d X

 bj Dj u − cu ,

(14.2.35)

j=1

we easily complete the proof of this step. Step 3. Now, we can make the induction argument work. We suppose that the assertion is true for every k ≤ 2m, and some m > 1, and prove it with k = 2m + 1 and k = 2m + 2. By k+1+α/2,2k+2+α assumptions u ∈ Cb ((0, T ) × Rd+ ). Hence, we can differentiate the differential equation Dt u = Au + g, m-times with respect to the spatial variables. We fix a multiindex β = (β1 , . . . , βd−1 , 0) with length at most 2m. Observing that Dt Dxβ u = Dxβ Dt u, we conclude that the function v = Dxβ u is a classical solution to the Cauchy problem  d   Dt v(t, x) = Av(t, x) + gβ (t, x), t ∈ [0, T ], x ∈ R+ , v(t, x) = 0, t ∈ [0, T ], x ∈ ∂Rd+ , (14.2.36)   β d v(0, x) = D f (x), x ∈ R+ , where d X  d X  X X β β Dβ−γ qij Dxγ Dij u + Dβ−γ bj Dxγ Dij u γ γ i,j=1 γ 0. Then, the following properties hold true. (i) There exists a positive constant C such that ||T1Ω (t)f ||C β (Ω;C) ≤ Ct b

θ−β 2

||f ||Cbθ (Ω;C) ,

t ∈ (0, T ],

(14.3.5)

for every f ∈ Cbθ (Ω; C) such that f = 0 on ∂Ω, if β ≤ 2, and f ≡ Af ≡ 0 on ∂Ω, otherwise. e such (ii) Let also Hypotheses 14.3.2 be satisfied. Then, there exists a positive constant C that θ−β t ∈ (0, T ], (14.3.6) ||T2Ω (t)f ||C β (Ω;C) ≤ Ct 2 ||f ||Cbθ (Ω;C) , b

for every f ∈

Cbθ (Ω; C),

if β ≤ 1, and f ∈ C θ (Ω; C) such that Bf ≡ 0 on ∂Ω, otherwise.

Proof The proof follows the same lines as that of Corollary 14.2.8. The only slight difference is in the proof of the embedding {f ∈ C 1 (Ω; C) : f|∂Ω ≡ 0} ,→ DAΩ0 (1/2, ∞). To show this property, we fix f ∈ C 1 (Ω; C), which vanishes on the boundary of Ω, and for every t > 0 we introduce the function Z gt (x) = %(y)(E1 f )(x − ty) dy, x ∈ Rd , Rd

where % ∈ Cc∞ (B(0, 1)) is such that ||%||L1 (Rd ) = 1 and E1 is the extension operator in Proposition B.4.1. The function gt belongs to C 2 (Ω; C) and t−1 ||f −gt ||C(Ω;C) +t||gt ||C 2 (Ω;C) ≤ C1 ||f ||C 1 (Ω;C) for every t > 0 and some positive constant C1 , which, as the other constants appearing in the proof, is independent of t and f . Finally, we consider the function ft = gt − E20 ((gt )|∂Ω ), where now E20 is the extension operator in Proposition B.4.9. As it is easily seen, ||ft ||C 2 (Ω;C) ≤ ||gt ||C 2 (Ω;C) + ||E20 ||L(C 2 (∂Ω;C);C 2 (Ω;C)) ||gt ||C 2 (∂Ω;C) ≤ C2 t−1 ||f ||C 1 (Ω;C) for every t > 0 and some positive constant C2 . Moreover, ft belongs to D0Ω . Similarly, since f vanishes on ∂Ω, we can estimate ||ft − f ||C(Ω;C) =||gt + E20 ((gt )|∂Ω ) − f − E20 (f|∂Ω )||C(Ω;C) ≤||gt − f ||C(Ω;C) + ||E20 ||L(C(∂Ω;C);C(Ω;C)) ||gt − f ||C(∂Ω;C) ≤ C3 t||f ||C 1 (Ω;C) for some positive constant C3 . Hence, √ K(t, f, C(Ω; C); D0Ω ) ≤ ||f − ft1/2 ||∞ + t||ft1/2 ||D0Ω ≤ C4 t||f ||C 1 (Ω;C) ,

t ∈ (0, 1],

for some positive constant C4 . This estimate yields immediately the embedding {f ∈ C 1 (Ω; C) : f|∂Ω = 0} ,→ DAΩ0 (1/2, ∞).  Remark 14.3.7 As already noticed in Remark 14.2.9, property (i) of Corollary 14.3.6 does not hold if function f does not satisfy the conditions on the boundary of Ω. Similarly, estimate (14.3.6) does not hold with θ > 1 without the extra condition Bf = 0 on ∂Ω. Using Corollary 14.3.6, we can prove the following regularity results for solutions to the Cauchy-Dirichlet problem (14.3.1) and the Cauchy problem with first-order boundary conditions (14.3.2).

Semigroups of Bounded Operators and Second-Order PDE’s

395

Theorem 14.3.8 Let Hypotheses 14.3.1 be satisfied and fix θ ∈ (0, 1). Suppose that f belongs to C 2+α−2θ (Ω; C), satisfies the condition f ≡ 0 on ∂Ω and Af ≡ 0 on ∂Ω (this latter condition if 2+α−2θ > 1) and g ∈ C((0, T ]×Ω; C) is such that supt∈(0,T ] tθ ||g(t, ·)||C α (Ω;C) < ∞ and g(t, ·) ≡ 0 on ∂Ω for every t ∈ (0, T ]. Then, the Cauchy problem (14.3.1) admits a unique classical solution u. In addition, u(t, ·) ∈ C 2+α (Ω; C) for every t ∈ (0, T ] and sup tθ (||u(t, ·)||C 2+α (Ω;C) + ||Dt u(t, ·)||C α (Ω;C) ) + ||u||C 0,2+α−2θ ([0,T ]×Ω;C)

t∈(0,T ]

  ≤C0 ||f ||C 2+α−2θ (Ω;C) + sup tθ ||g(t, ·)||C α (Ω;C) t∈(0,T ]

for some positive constant C0 , independent of u, f and g. In particular, if f and g belong to C 2+α (Ω; C) and C([0, T ] × Ω; C), respectively, f ≡ Af ≡ g(t, ·) ≡ 0 on ∂Ω for every t ∈ [0, T ] and supt∈[0,T ] ||g(t, ·)||C α (Ω;C) < ∞, then, u belongs to C 1,2 ([0, T ] × Ω; C), u(t, ·) ∈ C 2+α (Ω; C) for every t ∈ [0, T ] and sup ||u(t, ·)||C 2+α (Ω;C) + sup ||Dt u(t, ·)||C α (Ω;C) t∈[0,T ]

t∈[0,T ]



≤C1 ||f ||C 2+α (Ω;C) + sup ||g(t, ·)||C α (Ω;C)



t∈(0,T ]

for some positive constant C1 , independent of u, f and g. Proof The proof is completely similar to that of Theorem 7.4.1 with no relevant differences and is based on the characterization of the interpolation spaces DAΩ0 (α, ∞), the estimates in Corollary 14.3.6 and (1.1.6).  Also the proof of the following result can be obtained using the arguments in the proof of Theorem 7.4.1. Hence, the details are omitted, Theorem 14.3.9 Let Hypotheses 14.3.1 and 14.3.2 be satisfied and fix θ ∈ (0, 1). Suppose that f belongs to C 2+α−2θ (Ω; C) satisfies the condition Bf ≡ 0 on ∂Ω if 2 + α − 2θ > 1 and g ∈ C((0, T ] × Ω; C) is such that supt∈(0,T ] tθ ||g(t, ·)||C α (Ω;C) < ∞. Then, the Cauchy problem (14.3.2) admits a unique classical solution u, which is bounded in [0, T ] × Ω. In addition, u(t, ·) ∈ C 2+α (Ω; C) for every t ∈ (0, T ] and sup tθ (||u(t, ·)||C 2+α (Ω;C) + ||Dt u(t, ·)||C α (Ω;C) ) + ||u||C 0,2+α−2θ ([0,T ]×Ω;C)

t∈(0,T ]

  ≤C0 ||f ||C 2+α−2θ (Ω;C) + sup tθ ||g(t, ·)||C α (Ω;C) t∈(0,T ]

for some positive constant C0 , independent of u, f and g. In particular, if f and g belong to C 2+α (Ω; C) and C([0, T ] × Ω; C), respectively, and supt∈[0,T ] ||g(t, ·)||C α (Ω;C) < ∞, then, u belongs to C 1,2 ([0, T ] × Ω; C), u(t, ·) ∈ C 2+α (Ω; C) for every t ∈ [0, T ] and   sup ||u(t, ·)||C 2+α (Ω;C) + sup ||Dt u(t, ·)||C α (Ω;C) ≤ C1 ||f ||C 2+α (Ω) + sup ||g(t, ·)||C α (Ω;C) t∈[0,T ]

t∈[0,T ]

t∈(0,T ]

for some positive constant C1 , independent of u, f and g.

14.4

Exercises

1. Complete the proof of Theorem 14.2.3, adding the missing details.

396

Elliptic Operators and Analytic Semigroups

2. Complete the proof of property (ii) in Theorem 14.2.6. 3. Complete the proof of property (ii) in Corollary 14.2.8. 4. Add the missing details in the proof of Theorem 14.3.3. 3+α−h

h−1−α

2 2 ||f ||C 3+α for every f ∈ C 3+α (Rd+ ), h = 2, 3 5. Prove that ||f ||C h (Rd ) ≤ C||f ||C 1+α (Rd ) (Rd ) b

+

b

+

b

and a positive constant C, independent of f . 6. Complete the proof of Theorem 7.4.3.

+

Chapter 15 Kernel Estimates

As it has been noticed in Chapter 5, if f is a bounded and continuous function over Rd and {T (t)} is the Gauss-Weierstrass semigroup, then, for each t > 0, the function T (t)f can be expressed in an integral form as the convolution of the so-called heat kernel K and the function f , i.e., Z (T (t)f )(x) =

Kt (x, y)f (y) dy,

x ∈ Rd .

(15.0.1)

Rd

The function K is explicit, so that formula (15.0.1) greatly simplifies the analysis of the Gauss-Weierstrass semigroup and the heat-equation. This is the reason why the heat equation is used as the starting point for the analysis of more general parabolic equations. In the case when the Laplacian is replaced by an elliptic operator in divergence form Af = −

d X

Dj (qij Di f ) +

i,j=1

d X (bi Di f − Di (ci f )) + c0 f, i=1

subject to Dirichlet boundary conditions, we show that the associated semigroup {T (t)} still admits an integral representation of the form Z (T (t)f )(x) = kt (x, y)f (y) dy, x ∈ Ω, Ω

or a bounded open set of class C 2+α , under general for each f ∈ Cb (Ω), Ω being R , conditions on the coefficients qij bi , ci (i, j = 1, . . . , d) and c0 , see Hypotheses 15.3.1. Even if in general the kernel kt is unknown, some qualitative properties can be proved. We analyze the asymptotic behaviour of the function kt as |x| and |y| tend to ∞ and as t tends to 0. We use these estimates to prove the analyticity of the semigroup in Lp (Ω) for every p ∈ [1, ∞) ∪ {∞}. This chapter is based on the work by Arendt and ter Elst [2]. d

15.1

Rd+

Dunford-Pettis Criterion and Ultracontractivity

In this section, we first establish the Dunford-Pettis criterion for integral operators and provide sufficient conditions for a C0 -semigroup on Lp -spaces to be ultracontractive. Let Ω ⊂ Rd be an open set (not necessarily bounded) and k be a function in L∞ (Ω × Ω). Then, one can easily see that Z Tk f (x) := k(x, y)f (y) dy, f ∈ L1 (Ω), Ω

defines a bounded operator from L1 (Ω) into L∞ (Ω), which satisfies the condition ||Tk ||L(L1 (Ω),L∞ (Ω)) = ||k||L∞ (Ω×Ω) . 397

398

Kernel Estimates

Such integral operators are completely characterized by the following Dunford-Pettis criterion. Theorem 15.1.1 The mapping U : L∞ (Ω × Ω) → L(L1 (Ω), L∞ (Ω)), defined by Uk = Tk for every k ∈ L∞ (Ω × Ω) is an isometric isomorphism. Proof Consider the space E :=

X n

1



fj ⊗ gj : n ∈ N, fj , gj ∈ L (Ω) ,

j=1

where (fj ⊗ gj )(x, y) = fj (x)gj (y) for x, y ∈ Ω and j = 1, . . . n. It is well known that E is dense in L1 (Ω × Ω). Fix T ∈ L(L1 (Ω), L∞ (Ω)) and define the mapping Φ : E → R by setting Φ(u) =

n Z X j=1

(T gj )(y)fj (y) dx,

u=

Ω

n X

fj ⊗ gj , n ∈ N.

j=1

By the definition of the product measure it follows that the definition of Φ(u) is independent of the representation of u. So Φ is a well defined linear mapping on E. Moreover, if fj , gj ∈ L1+ (Ω) = {f ∈ L1 (Ω) : f ≥ 0 a.e.} for j = 1, . . . , n, then |Φ(u)| ≤ ||T ||L(L1 (Ω),L∞ (Ω))

n X

||fj ||L1 (Ω) ||gj ||L1 (Ω) = ||T ||L(L1 (Ω),L∞ (Ω)) ||u||L1 (Ω×Ω) .

j=1

Since the set of such functions is dense in the subset L1+ (Ω × Ω) of L1 (Ω × Ω), consisting of nonnegative functions, we can extend Φ to L1+ (Ω × Ω) and |Φ(u)| ≤ ||T ||L(L1 (Ω),L∞ (Ω)) ||u||L1 (Ω×Ω) ,

u ∈ L1+ (Ω × Ω).

Now, we fix u ∈ L1 (Ω × Ω), split u = u+ + u− and, using the linearity of Φ, we estimate |Φ(u)| ≤|Φ(u+ )| + |Φ(−u− )| ≤||T ||L(L1 (Ω),L∞ (Ω)) (||u+ ||L1 (Ω×Ω) + ||u− ||L1 (Ω×Ω) ) =||T ||L(L1 (Ω),L∞ (Ω)) ||u||L1 (Ω×Ω) . Since (L1 (Ω × Ω))0 = L∞ (Ω × Ω), there exists a function k ∈ L∞ (Ω × Ω) such that Z Z Φ(u) = dx k(x, y)u(x, y) dy Ω

Ω

for all u ∈ L1 (Ω × Ω) and ||k||L∞ (Ω×Ω) ≤ ||T ||L(L1 (Ω),L∞ (Ω)) . In particular, for f, g ∈ L1 (Ω) we have  Z Z Z Z Φ(f ⊗ g) = (T g)(x)f (x) dx = k(x, y)g(y) dy f (x) dx = (Tk g)(x)f (x) dx. Ω

Ω

Ω

Thus, T = Tk . This proves that U is surjective and isometric.

Ω



To go further in our analysis, we need to introduce the following definition of ultracontractivity.

Semigroups of Bounded Operators and Second-Order PDE’s

399

Definition 15.1.2 A C0 -semigroup {S(t)} on L2 (Ω) is called ultracontractive if each operator S(t) maps L1 (Ω) ∩ L2 (Ω) into L∞ (Ω) and ||S(t)f ||L∞ (Ω) ≤ at ||f ||L1 (Ω)

(15.1.1)

for all f ∈ L2 (Ω) ∩ L1 (Ω), t ≥ 0 and some positive constant at . It follows from (15.1.1) that each operator S(t) can be extended to a bounded linear operator from L1 (Ω) to L∞ (Ω). Applying Theorem 15.1.1 we easily deduce the following result. Proposition 15.1.3 If a C0 -semigroup {S(t)} on L2 (Ω) is ultracontractive, then for every t > 0 there exists an integral kernel kt ∈ L∞ (Ω × Ω) such that Z (S(t)f )(x) = kt (x, y)f (y) dy, t > 0, f ∈ L1 (Ω) ∩ L2 (Ω). Ω

Moreover, ||kt ||L∞ (Ω×Ω) ≤ at for every t > 0 and some positive constant at . For further use we also need the following concepts. Definition 15.1.4 Let 1 ≤ p1 < p2 ≤ ∞. A (not necessarily strongly continuous) semigroup {Sp (t)} which is defined on Lp (Ω) for p ∈ [p1 , p2 ] is called consistent if Sp (t)f = Sq (t)f for all t > 0, q ∈ [p1 , p2 ] and f ∈ Lp (Ω) ∩ Lq (Ω). Definition 15.1.5 Let 1 ≤ p1 ≤ 2 ≤ p2 ≤ ∞. We say that a C0 -semigroup {S(t)}, defined on L2 (Ω), interpolates on Lp (Ω) for p ∈ [p1 , p2 ], if there exists a consistent C0 -semigroup {Sp (t)} on Lp (Ω) for p ∈ [p1 , p2 ], if p 6= ∞, and a weakly∗ -continuous semigroup {S∞ (t)} otherwise, such that S(t) = S2 (t) for all t > 0. We recall that the semigroup {S∞ (t)} is weakly∗ -continuous if limt→0+ hS∞ (t)f, gi = hf, gi for all f ∈ L∞ (Ω) and g ∈ L1 (Ω). The difficulty in proving that a C0 -semigroup {S(t)} on L2 (Ω) interpolates is frequently the strong continuity at the endpoints p1 and p2 . Lemma 15.1.6 Assume that {S(t)} is a C0 -semigroup on L2 (Ω) satisfying the conditions S(t)(L2 (Ω) ∩ L1 (Ω)) ⊂ L1 (Ω) for all t > 0 and ||S(t)f ||L1 (Ω) ≤ eωt ||f ||L1 (Ω) for all t ≥ 0, f ∈ L2 (Ω)∩L1 (Ω) and some constant ω ∈ R. Then {S(t)} interpolates on Lp (Ω), 1 ≤ p ≤ 2. Proof By applying Riesz’s Thorin interpolation theorem, it follows that there exists a consistent semigroup {Sp (t)} on Lp (Ω) for p ∈ [1, 2]. Let us now prove the strong continuity. Fix p ∈ (1, 2), let θ = 2(p − 1)/p and apply H¨older’s inequality to estimate ||Sp (t)f − f ||Lp (Ω) ≤ ||S(t)f − f ||θL2 (Ω) ||S1 (t)f − f ||1−θ L1 (Ω) for every t ≥ 0 and f ∈ L2 (Ω) ∩ L1 (Ω). Therefore, the strong continuity in Lp (Ω) follows from the strong continuity in L2 (Ω). To prove the strong continuity in L1 (Ω), we suppose, without loss of generality, that ω = 0 and take f ∈ L2 (Ω) with compact support K. Using the strong continuity of {S(t)} in L2 (Ω) and H¨ older’s inequality, one obtains lim ||χK S(t)f − f ||L1 (Ω) ≤ lim ||χK S(t)f − f ||L2 (Ω) |K|1/2 = 0.

t→0+

t→0+

400

Kernel Estimates

On the other hand, ||S(t)f ||L1 (Ω) = ||χΩ\K S(t)f ||L1 (Ω) + ||χK S(t)f ||L1 (Ω) ≤ ||f ||L1 (Ω) . Thus, ||χΩ\K S(t)f ||L1 (Ω) converges to 0 as t tends to 0 and, therefore, lim+ ||S(t)f −f ||L1 (Ω) = t→0

0. The strong continuity of {S(t)} in L1 (Ω) follows by density.

15.2



Gaussian Estimates for Second-Order Elliptic Operators with Dirichlet Boundary Conditions

Let us begin by proving the Nash inequality on Rd which will play a central role to get Gaussian estimates. Proposition 15.2.1 For every f ∈ W 1,2 (Rd ; C) ∩ L1 (Rd ; C) the following estimate holds 2+ 4

4

2 d d ||f ||L2 (R d ;C) ≤ C|||∇f |||L2 (Rd ;C) ||f ||L1 (Rd ;C)

(15.2.1)

for some constant C > 0, independent of f . Proof Let us denote by d

F(f )(y) = (2π)− 2

Z

eiy·x f (x) dx,

y ∈ Rd ,

Rd

the Fourier transform of f ∈ L1 (Rd ; C). By Plancherel’s formula, we can estimate ||f ||L2 (Rd ;C) =||F(f )||L2 (Rd ;C) Z Z |F(f )(y)|2 dy + = |y|> √1ε

|F(f )(y)|2 dy

|y|< √1ε

√ |yF(f )(y)|2 dy + (2π)−d |B(0, 1/ ε)|||f ||2L1 (Rd ;C) Rd √ ≤ε|||∇f |||2L2 (Rd ) + (2π ε)−d |B(0, 1)|||f ||2L1 (Rd ;C) Z

≤ε

for every ε > 0. Minimizing the right-hand side of the above inequality with respect to ε one obtains (15.2.1) for some constant C > 0, independent of f .  The above Nash inequality remains true if Rd is replaced by any open subset Ω of Rd and W 1,2 (Rd ; C) by W01,2 (Ω; C). Corollary 15.2.2 For every f ∈ W01,2 (Ω; C) ∩ L1 (Ω; C) the following estimate holds 2+ 4

4

d ||f ||L2 (Ω;C) ≤ C||f ||2W 1,2 (Ω;C) ||f ||Ld 1 (Ω;C)

(15.2.2)

for some positive constant C > 0, independent of f . Proof Fix f ∈ W01,2 (Ω; C) ∩ L1 (Ω; C). Then, by Remark B.4.6, the trivial extension f of f to Rd belongs to W 1,2 (Rd ; C) ∩ L1 (Rd ; C). Writing the Nash inequality (15.2.1), with f being replaced by f , estimate (15.2.2) follows at once. 

Semigroups of Bounded Operators and Second-Order PDE’s

401

Also the following technical lemma will be used in this chapter. To state it, we need to introduce the signum function sign(f ), which is defined by    f (x) if f (x) 6= 0, |f (x)| (signf )(x) =   0 if f (x) = 0. for every f ∈ L2 (Ω; C). Lemma 15.2.3 Let V be W 1,2 (Ω; C) or W01,2 (Ω; C). If f ∈ V then the functions (Re f )+ , |f | and (1 ∧ |f |)sign(f ) belong to V and Dj (Re f )+ = Dj (Re f )χ{Re f >0} , Dj |f | = Re(sign(f )Dj f ), Dj ((1 ∧ |f |)sign(f )) = i

Im(sign(f )Dj f ) sign(f )χ{|f |>1} + χ{|f |≤1} Dj f. |f |

Moreover if g ∈ W 1,2 (Ω; C) then the mapping f 7→ f ∧ g is continuous from W 1,2 (Ω; C) into itself. Proof The easy proof is left to the reader.



In the rest of the chapter, by V we will denote a Hilbert space, densely and continuously embedded into L2 (Ω; C). Definition 15.2.4 A densely defined continuous sesquilinear form a : V × V → C, which satisfies the estimate |a(f, g)| ≤ M ||f ||V ||g||V for all f, g ∈ V and some constant M > 0, is called quasi-accretive, i.e., there exist ω ∈ R and ν > 0 such that Re[a(f, f )] + ω||f ||2L2 (Ω) ≥ ν||f ||2V ,

f ∈ V.

(15.2.3)

In the particular case when ω = 0, the form a is called accretive. To the sesquilinear form a, we associate the following unbounded linear operator ( D(A) = {f ∈ V : ∃g ∈ L2 (Ω; C) such that a(f, ϕ) = hg, ϕi, ∀ϕ ∈ V } Af = g, f ∈ D(A). The following result shows that −A generates an analytic semigroup on L2 (Ω; C). Theorem 15.2.5 Let a be a continuous sesquilinear quasi-accretive sesquilinear form. Then, the operator −A generates a C0 -semigroup {T (t)} on L2 (Ω; C) satisfying the growth bound ||T (t)|| ≤ eωt for all t ≥ 0. Moreover, {T (t)} can be extended to an analytic semigroup on L2 (Ω; C). Proof For every λ ∈ C such that Re λ ≥ ω we define the norm || · ||λ on V by setting ||ϕ||2λ = Re[a(ϕ, ϕ)] + Re λ||ϕ||L2 (Ω;C) for every ϕ ∈ V . From (15.2.3) it follows that || · ||λ is equivalent to the norm || · ||V . Thus, H := (V, || · ||λ ) is a Hilbert space. Fix g ∈ L2 (Ω; C), λ ∈ C such that Re λ ≥ ω and consider the sesquilinear form aλ and the functional `, defined by aλ (u, v) = a(u, v)+λhu, vi and `(u) := hu, gi for every u, v ∈ H. Using the Lax-Milgram theorem one deduces that there exists a unique f ∈ H such that `(u) = hu, gi = aλ (u, f )

402

Kernel Estimates

for all u ∈ H. Hence, hg, ui = a(u, f ) + λhu, f i = a(f, u) + hλf, ui for all u ∈ H. So, from the definition of operator A it follows that f ∈ D(A) and (λ+A)f = g. On the other hand, from (15.2.3) we infer that λ + A is injective and therefore invertible. Moreover, Re[hg, (λ + A)−1 gi] =Re[a((λ + A)−1 g, (λ + A)−1 g)] + Re λ||(λ + A)−1 g||2L2 (Ω;C) ≥ν||(λ + A)−1 g||2V + (Re λ − ω)||(λ + A)−1 g||2L2 (Ω;C) ≥(Re λ − ω)||(λ + A)−1 g||2L2 (Ω;C) . Thus, ||(λ + A)−1 g||L2 (Ω;C) ≤

1 ||g||L2 (Ω;C) , Re λ − ω

g ∈ L2 (Ω; C), Re λ > ω.

So the first assertion follows from the Lumer-Phillips theorem (see Theorem 2.3.6). To show the last assertion, we fix g ∈ L2 (Ω; C), λ ∈ C, with Re λ > ω, and set f = (λ + A)−1 g. Clearly, f belongs to V and from the definition of operator A we deduce that λhf, ui + a(f, u) = hg, ui,

u ∈ V.

(15.2.4)

This implies that ν||f ||2V + ω||f ||2L2 (Ω;C) ≤ Re a(f, f ) = Rehg, f i − (Re λ)||f ||2L2 (Ω;C) so that ν||f ||2V ≤ Rehg, f i − (Re λ − ω)||f ||2L2 (Ω;C) ≤ ||f ||L2 (Ω;C) ||g||L2 (Ω;C) . Using (15.2.4) once more, from the previous estimate we get   M + 1 ||f ||L2 (Ω;C) ||g||L2 (Ω;C) . |λ|||f ||2L2 (Ω;C) ≤ ||f ||L2 (Ω;C) ||g||L2 (Ω;C) + M ||f ||2V ≤ ν Therefore, −1

||λ(λ + A)

 ||L(L2 (Ω;C)) ≤

 M +1 , ν

λ ∈ C, Re λ > ω.

The analyticity of the semigroup {T (t)} follows from Proposition 3.2.8 and Theorem 3.2.2.  On W01,2 (Ω; C) × W01,2 (Ω; C) we consider the continuous sesquilinear form a0 , defined by a0 (f, g) =

d Z X i,j=1

+

qij (x)Di f (x)Dj g(x) dx +

Ω

i=1

d Z X i=1

d Z X

Ω

bi (x)Di f (x)g(x) dx

Ω

Z ci (x)f (x)Di g(x) dx +

c0 (x)f (x)g(x) dx

(15.2.5)

Ω

for every f, g ∈ W01,2 (Ω; C), under the following assumptions. Hypotheses 15.2.6 (i) The coefficients qij , bi , ci (i = 1, . . . , d) and c0 are real and belong to L∞ (Ω);

Semigroups of Bounded Operators and Second-Order PDE’s

403

(ii) the uniform ellipticity condition d X

qij (x)ξi ξj ≥ µ|ξ|2 ,

(15.2.6)

i,j=1

is satisfied for all ξ ∈ Cd , almost every x ∈ Ω and some positive constant µ. Lemma 15.2.7 Let Hypotheses 15.2.6 be satisfied. Then, the form a0 satisfies Psesquilinear d − 1 2 condition (15.2.3) with V = W01,2 (Ω; C), ν = µ2 and ω = 2µ i=1 ||bi + ci ||∞ + ||c0 ||∞ . Proof Fix f ∈ W01,2 (Ω; C). Using (15.2.6) and Young’s inequality, we can estimate µ|||∇f |||2L2 (Ω;C) ≤Re

d Z X i,j=1

qij (x)Di f (x)Dj f (x) dx

Ω

  Z d Z X c0 (x)|f (x)|2 dx =Re a0 (f, f ) − (bi (x) + ci (x))f (x)Di f (x) dx − ≤Re[a0 (f, f )] +

Ω

Ω

i=1

d Z X

Z |bi (x) + ci (x)||f (x)||Di f (x)| dx +

Ω

i=1

2 c− 0 (x)|f (x)| dx

Ω

d

µX ||Di f (x)||2L2 (Ω;C) 2 i=1   d 1 X − 2 ||bi + ci ||∞ + ||c0 ||∞ ||f ||2L2 (Ω;C) . + 2µ i=1

≤Re[a0 (f, f )] +

The assertion follows.



Let us denote by A0 the operator associated to the sesquilinear form a0 . Formally, A0 is given by A0 f = −

d X i,j=1

Dj (qij Di f ) +

d X

(bi Di f − Di (ci f )) + c0 f

i=1

with Dirichlet boundary conditions on ∂Ω. Applying Theorem 15.2.5, we obtain the following generation result. Proposition 15.2.8 Let Hypotheses 15.2.6 be satisfied. Then, the operator −A0 generates an analytic semigroup {T0 (t)} on L2 (Ω; C), which is holomorphic in the sector Σ0,θ0 − π2 for some θ0 ∈ (π/2, π) and satisfies the estimate ||T0 (z)|| ≤ eω|z| ,

z ∈ Σ0,θ0 − π2 .

(15.2.7)

Proof In view of Theorem 15.2.5 and Lemma 15.2.7, the operator −A0 is sectorial and generates a strongly continuous semigroup on L2 (Ω; C). To complete the proof, we just need to prove estimate (15.2.7). For this purpose, we fix θ ∈ 0, θ0 − π2 . By (15.2.6) it follows that Re

d X h,k=1

eiα ahk (x)ξh ξk = cos(α)

d X h,k=1

ahk (x)ξh ξk ≥ µ ˜|ξ|2

404

Kernel Estimates

for all α ∈ [−θ, θ], ξ ∈ Cd and almost every x ∈ Ω, where µ ˜ = µ sin θ0 . Thus, arguing as in the proof of Lemma 15.2.7, we can show that Re[eiα a0 (f, f )] + ω||f ||2L2 (Ω) ≥

µ ˜ |||∇f |||2L2 (Ω) , 2

for all α ∈ [−θ, θ] and f ∈ W01,2 (Ω; C), where ω is the positive constant defined in Lemma 15.2.7. Hence, from the proof of Theorem 15.2.5 we deduce that ||T (teiα )|| ≤ eωt for every t > 0, and (15.2.7) follows.  Definition 15.2.9 A C0 -semigroup {T (t)} on L2 (Ω; C) is L∞ -contractive ||T (t)f ||L∞ (Ω;C) ≤ ||f ||L∞ (Ω;C) . for every t ≥ 0 and f ∈ L2 (Ω; C) ∩ L∞ (Ω; C). Similarly, a C0 -semigroup {T (t)} on L2 (Ω; C) is L∞ -quasi contractive, if there exists a positive constant γ such that the semigroup {e−γt T (t)} is contractive. It is clear that {T (t)} is L∞ -contractive if and only if {T (t)} leaves invariant the closed convex set {f ∈ L2 (Ω) : |f | ≤ 1 a.e.}. To state the famous Beurling-Deny criteria, we need to recall the following well known result. Lemma 15.2.10 Let C be a non-empty closed and convex subset of a Hilbert space H. Then, for every x ∈ H there exists a unique element P x ∈ C such that d(x, C) = inf z∈C ||x − z|| = ||x − P x||. Moreover P x is characterized by the following property: y = P x ⇐⇒ y ∈ C and Rehx − y, z − yi ≤ 0 for all z ∈ C.

(15.2.8)

Proof Set d = d(x, C). Then d2 = inf z∈C ||x − z||2 and, hence, for every n ∈ N there exists yn ∈ C such that ||x − yn ||2 ≤ d2 + n1 . Using the parallelogram law, we get ||yn − ym ||2 =||(yn − x) − (ym − x)||2 =2||yn − x||2 + 2||ym − x||2 − ||yn + ym − 2x||2

2  

yn + ym

1 1

≤2 d2 + + 2 d2 + − 4 − x

n m 2 1  1 ≤4d2 + 2 + − 4d2 , n m where the last estimate follows from the convexity of C and the definition of d. Thus,   1 1 2 ||yn − ym || ≤ 2 + . n m Since C is a closed subspace of H, there exists y ∈ C to which yn converges as n tends to ∞. So, one can easily see that ||y − x|| = d. Assume now that there exists y 0 ∈ C, y 0 6= y such that ||y 0 − x|| = d. Since C is convex, (y + y 0 )/2 belongs to C. Using again the parallelogram law, we obtain

2

2

y + y0

1

1 0



2 − x = 2 (y − x) − 2 (x − y ) 1 1 1 = ||y − x||2 + ||y 0 − x||2 − ||y − y 0 ||2 2 2 4

Semigroups of Bounded Operators and Second-Order PDE’s

405

1 =d2 − ||y − y 0 ||2 < d2 , 4 which contradicts the definition of d. So, we have proved the uniqueness of the element y. Let us now prove the characterization in (15.2.8). Fix x ∈ H, z ∈ C and λ ∈ [0, 1]. Since C is convex, λz + (1 − λ)P x belongs to C. Therefore, ||x − (λz + (1 − λ)P x)||2 = ||x − P x + λ(P x − z)||2 ≥ d2 = ||x − P x||2 , and we deduce that 2λRehx − P x, P x − zi + λ2 ||P x − z||2 ≥ 0. Hence, for λ ∈ (0, 1] we obtain 2Rehx − P x, P x − zi + λ||P x − z||2 ≥ 0 and, by letting λ tend to 0, we obtain that Rehx − P x, z − P xi ≤ 0. For the converse, it suffices to show that ||z − x|| ≥ ||y − x|| for every z ∈ C. This follows from the following formula ||z − x||2 = ||(z − y) − (x − y)||2 = ||z − y||2 + ||x − y||2 − 2Rehx − y, z − yi ≥ ||x − y||2 . This concludes the proof.



The following criteria are known as the Beurling-Deny criteria and allow to characterize the positivity and the L∞ -contractivity of the semigroup {T0 (t)}. We set L2+ (Ω) = {f ∈ L2 (Ω) : f ≥ 0 a.e.}. Proposition 15.2.11 Let C be L2 (Ω) or L2+ (Ω) or {f ∈ L2 (Ω) : |f | ≤ 1 a.e.} and denote by P the projection of L2 (Ω; C) onto C. Let a : V × V → C be a densely defined, quasiaccretive and continuous sesquilinear form on L2 (Ω; C). If P V ⊆ V and Re[a(f, f − P f )] ≥ 0 for all f ∈ V,

(15.2.9)

then the semigroup {T (t)} associated to the form a leaves C invariant. Proof Let us apply (15.2.9) to λ(λ+A)−1 f for a given f ∈ C and a fixed λ > 0, where A denotes the operator associated to the sesquilinear form a. Recalling that λA(λ + A)−1 f = λf − λ2 (λ + A)−1 f , we can estimate 0 ≤Re[a(λ(λ + A)−1 f, λ(λ + A)−1 f − P λ(λ + A)−1 f )] =λRe[hf − λ(λ + A)−1 f, λ(λ + A)−1 f − P λ(λ + A)−1 f i] =λRe[hA(λ + A)−1 f, λ(λ + A)−1 f − P λ(λ + A)−1 f i] =λRe[hf − P λ(λ + A)−1 f, λ(λ + A)−1 f − P λ(λ + A)−1 f i] + λRe[hP λ(λ + A)−1 f − λ(λ + A)−1 f, λ(λ + A)−1 f − P λ(λ + A)−1 f i] ≤λRe[hf − P λ(λ + A)−1 f, λ(λ + A)−1 f − P λ(λ + A)−1 f i]. Since f ∈ C, from (15.2.8) and the previous chain of inequalities it follows that Re[hf − P λ(λ + A)−1 f, λ(λ + A)−1 f − P λ(λ + A)−1 f i] = 0, and, hence, the term λRe[hf − P λ(λ + A)−1 f, λ(λ + A)−1 f − P λ(λ + A)−1 f i] + λRe[hP λ(λ + A)−1 f − λ(λ + A)−1 f, λ(λ + A)−1 f − P λ(λ + A)−1 f i] in (15.2.10) is equal to zero, i.e., hλ(λ + A)−1 f − P λ(λ + A)−1 f, λ(λ + A)−1 f − P λ(λ + A)−1 f i

(15.2.10)

406

Kernel Estimates =Re[hf − P λ(λ + A)−1 f, λ(λ + A)−1 f − P λ(λ + A)−1 f i] ≤ 0.

Thus, ||λ(λ + A)−1 f − P λ(λ + A)−1 f || = 0, so that λ(λ + A)−1 C ⊆ C for all λ > 0 and the assertion follows from Corollary 2.3.9.  Remark 15.2.12 Assume that a : V × V → C is a densely defined, accretive, continuous sesquilinear form on L2 (Ω; C) such that P V ⊆ V and Re[a(P f, f − P f )] ≥ 0 for all f ∈ V.

(15.2.11)

Then (15.2.9) holds. Indeed, since a is accretive, we can write Re[a(f, f − P f )] = Re[a(P f, f − P f )] + Re[a(f − P f, f − P f )] ≥ 0. Actually, one can prove that the conditions (15.2.9) and (15.2.11) are equivalent. See, e.g., [28, Theorem 2.2]. Applying Proposition 15.2.11 and Lemma 15.2.3, we obtain the following result. Proposition 15.2.13 Under Hypotheses 15.2.6, the semigroup {T0 (t)} associated to the form a0 , defined by (15.2.5), is positive. Moreover, if d Z X j=1

cj Dj |f | dx ≥ 0,

f ∈ W01,2 (Ω; C)

Ω

then the semigroup {T0 (t)} is L∞ -quasi contractive on L2 (Ω; C). 1,2 1,2 Proof We introduce the sesquilinear form aω 0 : W0 (Ω; C) × W0 (Ω; C) → C, defined 1,2 ω by a0 (f, g) = a0 (f, g) + ωhf, giL2 (Ω;C) for every f, g ∈ V = W0 (Ω; C), where d

ω=

1 X ||bi + ci ||2∞ + ||c− 0 ||∞ . 2µ i=1

As it is immediately seen, aω 0 is an accretive form. We split the rest of the proof into two steps. In Step 1, we prove that the semigroup {e−ωt T0 (t)}, which is associated to the form aω 0 is positive (clearly, this will imply that the semigroup {T0 (t)} is positive as well) and, then, in Step 2, we show that it is L∞ -contractive. Step 1. Set C = L2+ (Ω; C). The projection P : L2 (Ω) → L2+ (Ω) is given by P f = (Re f )+ for every f ∈ L2 (Ω; C). From Lemma 15.2.3, it follows that P V ⊆ V and an easy computation shows that ω − Re[aω 0 (f, f − P f )] =Re[a0 (Re f + iIm f, −(Re f ) + iIm f )] − − ω + − ω =Re[aω 0 ((Ref ) , (Re f ) )] − Re[a0 ((Re f ) , (Re f ) )] + Re[a0 (Im f, Im f )] − − ω =Re[aω 0 ((Re f ) , (Re f ) )] + Re[a0 (Im f, Im f )] ≥ 0.

So, the positivity follows from Proposition 15.2.11. Step 2. Set C = {f ∈ L2 (Ω) : |f | ≤ 1 a.e.} and V = W01,2 (Ω). The projection P into C is given by P f = (1 ∧ |f |)sign(f ) for every f ∈ L2 (Ω). Applying Lemma 15.2.3 we obtain that P V ⊆ V . Moreover, note that f − (1 ∧ |f |)sign(f ) = (|f | − 1)+ sign(f ),

f ∈ L2 (Ω).

Semigroups of Bounded Operators and Second-Order PDE’s

407

By Lemma 15.2.3, it follows that Im(sign(f )Dj f ) sign(f )χ{|f |>1} + χ{|f |≤1} Dj f, |f |   Im(sign(f )Dj f ) Dj ((|f | − 1)+ sign(f )) = Dj |f | + i(|f | − 1) sign(f )χ{|f |>1} . |f | Dj ((1 ∧ |f |)sign(f )) = i

So, we deduce that ω + Re[aω 0 (P f, f − P f )] = Re[a0 ((1 ∧ |f |)sign(f ), (|f | − 1) sign(f ))] d Z X Im(sign(f )Dj f ) Im(sign(f )Dk f ) (|f | − 1)+ dx = qjk |f | |f | Ω j,k=1

+

d Z X

[cj Dj |f |χ{|f |>1} + (c0 + ω)(|f | − 1)+ ] dx ≥ 0.

Ω

j=1

Thus, the L∞ -contractivity follows from Remark 15.2.12 and Proposition 15.2.11.  R P d Concerning the L∞ -contractivity, the condition j=1 Ω cj Dj |f | ≥ 0 can be omitted if 1,∞ one assumes further that ci ∈ W (Ω) for all i = 1, . . . , d. Corollary 15.2.14 Besides Hypotheses 15.2.6, assume that the coefficients ci belong to W 1,∞ (Ω) for i = 1, . . . , d. Then there exists a positive constant ω b depending only on µ, ||qij ||∞ , ||bi ||∞ , ||ci ||W 1,∞ (Ω) and ||c0 ||∞ , such that ||T0 (t)f ||∞ ≤ eωb t ||f ||∞ ,

f ∈ L2 (Ω) ∩ L∞ (Ω).

Proof Integrating by parts, we can write a0 (f, g) =

d Z X

qij (x)Di f (x)Dj g(x) dx +

Ω

i,j=1

c0 (x) + Ω

d X

(bi (x) − ci (x))Di f (x)g(x) dx

Ω

i=1

Z  +

d Z X

 Di ci (x) f (x)g(x) dx

i=1

for every f, g ∈ W01,2 (Ω). Now, arguing as in the proof of Proposition 15.2.13, we can complete the proof.  Fix γ ∈ R and consider a real-valued function ψ ∈ W := {ψ ∈ Cb∞ (Rd ) : |∇ψ| ≤ 1}. Define the sesquilinear form with domain D(a0 ) = W01,2 (Ω; C) × W01,2 (Ω; C) by setting a0,γ (f, g) := a0 (eγψ f, e−γψ g),

f, g ∈ W01,2 (Ω; C).

As easy computation shows that a0,γ (f, g) =

d Z X i,j=1

+

qij (x)Di f (x)Dj g(x) dx +

Ω

i=1

d Z X i=1

d Z X

Ω

bi,γ (x)Di f (x)g(x) dx

Ω

Z ci,γ (x)f (x)Di g(x) dx +

c0,γ (x)f (x)g(x) dx Ω

408

Kernel Estimates

for each f, g ∈ W01,2 (Ω; C), where bi,γ = bi − γ

d X

qij Dj ψ,

ci,γ = ci + γ

j=1

c0,γ = c0 − γ 2

d X

qij Dj ψ,

i = 1, . . . , d,

j=1

d X

qij Di ψDj ψ + γ

i,j=1

d X (bi − ci )Di ψ. i=1

The following lemma shows that the sesquilinear form a0,γ is quasi-accretive. Lemma 15.2.15 Under Hypotheses 15.2.6, there exists a positive constant ω ˜ , depending only on µ, ||qij ||∞ , ||bi ||∞ , ||ci ||∞ and ||c0 ||∞ such that Re[a0,γ (f, f )] + ω ˜ (1 + γ 2 )||f ||2L2 (Ω;C) ≥

µ ||f ||2W 1,2 (Ω;C) 0 2

(15.2.12)

for all f ∈ W01,2 (Ω; C), γ ∈ R and ψ ∈ W . Proof Applying Lemma 15.2.7 to the sesquilinear form a0,γ we obtain that Re[a0,γ (f, f )] + ωγ ||f ||2L2 (Ω;C) ≥

µ ||f ||W 1,2 (Ω;C) , 0 2

f ∈ W01,2 (Ω; C),

d

where ωγ =

1 X − ||bi + ci ||2∞ + ||c− 0,γ ||∞ . Estimating ||c0,γ ||∞ yields 2µ i=1 − 2 ||c− 0,γ ||∞ ≤||c0 ||∞ + γ

d X

||qij ||∞ + |γ|

i,j=1 2 ≤||c− 0 ||∞ + γ

d X

|bi − ci |||Di ψ||∞

i=1

X d

d

||qij ||∞ +

i,j=1

1X ||bi − ci ||2∞ 2 i=1

 +

1 2

2

≤M (1 + γ ), n Pd 1 where M := max ||c− 0 ||∞ + 2 , i,j=1 ||qij ||∞ + by taking

1 2

Pd

i=1

o ||bi − ci ||2∞ . So the assertion follows

  d d d 1 X 1X 1 X − 2 2 ||bi + ci ||∞ + max ||c0 ||∞ + , ||qij ||∞ + ||bi − ci ||∞ . ω e= 2µ i=1 2 i,j=1 2 i=1  Let us denote by A0,γ the operator associated to the sesquilinear form a0,γ . Formally A0,γ is given by A0,γ f = −

d X i,j=1

Dj (qij Di f ) +

d X

(bi,γ Di f − Di (ci,γ f )) + c0,γ f,

i=1

with Dirichlet boundary conditions on ∂Ω. Since a0,γ has similar expression as a0 , the following L∞ -contractivity result follows by Lemma 15.2.15, see Proposition 15.2.13.

Semigroups of Bounded Operators and Second-Order PDE’s

409

Lemma 15.2.16 Besides Hypotheses 15.2.6, assume that the coefficients qij and ci belong to W 1,∞ (Ω) for all i, j = 1, . . . , d. Then, the operator A0,γ generates the analytic semigroup {T0,γ (t)} defined by T0,γ (t) = e−γψ(t) T0 (t)eγψ(t) , which satisfies the estimate ||T0,γ (t)f ||L∞ (Ω;C) ≤ eω˜ (1+γ

2

)t

||f ||L∞ (Ω;C)

for all f ∈ L2 (Ω; C) ∩ L∞ (Ω; C), γ ∈ R, ψ ∈ W and ω ˜ as in Lemma 15.2.15. Proof Since the sesquilinear form a0,γ has the same form as a0 , the proof follows from (15.2.12) and Corollary 15.2.14.  Remark 15.2.17 Since the form-adjoint of a0,γ is of the same form as a0,γ , it follows by 2 duality that ||T0,γ (t)f ||L1 (Ω;C) ≤ eω˜ (1+γ )t ||f ||L1 (Ω;C) for t ≥ 0 and f ∈ L2 (Ω; C) ∩ L1 (Ω; C), provided that bi , ci , qij ∈ W 1,∞ (Ω) for all i, j = 1, . . . , d. Applying Theorem 15.1.1 and Corollary 15.2.2 and using the above properties of the twisted semigroup {T0,γ (t)} we obtain Gaussian upper bounds. Theorem 15.2.18 Besides Hypotheses 15.2.6, assume that the coefficients qij , bi and ci belongs to W 1,∞ (Ω) for all i, j = 1, . . . , d. Then, the semigroup {T0 (t)} interpolates on Lp (Ω; C), for every p ∈ [1, ∞) ∪ {∞} and there exist a functions kt ∈ L∞ (Ω × Ω) and positive constants M and ω such that d

0 ≤ kt (x, y) ≤ M t− 2 eωt e−

|x−y|2 4ωt

a.e. (x, y) ∈ Ω × Ω,

,

and Z (T0 (t)f )(x) =

a.e. x ∈ Ω,

kt (x, y)f (y) dy, Ω

for all t > 0 and f ∈ L2 (Ω). Proof From Remark 15.2.17 (with γ = 0) and Lemma 15.1.6 it follows that {T0 (t)} interpolates on Lp (Ω; C) for every p ∈ [1, 2]. By duality {T0 (t)} interpolates on Lp (Ω; C) for p ∈ [2, ∞) ∪ {∞}. Let us prove now that d

||T0,γ (t)f ||L∞ (Ω) ≤ M t− 2 eω(1+γ

2

)t

||f ||L1 (Ω)

(15.2.13)

for every f ∈ L1 (Ω) ∩ L2 (Ω), t > 0, γ ∈ R, ψ ∈ W and some positive constants M and ω. For this purpose we fix f ∈ L1 (Ω) ∩ L2 (Ω) and t > 0. From the Nash inequality (15.2.2), estimate (15.2.12) and Remark 15.2.17 it follows that   2 2 d −eω(1+γ 2 )s ||e T0,γ (s)f ||2L2 (Ω) = − 2Re a0,γ (e−eω(1+γ )s T0,γ (s)f, e−eω(1+γ )s T0,γ (s)f ) ds 2 − 2e ω (1 + γ 2 )||e−eω(1+γ )s T0,γ (s)f ||2L2 (Ω) ≤ − µ||e−eω(1+γ

2

)s

T0,γ (s)f ||2W 1,2 (Ω) 0

4 2 2 µ 2+ d −4 ≤ − ||e−eω(1+γ )s T0,γ (s)f ||L2 (Ω) ||e−eω(1+γ )s T0,γ (s)f ||L1d(Ω) C 4 2 µ 2+ d −4 ||f ||L1d(Ω) ≤ − ||e−eω(1+γ )s T0,γ (s)f ||L2 (Ω) C

410

Kernel Estimates

for every s > 0. Hence,
2 d −4 ||e−eω(1+γ )s T0,γ (s)f ||L2d(Ω) ds 2 2 2 2µ −2− 4 d −4 = − ||e−eω(1+γ )s T0,γ (s)f ||L2 (Ω)d ||e−eω(1+γ )s T0,γ (s)f ||2L2 (Ω) ≥ ||f ||L1d(Ω) . d ds dC  d/4 f := dC and letting ε tend to 0, Integrating the above inequality in [ε, t], setting M 2µ gives ˜ t− d4 eω˜ (1+γ 2 )t ||f ||L1 (Ω) , ||T0,γ (t)f ||L2 (Ω) ≤ M t > 0. (15.2.14) Now, since the form-adjoint of a0,γ is of the same form as a0,γ , the above estimate remains true if one replaces T0,γ (t) by its adjoint T0,γ (t)∗ . Thus, ct− d4 eωb (1+γ 2 )t ||f ||L2 (Ω) , ||T0,γ (t)f ||L∞ (Ω) ≤ M

t > 0,

(15.2.15)

c, ω for some constants M b and all f ∈ L2 (Ω; C). Taking the semigroup property into account, we get d ct− d4 eωb (1+γ 2 )t ||T0,γ (t/2)f ||L2 (Ω) ||T0,γ (t)f ||L∞ (Ω) =||T0,γ (t/2)T0,γ (t/2)f ||L∞ (Ω) ≤ b 24 M d

d

fM ct− 2 e(bω+eω)(1+γ ≤2 2 M

2

)t

||f ||L1 (Ω)

for all f ∈ L1 (Ω)∩L2 (Ω), t > 0 and estimate (15.2.13) follows. Thus, the semigroup {T0,γ (t)} is ultracontractive and by Proposition 15.1.3 it follows that there exists an integral kernel kγ,t ∈ L∞ (Ω × Ω) such that Z T0,γ (t)f (x) = kγ,t (x, y)f (y) dy, t > 0, f ∈ L2 (Ω). Ω

Moreover, the inequality d

||kγ,t ||L∞ (Ω×Ω) ≤ M t− 2 eω(1+γ

2

)t

,

t > 0,

(15.2.16)

holds for some positive constants M , ω and all γ ∈ R. In particular, Z T0 (t)f (x) = kt (x, y)f (y) dy, t > 0, f ∈ L2 (Ω), Ω

where 0 ≤ kt ∈ L∞ (Ω × Ω) is given by kt (x, y) = eγ(ψ(x)−ψ(y)) kγ,t (x, y) for almost every (x, y) ∈ Ω × Ω and all ψ ∈ W . The positivity of kt follows from the positivity of the semigroup {T0 (t)}, see Proposition 15.2.13. Thus, by (15.2.16), we conclude that d

0 ≤ kt (x, y) ≤ M t− 2 eω(1+γ

2

)t γ(ψ(x)−ψ(y))

e

a.e. (x, y) ∈ Ω × Ω,

,

for every ψ ∈ W . Replacing γ by −γ in the above estimate, we deduce that d

0 ≤ kt (x, y) ≤ M t− 2 eω(1+γ

2

)t −γ|ψ(x)−ψ(y)|

e

for all ψ ∈ W and γ ∈ R. Choosing γ = ωγ 2 t − γ|ψ(x) − ψ(y)|, yields d

0 ≤ kt (x, y) ≤ M t− 2 eωt e−

,

a.e. (x, y) ∈ Ω × Ω

ψ(x) − ψ(y) to minimize the polynomial p(ω) = 2ωt

(ψ(x)−ψ(y))2 4ωt

,

a.e. (x, y) ∈ Ω × Ω,

Semigroups of Bounded Operators and Second-Order PDE’s

411

and hence d

0 ≤ kt (x, y) ≤ M t− 2 eωt e−

d(x,y)2 4ωt

,

a.e. (x, y) ∈ Ω × Ω,

for each t > 0 and ψ ∈ W , where d(x, y) := sup{|ψ(x)−ψ(y)| : ψ ∈ W } for all (x, y) ∈ Ω×Ω. Now, we conclude the proof by showing that d(x, y) = |x − y| for all x, y ∈ Rd . The inequality d(x, y) ≤ |x − y| follows from the fact that |∇ψ| ≤ 1 on Rd for all ψ ∈ W . To prove the converse inequality we fix ε > 0, y ∈ Rd and consider the function ψε : Rd → R, defined by p ε + |x − y|2 p ψε,y (x) = 1 + ε ε + |x − y|2 for every x ∈ Rd , which belongs to W . For every x ∈ Rd we can write p √ ε + |x − y|2 ε p √ ≤ d(x, y). |ψε,y (x) − ψε,y (y)| = − 1 + ε ε 1 + ε ε + |x − y|2 Letting ε tend to 0, we deduce that |x − y| ≤ d(x, y) holds for all x, y ∈ Rd .



As an applications of the previous results we obtain complex kernel bounds and analyticity of the semigroup in Lp (Ω) for p ∈ [1, ∞) ∪ {∞}. Theorem 15.2.19 Besides Hypotheses 15.2.6, assume that the coefficients aij , bi ci of the form a belong to W 1,∞ (Ω) for all i, j = 1, . . . , d. Then {T0 (t)} interpolates on Lp (Ω) for p ∈ [1, ∞) ∪ {∞} and {T0 (t)} is an analytic semigroup on Lp (Ω; C) for these values of p, holomorphic in the sector Σ0,θ0 − π2 for some θ0 ∈ (0, π). Moreover, there exist a function kz ∈ L∞ (Ω × Ω), and positive constants M , ω such that, for all θ ∈ (0, θ0 − π2 ), d

|kz (x, y)| ≤ M (Re z)− 2 eω|z| e−

|x−y|2 4ω|z|

,

a.e. (x, y) ∈ Ω × Ω,

(15.2.17)

for all z ∈ Σ0,θ and Z T0 (z)f (x) =

kz (x, y)f (y) dy,

a.e. x ∈ Ω,

Ω

for all z ∈ Σ0,θ and f ∈ L2 (Ω; C). Proof Let us first prove that T0 (z) has a kernel kz ∈ L∞ (Ω × Ω) and (15.2.17) holds for all z ∈ Σ0,θ with θ ∈ (0, θ0 − π2 ). Following the proof of Theorem 15.2.18, to prove (15.2.17) it suffices to show that d

||T0,γ (z)f ||L∞ (Ω;C) ≤ M (Re z)− 2 eω(1+γ

2

)|z|

||f ||L1 (Ω;C) ,

f ∈ L1 (Ω; C) ∩ L2 (Ω; C),

for some positive constants M , ω and all z ∈ Σ0,θ . For this purpose let us note that, since the sesquilinear form a0,γ has the same form as a0 , we can deduce from Proposition 15.2.8 that 2 ||T0,γ (z)f ||L2 (Ω;C) ≤ eω1 (1+γ )|z| ||f ||L2 (Ω;C) , f ∈ L2 (Ω; C), (15.2.18) for some M, ω1 > 0 and all z ∈ Σ0,θ . Let us fix z = t + is ∈ Σ0,θ and choose θ˜ ∈ (0, θ0 − π2 ). Then, there exists δ ∈ (0, 1) such that δt + is ∈ Σ0,θ˜ for all t + is ∈ Σ0,θ . Thus, by (15.2.15), (15.2.18) and (15.2.14), we obtain ||T0,γ (z)f ||L∞ (Ω;C) =||T0,γ ((1 − δ)t/2)T0,γ (δt + is)T0,γ ((1 − δ)t/2)f ||L∞ (Ω;C)

412

Kernel Estimates d

c((1 − δ)t/2)− 4 eωb (1+γ ≤M −d 4

≤M1 t

eωb (1+γ

2

2

)(1−δ)t/2

||T0,γ (δt + is)T0,γ ((1 − δ)t/2)f ||L2 (Ω;C)

2

)(1−δ)t/2 ω1 (1+γ )|δt+is|

e

||T0,γ ((1 − δ)t/2)f ||L2 (Ω;C)

≤M t

−d ω +˜ ω )(1+γ 2 )(1−δ)t/2 ω1 (1+γ 2 )|δt+is| 2 (b

≤M t

−d 2

e

e

eω(1+γ

2

)|z|

||f ||L1 (Ω;C)

||f ||L1 (Ω;C)

for all f ∈ L1 (Ω; C) ∩ L2 (Ω; C), z ∈ Σ0,θ , γ ∈ R and some positive constants M and ω. On the other hand, we know from Theorem 15.2.18 that the semigroup {T0 (t)} interpolates on Lp (Ω) for p ∈ [1, ∞) ∪ {∞}. Moreover, from (15.2.17) it follows that there exists ˜ a positive constant C such that ||T0 (eiθ t)||L(Lp (Ω;C)) ≤ Ceωt for all t > 0 and θ˜ ∈ [−θ, θ]. Hence the analyticity of the semigroup {T0 (t)} on Lp (Ω; C) follows from Corollary 3.2.9 and Remark 3.2.10 for every p ∈ [1, ∞) ∪ {∞}. 

15.3

Integral Representation for the Semigroups in Chapters 6, 7 and 9.

We conclude this chapter, showing that, under suitable of the coefficients of the operator A, defined on smooth functions ψ : Ω → C by Aψ = div(Q∇ψ) + hb, ∇ψi + cψ coincide with the semigroups introduced in Chapters 6, 7 and 9. Hypotheses 15.3.1 (i) Ω is Rd or Rd+ or a bounded domain of class C 2+α for some α ∈ (0, 1) (see Definition B.2.1); (ii) the coefficients qij = qji belong to Cb1 (Ω), the coefficients bj are Lipschitz continuous and bounded over Ω (i, j = 1, . . . , d) and c belongs to Cbα (Ω); (iii) there exists a positive constant µ such that hQ(x)ξ, ξi ≥ µ|ξ|2 for all x ∈ Ω and ξ ∈ Rd . Theorem 15.3.2 Under Hypotheses 15.3.1, let {T (t)} be either the semigroup in Section 6.4, of one of the semigroups in Sections 7.5 and 9.4, associated with the operator A with homogeneous Dirichlet boundary conditions. Then, for every t > 0 there exists a function kt ∈ L∞ (Ω × Ω) such that Z (T (t)f )(x) = kt (x, y)f (y) dy, t > 0, x ∈ Ω, f ∈ Cb (Ω). Ω

Moreover, there exist positive constants M and ω such that d

0 ≤ kt (x, y) ≤ M t− 2 eωt e−

|x−y|2 4ωt

(15.3.1)

for a.e. (x, y) ∈ Ω × Ω and all t > 0. Proof To fix the ideas we deal with the case Ω = Rd but the same arguments can be repeated to cover the remaining cases. Applying Proposition 3.2.8 and Theorem 10.2.23 it can be shown that the realization Ap of operator A in Lp (Rd ; C), i.e., the operator Ap : W 2,p (Rd ; C) → Lp (Rd ; C), defined by

Semigroups of Bounded Operators and Second-Order PDE’s

413

Ap u = Au for every u ∈ D(Ap ), generates a strongly continuous analytic semigroup {Tp (t)} on Lp (Rd ; C) for all p ∈ (1, ∞). Next, we observe that an integration by parts reveals that hA2 f, gi = a0 (f, g). for every f ∈ W 2,2 (Rd ; C) and g ∈ W 1,2 (Rd ; C). Hence, D(A2 ) is contained in D(A0 ) and A2 f = A0 f for all f ∈ D(A2 ). Since both A2 and A0 are generators of strongly continuous semigroups on L2 (Ω; C) we deduce that A2 and A0 coincides, see the end of the proof of Theorem 2.3.1. Hence, the semigroups {T0 (t)} and {T2 (t)} coincide. To conclude the proof, let us prove that T (t) = T0 (t) on Cc∞ (Rd ; C) for every t > 0. This will be enough for our purposes. Indeed, once this property is proved, we fix a function f ∈ Cb (Rd ; C) and approximate it with a sequence (fn ) of smooth and compactly supported functions, which is bounded with respect to the sup-norm. We write Z (T (t)fn )(x) = kt (x, y)fn (y) dy, x ∈ Rd , (15.3.2) Rd

for every t > 0. By Theorem 6.4.2 the left-hand side of the previous formula converges to T (t)f , pointwise on Rd , as n tends to ∞ for every n ∈ N. On the other hand, by estimate (15.3.1), also the right-hand side converges pointwise on Rd as n tends to ∞. Hence, letting n tend to ∞ in both sides of (15.3.2) the assertion follows. So, let us prove that T2 (t)f ≡ T (t)f for every t > 0 and every f ∈ Cc∞ (Rd ; C). By Proposition 10.2.25, the function R(λ, A2 )f belongs to W 2,p (Rd ; C) for every p > 1 and every λ ∈ ρ(A2 ). The Sobolev embedding theorems (see Theorem 1.3.6), show that R(λ, A2 )f ∈ Cbα (Rd ; C). Since the function AR(λ, A2 )f = λR(λ, A2 )f − f belongs to Cbα (Rd ; C), it follows that R(λ, A2 )f belongs to the domain of the sectorial operator A associated with the semigroup {T (t)}. Thus, R(λ, A2 )f = R(λ, A)f . Recalling that the semigroups {T (t)} and {T2 (t)} are defined through the Dunford integral (3.2.1), which converge in Cb (Rd ; C) and in L2 (Rd ; C), respectively, the equality T2 (t)f = T (t)f follows for every t > 0. 

15.4

Notes

Pd For uniformly elliptic operators of the form A = i,j=1 Dj (aij Di ) (on L2 (Rd )) it was Aronson [3] how first proved Gaussian upper bounds, see also Porper and Eidel’man [31]. The perturbation techniques that we used here was introduced by E.B. Davies [9] for proving Gaussian upper bounds symmetric operators on domains under Dirichlet or Neumann boundary conditions. For more details we refer to [10] and the references therein. For secondorder operators with drift terms acting on L2 (Rd ), Gaussian upper bounds have been proved by Stroock [35] and Norris and Stroock [27]. Concerning the general second-order elliptic operator that we consider here, Gaussian upper bounds were given by Arendt and ter Elst [2] for the kernel associated to such operators even with more general boundary conditions. For sharper Gaussian upper bounds and more general results we refer to the monograph by E.M. Ouhabaz [28] and the references therein.

414

15.5

Kernel Estimates

Exercises

1. Prove that if V is one of the spaces W 1,2 (Ω) or W01,2 (Ω) and f ∈ V then (Re f )+ , |f |, (1∧ |f |)sign(f ) ∈ V and Dj (Re f )+ = Dj (Re f )χ{Re f >0} , Dj |f | = Re(sign(f )Dj f ), Dj ((1 ∧ |f |)sign(f )) = i

Im(sign(f )Dj f ) sign(f )χ{|f |>1} + χ{|f |≤1} Dj f. |f |

2. Prove that if g ∈ W 1,2 (Ω) then the mapping f 7→ f ∧ g is continuous from W 1,2 (Ω) into W 1,2 (Ω). 3. Using the Phragmen-Lindel¨ of theorem and under the assumptions of Theorem 15.2.19, show that d

|kz (x, y)| ≤ M |z|− 2 eω|z| e−

|x−y|2 4ω|z|

,

for all z ∈ Σ0,θ and some constants M, ω > 0.

a.e. (x, y) ∈ Ω × Ω

Part IV

Appendices

Appendix A Basic Notions of Functional Analysis in Banach Spaces

In this appendix, we collect a few basic results of linear operators, elementary spectral theory and interpolation theory that we use in this textbook. For more details and for the proofs of the results that we present, we refer the reader mainly to [20, 37, 39].

A.1

Bounded and Closed Linear Operators

Definition A.1.1 Let X and Y be two Banach spaces. We denote by L(X, Y ) the vector space of linear and bounded operators T : X → Y . We endow it with the norm ||T ||L(X,Y ) = sup ||T x||Y = x∈X ||x||=1

||T x||Y . x∈X\{0} ||x||X sup

(A.1.1)

Remark A.1.2 The norm in (A.1.1) makes L(X, Y ) a Banach space. We introduce another class of linear operators which we use in this book. Definition A.1.3 Let D(A) be a vector subspace of X and let A : D(A) ⊂ X → Y be a linear operator. Operator A is called closed if its graph GA = {(x, y) ∈ X × Y : x ∈ D(A), y = Ax} is a closed subset of X × Y (endowed with the norm ||(x, y)||X×Y = ||x||X + ||y||Y for every x ∈ X and y ∈ Y ). The following proposition gives a useful characterization of closed operators. Proposition A.1.4 The linear operator A : D(A) ⊂ X → Y is closed if and only if for every sequence (xn ) ⊂ D(A) such that xn and Axn converge, respectively, to some elements x ∈ X and y ∈ Y , as n tends to ∞, then x ∈ D(A) and y = Ax. In general, a closed operator is not bounded in (X, || · ||X ) (see Exercise A.6.1). It turns out to be bounded if we endow D(A) with the graph norm ||x||D(A) = ||x||X + ||Ax||Y ,

x ∈ D(A).

(A.1.2)

Note that D(A) is a Banach space when it is endowed with the graph norm. Next, we introduce the closable operators. Definition A.1.5 A linear operator A : D(A) ⊂ X → Y is said to be closable if there exists a (closed) operator A whose graph coincides with the closure of GA . The operator A is called the closure of A. 417

418

Basic Notions of Functional Analysis in Banach Spaces

Remark A.1.6 Clearly, if A is a closable operator, then every x ∈ D(A) is the limit of a sequence (xn ) ⊂ D(A) which is a Cauchy sequence with respect to the graph norm (A.1.2). Moreover, Ax := limn→∞ Axn . A useful criterium to verify if a given linear operator is closable is provided by the following proposition. Proposition A.1.7 A linear operator A : D(A) ⊂ X → Y is closable if, for each sequence (xn ) ∈ D(A) converging to 0 in X and such that Axn converges to some y ∈ Y as n tends to ∞, then y = 0.

A.2

Vector Valued Riemann Integral

In this section, we define the Riemann integral for vector-valued functions. For more details and for the proof of the results that we present here, we refer the reader to [11, Chapter 2], [12, Chapter 3] and [19, Chapter 3]. Definition A.2.1 A bounded function f : [a, b] → X (−∞ < a < b < ∞) is said to be integrable on [a, b] if there exists x ∈ X with the following property: for each ε > 0 there exists δ > 0 such that, for every partition P = {a = t0 < t1 < . . . < tn = b} of [a, b], with maxi=1,...,n (ti − ti−1 ) < δ, and for every choice of the points ξi ∈ [ti−1 , ti ], it holds that

n X

x − f (ξi )(ti − ti−1 )

< ε.

i=1

In this case, we define Z

b

f (t) dt = x. a

Arguing as in the real-valued case, one can easily prove the following proposition. Proposition A.2.2 The following properties are satisfied. (i) The set of all integrable functions on [a, b] is a vector space and the integral over [a, b] is a linear operator on this vector space. (ii) Every continuous function f : [a, b] → X is integrable on [a, b]. (iii) If f is integrable on [a, b], then the function t 7→ ||f (t)||X is integrable on [a, b] as well and

Z b

Z b

f (t) dt ||f (t)|| dt.

≤ a

a

(iv) If f is integrable on [a, b], then it is integrable on every [c, d] ⊂ [a, b] and Z b Z c Z d Z b f (t) dt = f (t) dt + f (t) dt + f (t) dt. a

for every c ∈ (a, b).

a

c

d

Semigroups of Bounded Operators and Second-Order PDE’s

419

As in the real-valued case, the definition of the Riemann integral can be easily extended to the case of unbounded intervals or unbounded functions. Definition A.2.3 Let I ⊂ R be an interval with endpoints a and b (−∞ ≤ a < b ≤ ∞) with a and b not necessarily in I. Moreover, let f : I → X be Riemann integrable on [c, d] for every a < c < d < b. We say that f admits improper integral on I if, for each t0 ∈ I, the limits Z d Z t0 f (t) dt f (t) dt, lim− lim+ c→a

d→b

c

exist in X. In this case we set Z Z f (t) dt = lim+ c→a

I

c

t0

t0

Z f (t) dt + lim− d→b

d

f (t) dt. t0

Remark A.2.4 As it is easily checked, the previous definition is independent of the choice of t0 . To simplify the notation in the following proposition we simply write “f is integrable on I” to mean indifferently, that f is Riemann integrable on I or that it admits improper integral on I. Proposition A.2.5 Let f : I → X be integrable on I. Then, the following properties are satisfied. (i) For each bounded operator T ∈ L(X, Y ), the function T f is integrable on I and Z Z T f (t) dt = T f (t) dt. I

I

(ii) If A : D(A) ⊂ X → Y is a closed operator and f (I) ⊂ D(A) such that Af is integrable on I, then Z Z Z f (t) dt ∈ D(A) and A f (t) dt = Af (t) dt. I

I

I

Proof We limit ourselves to proving the last part of the proposition, since the first one is similar and even simpler. (ii) We first assume that f is Riemann integrable on I = [a, b] for some a, b ∈ R with a < b. We fix n ∈ N and consider the partition Pn = {a = t0 < . . . < tn = b} where tk = a + (b − a)k/n for every k = 0, . . . , n. Moreover, for every i = 0, . . . , n − 1 we fix ξi ∈ [ti , ti+1 ] and set Sn =

n X

f (ξi )(ti − ti−1 ),

n ∈ N.

i=1

Of course, Sn belongs to D(A) for each n ∈ N and ASn =

n X i=1

Af (ξi )(ti − ti−1 ),

n ∈ N.

420

Basic Notions of Functional Analysis in Banach Spaces

Since both f and Af are integrable, Sn and ASn converge, respectively, to Z b Z b Af (t) dt f (t) dt, and y= x= a

a

as n tends to ∞. Using the closedness of operator A, we conclude that x belongs to D(A) and Ax = y. Now, suppose that f admits improper integral on I. To fix the ideas we assume that I = [a, ∞). Then, b

Z

Z

b→∞

a

b > a.

a

a

By hypothesis Z b Z lim Af (t) dt =

b

Af (t) dt,

f (t) dt =

A

∞

Z Af (t) dt

and

lim

b→∞

a

Since A is closed, Z ∞ f (t) dt ∈ D(A)

Z and

a

b

Z f (t) dt =

a

∞

A

∞

f (t) dt. a

Z f (t) dt =

a

This completes the proof.

∞

Af (t) dt. a



Next, we state the fundamental theorem of calculus for X-valued functions. For this purpose we first recall the definition of Fr´echet derivative. Let I ⊂ R be an (open) interval and let t0 ∈ I. The function f : I → X is Fr´echet differentiable at t0 ∈ I if the limit lim

t→t0

f (t) − f (t0 ) t − t0

exists in X. Such a limit, when existing, is denoted by f 0 (t0 ) and is called the Fr´echet derivative of f at t0 . In an analogous way the right- and left-derivatives can be defined. Theorem A.2.6 Let f : [a, b] → X be continuous. Then, the integral function F : [a, b] → X defined by Z t F (t) = f (s) ds, t ∈ [a, b], a 0

is (Fr´echet) differentiable, and F (t) = f (t) for each t ∈ [a, b]. Using Theorem A.2.6, we can prove the following result. Proposition A.2.7 Suppose that f : [a, b) → X (a, b ∈ R, a < b) is a continuous function which admits right-derivative at each point of [a, b) and the right-derivative is continuous in [a, b). Then, f is Fr´echet differentiable in [a, b). Proof We denote by g the right-derivative of f . Since it is a continuous function, Theorem A.2.6 implies that the function h : [a, b) → X, defined by Z t h(t) = f (a) + g(s) ds, t ∈ [a, b), a

is Fr´echet differentiable in [a, b). In particular, h0 = g so that the right-derivative of function

Semigroups of Bounded Operators and Second-Order PDE’s

421

f − h identically vanishes in [a, b). Therefore, we just need to show that each function ψ : [a, b) → X, with identically vanishing right-derivative, is constant in [a, b). For this purpose, we arbitrarily fix a real valued functional x0 : X → R (called also continuous linear form) and introduce the function Rψ : [a, b) → R defined by (Rψ)(a) = 0,

(Rψ)(t) =

x0 (ψ(t)) − x0 (ψ(a)) , t−a

t ∈ (a, b),

and, for every ε > 0, the set Dε = {t ∈ [a, b) : |(Rψ)(s)| < ε for each s ∈ [a, t]}. Such a set is not empty since it contains a. Moreover, it is an interval and it is both closed and not reduced to a single point since the function Rψ is continuous in [a, b). To prove that Dε = [a, b), we assume by contradiction that Dε = [a, c] for some c < b. Then, |(Rψ)(c)| = ε. To fix the ideas, we assume that (Rψ)(c) = ε. From the definition of the set Dε it follows easily that there exists a sequence (sn ) converging to c from the right, such that (Rψ)(sn ) > ε. From the conditions (Rψ)(c) = ε and (Rψ)(sn ) > ε for each n ∈ N, it follows that x0 (ψ(c)) = x0 (ψ(a)) + ε(c − a), x0 (ψ(sn )) > x0 (ψ(a)) + ε(sn − a),

n ∈ N.

As a byproduct, we infer that x0 (ψ(sn )) − x0 (ψ(c)) > ε, sn − c

n ∈ N.

As n tends to ∞ the previous inequality implies that x0 (g(c)) ≥ ε, where g denotes the right-derivative of ψ. So, this leads to a contradiction. We have so proved that Eε = [a, b) for every ε > 0. Letting ε tend to 0 we conclude that Rψ ≡ 0, i.e., x0 (ψ(t)) = x0 (ψ(a)) for every t ∈ [a, b). Now, if x0 is a complex-valued functional, we split it into the sum x0 = y10 + iy20 , where y10 and y20 are real-valued functionals. Applying the above result to y10 and y20 , we obtain that x0 (f (t)) = x0 (f (a)). By the arbitrariness of x0 ∈ X 0 , we conclude that f (t) = f (a) for each t ∈ [a, b).  Now, we recall the definition of the integral of vector-valued functions of a complex variable, along a smooth curve γ, which we use in the next section and in Chapter 3. Definition A.2.8 Let Ω be an open subset of C, f : Ω → X be a continuous function and γ : [a, b] → Ω be a piecewise C 1 -curve. The integral of f along γ is defined as follows: Z Z b f (z) dz = f (γ(t))γ 0 (t) dt. γ

a

As in the case of vector-valued functions defined on a real interval, we can define the improper complex integrals in an obvious way. Definition A.2.9 Let Ω ⊂ C be a (possibly) unbounded open set. Moreover, let I = (a, b) be a (possibly) unbounded interval and γ : I → C be a (piecewise) C 1 -curve in Ω. We say that the continuous function f : Ω → X admits an improper integral along γ if for each t0 ∈ (a, b) the limits Z t0 Z b lim+ f (γ(τ ))γ 0 (τ ) dτ and lim− f (γ(τ ))γ 0 (τ ) dτ s→a

s→b

s

t0

exist in X. In such a case, we set Z Z t0 Z 0 f (z) dz = lim+ f (γ(τ ))γ (τ ) dτ + lim− γ

s→a

s

s→b

s

t0

f (γ(τ ))γ 0 (τ ) dτ.

422

Basic Notions of Functional Analysis in Banach Spaces

Remark A.2.10 As it is easily checked, the definition of the improper integral is independent of the choice of t0 . Moreover, if I is bounded and the integral of f along γ exists, then f admits an improper integral along γ and the two integrals coincide.

A.3

Holomorphic Functions

In this section, we briefly deal with holomorphic functions with values in a Banach space. Definition A.3.1 Let Ω be an open subset of C. The function f : Ω → X is called (i) holomorphic in Ω if for each z0 ∈ Ω the limit lim

z→z0

f (z) − f (z0 ) = f 0 (z0 ) z − z0

exists in X; (ii) weakly holomorphic in Ω if it is continuous in Ω and, for every x0 ∈ X 0 , the complexvalued function z 7→ x0 (f (z)) is holomorphic in Ω. As it is immediately seen, each holomorphic function f : Ω → X is also weakly holomorphic in Ω. As the following proposition shows, also the converse is true. Theorem A.3.2 Each weakly holomorphic function f : Ω → X is holomorphic in Ω. Proof We split the proof into two steps. In the first one, we prove that Z 1 f (ξ) f (z) = dξ 2πi ∂B(z0 ,r) ξ − z

(A.3.1)

for every z ∈ B(z0 , r), every z0 ∈ Ω and every r > 0 such that the ball B(z0 , r) is contained in Ω. Then, in Step 2, using this formula, we will show that f is differentiable in Ω, obtaining the assertion. Step 1. Fix z0 ∈ Ω and r > 0 such that B(z0 , r) ⊂ Ω. To begin with, we observe that the results in Section A.2 guarantee that the integral in the right-hand side of (A.3.1) is well defined. Next we fix x0 ∈ X 0 . Since f is weakly holomorphic in Ω, from the classical Cauchy integral formula we obtain that Z  Z 1 x0 (f (ξ)) 1 0 f (ξ) 0 x (f (z)) = dξ = x dξ . 2πi ∂B(z0 ,r) ξ − z 2πi ∂B(z0 ,r) ξ − z By the arbitrariness of x0 ∈ X 0 , formula (A.3.1) follows at once. Step 2. Now, we can complete the proof. Using formula (A.3.1), we will show that f is differentiable at each z ∈ B(z0 , r). For this purpose, we fix such a z and w ∈ C \ {0} such that z + w ∈ B(z0 , r). Then, we can write Z Z 1 1 f (z + w) − f (z) f (ξ) f (ξ) = dξ − dξ w 2πiw ∂B(z0 ,r) ξ − z − w 2πiw ∂B(z0 ,r) ξ − z Z 1 f (ξ) = dξ 2πi ∂B(z0 ,r) (ξ − z)(ξ − z − w)

Semigroups of Bounded Operators and Second-Order PDE’s Z 2π f (z0 + reiθ ) r = eiθ dθ. 2π 0 (z0 − z + reiθ )(z0 − z − w + reiθ )

423

Since f is continuous in Ω and both z and z + w lie in the interior of the ball centered at z0 with radius r, the function under the integral sign in the last side of the previous chain of equalities is bounded, uniformly with respect to w. Moreover, it converges to (z0 − z + reiθ )−2 f (z0 + reiθ )eiθ as w tends to 0. Hence, we can apply the dominated convergence theorem and infer that function f is differentiable at z and Z 1 f (ξ) 0 dξ. f (z) = 2πi ∂B(z0 ,r) (ξ − z)2 Due to the arbitrariness of z0 , we have proved that f is differentiable in Ω. The proof is complete.  Remark A.3.3 Iterating the arguments in the proof of Theorem A.3.2 it can be easily shown that Z f (ξ) n! (n) dξ, z ∈ B(z0 , r), n ∈ N, f (z) = 2πi ∂B(z0 ,r) (ξ − z)n+1 for every B(z0 , r) ⊂ Ω. Now, we collect some remarkable property of holomorphic functions. Theorem A.3.4 Let Ω be an open subset of C and f : Ω → X be a given function. Then, the following properties are satisfied. (i) Function f is holomorphic in Ω if and only if it admits a power series expansion around every point of Ω, i.e., for every z0 ∈ Ω there exist r > 0 and a sequence (an ) ⊂ X such that ∞ X f (z) = an (z − z0 )n , z ∈ B(z0 , r). (A.3.2) n=0

In such a case, 1 an = 2πi

Z

f (ξ) dξ, (ξ − z0 )n+1

∂B(z0 ,r)

n ∈ N ∪ {0}. Z

(ii) If f is holomorphic in Ω and D ⊂ Ω is a regular domain contained, then

f (z) dz = ∂D

0.

(iii) If Ω = B(z0 , R) \ B(z0 , r) for some z0 ∈ C and 0 < r < R, and f is holomorphic in Ω, then f admits the Laurent expansion f (z) =

∞ X

an (z − z0 )n ,

z ∈ D,

n=−∞

where an =

1 2πi

Z ∂B(z0 ,r0 )

f (ξ) dz (ξ − z0 )n+1

and r0 is arbitrarily fixed in the interval (r, R).

424

Basic Notions of Functional Analysis in Banach Spaces

Proof (i) Assume that f is holomorphic in Ω and fix x0 ∈ X 0 , z0 ∈ Ω and r > 0 such that B(z0 , r) ⊂ Ω. Since the complex-valued function x0 (f ) is holomorphic in Ω, it admits a power series expansion around z0 , i.e.,  ∞ Z x0 (f (ξ)) 1 X x (f (z)) = dξ (z − z0 )n , n+1 2πi n=0 ∂B(z0 ,r) (ξ − z0 ) 0

z ∈ B(z0 , r).

(A.3.3)

Since the series ∞ Z X n=0

∂B(z0 ,r)

 f (ξ) dξ (z − z0 )n (ξ − z0 )n+1

converges for each z ∈ B(z0 , r), from (A.3.3) we easily infer that x0 (f (z)) = x0



 Z ∞ 1 X f (ξ) (z − z0 )n dξ , n+1 2πi n=0 ∂B(z0 ,r) (ξ − z0 )

z ∈ B(z0 , r).

The arbitrariness of x0 ∈ X 0 yields the assertion. Conversely, suppose that f admits a power series expansion around each point z0 ∈ Ω. For each z0 ∈ Ω and x0 ∈ X 0 , from (A.3.2) we deduce that 0

x (f (z)) =

∞ X

x0 (an )(z − z0 )n ,

z ∈ B(z0 , r),

n=0

for a suitable r > 0. Hence, the complex-valued function x0 (f ) is holomorphic in Ω and, due to the arbitrariness of x0 ∈ X 0 , Theorem A.3.2 shows that f itself is holomorphic in Ω. (ii) & (iii) It suffices to argue as in the proof of (A.3.1), taking the classical Cauchy theorem and the classical Laurent expansion of complex valued functions into account. 

A.4

Spectrum and Resolvent

Here, we introduce the definitions of spectrum/resolvent set of a linear operator and of resolvent operator. We also study some of their basic properties. Definition A.4.1 Let A : D(A) ⊂ X → X be a closed linear operator. (i) The set σ(A) = {λ ∈ C : λI − A : D(A) → X is not bijective} is called the spectrum of A. Its complement in C is called the resolvent set of A and will be denoted by ρ(A). (ii) For each λ ∈ ρ(A), the operator R(λ, A) = (λI − A)−1 is called the resolvent of A at the point λ. Remark A.4.2 (i) The closed graph theorem shows that, if ρ(A) 6= ∅, then R(λ, A) is a bounded operator in X for every λ ∈ ρ(A). (ii) We stress that the resolvent set of a closed operator may be empty. See Exercise A.6.1.

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Proposition A.4.3 The resolvent set of a closed operator A : D(A) ⊂ X → X is an open subset of C. Moreover, the function R(λ, A) : ρ(A) → L(X) is holomorphic. Finally, the family of operators {R(λ, A) : λ ∈ ρ(A)} satisfies the so-called resolvent identity, i.e., R(λ, A) − R(µ, A) = (µ − λ)R(µ, A)R(λ, A),

λ, µ ∈ ρ(A).

(A.4.1)

In particular, R(λ, A) commutes with R(µ, A) for every λ, µ ∈ ρ(A). In the proof of Proposition A.4.3, we will use the following lemma. Lemma A.4.4 (Perturbation of the identity) Let X be a Banach space and T ∈ L(X) with ||T ||L(X) < 1. Then, the operator I − T is invertible and its inverse is given by the Von Neumann series, i.e., ∞ X (I − T )−1 = T n. (A.4.2) n=0

Proof As a first step, we observe that the series in the right-hand side of (A.4.2) converges in L(X). Indeed,

m+p

X n

T

n=m

≤

L(X)

m+p X

||T ||nL(X)

n=m

P∞ for every m, p ∈ N. Since ||T ||L(X) < 1, the series n=0 ||T ||nL(X) converges. In particular it P∞ is a Cauchy sequence and this implies that the series n=0 T n converges in L(X). To complete the proof, we observe that, for every m ∈ N, it holds that X m

T

n

 (I − T ) = (I − T )

m X n=0

n=0

Tn =

m X

Tn −

n=0

m X

T n+1 =

m X

Tn −

n=0

n=0

m+1 X

T n = I − T m+1 .

n=1

Since T m+1 tends to 0 in L(X) as m tends to ∞, from the previous chain of equalities, it follows that  X ∞ m X n T (I − T ) = (I − T ) T n = I, n=0

n=0

thus completing the proof.



Proof of Proposition A.4.3 Fix λ0 ∈ ρ(A), λ ∈ C and observe that λI − A = (I−(λ0 −λ)R(λ0 , A))(λ0 I−A). Thus, λ ∈ ρ(A) if and only if the operator I−(λ0 −λ)R(λ0 , A) is invertible and, by Lemma A.4.4, this is the case if λ ∈ B(λ0 , r), where r = ||R(λ0 , A)||−1 L(X) . Hence, ρ(A) is open. Moreover, for each λ ∈ B(λ0 , r), R(λ, A) = R(λ0 , A)(I − (λ0 − λ)R(λ0 , A))

−1

=

∞ X

(−1)n (λ − λ0 )n (R(λ0 , A))n+1 ,

n=0

and this formula together with Theorem A.3.4(i) show that the function R(·, A) is holomorphic in ρ(A) with values in L(X). The resolvent identity follows from the following chain of equalities: R(λ, A) − R(µ, A) =R(µ, A)[(µI − A) − (λI − A)]R(λ, A) = (µ − λ)R(µ, A)R(λ, A) for every λ, µ ∈ ρ(A). The last assertion is straightforward to prove.



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Basic Notions of Functional Analysis in Banach Spaces

Remark A.4.5 The proof of Proposition A.4.3 shows that ||R(λ, A)||L(X) ≥

1 , dist(λ, σ(A))

λ ∈ ρ(A),

and this estimate implies that the operator norm of R(λ, A) blows up as λ approaches the boundary of ρ(A). In particular, the function λ 7→ R(λ, A) cannot be continued outside ρ(A) with a continuous function. Now, we prove the following important result, which roughly speaking shows that the unique families of operators which satisfy the resolvent identity (A.4.1) are resolvent families of closed operators. Proposition A.4.6 Let Ω ⊂ C be an open set, and let {F (λ) : λ ∈ Ω} ⊂ L(X) be a family of linear operators verifying the resolvent identity F (λ) − F (µ) = (µ − λ)F (λ)F (µ),

λ, µ ∈ Ω.

If the operator F (λ0 ) is injective, for some λ0 ∈ Ω, then there exists a closed linear operator A : D(A) ⊂ X → X such that ρ(A) contains Ω and R(λ, A) = F (λ) for each λ ∈ Ω. Proof Fix λ0 ∈ Ω, and set D(A) = RangeF (λ0 ) and Ax = λ0 x − F (λ0 )−1 x for every x ∈ D(A). By the resolvent identity one can easily see that Range F (λ) = Range F (λ0 ) for every λ ∈ Ω. Now, for λ ∈ Ω and y ∈ X the resolvent equation λx − Ax = y is equivalent to (λ − λ0 )x + F (λ0 )−1 x = y. Applying F (λ) to both sides of the previous equation, we obtain (λ − λ0 )F (λ)x + F (λ)F (λ0 )−1 x = F (λ)y and, using the resolvent identity, we deduce that F (λ)F (λ0 )−1 x = F (λ0 )−1 F (λ)x for every x ∈ Range F (λ0 ) and F (λ0 )−1 F (λ)x = (λ0 − λ)F (λ)x + x for every x ∈ X. Hence, if x is solution of the resolvent equation, then x = F (λ)y. Let us check that x = F (λ)y is actually a solution. In fact, (λ − λ0 )F (λ)y + F (λ0 )−1 F (λ)y = y and, therefore, λ belongs to ρ(A) and the equality R(λ, A) = F (λ) holds. 

A.5

A Few Basic Notions from Interpolation Theory

To define the interpolation space (X, Y )α,∞ between two Banach spaces X and Y , with Y ,→ X, we introduce the function K : (0, 1] × X → [0, ∞), defined by K(t, x, X, Y ) = inf{||z||X + t||y||Y : t ∈ (0, 1], x = z + y, z ∈ X, y ∈ Y }. When there is no risk of confusion, we simply write K(t, x) instead of K(t, x, X, Y ). Definition A.5.1 For α ∈ (0, 1) the space (X, Y )α,∞ is the set of all x ∈ X such that [x](X,Y )α,∞ = sup t−α K(t, x) < ∞. t∈(0,1]

It is a Banach space when endowed with the norm ||x||(X,Y )α,∞ = ||x|| + [x](X,Y )α,∞ . Remark A.5.2 Note that the function t 7→ t−α K(t, x) is bounded on (0, 1] if and only if

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427

it is bounded on (0, ∞). Indeed, if t > 1, then K(t, x) ≤ ||x||. Therefore, t−α K(t, x) ≤ ||x||. It thus follows that   −α −α −α sup t K(t, x) ≤ max sup t K(t, x), sup t K(t, x) t>0

t∈(0,1]

t>1

≤ max{[x](X,Y )α,∞ , ||x||} ≤ ||x||(X,Y )α,∞ . In particular, the norm x 7→ ||x|| + sup t−α K(t, x) on (X, Y )α,∞ is equivalent to the norm t>0

|| · ||(X,Y )θ,∞ . Example A.5.3 From the very definition of the interpolation spaces and the characterization of the H¨ older spaces in Remark 5.3.7, it follows immediately that (Cbk (Rd ; K); Cbk+2 (Rd ; K))θ,∞ = Cbk+2θ (Rd ; K) for every θ ∈ (0, 1) \ {1/2} and k ∈ N ∪ {0}, with equivalence of the corresponding norms. Indeed, fix k ∈ N ∪ {0}, u ∈ (Cbk (Rd ; K); Cbk+2 (Rd ; K))θ,∞ and split it as u = ψ1 + ψ2 , where ψ1 ∈ Cbk (Rd ; K) and ψ2 ∈ Cbk+2 (Rd ; K). Note that for every multi-index β with length k and x, y ∈ Rd arbitrarily fixed, we can estimate |Dβ ψ2 (x + y) − 2Dβ ψ2 (x) + Dβ ψ2 (x − y)| Z 1 ≤ |h∇Dβ ψ2 (x + sy) − ∇Dβ ψ2 (x − sy), yi|ds 0

≤C||ψ2 ||C k+2 (Rd ;K) |y|2 , b

and, hence, |Dβ ψ2 (x) − 2Dβ ψ2 ((x + y)/2) + Dβ ψ2 (y)| ≤ C1 ||ψ2 ||C k+2 (Rd ;K) |x − y|2 , where b

C1 = C/4. Moreover, |Dβ ψ1 (x) − 2Dβ ψ1 ((x + y)/2) + Dβ ψ1 (y)| ≤ 4||Dβ ψ1 ||∞ . Therefore, |Dβ u(x) − 2Dβ u((x + y)/2) + Dβ u(y)| ≤ (4 ∨ C1 )(||ψ1 ||Cbk (Rd ;K) + |x − y|2 ||ψ2 ||C k+2 (Rd ;K) ). b

Minimizing over all the splittings of u into the sum of a function in Cbk (Rd ; K) and a function in Cbk+2 (Rd ; K), we conclude that |Dβ u(x) − 2Dβ u((x + y)/2) + Dβ u(y)| ≤(4 ∨ C1 )K(|x − y|2 , u) ≤(4 ∨ C1 )[u](C k (Rd ;K);C k+2 (Rd ;K))θ,∞ |x − y|2θ b

b

d

for every x, y ∈ R . Since, clearly, ||u||Cbk (Rd ;K) ≤ [u](C k (Rd ;K);C k+2 (Rd ;K))θ,∞ , from the b

b

previous chain of inequalities and Remark 5.3.7, we infer that u ∈ Cbk+2θ (Rd ; K) and ||u||C k+2θ (Rd ;K) ≤ C2 ||u||(C k (Rd ;K);C k+2 (Rd ;K))θ,∞ for some positive constant C2 , independent b b b of u as all the other constants Cj that appear in the proof (which are also independent of t). The embedding (Cbk (Rd ; K); Cbk+2 (Rd ; K))θ,∞ ,→ Cbk+2θ (Rd ; K) follows. To prove the other embedding, we fix u ∈ Cbk+2θ (Rd ; K) and a function % ∈ Cc∞ (Rd ) which is even with respect to all its variables and satisfies the conditions 0 ≤ % ≤ 1 and R %(x) dx = 1. For every t ∈ (0, ∞), we denote by ψt the convolution of u with the d R function %t , which is defined by %t (x) = t−d %(t−1 x) for every x ∈ Rd . Function ψt belongs to Cb∞ (Rd ; K) and ||ψt ||Cbk (Rd ;K) ≤ C3 ||u||Cbk (Rd ;K) . Moreover, if β is a multi-index with length greater than k, then we can split it into the sum of two multi-index β1 and β2 with |β1 | = k. Then, Z Dβ ψt (x) = tk−|β| Dβ2 %(y)Dβ1 u(x + ty)dy Rd

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Basic Notions of Functional Analysis in Banach Spaces

for every x ∈ Rd and every nontrivial multi-index β ∈ (N ∪ {0})d . Since % is even with respect to all its variables and, clearly, the integral of Dβ % over Rd vanishes, we can split Z Z 1 1 Dβ2 %(y)Dβ1 u(x + ty)dy + |β | Dβ2 %(y)Dβ1 u(x − ty) dy Dβ ψt (x) = |β|−k 2t 2t 2 Rd Rd Z 1 = |β|−k Dβ2 %(y)[Dβ1 u(x + ty) − 2Dβ1 u(x) + Dβ1 u(x − ty)] dy 2t Rd for every x ∈ Rd , so that ||Dβ ψt ||∞ ≤ C4 t2θ+k−|β| [Dβ1 u]Cb2θ (Rd ) . As a byproduct we infer that ||ψt ||C k+2 (Rd ;K) ≤ C5 t2θ−2 ||u||C k+2θ (Rd ;K) for t ∈ (0, 1]. Arguing in the same way, we can b

b

easily show that ||u − ψt ||Cbk (Rd ;K) ≤ C5 t2θ ||u||C k+2θ (Rd ;K) for the same values of t. Summing b up, we have proved that K(t, u) ≤ ||u − ψt1/2 ||Cbk (Rd ;K) + t||ψt1/2 ||C k+2 (Rd ;K) ≤ (C4 ∨ C5 )tθ ||u||C k+2θ (Rd ) b

for every t ∈ (0, 1], so that u belongs to

b

(Cbk (Rd ; K); C k+2 (Rd ; K))θ,∞

and

[u](C k (Rd ;K);C k+2 (Rd ;K))θ,∞ ≤ (C4 ∨ C5 )||u||C k+2θ (Rd ;K) . b

The embedding

Cb2θ (Rd ; K)

b

b

d

2

d

,→ (Cb (R ; K); C (R ; K))θ,∞ follows.

Example A.5.4 Using the extension operator Ek+2 in Proposition B.4.1, it can be shown that (Cbk (Ω; K); Cbk+2 (Ω; K))θ,∞ = Cbk+2θ (Ω; K) with equivalence of the corresponding norms, for every k ∈ N ∪ {0}, when Ω is Rd+ or a bounded open set of class C k+2 and θ ∈ (0, 1) \ {1/2}. Indeed, the operator Ek+2 is bounded from Cb (Ω; K) into Cb (Rd ; K) and from Cbβ (Ω; K) into Cbβ (Rd ; K) for β ∈ {k, k + 2θ, k + 2}. Thus, applying the forthcoming Proposition A.5.6, we infer that Ek+2 is bounded from (Cbk (Ω; K), Cbk+2 (Ω; K))θ,∞ into Cbk+2θ (Rd ; K). Hence, the inclusion (Cbk (Ω; K), Cbk+2 (Ω; K))θ,∞ ,→ C k+2θ (Ω; K) follows. Vice versa, fix ψ ∈ Cbk+2θ (Ω; K). Then, the function Ek+2 ψ belongs to Cbk+2θ (Rd ; K) = (Cbk (Rd ; K); Cbk+2 (Rd ; K))θ,∞ and thus, for every t > 0, we can determine two functions ϕ1,t ∈ Cbk (Rd ; K) and ϕ2,t ∈ Cbk+2 (Rd ; K) such that Ek+2 ψ = ϕ1,t + ϕ2,t and t−θ (||ϕ1,t ||C k (Ω;K) + t||ϕ2,t ||C k+2 (Ω;K) ) ≤t−θ (||ϕ1,t ||Cbk (Rd ;K) + t||ϕ2,t ||C k+2 (Rd ;K) ) b

b

b

≤2||Ek+2 ψ||(C k (Rd ;K),C k+2 (Rd ;K))θ,∞ b

b

≤C||Ek+2 ψ||C k+2θ (Rd ;K) b

≤C||ψ||C k+2θ (Ω;K) . b

This chain of inequalities shows that ψ belongs to (Cbk (Ω; K); Cbk+2 (Ω; K))θ,∞ and the inclusion Cbk+2θ (Ω; K) ,→ (Cbk (Ω; K); Cbk+2 (Ω; K))θ,∞ follows. Proposition A.5.5 Let X and Y two Banach spaces, with Y ,→ X. Then, the following properties are satisfied. (i) Y ,→ (X, Y )α,∞ ,→ X for every α ∈ (0, 1). (ii) (X, Y )α2 ,∞ ,→ (X, Y )α1 ,∞ for every α1 , α2 ∈ (0, 1), with α1 < α2 . Proof (i) Fix α ∈ (0, 1). The embedding (X, Y )α,∞ ,→ X follows immediately from the definition of the interpolation space (X, Y )α,∞ . Similarly, if x ∈ Y , then K(t, x) ≤ t||x||Y for every t ∈ (0, 1]. Consequently, [x](X,Y )α,∞ ≤ ||x||Y and the other embedding follows immediately. (ii) Fix α1 and α2 as in the statement and x ∈ (X, Y )α2 ,∞ . Observing that t−α1 K(t, x) ≤ −α2 t K(t, x) for every t ∈ (0, 1], we get immediately the assertion. 

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429

Proposition A.5.6 Let X1 , X2 , Y1 and Y2 be four Banach spaces with Y1 ,→ X1 and Y2 ,→ X2 . Further, let T : X1 → X2 be a linear bounded operator such that its restriction to Y1 belongs to L(Y1 , Y2 ). Then, for every α ∈ (0, 1), T is bounded from (X1 , Y1 )α,∞ into (X2 , Y2 )α,∞ and 1−α ||T ||L((X1 ,Y1 )α,∞ ,(X2 ,Y2 )α,∞ ) ≤ ||T ||α L(Y1 ,Y2 ) ||T ||L(X1 ,X2 ) .

Proof We assume that T 6≡ 0, fix t ∈ (0, 1] and split x ∈ (X1 , Y1 )α,∞ into the sum x = z + y, where z ∈ X1 and y ∈ Y1 . Then, T x = T z + T y, where T z ∈ X2 and T y ∈ Y2 . Moreover, K(t, T x, X2 , Y2 ) ≤ ||T z||X2 + t||T y||Y2 ≤ ||T ||L(X1 ,X2 ) ||z||X1 + t||T ||L(Y1 ,Y2 ) ||y||Y1 . Minimizing with respect to every splitting of x into the sum of an element of X1 and an element of Y1 , we get K(t, T x, X2 , Y2 ) ≤ ||T ||L(X1 ,X2 ) K(||T ||−1 L(X1 ,X2 ) ||T ||L(Y1 ,Y2 ) t, x, X1 , Y1 ). It thus follows that α −α t−α K(t, T x, X2 , Y2 ) ≤||T ||1−α K(s, x, X1 , Y1 ) L(X1 ,X2 ) ||T ||L(Y1 ,Y2 ) s α ≤||T ||1−α L(X1 ,X2 ) ||T ||L(Y1 ,Y2 ) [x](X1 ,Y1 )α,∞ ,

where s = ||T ||−1 L(X1 ,X2 ) ||T ||L(Y1 ,Y2 ) t. The assertion follows taking the supremum of both the sides of the previous inequality with respect to t ∈ (0, ∞) and taking Remark A.5.2 into account.  Remark A.5.7 Applying Proposition A.5.6 to the operator T : R → X defined by T s = sx for every s ∈ R and x ∈ X, we easily infer that α ||x||(X,Y )α,∞ ≤ ||x||1−α X ||x||Y ,

x ∈ Y.

The interpolation space (X, Y )α,∞ can be defined in an equivalent way through the so-called trace-method. Theorem A.5.8 Let X and Y be two Banach spaces with Y ,→ X. Then, for every α ∈ (0, 1), x belongs to (X, Y )α,∞ if and only if there exists a function u ∈ C([0, ∞); X) ∩ C 1 ((0, ∞); Y ) such that u(0) = x and   sup t1−α ||u(t)||Y + ||u0 (t)||X + t||u0 (t)||Y < ∞. t>0

Moreover, denoting by ||x||(X,Y )α,∞ the infimum of the sum of the sup-norm of the functions t 7→ t1−α ||u0 (t)||X and t 7→ t1−α ||u(t)||Y over all the functions u, we define a norm on (X, Y )α,∞ (say || · ||∗(X,Y )α,∞ ), which is equivalent to the norm || · ||(X,Y )α,∞ . Proof Let u ∈ C([0, ∞); X) ∩ C 1 ((0, ∞); Y ) be a function satisfying the estimate t (||u0 (t)||X + ||u(t)||Y ) ≤ C1 for every t > 0 and some positive constant C1 , independent of t. Let us prove that x = u(0) belongs to (X, Y )α,∞ . For this purpose, we apply Theorem A.2.6 in the interval [ε, t] and then let ε tend to 0+ to infer that 1−α

Z x = u(t) −

t

u0 (s) ds,

t > 0.

0

By assumption u(t) ∈ Y for every t > 0; moreover,

Z t

Z t

0

≤ u (s) ds ||u0 (s)||X ds ≤ C2 tα sup ||s1−α u0 (s)||X ,

0

X

0

s>0

t > 0,

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Basic Notions of Functional Analysis in Banach Spaces

for some positive constant C2 , independent of t. Therefore,

  Z t

0 −α −α

u (s) ds + t||u(t)||Y t K(t, x, X, Y ) ≤t

0 X   ≤C3 sup ||s1−α u0 (s)||X + sup ||s1−α u(s)||Y , s>0

s>0

where the constant C3 is independent of t. It thus follows that x ∈ (X, Y )α,∞ and   [x](X,Y )α,∞ ≤ C3 sup ||s1−α u0 (s)||X + sup ||s1−α u(s)||Y . s>0

s>0

Minimizing with respect to u, we conclude that [x](X,Y )α,∞ ≤ C3 || · ||∗(X,Y )α,∞ . Vice versa, let us suppose that x ∈ (X, Y )α,∞ . For every n ∈ N, we fix x0n ∈ X and yn ∈ Y such that ||x0n ||X + n−1 ||yn ||Y ≤ 2K(n−1 , x, X, Y ). Note that x0n tends to zero as n tends to ∞ since the function K(·, x, X, Y ) vanishes as t tends to 0+ . Let ψ : [0, ∞) → X be the function defined by Z t ∞ 1X yn+1 t ∈ (0, ∞). χ( 1 , 1 ] (s) ds, ψ(t) = n+1 n t n=1 0 Clearly,

Z t ∞ 1X 0 x x − ψ(t) = χ 1 1 (s) ds, t n=1 n+1 0 ( n+1 , n ]

t ∈ (0, 1],

(A.5.1)

and from this formula we infer that ψ(t) converges to x in X as t tends to 0. Indeed, a straightforward computation shows that ||x − ψ(t)||X ≤

1 X ||x0n+1 ||X + ||x0n0 +1 ||X ≤ sup ||x0n+1 ||X + ||x0n0 +1 ||X , t n(n + 1) n≥t−1 −1 n≥t

where n0 is the unique integer in the interval (t−1 − 1, t−1 ]. Since x0n vanishes as n tends to ∞, the right-hand side of the previous inequality converges to zero as n tends to ∞. Function ψ is also continuous in (0, ∞) with values in Y (and, hence, with values in X). Indeed, for every t0 > 0 the series defining ψ(t) consists of a finite number of terms if t belongs to a sufficiently small neighborhood of t0 , and all these terms are continuous functions on (0, ∞) with values in Y . Since ||yn ||Y ≤ 2nK(n−1 , x, X, Y ) ≤ 2n1−α ||x||(X,Y )α,∞ for every n ∈ N, we can estimate Z t ∞ X t1−α ||ψ(t)||Y ≤2t−α ||x||(X,Y )α,∞ (n + 1)1−α χ( 1 , 1 ] (s) ds ∞ Z t X

≤22−α t−α ||x||(X,Y )α,∞

n=1 0 Z t∧1

=22−α t−α ||x||(X,Y )α,∞

n+1 n

0

n=1

sα−1 χ(

1 1 n+1 , n

] (s) ds

sα−1 ds

0

≤22−α α−1 ||x||(X,Y )α,∞

(A.5.2)

for every t ∈ (0, ∞). Arguing similarly and using (A.5.1), we can show that Z t ∞ X t−α ||ψ(t) − x||X ≤2t−1−α ||x||(X,Y )α,∞ (n + 1)−α χ( 1 , 1 ] (s) ds n=1

0

n+1 n

Semigroups of Bounded Operators and Second-Order PDE’s ∞ Z t X ≤2t−1−α ||x||(X,Y )α,∞ sα χ( 1 , 1 ] (s) ds ≤2t−1−α ||x||(X,Y )α,∞

n=1 Z t

0

431

n+1 n

sα ds

0

≤2(α + 1)−1 ||x||(X,Y )α,∞

(A.5.3)

for every t ∈ (0, 1]. Rt We now define the function u : [0, ∞) → X by setting u(t) = t−1 0 ψ(s) ds for every t > 0. Since ψ ∈ C([0, ∞); X) ∩ C 1 ((0, ∞); Y ), u is continuous in [0, ∞) with values in X and differentiable in (0, ∞) with values in Y . Moreover, u(0) = ψ(0) = x. From (A.5.2) we deduce that Z t Z t 1−α −α 2−α −1 −α t ||ψ(s)||Y ds ≤ 2 sα−1 ds ||u(t)||Y ≤t α ||x||(X,Y )α,∞ t 0

0

=22−α α−2 ||x||(X,Y )α,∞ . Since Y ,→ X, it follows that ||u(t)||X ≤ C4 tα−1 ||x||(X,Y )α,∞ for every t > 0 and some positive constant C4 , which is independent of t and x as the other forthcoming constants. Next, we observe that u is differentiable in (0, ∞) with values in X and 1 u (t) = − 2 t 0

Z 0

t

1 1 ψ(s) ds + ψ(t) = − 2 t t

Z 0

t

1 (ψ(s) − x) ds + (ψ(t) − x) t

for every t ∈ (0, ∞). Hence, using (A.5.3) we conclude that t1−α ||u0 (t)||X ≤ C5 ||x||(X,Y )α,∞ for every t ∈ (0, 1]. Finally, using estimate (A.5.2) we can also show that t2−α ||u0 (t)||Y ≤ C6 ||x||(X,Y )α,∞ for every t > 0. Using again the embedding Y ,→ X, we complete the proof.  Corollary A.5.9 Let X and Y be two Banach spaces, with Y ,→ X, and α ∈ (0, 1). Then, (X, Y )α,∞ is contained in the closure of Y into X. Proof Fix x ∈ (X, Y )α,∞ . By Theorem A.5.8, x = limt→0 u(t), where the limit is meant in X and u is a continuous function in (0, ∞) with values in Y . Hence, x belongs to the closure of Y in X.  Remark A.5.10 We stress that the first part of the proof of Theorem A.5.8 shows that, if x is the value at zero of a function u ∈ C([0, ∞); X) ∩ C((0, ∞); Y ) ∩ C 1 ((0, ∞); X) such that supt>0 (t1−α ||u0 (t)||X + t||u(t)||Y ) < ∞, then x ∈ (X, Y )α,∞ and ||x||(X,Y )α,∞ ≤ C sup(t1−α ||u0 (t)||X + ||u(t)||Y ) t>0

for some positive constant C, independent of x and u. Example A.5.11 Using Theorem A.5.8 and Example A.5.4, we can show that (Cb (Ω; K), C∗2 (Ω; K))θ,∞ = (Cb (Ω; K), C∗2 (Ω; K))θ,∞ = C∗2θ (Ω; K),

(A.5.4)

with equivalence of the corresponding norms, for every θ ∈ (0, 1) \ {1/2}. Here, Ω is Rd+ or a bounded open set of class C 2 and C∗β (Ω; K) := Cbβ (Ω; K) ∩ C0 (Ω; K) for β ∈ (0, 2), is endowed with the norm of Cbβ (Ω; K). We begin by proving the second equality in (A.5.4). Fix f ∈ (Cb (Ω; K), C∗2 (Ω; K))θ,∞ .

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Since C∗2 (Ω; K) is contained in Cb2 (Ω; K) and these two spaces are endowed with the same norm, it follows easily that K(t, f, Cb (Ω; K); C∗2 (Ω; K)) ≥ K(t, f, Cb (Ω; K); Cb2 (Ω; K)) for every t ∈ (0, 1] and this implies that (Cb (Ω; K), C∗2 (Ω; K))θ,∞ ,→ (Cb (Ω; K), Cb2 (Ω; K))θ,∞ = Cb2θ (Ω; K), where the last equality follows from Example A.5.4. Corollary A.5.9 shows that f vanishes on ∂Ω. To prove the other inclusion, we fix f ∈ C∗2θ (Ω; K) ⊂ Cb2θ (Ω; K). By Theorem A.5.8, there exists a function u ∈ C([0, ∞); Cb (Ω; K)) ∩ C 1 ((0, ∞); Cb2 (Ω; K)) such that   sup t1−θ ||u(t)||C 2 (Ω;K) + ||u0 (t)||∞ + t||u0 (t)||C 2 (Ω;K) < ∞ b

t>0

b

and t1−θ (||u0 (t)||∞ + ||u(t)||C 2 (Ω;K) ) ≤ C1 ||f ||C∗2θ (Ω;K) for every t > 0 and a positive C1 > 0. b We now introduce the function v, defined by v(t) = u(t) − E20 (u(t)|∂Ω ) for every t > 0, where E20 is the extension operator in Proposition B.4.9. Clearly, v(t) vanishes on ∂Ω for every t > 0 and v 0 (t) = u0 (t) − E20 (u0 (t)|∂Ω ) for every t > 0, since E20 is a bounded operator. Hence, v ∈ C([0, ∞); Cb (Ω; K)) ∩ C 1 ((0, ∞); C∗2 (Ω; K)) and it is compactly supported in [0, 1]. Moreover, ||v(t)||C 2 (Ω;K) ≤ C||u(t)||C 2 (Ω;K) for every t > 0 and some positive constant b b C2 , independent of t and u. Hence,   sup t1−θ ||v(t)||C 2 (Ω;K) + ||v 0 (t)||∞ + t||v 0 (t)||C 2 (Ω;K) < ∞ b

b

t>0

and t1−θ (||v 0 (t)||∞ + ||v(t)||C 2 (Ω;K) ) ≤ C3 ||f ||C∗2θ (Ω;K) for every t > 0 and a positive constant b

C3 , independent of t. Clearly, v(0) = f . It follows that f ∈ (Cb (Ω; K), C∗2 (Ω; K))θ,∞ and ||f ||(Cb (Ω;K),C 2 (Ω;K))θ,∞ ≤ C3 ||f ||C 2θ (Ω;K) . b

∗

Finally, we show that (Cb (Ω; K); C∗2 (Ω; K))θ,∞ = (Cb (Ω; K); C∗2 (Ω; K))θ,∞ , with equivalence of the corresponding norms. Arguing as in the first part of the proof, the embedding “,→” can be easily proved. To prove the other embedding, we observe that Corollary A.5.9 shows that (Cb (Ω; K); C∗2 (Ω; K))θ,∞ is contained in the closure of C∗2 (Ω; K), with respect to the sup-norm. Let f ∈ (Cb (Ω; K); C∗2 (Ω; K))θ,∞ . Hence, f ∈ Cb (Ω; K). As a consequence, it follows that, if f = ψ1 + ψ2 , with ψ1 ∈ Cb (Ω; K) and ψ2 ∈ C∗2 (Ω; K), then actually ψ1 ∈ Cb (Ω; K). Hence, K(t, f, Cb (Ω; K), C∗2 (Ω; K)) ≤ K(t, f, Cb (Ω; K), C∗2 (Ω; K)) for every t ∈ (0, 1] and the embedding (Cb (Ω; K); C∗2 (Ω; K))θ,∞ ,→ (Cb (Ω; K); C∗2 (Ω; K))θ,∞ follows. Based on the previous theorem, we can state and prove the famous reiteration theorem. Theorem A.5.12 Let X, Y , Z1 and Z2 be four Banach spaces, with Y ,→ Z2 ,→ Z1 ,→ X, such that i i ||x||Zi ≤ Ci ||x||1−α ||x||α Zi ,→ (X, Y )αi ,∞ (A.5.5) X Y , for every x ∈ Y , i = 1, 2, some positive constants C1 , C2 and 0 ≤ α1 < α2 ≤ 1. Then, (Z1 , Z2 )θ,∞ = (X, Y )(1−θ)α1 +θα2 ,∞ , with equivalence of the corresponding norms. Proof To begin with, we prove that (Z1 , Z2 )θ,∞ ,→ (X, Y )(1−θ)α1 +θα2 ,∞ . For this purpose, we fix x ∈ (Z1 , Z2 )θ,∞ and split into the sum z1 + z1 , where z1 ∈ Z1 and z2 ∈ Z2 . As it is easily seen, the function K(t, ·, X, Y ) is sublinear for every t > 0, so that K(t, x, X, Y ) ≤ K(t, z1 , X, Y ) + K(t, z2 , X, Y ). Since Zi (i = 1, 2) satisfies the secei ||zi ||Z for every t ∈ (0, 1], ond condition in (A.5.5), it follows that t−αi K(t, zi , X, Y ) ≤ C i e e i = 1, 2 and some positive constants C1 and C2 , independent of t and zi . We can thus e1 tα1 ||z1 ||Z + C e2 tα2 ||z2 ||Z for every t ∈ (0, 1]. Minimizing with estimate K(t, x, X, Y ) ≤ C 1 2 respect to all the splittings of x into the sum of an element of Z1 and an element of Z2 , we

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e1 ∨ C e2 . deduce that K(t, x, X, Y ) ≤ C3 tα1 K(tα2 −α1 , x, Z1 , Z2 ) for t ∈ (0, 1], where C3 = C It follows that the function t 7→ t−(1−θ)α1 −θα2 K(t, x, X, Y ) is bounded on (0, ∞) and sup t−(1−θ)α1 −θα2 K(t, x, X, Y ) ≤ C3 sup t−θ(α2 −α1 ) K(tα2 −α1 , x, Z1 , Z2 ). t∈(0,1]

t∈(0,1]

The embedding (Z1 , Z2 )θ,∞ ,→ (X, Y )(1−θ)α1 +θα2 ,∞ follows. To prove the other embedding, we fix x ∈ (X, Y )(1−θ)α1 +θα2 ,∞ . By Theorem A.5.8 there exists a function v ∈ C([0, ∞); X) ∩ C 1 ((0, ∞); Y ) such that v(0) = x and t1−(1−θ)α1 −θα2 (||v 0 (t)||X + t||v 0 (t)||Y ) ≤ C||x||(X,Y )(1−θ)α1 +θα2 ,∞ ,

t ∈ (0, 1].

Since Z1 satisfies the first condition in (A.5.5), from the previous estimate we infer that 1−α1 2−(1−θ)α1 −θα2 0 1 v (t)||α ||t ||v 0 (t)||Z1 ≤C1 tθ(α2 −α1 )−1 ||t1−(1−θ)α1 −θα2 v 0 (t)||X Y

≤C4 tθ(α2 −α1 )−1 ||x||(X,Y )(1−θ)α1 +θα2 ,∞ ,

(A.5.6)

where C4 , as all the other forthcoming estimates, are independent of t and x. Arguing similarly and using the fact that Z2 satisfies the first condition in (A.5.5), we can show that ||v 0 (t)||Z2 ≤ C5 t−1−(1−θ)(α2 −α1 ) ||x||(X,Y )(1−θ)α1 +θα2 ,∞ . From (A.5.6) and (A.5.7) it follows that Z ∞ v(t) = − v 0 (s) ds,

(A.5.7)

t > 0,

t

where the integral converges in Z2 . In particular, t(1−θ)(α2 −α1 ) ||v(t)||Z2 ≤ C6 ||x||(X,Y )(1−θ)α1 +θα2 ,∞ ,

t > 0.

(A.5.8)

Moreover, since Z v(t) = x +

t

v 0 (s) ds,

t ≥ 0,

0

where the integral term converges in X, using estimate (A.5.6) we easily infer that the integral term in the previous formula converges also in Z1 and it vanishes as t tends to 0. This is enough to conclude that v(t) converges in x also in Z1 as t tends to 0+ . Let u : [0, ∞) → Z2 be the function defined by u(t) = v(t1/(α2 −α1 ) ) for every t ∈ [0, ∞). From the embedding Y ,→ Z2 , it follows that u ∈ C 1 ((0, ∞); Z2 ). Moreover, u is continuous up to t = 0 and u(0) = x Estimate (A.5.8) shows that t1−θ ||u(t)||Z2 ≤ C6 ||x||(X,Y )(1−θ)α1 +θα2 ,∞ ,

t > 0.

(A.5.9)

Moreover, a straightforward computation reveals that t1−θ u0 (t) =

1−θ(α2 −α1 ) 1 1 t α2 −α1 v 0 (t α2 −α1 ), α2 − α1

t > 0,

so that, from (A.5.6) it follows that ||t1−θ u0 (t)||Z1 ≤ C7 ||x||(X,Y )(1−θ)α1 +θα2 ,∞ ,

t > 0.

(A.5.10)

From Remark A.5.10 and estimates (A.5.9), (A.5.10), we conclude that x ∈ (Z1 , Z2 )θ,∞ and ||x||(Z1 ,Z2 )θ,∞ ≤ C8 ||x||(X,Y )(1−θ)α1 +θα2 ,∞ . We have so proved that (X, Y )(1−θ)α1 +θα2 ,∞ is continuously embedded into (Z1 , Z2 )θ,∞ . 

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Example A.5.13 As an application of the reiteration theorem, we can show that (2+α)θ (Ω; K) for every α, θ ∈ (0, 1) such that (2 + α)θ 6∈ N (Cb (Ω; K); C 2+α (Ω; K))θ = Cb when Ω is Rd or Rd+ or a bounded open set of class C 2+α . We begin by considering the case Ω = Rd . The main steps consists in proving that (Cb (Rd ; K); Cb3 (Rd ; K))γ,∞ = Cb3γ (Rd ; K), with equivalence of the corresponding norms, for every γ ∈ (0, 1) \ {1/3, 2/3}. Once this property is proved, it suffices to apply the reiteration theorem to write (Cb (Rd ; K), Cb2+α (Rd ; K))θ,∞ =(Cb (Rd ; K), (Cb (Rd ; K), Cb3 (Rd ; K)) 2+α ,∞ )θ,∞ 3

=(Cb (Rd ; K), Cb3 (Rd ; K)) (2+α)θ ,∞ 3

(2+α)θ =Cb (Rd ; K),

with equivalence of the corresponding norms. So, let us prove that (Cb (Rd ; K); Cb3 (Rd ; K))γ,∞ = Cb3γ (Rd ; K). To fix the ideas, we assume that γ ∈ (1/3, 1) \ {2/3}. The case γ < 1/3 is completely similar and, thus, left to the reader. We first prove that 2

1

Cb1 (Rd ; K) ,→ (Cb (Rd ; K), Cb3 (Rd ; K))1/3,∞ (A.5.11) for every f ∈ Cb3 (Rd ; K). The first property follows immediately from Proposition 1.1.4. To prove the other property, we fix g ∈ Cb1 (Rd ; K) and, for every t > 0, we consider the convolution gt of g with the function %t , defined by %t (x) = t−d %(t−1 x) for every x ∈ Rd , where % ∈ Cc∞ (Rd ) is an even function, with respect to all its entries such that 0 ≤ % ≤ 1 and R %(x) dx = 1. Function gt belongs to Cb3 (Rd ; K). Moreover, arguing as in Example A.5.3, Rd it can be easily checked that ||g − gt ||∞ ≤ C1 t||g||Cb1 (Rd ;K) and t2 ||gt ||Cb3 (Rd ;K) ≤ C2 ||g||Cb1 (Rd ;K) for some positive constants C1 and C2 , independent of g and t ∈ (0, 1]. Thus, 3 ||f ||C3 3 (Rd ;K) , ||f ||Cb1 (Rd ;K) ≤ C||f ||∞ b

1

K(t, g, Cb (Rd ; K), Cb3 (Rd ; K)) ≤||g − gt1/3 ||∞ + t||gt1/3 ||Cb3 (Rd ;K) ≤ 2C3 t 3 ||g||Cb1 (Rd ;K) for every t ∈ (0, 1] and some positive constant C3 , independent of t and g. The embedding in (A.5.11) follows. Applying the reiteration theorem, we conclude that (Cb (Rd ; K); Cb3 (Rd ; K))γ,∞ = (Cb1 (Rd ; K); Cb3 (Rd ; K)) 3γ−1 ,∞ = Cb3γ (Rd ; K), 2

with equivalence of the corresponding norms, thanks to Example A.5.3. When Ω is Rd+ or a bounded open set of class C 2+α , it suffices to repeat the same arguments as in Example A.5.4. Example A.5.14 Here, we characterize other interpolation spaces. More precisely, we prove that, for every θ ∈ (0, 1), it holds that (Cbk (Ω; K), Cbk+1 (Ω; K))θ,∞ = (Cbk (Ω; K), Cbk+1 (Ω; K))θ,∞ = Cbk+θ (Ω; K), (CB1 (Ω; K), CB2 (Ω; K))θ,∞ = (CB1 (Ω; K), CB2 (Ω; K))θ,∞ = CB1+θ (Ω; K),

k = 0, 1, (A.5.12) (A.5.13)

with equivalence of the corresponding norms. Here, Ω is either Rd+ or a bounded open set of class C 2 and CBk (Ω; K) is the set of functions u ∈ Cbk (Ω; K) such that Bu vanishes on ∂Ω. ∂ Here, B is the boundary differential operator, defined by B = a + , whose coefficients ∂η 1 1 d belong to Cb (∂Ω) and Cb (∂Ω; R ), respectively. Moreover, inf ∂Ω hη, νi > 0, where ν(x) denotes the unit exterior normal vector to ∂Ω at x.

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435

To prove (A.5.12), with k = 0, we first fix f ∈ (Cb (Ω; K), Cb1 (Ω; K))θ,∞ and split f = ψ1 + ψ2 , with ψ1 ∈ Cb (Ω; K) and ψ2 ∈ Cb1 (Ω; K). Then, for every x, y ∈ Ω, we can estimate |f (x) − f (y)| ≤ |ψ1 (x) − ψ1 (y)| + |ψ2 (x) − ψ2 (y)| ≤ ||ψ1 ||∞ + |x − y|||ψ2 ||C 1 (Ω;C) . b

The arbitrariness of ψ1 and ψ2 yields |f (x) − f (y)| ≤ K(|x − y|, f, Cb (Ω; K), Cb1 (Ω; K)) = K(|x − y|, f, Cb (Ω; K), Cb1 (Ω; K)). Hence, function f belongs to Cbθ (Ω; K) and [f ]Cbθ (Ω;K) ≤ [f ](Cb (Ω;K),C 1 (Ω;K))θ,∞ = [f ](Cb (Ω;K),C 1 (Ω;K))θ,∞ . b b Vice versa, suppose that f ∈ Cbθ (Ω; K) and, for every t > 0, set Z ft (x) = %(y)(E2 f )(x − ty) dy, x ∈ Rd , Rd

where E2 is the extension operator in Proposition B.4.1, which is bounded from Cbθ (Ω; K) into Cbθ (Rd ; K), and % ∈ Cc∞ (Rd ) has L1 (Rd )-norm equal to one. As it is easily seen, Z Z |y|θ %(y) dy %(y)[(E2 f )(x − ty) − (Eθ f )(x)] dy ≤ [E2 f ]Cbθ (Rd ;K) tθ |ft (x) − f (x)| = Rd

Rd

for every x ∈ Ω. Moreover, Z Z Dj ft (x) = t−1 Dj %(y)(E2 f )(x − ty) dy = t−1 Rd

Dj %(y)[(E2 f )(x − ty) − (E2 f )(x)] dy

Rd

for every x ∈ Ω and j = 1, . . . , d. Hence, arguing as above, we deduce that Z |Dj ft (x)| ≤ tθ−1 |Dj %(y)||y|θ dy, x ∈ Ω. Rd

Clearly, ||ft ||∞ ≤ ||E2 f ||∞ ≤ C1 ||f ||∞ for some positive constant C1 , independent of f and t. Therefore, K(t, f, Cb (Ω; K); Cb1 (Ω; K)) ≤ ||f − ft ||∞ + t||ft ||C 1 (Ω;K) ≤ C2 tθ ||f ||Cbθ (Ω;K) ,

t > 0.

b

This shows that f belongs to (Cb (Ω; K), Cb1 (Ω; K))θ,∞ and ||f ||(Cb (Ω;K),C 1 (Ω;K))θ,∞ is bounded b from above by (1 + C2 )||f ||Cbθ (Ω;K) . The proof of (A.5.12), with k = 1, can be obtained arguing in the same way. Indeed, |Dj f (x) − Dj f (y)| ≤ 2||Dj ψ1 ||∞ + |x − y|||ψ2 ||Cb2 (Rd ;C) ,

x, y ∈ Rd+ , j = 1, . . . , d.

Moreover, if f ∈ Cb1+θ (Ω; K), then we define the function ft as above. Since Z Dj ft (x) = %(y)(Dj E2 f )(x − ty) dy, x ∈ Rd+ , Rd

the same computations can be used to show that ||ft − f ||Cb1 (Rd+ ;K) ≤ C3 ||f ||Cbθ (Ω;K) and ||ft ||Cb2 (Rd+ ;K) ≤ C3 t−1 ||f ||C 1+θ (Ω;K) for every t > 0 and some positive constant C3 , indepenb dent of t and f . Finally, to prove formula (A.5.13), we argue as in Example A.5.11. The embedding (CB1 (Ω; K); CB2 (Ω; K))θ,∞ ,→ CB1+θ (Ω; K) can be proved with no particular differences, using estimate (A.5.12), with k = 1. Also the proof of the other embedding follows the lines in the quoted example. The only difference is in the definition of function v, which is now given by v(t) = u(t) − E(Bu(t)) for every t ≥ 0, where E is the operator in Proposition B.4.14, if Ω is bounded, and the operator in Proposition B.4.12(i), if Ω = Rd+ , which satisfies the ∂Eψ = ψ on ∂Ω for every ψ ∈ Cb (∂Ω; K). condition ∂η

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Basic Notions of Functional Analysis in Banach Spaces

Finally, we consider the following interpolation result that we use in Chapter 8. For this purpose, for every β ∈ (0, 3) \ {1, 2}, every T > 0 and every domain Ω ⊂ Rd , we denote by Zβ (T, Ω; K) the set of all f ∈ C β/2,β ((0, T ) × Ω; K) such that f (0, ·) ≡ 0 if β < 2 and f (0, ·) = Dt f (0, ·) = 0, otherwise. Proposition A.5.15 Let Ω be Rd or Rd+ or a bounded open set of class C 2+α for some α ∈ (0, 1). Then, it holds that Z1+α (T, Ω; K) = (Cb ([0, T ] × Ω; K), Z2+α (T, Ω; K))θ,∞ ,

(A.5.14)

with equivalence of the corresponding norms. Here, θ = (2 + α)−1 (1 + α). (1+α)/2

Proof In the proof, we denote by C∗ ((0, T ); K) the set of functions u ∈ 1+α/2 C ((0, T ); K) such that u(0) = 0. Similarly, we denote by C∗∗ ((0, T ); K) the set of 1+α/2 0 functions u ∈ C ((0, T ); K) such that u(0) = u (0) = 0. Moreover, to lighten the notation, we simply write Zβ instead of Zβ (T, Ω; K), we set Z0 = Cb ([0, T ]×Ω; K) and denote by (1+α)/2,1+α 1+α/2,2+α || · ||1+α and || · ||2+α the norms of Cb ((0, T ) × Ω; K) and Cb ((0, T ) × Ω; K), respectively. Step 1. For each t ∈ [0, T ] we consider the operator Πt : Z0 → Cb (Ω; K), defined by Πt f = f (t, ·) for every f ∈ Z0 . This operator and its restriction to Z2+α , which maps this space into Cb2+α (Ω; K), are contractions. Thus, by Proposition A.5.6, Πt is a contraction mapping (Z0 , Z2+α )θ,∞ into (Cb (Ω; K); Cb2+α (Ω; K))θ,∞ . Fix f ∈ (Z0 , Z2+α )θ,∞ . Taking Example A.5.13 into account, it follows that f (t, ·) ∈ Cb1+α (Ω; K) and ||f (t, ·)||C 1+α (Rd ;K) ≤ b C1 ||f ||(Z0 ,Z2+α )θ,∞ for every f ∈ (Z0 , Z2+α )θ,∞ and some positive constant C1 , independent of t. Repeating the same argument with Πt replaced by the operator Π0x , defined by Π0x f = (1+α)/2 f (·, x) for every f ∈ Z0 and x ∈ Ω, we deduce that f (·, x) belongs to Cb ((0, T ); K) (1+α)/2 and its Cb ((0, T ); K)-norm can be bounded from above by a positive constant C2 , independent of f and x, times the norm of f in (Z0 , Z2+α )θ,∞ . So, to prove that f ∈ Z1+α , it suffices to show that f (0, ·) ≡ 0. This property follows from Corollary A.5.9. Step 2. We now prove the other inclusion in (A.5.14). For this purpose, we fix f ∈ Z1+α , a nonnegative function ϑ ∈ Cc∞ ([0, ∞) × Rd ), such that ||ϑ||L1 (Rd+1 ) = 1, ϑ(t, x) = ϑ(t, −x) for every (t, x) ∈ Rd+1 , and set Z fr (t, x) = ϑr (t − s, x − y)f (s, y) ds dy, (t, x) ∈ [0, T ] × Ω, (1+α)/2

Rd+1 −d−2

where ϑr (t, x) = r ϑ(r−2 t, r−1 x) for every (t, x) ∈ Rd+1 , r > 0 and f is the extension d+1 to R of the function f , defined as follows: if Ω = Rd , then we set f = 0 in (−∞, 0] × Rd and f = f (T, ·) in [T, ∞) × Rd , if Ω 6= Rd , we first extend the function f to [0, T ] × Rd using the extension operator E1+α and then extend this function to Rd+1 as illustrated (1+α)/2,1+α above. Clearly, f belongs to Cb (Rd+1 ; K) and its norm is bounded from above by a positive constant times the Z1+α -norm of f . If not otherwise specified, in what follows we assume that r is arbitrarily fixed in (0, ∞). Moreover, by C we denote a positive constant, which is independent of f and r and may vary from line to line. As it is immediately seen, function fr belongs to C ∞ ([0, T ] × Ω; K), and is bounded, together with all its derivatives. Since fr (0, ·) = Dt fr (0, ·) = 0, it follows that fr ∈ Z2+α . In particular, ||fr ||∞ ≤ C||f ||∞ . Moreover, we can write Z −1 Dij fr (t, x) = r Di ϑ(s, y)[Dj f (t − r2 s, x − ry) − Dj f (t, x)] ds dy, Rd+1

Dtk fr (t, x) = r−2k

Z Rd+1

Dt ϑ(s, y)[f (t − r2 s, x − ry) − f (t, x)] ds dy,

Semigroups of Bounded Operators and Second-Order PDE’s Z Dtk Dj fr (t, x) = r−2k Dt ϑ(s, y)[Dj f (t − r2 s, x − ry) − Dj f (t, x)] ds dy,

437

Rd+1

for every (t, x) ∈ [0, T ] × Ω, i, j ∈ {1, . . . , d} and k = 1, 2 since the integrals over Rd+1 of the derivatives of ϑ vanish. These formulas allow us to infer that r1−α ||Dij fr ||∞ + r[Dij fr ]C α/2,α ((0,T )×Ω;K) ≤ C||f ||1+α , b

r

2−α

||Dj Dt fr ||∞ ≤ C||f ||1+α ,

r2k−1−α ||Dtk fr ||∞ ≤ C||f ||1+α . The third inequality (with k = 1, 2) together with Proposition 1.1.4 yield [Dt fr (·, x)]C α/2 ((0,T );K) ≤ Cr−1 ||f ||1+α ,

x ∈ Ω.

b

Similarly, the second inequality and the third one (with k = 1), again together with Proposition 1.1.4, show that [Dt fr (t, ·)]Cbα (Ω;K) ≤ Cr−1 ||f ||1+α for every t ∈ (0, T ). Taking Proposition 1.2.6 into account, from the above estimates we infer that ||fr ||2+α ≤ Cr−1 ||f ||1+α ,

r ∈ (0, 1].

(A.5.15)

Finally, we observe that Z fr (t, x) − f (t, x) = ϑ(s, y)[f (t − r2 s, x − ry) − f (t, x − ry)] ds dy Rd+1 Z + ϑ(s, y)[f (t, x − ry) − f (t, x)] ds dy Rd+1 Z = ϑ(s, y)[f (t − r2 s, x − ry) − f (t, x − ry)] ds dy Rd+1 Z 1 + ϑ(s, y)[f (t, x − ry) − 2f (t, x) + f (t, x + ry)] ds dy, 2 Rd+1 where we have used the fact that ϑ is an even function with respect to the spatial variables. Observing that |f (t, x + ry) − 2f (t, x) + f (t, x − ry)| ≤ C||f ||1+α |ry|1+α for every x, y ∈ Rd , t ∈ R, r > 0, we conclude that ||fr − f ||∞ ≤ Cr1+α ||f ||C (1+α)/2,1+α ((0,T )×Ω;K) .

(A.5.16)

b

Summing up, from (A.5.15) and (A.5.16) and taking ξ, r ∈ (0, 1], we deduce that 1+α

1+α

1

1

ξ − 2+α ||f − fr ||∞ + ξ 2+α ||fr ||2+α ≤ (ξ − 2+α r1+α + ξ 2+α r−1 )||f ||1+α . 1+α

Taking r = ξ 1/(2+α) , we obtain that ξ − 2+α K(ξ, f, Z0 , Z2+α ) ≤ C||f ||1+α for ξ ∈ (0, 1]. Hence, f belongs to (Z0 , Z2+α )θ,∞ and ||f ||(Z0 ,Z2+α )θ,∞ ≤ C||f ||1+α . 

A.5.1

Marcinkiewicz’s Interpolation Theorem

Here, we introduce the well-known Marcinkiewicz’s interpolation theorem that we use in Chapter 10. Throughout the section, we denote by |A| the Lebesgue measure of the set A ⊂ Rd . We begin with a preliminary lemma.

438

Basic Notions of Functional Analysis in Banach Spaces

Lemma A.5.16 Let f : Ω → K be a Lebesgue measurable function. Then, for every p ∈ [1, ∞) it holds that Z Z ∞ |f (x)|p dx = p rp−1 λf (r) dr, Ω

0

where λf (r) is the Lebesgue measure of the set {x ∈ Ω : |f (x)| > r}, for every r ≥ 0. Proof Note that it suffices to prove the assertion with p = 1. Indeed, if p > 1, then we can apply the result to the function |f |p observing that λ|f |p (r) = |{x ∈ Ω : |f (x)|p > r}| = |{x ∈ Ω : |f (x)| > r1/p }| = λf (r1/p ) for every r ≥ 0. Thus, performing the change of variable s = r1/p , we can write Z ∞ Z ∞ Z Z ∞ sp−1 λf (s) ds. λf (r1/p ) dr = p λ|f |p (r) dr = |f |p dx = 0

0

0

Ω

So, let us consider the case p = 1 and set C = {(r, x) ∈ [0, ∞) × Ω : |f (x)| > r}. The assertion follows from the following chain of equalities, where we use Tonelli theorem: Z

Z |f (x)| dx =

Ω

Z dx

Ω

|f (x)|

Z dr =

0

Z dr dx =

C

∞

Z dr

0

Z dr =

|f (x)|>r

∞

λ(r) dr. 0

 Next, we introduce the definitions of weak Lp -spaces and of operators of weak type (p, p). Definition A.5.17 Let Ω be an open subset of Rd . For every p ∈ [1, ∞), we denote by Lpw (Ω; K) the set of all equivalence classes of measurable functions f : Ω → K such that λf (r) ≤ r−p ap for every r > 0 and some positive constant a. Remark A.5.18 For every p ∈ [1, ∞), Lpw (Ω; K) is a proper subspace of Lp (Ω; K). Indeed, Chebishev inequality shows that λ(r) ≤ r−p ||f ||pLp (Ω;K) for every r > 0, so that we can take a = ||f ||pLp (Ω;K) . On the other hand, the function x 7→ f (x) = x−1/p belongs to Lpw ((0, 1); K) for every p ∈ [1, ∞) since λf (r) = |{x ∈ (0, 1) : x−1/p > r}| = |{x ∈ (0, 1) : x < r−p }| = r−p ∧ 1 for every r > 0. Definition A.5.19 Let M(Ω; K) denote the set of all (the equivalence classes of ) measurable functions f : Ω → K and let T be an operator (not necessarily linear) mapping Lp (Ω; K) into M(Ω; K) for some p ∈ [1, ∞) ∪ {∞}. T is said to be an operator of weak type (p, p) (p < ∞) if there exists a positive constant Ap such that λT (f ) (r) ≤ App r−p ||f ||pLp (Ω;K) ,

f ∈ Lp (Ω; K).

T is said to be an operator of weak type (∞, ∞) if T f ∈ L∞ (Ω; K) for every f ∈ L∞ (Ω; K) and there exists a positive constant A∞ such that ||T (f )||∞ ≤ A∞ ||f ||∞ ,

f ∈ L∞ (Ω; K).

Semigroups of Bounded Operators and Second-Order PDE’s

439

Remark A.5.20 If T is linear and of weak type (∞, ∞), then T is a bounded operator mapping L∞ (Ω; K) into itself. Now, we can state and prove Marcinkiewicz’s interpolation theorem. Theorem A.5.21 (Marcinkiewicz’s interpolation theorem) Let p1 , p2 be fixed in [1, ∞) ∪ {∞}, with p1 < p2 , and let T : Lp1 (Ω; K) ∩ Lp2 (Ω; K) → M(Ω; K) be an operator which satisfies the following properties: (i) T is sublinear, i.e., |T (f + g)| ≤ |T (f )| + |T (g)| for every f, g ∈ Lp1 (Ω; K) ∩ Lp2 (Ω; K); (ii) T is of weak type (p1 , p1 ) and (p2 , p2 ). Then, for every q ∈ (p1 , p2 ), T maps Lp1 (Ω; K) ∩ Lp2 (Ω; K) into Lq (Ω; K) and there exists a positive constant Aq such that f ∈ Lp1 (Ω; K) ∩ Lp2 (Ω; K).

||T (f )||Lq (Ω;K) ≤ Aq ||f ||Lq (Ω;K) ,

(A.5.17)

Remark A.5.22 Let T be a linear operator of weak type (p1 , p1 ) and (p2 , p2 ). Since Lp1 (Ω; K) ∩ Lp2 (Ω; K) is dense in Lq (Ω; K) (it contains Cc∞ (Ω; K)) for every q ∈ (p1 , p2 ), we can use estimate (A.5.17) to extend T to Lq (Ω; K) with a bounded operator. Proof of Theorem A.5.21 We split the proof into two steps. Step 1. Here, we assume that p2 < ∞. We fix f ∈ Lp1 (Ω; K) ∩ Lp2 (Ω; K), r > 0 and split f = f1,r + f2,r , where ( f (x), |f (x)| > r, f1,r (x) = 0, |f (x)| ≤ r. Since |f1,r (x)| ≤ |f (x)| for x ∈ Ω, then f1 belongs to Lp1 (Ω; K) ∩ Lp2 (Ω; K). On the other hand, f2,r belongs to L∞ (Ω; K) and ||f2,r ||∞ ≤ r. In particular, it belongs to Lp1 (Ω; K) ∩ Lp2 (Ω; K). Since T is sublinear, it follows that λT (f ) (r) ≤ λT (f1,r ) (r/2) + λT (f2,r ) (r/2). Recalling that T is an operator of weak type (p1 , p1 ) and (p2 , p2 ), we can estimate λT (f1,r ) ≤ 2p1 App11 r−p1 ||f1,r ||pL1p1 (Ω;K) ,

λT (f2,r ) ≤ 2p2 App22 r−p2 ||f1,r ||pL2p2 (Ω;K)

(A.5.18)

for some positive constants Ap1 and Ap2 , independent of f and r. Thus, λT (f ) (r) ≤ 2p1 App11 r−p1 ||f1,r ||pL1p1 (Ω;K) + 2p2 App22 r−p2 ||f1,r ||pL2p2 (Ω;K) .

(A.5.19)

Using (A.5.19) and Lemma A.5.16, we can estimate Z Z ∞ q |T f | dx =q rq−1 λT f (r) dr Ω 0 Z ∞ Z ∞ p1 p1 ≤2 Ap1 q rq−p1 −1 ||f1,r ||pL1p1 (Ω;K) dr + 2p2 App22 q rq−p2 −1 ||f1,r ||pL2p2 (Ω;K) dr 0 Z0 ∞ Z p1 p1 q−p1 −1 p1 =2 Ap1 q r dr |f (x)| dx {x∈Ω:|f (x)|>r}

0

+ 2p2 App22 q =2p1 App11 q

Z Ω

Z

∞

rq−p2 −1 dr

|f (x)|p1 dx

{x∈Ω:|f (x)|≤r}

0

|f (x)|p1 dx

Z

Z 0

|f (x)|

rq−p1 −1 dr

440

Basic Notions of Functional Analysis in Banach Spaces Z Z ∞ rq−p2 −1 dr |f (x)|p1 dx + 2p2 App22 q |f (x)|

Ω

2p1 App11 2p2 App22 = + q − p1 p2 − q 

Z

|f (x)|q dx.

Ω

2p1 App11 2p2 App22 + . q − p1 p2 − q Step 2. Here, we complete the proof, dealing with the case p2 = ∞. In such a case, ||T (f2,r )||∞ ≤ A∞ ||f2,r ||∞ ≤ A∞ r for some constant A∞ , independent of f and r. Therefore, Hence, the assertion follows with Aqq =

λT (f ) (2A∞ r) ≤ λT (f1,r ) (A∞ r) + λT (f2,r ) (A∞ r) = λT (f1,r ) (A∞ r) for every r > 0. Taking (A.5.18) and Lemma A.5.16 into account, we can write Z Z ∞ |T (f )|q dx =q sq−1 λT (f ) (r) dr Ω 0 Z ∞ rq−1 λT (f ) (2A∞ r) dr =(2A∞ )q q 0 Z ∞ Z p1 q−p1 −1 1 ≤2q Aq−p A q r dr |f (x)|p1 dx ∞ p1 {x∈Ω:|f (x)|>r}

0

1 App11 q =2q Aq−p ∞

=

q−p1 p1 Ap 1 q 2q A∞ q − p1

and the assertion follows with Aqq =

A.6

Z

|f (x)|p1 dx

Ω

Z

Z

|f (x)|

rq−p1 −1 dr

0

|f (x)|q dx

Ω

q−p1 p1 Ap 1 q 2q A∞ . q − p1



Exercises

1. Let X = C([0, 1]). Prove that (i) the spectrum of the operator A1 : {f ∈ C 1 ([0, 1]) : f (0) = 0} ⊂ X → X, defined by A1 f = f 0 for f ∈ D(A1 ), is empty; (ii) the resolvent set of the operator A2 : {f ∈ C 1 ([0, 1]) : f (0) = f (1) = 0} ⊂ X → X, defined by A2 f = f 0 for f ∈ D(A2 ), is empty; (iii) operators A1 and A2 are closed but not bounded. 2. Prove that if a function u : [a, b] → X is differentiable from the right in [a, b) and the right-derivative can be extended by continuity to [a, b], then u ∈ C 1 ([a, b]; X). 3. Prove Proposition A.1.7. 4. Complete the proof of Theorem A.3.4. 5. Complete the proof of Example A.5.13 in the case γ < 1/3.

Appendix B Smooth Domains and Extension Operators

This appendix is devoted to introducing some definitions and tools that are used throughout the book. In Section B.1, we prove that for every compact set K ⊂ Rd and every finite covering {Ωi : i = 1, . . . , N } of K consisting of open sets, there exist smooth functions ηi ∈ Cc∞ (Ωi ) (i = 1, . . . , N ) those sum is identically equal to one on K. This result is crucial to prove, in Section B.4, the existence of some extension operators. We consider both the problem of extending to the whole Rd a function defined on Ω, when Ω = Rd+ or Ω is a sufficiently smooth bounded open set, obtaining a function which has the same degree of smoothness and the problem of the extending a function defined on ∂Ω to Ω. More precisely, we prove (i) the existence of bounded linear operators mapping C α (Ω; K) into Cbα (Rd ; K) (α ≥ 0) and W k,p (Ω; K) into W k,p (Rd ; K) (k ∈ N, p ∈ [1, ∞)), whose restriction to Ω is the identity operator, (ii) the existence of a bounded linear operator mapping B k−1/p,p (∂Ω; K) into W k,p (Ω; K), for k and p as above, which is a right inverse of the trace operator. The definition of smooth domain is provided in Section B.2, see Definition B.2.1, where we also give two equivalent characterizations of smooth bounded domains (see Propositions B.2.5 and B.2.11). Finally, in Section B.3 we define the trace of a function u ∈ W 1,p (Ω; K). Notation. Throughout the chapter, by K we denote both the sets R and C.

B.1

Partition of Unity

This section is devoted to proving the following proposition which is crucially used in Chapters 1, 9, 13, 14 and in the next Section B.4. Proposition B.1.1 Let K ⊂ Rd be a compact set and {Ωi : i = 1, . . . , N } be an open SN covering of K (i.e., K ⊂ i=1 Ωi , Ωi being open subsets of Rd ). Then, there exist functions PN ηi ∈ Cc∞ (Ωi ) (i = 1, . . . , N ) such that 0 ≤ ηi ≤ 1 on Ωi and i=1 ηi ≡ 1 on K. Proof To begin with, we show that there exists a covering {Ω0i : i = 1, . . . , N } of K such that Ω0i ⊂ Ωi for every i = 1, . . . , N . For this purpose, for every x ∈ Ωi , we consider a ball B(x, rx ) ⊂ Ωi . Clearly, {B(x, rx /2) : x ∈ Ωi , i = 1, . . . , N } is an open covering of K, from which we can extract a finite subcovering {B(xj , rxj /2) : j = 1,S. . . , M }. For every i ∈ {1, . . . , N }, we set Ii = {j ∈ {1, . . . , M } : B(xj , rxj ) ⊂ Ωi } and Ω0i = k∈Ii B(xk , rxk /2). Each Ω0i is an open subset of Rd and [ [ Ω0i = B(xk , rxk /2) ⊂ B(xk , rxk ) ⊂ Ωi , i = 1, . . . , N. k∈Ii

k∈Ii

441

442

Smooth Domains and Extension Operators

By Exercise 6.7.1, for every i = 1, . . . , N we can determine a function %i ∈ Cc∞ (Rd ) compactly supported in Ωi and such that %i ≡ 1 in Ω0i . Note that we can also assume that 0 ≤ %i ≤ 1 on Rd To complete the proof, it suffices to set η1 = %1 , η2 = (1 − %1 )%2 and, more generally, Qk−1 ηk = %k j=1 (1 − %j ) for k = 2, . . . , N . Each function ηi is compactly supported in Ωi and its image is contained in [0, 1]. Moreover, as it is easily seen, N X

ηi (x) = 1 −

i=1

N Y

(1 − %i (x)),

x ∈ Rd .

i=1

PN From this formula, it follows immediately that i=1 ηi ≡ 1 on K. Indeed, if x ∈ K, then x ∈ Ω0i for some i ∈ {1, . . . , N } and on Ω0i the function 1 − %i identically vanishes.  PN Remark B.1.2 The proof of Proposition B.1.1 shows that i=1 ηi ≡ 1 on a neighborhood PN SM of K. Indeed, i=1 ηi ≡ 1 on j=1 B(xj , rxj /2) and this set contains the set Kδ = {x ∈ Rd : dist(x, K) ≤ δ}. By contradiction, suppose that there exists a sequence (yn ) ∈ Rd such SM that dist(yn , K) ≤ n−1 and yn ∈ / j=1 B(xj , rxj /2) for every n ∈ N. Since this sequence is y ∈ K, which belongs to the open set bounded, up to a subsequence, it converges to some S S M M B(x , r /2). Hence, for large n, y belongs to j x n j j=1 j=1 B(xj , rxj /2): a contradiction. ∞ d Now, if we introduce a function ζ ∈ Cc (R ) such that χKδ/2 ≤ ζ ≤ χKδ (whose existence follows from Exercise 6.7.1) and set ηei = ηi ζ for every i = 1, . . . , N , then we can claim that (i) ηei ∈ Cc∞ (Rd ), ηei ≤ χΩi for every i = 1, . . . , N ; PN PN (ii) 0 ≤ i=1 ηei ≤ 1 on Rd and i=1 ηei = 1 on Kδ/2 . We will use this remark in the proof of Proposition B.2.5. In Chapters 9 and 13, we shall make use of the following corollary of the Proposition B.1.1. Corollary B.1.3 Let K ⊂ Rd be a compact set and {Ωi : i = 1, . . . , N } be an open covering of K. Then, there exist functions ϑi ∈ Cc∞ (Ωi ) (i = 1, . . . , N ) such that 0 ≤ ϑi ≤ 1 on Ωi PN and i=1 ϑ2i ≡ 1 on K. Proof Let ηi (i = 1, . . . , N ) be the same functions as in the proof of Proposition B.1.1. PN We claim that there exists a positive constant δ0 such that i=1 ηi2 ≥ δ0 on K. Indeed, the PN function i=1 ηi2 is continuous on K. Hence, it admits a minimum over K attained at some point x0 . If such a minimum were zero, then each function ηi (i = 1, . . . , N ) would vanish PN at x0 and this would clearly contradict the condition i=1 ηi (x0 ) = 1. PN 2 Next, we claim that there exists ε > 0 such that i=1 ηi ≥ δ0 /2 on K + B(0, ε). By contradiction, suppose that this is not the case. Then, for every n ∈ N we may find PN 2 a point xn ∈ K + B(0, 1/n) such that (η (x )) < δ 0 /2. Since the sequence (xn ) i=1 i n is bounded, up to a subsequence it converges to some point x ∈ K, which satisfies the PN condition i=1 (ηi (x))2 ≤ δ0 /2: a contradiction. Now, we are almost done. By Exercise 6.7.1 we can determine a function ψ ∈ Cc∞ (Rd ) such that χK ≤ ψ ≤ χK+B(0,ε) . Setting ϑi =

X N

ηj2

− 21 ηi ψ,

i = 1, . . . , N,

j=1

we obtain the functions that we are looking for.



Semigroups of Bounded Operators and Second-Order PDE’s

B.2

443

Smooth Domains

Definition B.2.1 A bounded domain Ω ⊂ Rd is called a domain of class C α (α ∈ (0, ∞)) if there exist a positive constant r and, for every x0 ∈ ∂Ω, an open neighborhood Ux0 of x0 and a bijective function ψx0 : Ux0 → B(0, r) of class C α , together with its inverse function, such that ψx0 (Ω ∩ Ux0 ) = B+ (0, r) := {x ∈ B(0, r) : xd > 0}, ψx0 (∂Ω ∩ Ux0 ) = {x ∈ B(0, r) : xd = 0}. Remark B.2.2 Some authors give a seemingly different definition of domains of class C α , replacing the ball B(0, r) with the unit cube C(0, 1) centered at the origin. This difference is immaterial, since the above definition has a local nature: we may replace the ball B(0, r) by the cube C(0, 1) in this simple way: (i) we consider r0 sufficiently small such that C(0, r0 ) ⊂ B(0, r), (ii) we introduce the dilation δ1/r : Rd → Rd , defined by δ1/r (y) = r−1 y for every y ∈ Rd ; (iii) fixed x0 ∈ ∂Ω, we introduce the function ψex0 = δ1/r ◦ ψx0 which is as smooth as ψx0 is and its inverse is of class C α as well; (iv) finally, we set Ux0 0 = ψex−1 (C(0, 1)). It is 0 now easy to check that ψex0 (Ω ∩ Ux0 0 ) = C+ (0, 1) = {x ∈ C(0, 1) : xd > 0}, ψex (∂Ω ∩ U 0 ) = {x ∈ C(0, 1) : xd = 0}. 0

x0

Vice versa, if ψx0 : Ux0 0 → C(0, 1) is a function of class C α , together with its inverse function, and the above two conditions are satisfied, then the function ψx0 = δr ◦ ψex0 satisfies all the conditions in Definition B.2.1. Modulo a dilation, we can also replace the cube C(0, 1) with the parallelepiped C(0, α)× (−β, β) = {x ∈ Rd : x0 ∈ C(0, α), |xd | < β} for every choice of the positive constants α and β. Remark B.2.3 The argument in Remark B.2.2 also show that in Definition B.2.1 r can be equivalently taken equal to one. Remark B.2.4 Repeating the same arguments as in Remark B.1.2 and observing that ∂Ω is a compact set, it can be easily shown that from {Ux : x ∈ ∂Ω} we can extract a finite subcovering, which is a covering also of the set Ωδ := {x ∈ Ω : dist(x, ∂Ω) ≤ δ} for some δ > 0. The following proposition provides us with an equivalent condition for a bounded domain to be a domain of class C 2+α , which is useful to find out concrete examples. Proposition B.2.5 A bounded domain Ω is a domain of class C 2+α (α ∈ [0, 1]) if and only if there exists a function g ∈ C 2+α (Rd ) such that Ω = {x ∈ Rd : g(x) > 0}, ∂Ω = {x ∈ Rd : g(x) = 0}, Rd \ Ω = {x ∈ Rd : g(x) < 0} and ∇g 6= 0 on ∂Ω.

444

Smooth Domains and Extension Operators

Proof of Proposition B.2.5 Since it is rather long, we split the proof into two steps. Step 1. Here, we prove that, if Ω is of class C 2+α , then there exists a function g as in the statement. For this purpose, for every x ∈ ∂Ω we denote by ϕx the last component of the function ψx in Definition B.2.1. It is easy to check that Ω ∩ Ux = {y ∈ Ux : ϕx (y) > 0}, ∂Ω ∩ Ux = {y ∈ Ux : ϕx (y) = 0}, (Rd \ Ω) ∩ Ux = {y ∈ Ux : ϕx (y) < 0}. We now consider a finite subcovering {Uxi : i = 1, . . . , N } of ∂Ω and observe that, by Remarks B.1.2 and B.2.4, there exist δ > 0, such that the set Ωδ = {x ∈ Rd : dist(x, ∂Ω) ≤ δ} is contained in the union of the open sets Uxi , and nonnegative functions ηei ∈ Cc∞ (Rd ) PN (i = 1, . . . , N ) such that supp(e ηi ) ⊂ Ui , for every i, and χΩδ ≤ ηe := i=1 ηei ≤ 1 on Rd . Let g : Rd → R be the function defined on Rd by g(x) = (e η (x) − 1)(1 − 2χΩ (x)) +

N X

ηei (x)ϕxi (x),

x ∈ Rd ,

i=1

where ϕxi is the trivial extension to Rd of the function ϕxi (i = 1, . . . , N ). It is clear that each function ηei ϕxi belongs to Cb2+α (Rd ) since it is compactly supported in Uxi . Similarly, the function (e η − 1)(1 − 2χΩ ) belongs to Cb2+α (Rd ) since it vanishes on a neighborhood of ∂Ω and it is trivially smooth elsewhere. We are going to prove that function g has all the properties in the statement of the proposition. To begin with, we observe that it vanishes on ∂Ω. Indeed, ηe ≡ 1 on ∂Ω (since ∂Ω ⊂ Ωδ ) and each function ηei ϕxi vanishes on ∂Ω. This is clear, if x ∈ / Ui . On the other hand, if x ∈ Ui ∩ ∂Ω, then ϕxi = 0 as it has been shown in the first part of the proof. Suppose now that x ∈ Ω. Then, g(x) = 1 − ηe(x) +

N X

ηei (x)ϕxi (x).

i=1

Note that ηei (x)ϕxi (x) ≥ 0 for every i. Indeed, if x ∈ Uxi , then ϕxi (x) > 0 and ηei (x) ≥ 0, whereas, if x ∈ / Uxi , then ϕxi (x) = 0, so that ηei (x)ϕxi (x) = 0. Next, we observe that we have two possibilities: (i) ηei (x) = 0 for every i = 1, . . . , N ; (ii) there exists i ∈ {1, . . . , N } such that ηei (x) 6= 0. PN In the first case 0 = ηe(x) = i=1 ηei (x) and g(x) ≥ 1, On the other hand, if (ii) holds, then x ∈ Uxi . Since ϕxi (x) > 0, it follows that g(x) ≥ ηei (x)ϕxi (x) > 0. Finally, we suppose that x ∈ / Ω. In such a case, g(x) = ηe(x) − 1 +

N X

ηei (x)ϕxi (x).

i=1

Arguing as above, we can easily show that ηei (x)ϕxi (x) ≤ 0 for every i = 1, . . . , N . We have also the same cases (i) and (ii) as above. In the first case g(x) = −1, whereas in the latter one, g(x) ≤ ηei (x)ϕxi (x) < 0, since ϕxi < 0 on (Rd \ Ω) ∩ Uxi .

Semigroups of Bounded Operators and Second-Order PDE’s

445

To complete the proof, let us show that the gradient of the function g nowhere vanishes on ∂Ω. For this purpose, we observe that g(x) =

N X

ηei (x)ϕxi (x),

x ∈ Ωδ .

i=1

From this formula, it follows immediately that ∇g(x) =

N X

∇e ηi (x)ϕxi (x) +

i=1

N X

ηei (x)∇ϕxi (x),

x ∈ Ωδ .

i=1

Hence, if x ∈ ∂Ω, then ∇g(x) =

X

ηei (x)∇ϕxi (x) =

i∈Jx

X

ηei (x)|∇ϕxi (x)|νi (x),

i∈Jx

where Jx = {i ∈ {1, . . . , N } : x ∈ Uxi } and νi (x) = |∇ϕxi (x)|−1 ∇ϕxi (x). We claim that νi (x) = νj (x) for every i, j ∈ Jx . Denoting by ν(x) the common value, from the claim it will follow that X ∇g(x) = ν(x) ηei (x)|∇ϕxi (x)|. i∈Jx

P

Since i∈Jx ηei (x) = 1, |∇ϕxi (x)| > 0 for every i ∈ Jx and the functions ηei are all nonnegative, we will easily conclude that |∇g(x)| > 0. So, let us prove the claim. Fix i, j ∈ Jx . Since νi (x) and νj (x) are the normal vectors to Ω ∩ Uxi at x, it follows that νi (x) = νj (x) or νi (x) = −νj (x). To prove that the second possibility cannot occur, we apply Taylor formula to write ϕxi (x + tνi (x)) = ϕxi (x) + th∇ϕxi (x), νi (x)i + o(t, 0) = t(|∇ϕxi (x)| + o(1, 0)), ϕxj (x + tνj (x)) = ϕxj (x) + th∇ϕxj (x), νj (x)i + o(t, 0) = t(|∇ϕxj (x)| + o(1, 0)) for t ∈ (−δ0 , δ0 ) and δ0 small enough. From these formulas it follows that, up to replacing δ0 with a smaller value, if needed, x + tνi (x) belongs to Ω for every t ∈ [0, δ0 ) whereas x − tνj (x) does not belong to Ω if t ∈ [0, δ0 ). We thus conclude that νj (x) = νi (x). Step 2. We now show that, if Ω = {x ∈ Rd : g(x) > 0} for some function g ∈ C 2+α (Rd ) such that ∇g does not vanish on ∂Ω = {x ∈ Rd : g(x) = 0} and Ω is bounded, then it is a domain of class C 2+α . For this purpose, we fix x0 ∈ ∂Ω and observe that, since ∇g does not vanish on ∂Ω, there exists i ∈ {1, . . . , d} such that Di g(x0 ) 6= 0. Let ψex0 : Rd → Rd be the function defined by ψex0 (x) = (x1 − x0,1 , . . . , xi−1 − x0,i−1 , g(x), xi+1 − x0,i+1 , . . . , xd − x0,d ) for every x ∈ Rd . As it is immediately seen, ψex0 belongs to C 2+α (Rd ; Rd ) and its Jacobian determinant at x0 is positive. Hence, there exists r > 0 such that the function ψex0 is invertible in B(x0 , r). Since ψex0 (x0 ) = 0, we can fix δ > 0 such that B(0, δ) ⊂ ψex0 (B(x0 , r)). Hence, if we set Ux0 = ψex−1 (B(0, δ)), then ψex0 is a diffeomorphism of class C 2+α between 0 Ux0 and B(0, δ). We are going to prove that ψex0 (Ω ∩ Ux0 ) = {x ∈ B(0, δ) : xi > 0} and ψex0 (∂Ω∩Ux0 ) = {x ∈ B(0, δ) : xi = 0}. For this purpose, we recall that x ∈ Ω (resp. x ∈ ∂Ω) if and only if g(x) > 0 (resp. g(x) = 0). Hence, if x ∈ Ω ∩ Ux0 then y = ψex0 (x) ∈ B(0, δ) and yi > 0. Similarly, if x ∈ ∂Ω ∩ Ux0 then y = ψex0 (x) ∈ B(0, δ) and yi = 0. Vice versa, let us assume that y ∈ B(0, δ) is such that yd > 0. Then, y = ψex0 (x), with x = ψex−1 (y) ∈ Ux0 . 0 Moreover, g(x) = yd > 0. Hence, x belongs also to Ω. In the same way, it can be proved

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that each y ∈ B(0, δ) such that yd = 0 is the image under ψex0 of an element of Ω ∩ Ux0 and we are done. Now, we set ψx0 = T ψex0 , where T (y) = (δ −1 y1 , . . . , δ −1 yi−1 , δ −1 yd , δ −1 yi+1 , . . . , δ −1 yd−1 , δ −1 yi ) for every y ∈ Rd . Function ψx0 belongs to C 2+α (Ux0 ), ψx0 (Ω∩Ux0 ) = B+ (0, 1) and ψx0 (∂Ω∩ Ux0 ) = {y ∈ B(0, 1) : yd = 0}. This completes the proof.  Remark B.2.6 In view of the previous proposition examples of bounded domains of class C 2+α are easily provided. For instance every open ball has this property (actually, it is a Pd C ∞ -smooth domain). Indeed, B(x, r) = {x ∈ Rd : r2 − i=1 (xi −xi )2 > 0} for every x ∈ Rd P d and the function g : Rd → R, defined by g(x) = r2 − i=1 (xi − xi )2 > 0 for every x ∈ Rd , ∞ d belongs to C (R ). Remark B.2.7 Some remarks are in order. (i) The last part of Step 1 of the proof of Proposition B.2.5 shows that, if x ∈ ∂Ω, then the unit outer normal vector to ∂Ω at x is ν(x) = −|∇ϕx (x)|−1 ∇ϕx . From Step 1, we also deduce that we can assume that ν(x) = −|∇g(x)|−1 ∇g(x) for every x ∈ ∂Ω. (ii) Still from the last part of Step 2 of the proof of the quoted proposition, we deduce that, for each x ∈ ∂Ω, the set x − tν(x) belongs to Ω if t ∈ (0, δ0 ) and δ0 is sufficiently small. The smoothness of the boundary of a domain can be defined in an equivalent way as the following proposition shows. Proposition B.2.8 Ω is a bounded domain of class C 2+α (α ∈ [0, 1]) if and only if for every x0 ∈ ∂Ω there exist positive an open neighborhood Ux0 0 of x0 , numbers αx0 and βx0 and a smooth function φx0 : C(0, αx0 ) ⊂ Rd−1 → R of class C 2+α such that, up to a rotation and a translation, (i) Ux0 0 ∩ ∂Ω = {x ∈ Rd : x0 ∈ C(0, αx0 ), xd = φx0 (x0 )}, (ii) Ux0 0 ∩ Ω = {x ∈ Rd : x0 ∈ C(0, αx0 ), φx0 (x0 ) < xd < φx0 (x0 ) + βx0 }; (iii) Ux0 0 ∩ (Rd \ Ω) = {x ∈ Rd : x0 ∈ C(0, αx0 ), φx0 (x0 ) − βx0 < xd < φx0 (x0 )}. Proof We first assume that Ω is a domain of class C 2+α and prove that properties (i)(iii) are satisfied. We fix x0 = (x00 , x0,d ) ∈ ∂Ω. From the first part of the proof of Proposition B.2.5 we know that Ω ∩ Ux0 = {y ∈ Ux0 : ϕx0 (y) > 0},

∂Ω ∩ Ux0 = {y ∈ Ux0 : ϕx0 (y) = 0}.

(B.2.1)

Since function ψx0 is invertible in Ux0 , the gradient of ϕx0 never vanishes in Ux0 . We first assume that Dd ϕx0 (x0 ) > 0. Then, by the implicit function theorem it follows that there exist r1 , r2 > 0, such that Vx0 := C(x00 , r1 ) × (x0,d − 2r2 , x0,d + 2r2 ) ⊂ Ux0 , and a function φx0 : C(x00 , r1 ) → (x0,d − 2r2 , x0,d + 2r2 ) of class C 2+α such that Dd ϕx0 > 0 in Vx0 and {y ∈ Vx0 : ϕx0 (y) = 0} = graph(φx0 ).

(B.2.2)

Next, we observe that, since (x0 , φx0 (x0 )) tends to x0 as x0 tends to x00 , we can choose r0 < r1 small enough such that φx0 (C(x00 , r0 )) ⊂ (x0,d − r2 , x0,d + r2 ) and, consequently,

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the set Ux0 0 = {x ∈ Rd : x0 ∈ C(x00 , r0 ), φx0 (x0 ) − r2 < xd < φx0 (x0 ) + r2 } is an open neighborhood of x0 which is contained in Vx0 and, hence, into Ux0 . It is immediate to check that Ux0 0 ∩ ∂Ω = {x ∈ Rd : x0 ∈ C(x00 , r0 ), xd = φx0 (x0 )}. The inclusion “⊂” follows from (B.2.2) and the fact that Ux0 0 ⊂ Vx0 ; the other one follows from observing that (x0 , φx0 (x0 )) ∈ C(x00 , r0 ) × (x0,d − r2 , x0,d + r2 ) ⊂ Vx0 for every x0 ∈ C(x00 , r0 ) and taking again (B.2.2) into account. Similarly, we can check that Ux0 0 ∩ Ω = {x ∈ Rd : x0 ∈ C(x00 , r0 ), φx0 (x0 ) < xd < φx0 (x0 ) + r2 }. Indeed, since Dd ϕx0 (x) > 0 for every x ∈ C(x00 , r0 ) × (x0,d − r2 , x0,d + r2 ) and ϕx0 (x0 , φx0 (x0 )) = 0, it follows that ϕx0 (x0 , xd ) > 0 if and only if xd > φx0 (x0 ) and the first formula in (B.2.1) yields the above set equality. If Dd ϕ < 0 on Ux000 , then Ux0 0 ∩Ω = {x ∈ Rd : x0 ∈ C(x00 , r0 ), φx0 (x0 )−r2 < xd < φx0 (x0 )}. Applying the rotation x 7→ (x0 , −xd ) and replacing the function φx0 by the function −φx0 , we reduce to the above first case. Finally, if Dd ϕx0 (x0 ) vanishes, then D j < d. Repeating the same Pj ϕx0 (x0 ) 6= 0 for someP arguments as above and setting x ˆj = x e and x ˆ = i i 0,j i6=j i6=j x0,i ei , x0,i being the ˆj ∈ C(ˆ x0,j , r0 ), xj = φx0 (ˆ xj )} component of x0 , we conclude that Ux0 0 ∩ ∂Ω = {x ∈ Rd : x and Ux0 0 ∩ Ω = {x ∈ Rd : x ˆj ∈ C(ˆ x0,j , r0 ), φx0 (ˆ xj ) < xj < φx0 (ˆ xj ) + r2 } or Ux0 0 ∩ Ω = {x ∈ Rd : x ˆj ∈ C(ˆ x0,j , r0 ), φx0 (ˆ xj ) − r2 < xj < φx0 (ˆ xj )} depending on the sign of Dj ϕx0 on Ux000 . Again a rotation reduces us to the first case that we considered. In all the cases if x ˆ0,j = 0 then we are done. On the other hand, if x ˆ0,j 6= 0, properties (i)–(iii) follow applying the translation x0 7→ x0 − x00 and replacing the function φx0 with the function φx0 (· + x00 ). Vice versa, suppose that properties (i)–(iii) hold true. We fix x0 ∈ ∂Ω and first assume that no rotations are needed to write (i)–(iii). We set Ux0 = {x ∈ Rd : x0 ∈ C(x00 , αx0 ), |xd − φx0 (x0 − x00 )| < βx0 } and observe that Ux0 is a neighborhood of x0 . On Ux0 we define the function ϕx0 by setting ϕx0 (x) = (x0 , xd − φx0 (x0 − x00 )) for every x ∈ Ux0 . As it is immediately seen, ϕx0 is as smooth as φx0 is. Moreover, ϕx0 (Ux0 ) = {x ∈ Rd : x0 ∈ C(x00 , αx0 ), xd ∈ (−βx0 , βx0 )}, ϕx0 (Ux0 ∩ Ω) = {x ∈ Rd : x0 ∈ C(x00 , αx0 ) × (0, βx0 )}, ϕx0 (Ux0 ∩ ∂Ω) = {x ∈ Rd : x0 ∈ C(x00 , αx0 ), xd = 0}. By Remark B.2.2, this is enough to infer that Ω is a domain of class C 2+α . The general case can be handled by composing ϕx0 with an affine transformation.



Remark B.2.9 The implicit function theorem shows that, if x0 ∈ ∂Ω and Ux0 0 ∩ ∂Ω = {x ∈ Rd : x ˆj ∈ C(ˆ x00 , r0 ), xj = φx0 (ˆ xj )}, then (∇φx0 (ˆ xj ), −1) is a multiple of the outer unit normal vector to ∂Ω at x0 . Indeed, Dk φx0 (ˆ x0,j ) = −

Dk g(x0 ) , Dj g(x0 )

k 6= j,

so that (∇φx0 (ˆ x0,j ), 1) = −(Dj g(x0 ))−1 ∇g(x0 ) and we conclude thanks to Remark B.2.7(i).

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Definition B.2.10 A bounded domain Ω satisfies the interior sphere condition if there exists r0 > 0 such that for every y ∈ ∂Ω there exists a ball B of radius r0 contained in Ω such that B ∩ ∂Ω = {y}. Proposition B.2.11 Let Ω be a (bounded) domain of class C 2 . Then, it satisfies the interior sphere condition. Proof In the proof, we will take advantage of the proof of Proposition B.2.8, using the same notation therein introduced. Fix y ∈ ∂Ω. Then, there exist positive constants αy , βy , an index j = j(y) ∈ {1, . . . , d} yj , αy ) ⊂ Rd−1 → [yj − βy , yj + βy ] of class C 2 such that Uy0 = and a function φy : C(ˆ d 0 {x ∈ R : x ∈ C(ˆ yj , αy ), |φy − xj | < βy }, where x ˆj = (x1 , . . . , xj−1 , xj+1 , . . . , xd ) for each x ∈ Rd . Moreover, Uy0 ∩ ∂Ω = {x ∈ Rd : x ˆj ∈ C(ˆ yj , αy ), xj = φy (ˆ xj )} and either Uy0 ∩ Ω = {x ∈ Rd : x ˆj ∈ C(ˆ yj , αy ), φy (ˆ xj ) < xj < φy (ˆ x j ) + βy } or Uy0 ∩ Ω = {x ∈ Rd : x ˆj ∈ C(ˆ yj , αy ), φy (ˆ xj ) − βy < xj < φy (ˆ xj )}. For each y ∈ ∂Ω, we set 3ry = min{αy , (1 + Ly )−1 βy , L−1 y } where Ly denotes the C -norm of the function φy . Further, for every z ∈ B(y, ry ) ∩ ∂Ω, we set ηy (z) = (D1 φy (ˆ zj ), . . . , Dj−1 φy (ˆ zj ), −1, Dj+1 φy (ˆ zj ), . . . , Dd φy (ˆ zj )) and γy (z) = (1 +|∇φy (ˆ zj )|2 )1/2 , −1 so that γy (z) ηy (z) is a unit vector (actually, due to Remark B.2.9, this vector is the interior unit normal vector to ∂Ω at z). We are going to prove that either the ball B+ = B(z + ry γy (z)−1 ηy (z), ry ) or the ball B− = B(z − ry γy (z)−1 ηy (z), ry ) is contained in Ω. For this purpose, we begin by observing that, since Z 1 φy (z2 ) − φy (z1 ) = h∇φy (z1 + t(z2 − z1 )), z2 − z1 i dt, 2

0

Z φy (z2 ) − φy (z1 ) − h∇φy (z1 ), z2 − z1 =

1

h(∇φy (z1 + t(z2 − z1 )) − ∇φy (z1 ), z2 − z1 i dt 0

for every z1 , z2 ∈ C(ˆ yj , αy ), it follows straightforwardly that |φy (z2 ) − φy (z1 )| ≤ Ly |z2 − z1 |, |φy (z2 ) − φy (z1 ) − h∇φy (z1 ), z2 − z1 i| ≤

(B.2.3) Ly |z2 − z1 |2 . 2

(B.2.4)

From (B.2.3), it follows that B± ⊂ Uy0 . Indeed, as it is easily seen, B± ⊂ B(z, 2ε) ⊂ B(y, 3ry ). Thus, if x ∈ B± , then |ˆ xj − yˆj | < 3ry < αy and |xj − yj | < 3ry . Moreover, we can estimate |xj − φy (ˆ xj )| ≤|xj − φy (ˆ yj )| + |φy (ˆ yj ) − φy (ˆ xj )| ≤ |xj − yj | + |φy (ˆ yj ) − φy (ˆ xj )|

γy (z) |x − z|2 2ry

(B.2.5)

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and x ∈ B− if and only if hz − x, ηy (z)i >

γy (z) |x − z|2 . 2ry

(B.2.6)

Using (B.2.4) and (B.2.5), we can write xj − φy (ˆ xj ) =xj − zj − φy (ˆ xj ) + φy (ˆ zj ) =hz − x, ηy (z)i − φy (ˆ xj ) + φy (ˆ zj ) + h∇φy (ˆ zj ), x ˆj − zˆj i   γy (z) 1 |ˆ xj − zˆj |2 < 0 ≤ Ly − 2 ry for every x ∈ B+ , due to the fact that 3ry Ly ≤ 1 < γy (z). Similarly, using (B.2.6) we can show that xj − φy (ˆ xj ) > 0 if x ∈ B− . It thus follows that either B+ or B− is contained in Ω. Clearly, z ∈ ∂B± . If ∂B± contains other points of the boundary of ∂Ω, then it suffices to replace ry with any arbitrary smaller radius. Finally, we observe that {B(y, ry ) : y ∈ ∂Ω} is an open covering of ∂Ω. Hence, we can extract a finite subset J of ∂Ω such that {B(y, ry ) : y ∈ J } is a covering of ∂Ω. Now, taking r0 = miny∈J ry , the assertion follows.  Remark B.2.12 Some remarks are in order. (i) Actually, the proof of Proposition B.2.11 shows that for each point z of the boundary of a (bounded) open set of class C 2 we can determine two open balls, one contained in Ω and the other in Rd \ Ω which are tangent at z and do not contain any other points of ∂Ω. (ii) An insight in the proof of Proposition B.2.11 also shows that it is enough to assume that the gradient of φy is a Lipschitz continuous function. This assumption is sharp. Indeed, it can be shown that, if Ω satisfies the interior sphere condition, then, locally, ∂Ω is the graph of a function which is differentiable and its gradient is Lipschitz continuous. We refer the interested reader to e.g., [8, Theorem 2.7]. (iii) For each y ∈ ∂Ω, the center of the ball B, constructed in the proof of Proposition B.2.11, lies on the line passing through y and having the direction of the inner unit normal vector to ∂Ω at y. In Chapter 13, we need to deal with a partition of the unity subordinated to a finite covering of the boundary of a bounded open set of class C 2 , which satisfies an additional condition as explained in the following proposition those proof is postponed to Subsection B.4.2. Proposition B.2.13 Let Ω be a bounded domain of class C 2+α for some α ∈ [0, 1) and let {Ui : i = 1, . . . , N − 1} be finite open covering of ∂Ω, where the open sets Ui are as in Definition B.2.1. Finally, let η : ∂Ω → Rd be a continuous function such that hη(x), ν(x)i > 0 for every x ∈ ∂Ω, where ν(x) denotes the exterior norm unit vector to ∂Ω at x. Then, there exist an open set UN b Ω and functions φi , ζi ∈ Cc2+α (Ui ) (i = 1, . . . , N −1), φN , ζN ∈ PN ∂ζi = 0 on ∂Ω for i = 1, . . . , N − 1 and i=1 φi ζi = 1 on Ω. Cc∞ (UN ) such that ∂η We conclude this section with the following useful result which, under suitable assumptions, shows that the distance function d from the boundary of Ω, i.e., the function d : Ω → R defined by d(x) = d(x, ∂Ω) for every x ∈ Ω, is sufficiently smooth “close” to the boundary of Ω.

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Proposition B.2.14 Let Ω be a bounded domain of class C 2 . Then, there exists δ > 0 such that the function d belongs to C 2 (Ωδ ), where Ωδ = {x ∈ Ω : d(x) ≤ δ}. Moreover, ∇d(x) = −ν(x) for every x ∈ ∂Ω, where ν(x) denotes the unit outward vector to ∂Ω at x, i.e., ∇d(x) coincides with the unit inner vector to ∂Ω at x. For further results on smooth domains, we refer the reader to [38, Chapter 1].

B.3

Traces of Functions in Sobolev Spaces

In this section, we define the concept of traces of functions which belong to the Sobolev space W 1,p (Ω; K), when Ω is Rd+ or a smooth bounded open subset of Rd . An exhaustive treatment of the theory of traces is beyond our scopes. Here, we present just those basic results that we need in this book. For a much detailed description of the topic, we refer the interested reader, e.g., to the monographs [24, 37]. If p > d, then W 1,p (Ω; K) ,→ Cb (Ω; K) by the Sobolev embedding theorem (see Theorem 1.3.6). Hence, we can define the trace of u in the classical way as the value of u on ∂Ω. Things are different and much more complicated if p ≤ d, as the following theorem shows. In such a theorem we identify ∂Rd+ and Rd−1 . Theorem B.3.1 Let Ω be Rd+ or a bounded open subset of class C 1 . For each p ∈ (1, ∞) there exists a bounded linear operator Tr : W 1,p (Ω; K) → B 1−1/p,p (∂Ω; K) such that Tr(u) coincides with the usual pointwise trace of u on ∂Ω when u ∈ W 1,p (Ω; K) ∩ Cb (Ω; K). In particular, Tr is bounded from W 1,p (Ω; K) into Lq (∂Ω; Hd−1 ) where q = p(d − 1)/(d − p), if p < d, and q is any number in [p, ∞), otherwise. Moreover, if Ω is Rd+ or a bounded open set of class C k for some k ∈ N, k ≥ 2, then Tr is bounded from W k,p (Ω; K) into B k−1/p,p (∂Ω; K). Proof We split the proof into two steps, first considering the case Ω = Rd+ . Based on this result, in Step 2 we complete the proof. Throughout the proof, the constants which appear are all independent of the functions that we consider. Step 1. We fix u ∈ Cc∞ (Rd ; K), q as in the statement and observe that Z ∞ Z ∞ |u(x0 , 0)|q = − Dd |u(x0 , y)|q dy = −q |u(x0 , y)|q−2 Re(hDd u(x0 , y), u(x0 , y)i) dy 0

0

for every x0 ∈ Rd−1 . Therefore, Z ∞ 0 q |u(x , 0)| ≤q |u(x0 , y)|q−1 |Dd u(x0 , y)| dy 0

Z ≤q

∞ 0

p0 (q−1)

|u(x , y)|

1− p1  Z

∞ 0

p

|Dd u(x , y)| dy

dy

0

 p1 ,

0

where 1/p + 1/p0 = 1. Integrating both sides of the previous inequality over Rd−1 and using H¨older’s inequality, we infer that Z |u(x0 , 0)|q dx0 ≤ ||u||q−1 ||Dd u||Lp (Rd+ ;K) . Lp0 (q−1) (Rd ;K) Rd−1

+

Note that p0 (q−1) is the Sobolev exponent p∗ , if p < d, whereas it is a number larger than p, if

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p ≥ d. Therefore, by the Sobolev embedding theorems we conclude that ||u(·, 0)||Lq (Rd−1 ;K) ≤ Cp,q ||u||W 1,p (Rd+ ;K) . Using this inequality, we can extend the usual trace operator, defined on Cc∞ (Rd ; K) with a bounded operator mapping W 1,p (Rd+ ; K) into Lq (Rd−1 ; K). Let us prove that actually the trace of u ∈ W 1,p (Rd+ ; K) on ∂Rd+ belongs to the space 1−1/p,p B (Rd−1 ; K). For this purpose, we fix u ∈ Cc∞ (Rd+ ; K), j = 1, . . . , d − 1, and write Z xd Z y Dj u(x + σej ) dσ + Dd u(x0 , σ) dσ u(x0 + yej , 0) − u(x0 , 0) = 0 0 Z xd − Dd u(x0 + yej , σ) dσ 0

and estimate Z y Z Z y 1 y dxd |Dj u(x + σej )| dσ + dxd |Dd u(x0 , σ)| dσ y 0 0 0 0 Z Z y 1 y dxd |Dd u(x0 + yej , σ)| dσ + y 0 0 Z Z y Z y 1 y dσ = |Dj u(x + σej )| dxd + |Dd u(x0 , σ)| dσ y 0 0 0 Z y + |Dd u(x0 + yej , σ)| dσ

|u(x0 + yej , 0) − u(x0 , 0)| ≤

1 y

Z

y

0 0

for every y > 0 and x = (x , xd ) ∈ Rd+ . Raising the first and last side of the previous chain of inequalities to the power p, using both Jensen and H¨older inequalities, and observing that p  Z y Z y p Z y p−1 Z y −ε ε −εp0 ≤ = σ σ v(σ) dσ σ v(σ) dσ dσ σ εp |v(σ)|p dσ 0 0 0 0 Z y p−1−εp εp p ≤Cp y σ |v(σ)| dσ 0 0

∞

for every ε ∈ (0, 1/p ), v ∈ L ((0, y)) and some positive constant Cp , which is independent also of y, we can write |u(x0 + yej , 0) − u(x0 , 0)|p p Z y p Z Z Cp0 y y 0 0 |Dj u(x + σej )| dxd dσ + Cp |Dd u(x , σ)| dσ ≤ y 0 0 Z y0 p + Cp0 |Dd u(x0 + yej , σ)| dσ 0 Z y Z y Z y εp 00 p−2−εp p 00 p−1−εp 0 p ≤Cp y dσ xd |Dj u(x + σej )| dxd + Cp y xεp d |Dd u(x , xd )| dxd 0 0 0 Z y 0 p + Cp00 y p−1−εp xεp d |Dd u(x + yej , xd )| dxd 0

for some positive constants Cp0 and Cp00 , which are independent also of y. Integrating with respect to x0 over Rd−1 gives Z |u(x0 + yej , 0) − u(x0 , 0)|p dx0 d−1 R Z y Z y Z p 0 dxd ≤Cp00 y p−2−εp dσ xεp d |Dj u(x + σej )| dx 0

0

Rd−1

452

Smooth Domains and Extension Operators Z y Z 0 p 0 dxd + Cp00 y p−1−εp xεp d |Dd u(x , xd )| dx d−1 Z0 y ZR 00 p−1−εp 0 p 0 + Cp y dxd xεp d |Dd u(x + yej , xd )| dx 0 Rd−1 Z y Z 00 p−1−εp p 0 =Cp y dxd xεp d |Dj u(x)| dx 0 Rd−1 Z y Z 00 p−1−εp p 0 dxd + 2Cp y xεp d |Dd u(x)| dx . 0

Rd−1

Therefore, Z ∞

Z |u(x0 + yej , 0) − u(x0 , 0)|p 0 dy dx yp 0 Rd−1 Z ∞ Z y dy ≤2Cp00 xεp (||Dj u(·, xd )||pLp (Rd−1 ;K) + ||Dd u(·, xd )||pLp (Rd−1 ;K) ) dxd y 1+εp 0 d 0 Z ∞ Z ∞ 1 εp p p 00 =2Cp xd (||Dj u(·, xd )||Lp (Rd−1 ;K) + ||Dd u(·, xd )||Lp (Rd−1 ;K) ) dxd dy 1+εp y 0 xd bp (||Dj u||p p d ≤C + ||Dd u||p p d ). L (R+ ;K)

L (R+ ;K)

bp∗ |||∇u|||Lp (Rd ;K) . By density, we Hence, u(·, 0) belongs to B 1−1/p (Rd−1 ; K) and [u(·, 0)]p ≤ C + infer that the trace of u on ∂Rd+ belongs to B 1−1/p,p (Rd−1 ; K) for every u ∈ W 1,p (Rd+ ; K) and satisfies the previous estimate. The proof is complete if k = 1. The same arguments in the first part of the proof of this step also show that ∗ ||u(·, 0)||W k−1,p (Rd−1 ;K) + [Dβ u(·, 0)]p ≤ Cp,k ||u||W k,p (Rd ;K) for every multi-index β ∈ (N ∪ d−1 {0}) of order k − 1. By density, we conclude that Tr is bounded from W k,p (Rd+ ; K) into k−1/p,p B (Rd−1 ; K). Step 2. Fix u ∈ Cc∞ (Rd ; K) and k ∈ N. Further, let {Uxj : j = 1, . . . , N } be a finite covering of ∂Ω, ψxj : Uxj → B(0, r) be bijective functions of class C k together with their inverse functions and let {ηj : j = 1, . . . , N } be a partition of the unity associated with the above covering of ∂Ω. Since u ∈ W k,p (Ω), the function vj = (uηj ) ◦ ψx−1 belongs j to W k,p (B+ (0, r); K) for every j = 1, . . . , N , where B+ (0, r) = B(0, r) ∩ Rd+ , actually it belongs to W k,p (Rd+ ; K) since it is compactly supported in B+ (0, r). Therefore, from Step 1, we infer that 0 00 ||vh (·, 0)||B k−1/p,p (Rd−1 ;K) ≤ Cp,h ||vh ||W k,p (Rd+ ;K) ≤ Cp,h ||u||W k,p (Ω;K) ,

h ∈ N.

Define Tr(u) as the classical value of u on ∂Ω. From the definition of the space B k−1/p,p (∂Ω; K) and the above estimate, it follows that Tr(u) belongs to B k−1/p,p (∂Ω; K) e ∗ ||u||W k,p (Ω;K) . and ||Tr(u)||B k−1/p,p (Rd−1 ;K) ≤ N C p,h 0 ep,q Next, we observe that ||vi (·, 0)||Lq (Rd+ ;K) ≤ Cp,q ||vi ||W 1,p (Rd+ ;K) ≤ C ||u||W 1,p (Ω;K) , where q is as in the statement of the theorem (see Step 1). Applying the area-formula we infer that Z Z q d−1 |ui (x)| dH (x) = |vi (x0 , 0)|q gi (x0 ) dx0 , ∂Ω∩Uxi

B 0 (0,r)

where B 0 (0, r) denotes the ball of Rd−1 , centered at zero and with radius r, and gi (x0 ) is the square root of the determinant of the matrix A∗i (x0 )Ai (x0 ), where Ai (x0 ) is the Jacobian

Semigroups of Bounded Operators and Second-Order PDE’s

453

matrix of the function ψi−1 (·, 0) evaluated at x0 ∈ B 0 (0, r). In particular, the function g is bounded on B 0 (0, r). Since ui is compactly supported in ∂Ω ∩ Uxi , we can estimate Z Z q d−1 |ui (x)| dH (x) = |ui (x)|q dHd−1 (x) ∂Ω

∂Ω∩Uxi

Z ≤||gi ||∞ B 0 (0,r)

bq0 ||u||q 1,p |vi (x0 , 0)|q dx0 ≤ C W (Ω;K) .

PN bq00 ||u||W 1,p (Ω;K) . Observing that u = i=1 ui on ∂Ω, we conclude that ||u||Lq (∂Ω,Hd−1 ;K) ≤ C ∞ d k,p The density of Cc (R ; K) in W (Ω; K) allows us to extend the trace operator to W k,p (Ω; K) and complete the proof.  Remark B.3.2 From Theorem B.3.1 and the forthcoming Proposition B.4.15, it follows that the image of the trace operator, defined in W 1,p (Ω; K) is the Besov space B 1−1/p,p (∂Ω; K) for every p ∈ (1, ∞) and the image of its restriction to W k,p (Ω; K) is B k−1/p,p (∂Ω; K). It can be proved that if p = 1, the trace operator is bounded and surjective from W 1,1 (Ω; K) into L1 (∂Ω, Hd−1 ; K). We conclude this subsection, showing that W01,p (Ω; K) can be characterized as the kernel of the trace operator Tr. Theorem B.3.3 Fix p ∈ (1, ∞) and let Ω be Rd+ or a bounded open set of class C 1 . Then, a function u ∈ W 1,p (Ω; K) belongs to W01,p (Ω; K) if and only if Tr(u) = 0. Proof One inclusion is straightforward. Indeed, if u ∈ W01,p (Ω; K), then there exists a sequence (un ) contained in Cc∞ (Ω; K) and converging to u in W 1,p (Ω; K). Clearly, Tr(un ) = 0 for every n ∈ N and letting n tend to ∞, we conclude that Tr(u) = 0 as well. To prove the other part of the assertion, as in the proof of Theorem B.3.1 we first consider the case Ω = Rd+ . Take u ∈ W 1,p (Rd+ ; K) with null trace on ∂Rd+ and fix a sequence (un ) ⊂ Cc∞ (Rd ; K) converging to u in W 1,p (Rd ; K). Extend u by zero to Rd and denote by u the so obtained function. Then, for every ϕ ∈ Cc∞ (Rd ; K) and j ∈ {1, . . . , d} we get Z Z Z Z un Dj ϕ dx = un Dj ϕ dx = − un (x0 , 0)ϕ(x0 , 0) dx0 − ϕDj un dx. Rd +

Rd

Rd +

Rd−1

Since un (·, 0) converges to 0 in Lp (Rd−1 ; K), as n tends to ∞, from the previous formula we infer that Z Z Z uDj ϕ dx = uDj ϕ dx = − ϕDj u dx, Rd

Rd +

Rd +

so that the distributional derivative Dj u is the trivial extension to Rd of Dj u. Thus, u ∈ W 1,p (Rd ; K). Next, for every n ∈ N, let vn ∈ W 1,p (Rd ; K) be the function defined by vn (x) = u(x − −1 n ed ) for x ∈ Rd . The support of vn is contained in Rd−1 × [n−1 , ∞) and the sequence (vn ) converges to u in W 1,p (Rd+ ; K) as n tends to ∞, since the translations are continuous on Lp (Rd ; K). We regularize each function vn by convolution with a sequence (vm,n ), defined as in the proof of Theorem 1.3.4, which converges to vn in W 1,p (Rd+ ; K). Each function vm,n is compactly supported in Rd+ if m > n. This is enough to infer that u ∈ W01,p (Rd+ ; K). Let us now consider the case when Ω is a bounded open set of class C 1 . By Remark B.2.4, we can consider a finite covering {Uxj : j = 1, . . . , N − 1} of ∂Ω. which is also a covering of the set Ωδ := {x ∈ Ω : dist(x, ∂Ω) ≤ δ} for some δ > 0. Therefore, if we set

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Ωj = Uxj for each j = 1, . . . , N − 1 and ΩN = Ω \ Ωδ/2 , then the family {Ωj : j = 1, . . . , N } is an open finite covering of Ω. Now, we consider a function u ∈ W 1,p (Ω; K) with null trace on ∂Ω. By Proposition B.1.1, we can determine N smooth functions η1 , . . . , ηN such that supp(ηi ) ⊂ Ωi (i = 1, . . . , N ) PN PN and i=1 ηi ≡ 1 on Ω. Clearly, u = i=1 ui , where ui = uηi . Since uN is compactly supported in Ω it can approximated in W 1,p (Ω; K) by a sequence of functions which belong ∈ to Cc∞ (Ω; K). Hence, uN belongs to W01,p (Ω; K). If i < N , then the function vi = ui ◦ ψx−1 i W 1,p (B+ (0, r); K) and its support is contained in B+ (0, r0 ) for some r0 < r. Here, ψxi is the diffeomorphism of class C 1 , associated with the set Uxi (see Definition B.2.1). Extending vi by zero outside B+ (0, r) we get a function, still denoted by vi , which belongs to W 1,p (Rd+ ; K). Moreover, Z Z 0= |ui (x)|p dH(x) = |vi (x0 , 0)|p gi (x0 ) dx0 , B 0 (0,r)

∂Ω∩Ωj

where B 0 (0, r) is the ball in Rd−1 , centered at zero and with radius R, and gi (x0 ) is the square root of the determinant of the matrix A∗ (x0 )A(x0 ), A(x0 ) being the Jacobian matrix (·, 0) evaluated at x0 ∈ B 0 (0, r). In particular, there exists a positive of the function ψx−1 i constant C such gi (x0 ) ≥ C for every x0 ∈ B(0, R). Therefore, Z Z Z |vi (x0 , 0)|p dx0 . 0= |vi (x0 , 0)|p g(x0 ) dx0 ≥ C |vi (x0 , 0)|p dx0 = C B 0 (0,r)

B 0 (0,r)

Rd +

Hence, vi has null trace on ∂Rd+ . From the first part of the proof, it follows that vi ∈ W01,p (Rd+ ; K). We can thus determine a sequence (vn,i ) ⊂ Cc∞ (B+ (0, r); K) which converges to vi in W 1,p (Rd+ ; K) as n tends to ∞. The sequence (vn,i ◦ ψxi ) is contained in Cc∞ (Ω ∩ Ω ; K) and converges to ui in W 1,p (Ω; K). To complete the proof it suffices to set un = Pi N −1 i=1 vn,i ◦ ψxi for every n ∈ N, where vn,i ◦ ψxi is the trivial extension outside Ω ∩ Ωi of the function vn,i ◦ ψxi . The sequence (un ) is contained Cc∞ (Ω; K) and converges to u − uN in W 1,p (Ω; K). Hence, u belongs to W01,p (Ω; K). 

B.4

Extension Operators

In this section, we introduce some extension operators. We split the section into two subsections. In the first one, we extend functions defined on open sets Ω ⊂ Rd for some d > 1 to the whole Rd . In the second subsection we extend functions defined on the boundary of Rd+ or on a boundary of a smooth bounded open Ω.

B.4.1

Extending functions defined on open sets

We begin this subsection with the following proposition. Proposition B.4.1 Let Ω = Rd+ or Ω ⊂ Rd (d ≥ 2) be a smooth bounded open set of class C α for some α ∈ (0, ∞). Then, there exists a bounded linear operator Eα mapping Cb (Ω; K) into Cb (Rd ; K) with the following two properties: (i) Eα f = f in Ω for each f ∈ Cb (Ω; K); (ii) Eα is bounded from C β (Ω; K) into Cbβ (Rd ; K) for each β ∈ [0, α].

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Proof We split the proof into two steps. In the first one, we consider the easiest case Ω = Rd+ and in the latter step, we address the case of a domain of class C α . Step 1. Fix α ∈ (0, ∞), f ∈ Cb (Rd+ ; K) and set m = [α] + 1. We define Eα f on Rd by setting  f (x), xd ≥ 0,   m X (Eα f )(x) = γj f (x0 , −jxd ), xd < 0,   j=1

where the constants γ1 , . . . , γm are solutions to the system      1 γ1 1 1 1 ... ...1     1 2 3 ... ...m    γ2   −1         .. .. . .. .. .. ..   ..   . . . . . .        1 2j 3j . . . . . . mj   γj  =  (−1)j−1  .       . .. .. .. .. ..   ..   .  . . . . . .  .   (−1)m−1 γm 1 2m 3m . . . . . . mm

(B.4.1)

The matrix of the above linear system is the transpose of a Vandermonde matrix. In particular, it is invertible so that the system admits a unique solution (γ1 , . . . , γm ). Clearly, the function Eα f belongs to Cbβ (Rd ; K) for every f ∈ C β (Rd+ ; K), β ∈ [0, α] and ||Eβ f ||C β (Rd ;K) ≤ Cm ||f ||C β (Rd ;K) . b

b

+

Step 2. Let {Ux : x ∈ ∂Ω} be as in Definition B.2.1. Since {Ux : x ∈ ∂Ω} is an open covering of ∂Ω, which is a compact set, there exists a finite subcovering {Uxj : j = 1, . . . , N − SN −1 1}. Note that there exists δ > 0 such that j=1 Uxj ⊃ Ωδ := {x ∈ Ω : dist(x, ∂Ω) ≤ δ} (see Remark B.2.4). To simplify the notation, we write Uj and ψj instead of Uxj and ψxj . Moreover, we set UN = Ω \ Ωδ/2 . The above arguments show that the family {Uj : j = 1, . . . , N } is an open finite covering of Ω. Let us fix now a function f ∈ C α (Ω; K). By Proposition B.1.1, we can determine N PN smooth functions η1 , . . . , ηN such that supp(ηj ) ⊂ Uj (j = 1, . . . , N ) and j=1 ηj2 ≡ 1 on PN Ω. Clearly, f = j=1 ηj fj , where fj = ηj f for each j = 1, . . . , N . We are going to prove that each function fk can be extended to Rd with a function which belongs to Cbα (Rd ; K). For this purpose, we treat separately the cases k < N and k = N. The second case is easier. Indeed, since the function ηN is compactly supported in Ω, the trivial extension fN of function fN belongs to Cbα (Rd ; K) and ||fN ||Cbα (Rd ;K) = ||fN ||C α (Ω;K) . Let us consider the trickier case k < N . We note that the function fj = fj ◦ ψj−1 belongs to C α (B+ (0, R); K) and ||fj ||C α (B+ (0,R);K) ≤ C||fj ||C α (Ω∩Uj ;K) . Since ηj is compactly supported in Uj , function fj is supported in B+ (0, r) for some r < R. Hence, we can extend it in the trivial way to Rd+ with a function efj which belongs to Cbα (Rd+ ; K) with norm which can be bounded from above in terms of the C α -norm of f . We extend efj to the whole Rd as explained in Step 1 and we denote by Eα0 efj the so extended function. Clearly, the function (Eα0 efj ) ◦ ψj belongs to C α (Uj ; K) and ||(Eα0 efj ) ◦ ψj ||C α (Uj ;K) ≤ C||f ||C α (Ω;K) . We can now define the function Ef by setting Eα f =

N −1 X j=1

ηj (Eα0 efj ) ◦ ψj + ηN fN .

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Smooth Domains and Extension Operators

It is immediate that operator Eα has the properties (i) and (ii) in the statement of the proposition.  Remark B.4.2 If α ∈ (0, 1), then each function f ∈ Cbα (Ω; K) can be extended uniquely up to the boundary of Ω with a function which belongs to C α (Ω; K), see Exercise 1.5.4, so that the spaces Cbα (Ω; K) and C α (Ω; K) can be identified, with no requirement of the smoothness of Ω. The previous (topological) equivalence still holds for not integers α > 1 if Ω is a bounded open set of class C α , as revealed by the proof of Proposition B.4.1. Indeed, suppose that f ∈ C α (Ω; K). Since the function fN is compactly supported in Ω, its trivial extension to Rd clearly belongs to Cbα (Rd ; K). On the other hand, if k < N , then the function fk belongs to C α (B+ (0, R); K). Since B+ (0, R) is convex, then, using the mean value theorem, it is easy to check that Cb1 (B+ (0, R); K) ,→ C α (B+ (0, R); K) and this shows that all the derivatives of fk can be extended by continuity up to the boundary of B+ (0, R), so that, actually, fk ∈ C α (B+ (0, R); K). As a byproduct, the function ηk fk belongs to C α (Ω; K). PN Since f = k=1 ηk fk , we conclude that f ∈ C α (Ω; K) as claimed. Remark B.4.3 The arguments in Remark B.4.2 show in particular that Cb1 (Ω) continuously embeds into C α (Ω) for every α ∈ (0, 1). Note that without any requirement on the geometry of Ω this result is false, in general. For instance, if Ω = {(x, y) ∈ B((0, 0), 1) : y < p 2 |x|} and f : Ω → R is the function defined by ( sign(x)y 3/2 , (x, y) ∈ (R \ {0}) × (0, ∞), f (x, y) = 0, otherwise. Function f clearly belongs to Cb1 (Ω) but it does not belong to C α (Ω) if α > 3/4. Indeed, for |x| small enough it holds that |f (x, |x|1/2 ) − f (−x, |x|1/2 )| = 2|x|3/4 , whereas |(x, |x|1/2 ) − (−x, |x|1/2 )|α = 2|x|α . As x tends to zero, we are led to a contradiction. Remark B.4.4 Proposition B.4.1 holds true also in the case d = 1 and the proof is easier. Indeed, if Ω is a right halfline, up to a translation we can assume that it coincides with [0, ∞). We can define the operator Eα by setting (Eα f )(x) =

m X

γj f (−jx),

x≤0

j=1

for every f ∈ Cb ([0, ∞); K), where m denotes the integer part of α and the constants γ1 , . . . , γm are solutions to the linear system (B.4.1). If Ω is a left-halfline, then the construction of the operator Eα is completely similar. Finally, if Ω is a bounded interval, say [a, b], we first extend f : [a, b] → C to [a, 2b − a] by setting  f (x), x ∈ [a, b],   m X e f (x) = γj f (−j(2b − x)), x ∈ (b, 2b − a],   j=1

where the constants γj (j = 1, . . . , m) are determined through system (B.4.1). Next we consider a compactly supported function η ∈ C ∞ (R) such that η ≡ 1 in [a, b] and η ≡ 0 outside [a, 2b−a]. Function η fe is defined in [a, ∞). Hence, we can extend it to R as explained at the beginning of the remark. It is easy to check that the extension operator so constructed has all the properties listed in the statement of Proposition B.4.1.

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As the following proposition shows, we can define an extension operator also in the Sobolev space W k,p (Ω; K) provided that Ω is sufficiently smooth. Actually, as the proof shows, this extension operator coincides with the operator defined in Proposition B.4.1. Proposition B.4.5 Let Ω = Rd+ or Ω ⊂ Rd (d ≥ 2) be a smooth domain of class C k for some k ∈ N. Then, there exists a bounded linear operator Ek mapping Lp (Ω; K) into Lp (Rd ; K) for every p ∈ [1, ∞] with the following two properties: (i) Ek f = f in Ω for each f ∈ Lp (Ω; K); (ii) Ek is bounded from W j,p (Ω; K) into W j,p (Rd ; K) for every j = 0, . . . , k. Proof We first consider the case Ω = Rd+ . We fix k ∈ N, f ∈ Lp (Rd+ ; K) and, as in the proof of Proposition B.4.1, we set  f (x), xd ≥ 0,    k X Ek f (x) =  γj f (x0 , −jxd ), xd < 0,   j=1

where the constants γ1 , . . . , γk are defined by (B.4.1). We fix a multi-index β with length at most k and split β = (β 0 , βd ), with βd ∈ N ∪ {0}. Then, we observe that for every u ∈ W k,p (Rd+ ; K) and ϕ ∈ Cc∞ (Rd ; K), it holds that Z

Z

β

β

(Ek f )(x)(D ϕ)(x) dx =

f (x)D ϕ(x) dx + Rd +

Rd

k X

Z

f (x0 , −jxd )Dβ ϕ(x) dx

γj

j=1

Rd −

  k X β −1 β 0 −1 f (x) D ϕ(x) + j γj D ϕ(x , −j xd ) dx

Z = Rd +

Z

j=1

f (x)Dβ ψ(x) dx,

= Rd +

where ψ(x) = ϕ(x) + (−1)

βd

k X

j βd −1 γj ϕ(x0 , −j −1 xd ) for every x ∈ Rd+ . Now, for every

j=1

n ∈ N and s ≥ 0, we set ϑn (s) = ϑ(ns), where ϑ ∈ Cb∞ ([0, ∞)) satisfies the conditions ϑ(s) = 0 for every s ∈ [0, 1/2] and ϑ(s) = 1 for every s ≥ 1. As it is easily seen, β

ϑn D ψ =

0 ϑn Ddβd Dxβ0 ψ

β

= D (ϑn ψ) −

βX d −1  h=0

 βd (βd −h) β 0 h ϑn Dx0 Dd ψ, h

(h) ϑn

where denotes the h-th derivative of the function ϑn . Therefore, Z Z β (Ek f )D ϕ dx = lim f ϑn Dβ ψ dx n→∞

Rd

Rd +

Z = lim

n→∞

Rd +

f Dβ (ϑn ψ) dx −

βX d −1  h=0

βd h



Z lim

n→∞

Rd +

0

f ϑn(βd −h) Dxβ0 Ddh ψ dx.

Since ϕ is compactly supported in Rd+ , the function ϑn ψ is compactly supported in Rd+ so that Z Z lim f Dβ (ϑn ψ) dx =(−1)|β| lim (Dβ f )ϑn ψ dx n→∞

Rd +

n→∞

Rd +

458

Smooth Domains and Extension Operators Z Z =(−1)|β| (Dβ f )ψ dx = (−1)|β| Rd +

gβ ϕ dx,

Rd

where  Dβ f (x), xd > 0,    k X gβ (x) =  (−1)βd j βd γj Dβ f (x0 , −jxd ), xd < 0.   j=1

Function gβ belongs to Lp (Rd ; K) and ||gβ ||Lp (Rd ;K) ≤ Ck ||Dβ f ||Lp (Rd+ ;K) , where the positive constant Ck is independent of f . Hence, if βd = 0, then gβ is the distributional derivative Dβ f . So, let us suppose that βd ≥ 1. We claim that Z 0 d −h) lim f ϑ(β Dxβ0 Ddh ψ dx = 0, h = 0, . . . , βd − 1. (B.4.2) n n→∞

Rd +

(β −h)

For this purpose, we observe that |ϑn d Moreover, 0

(xd )| = nβd −h |ϑ(βd −h) (nxd )| for every xd > 0.

Dxβ0 Ddm ψ(x) = Dxβ0 Ddm ϕ(x) + (−1)βd +m

k X

0

j βd −1−m γj Dxβ0 Ddm ϕ(x0 , −j −1 xd )

j=1 0

for every x ∈ Rd+ and m ∈ N. Note that (B.4.1) implies that Ddi Dxβ0 Ddh ψ identically vanishes on ∂Rd+ for every i = 0, . . . , βd − h − 1. Using the Taylor expansion of the 0 e such that function Dxβ0 Ddh ψ(x0 , ·), we can show that there exists a positive constant C 0 β β −h d h d e |D 0 D ψ(x)| ≤ Cx for every x ∈ R+ . Thus, we can estimate x

d

d

0

e (x)||nxd |βd −h |ϑ(βd −h) (nxd )| |f (x)ϑn(βd −h) (xd )Dxβ0 Ddh ψ(x)| ≤C|f e sup |sβd −h ϑ(βd −h) (s)||f (x)| ≤C s≥0 (β −h)

0

Dxβ0 Ddh ψ vanishes pointwise on Rd+ as n tends to ∞. for every x ∈ Rd+ . Moreover, f ϑn d Therefore, (B.4.2) follows by dominated convergence. As a byproduct, also in this situation the distributional derivative Dβ Ek f belongs to Lp (Rd ; K). Summing up, we have proved that Ek f belongs to W k,p (Rd ; K) and properties (i) and (ii) follows. The general case can be addressed arguing as in Step 2 of the proof of Proposition B.4.1. As a starting point, we fix a finite open subcovering {Uj : j = 1, . . . , N } of Ω, where the sets U1 , . . . , UN −1 intersect ∂Ω, whereas UN b Ω. Associated with this covering, we consider PN functions ηj ∈ Cc∞ (Uj ) such that i=1 ηi2 ≡ 1 on Ω. Finally, for every j = 1, . . . , N − 1 we denote by ψj : Uj → B(0, R) a bijective function of class C k together with its inverse, such that ψj (Uj ∩ Ω) = B+ (0, R) and ψj (Uj ∩ ∂Ω) = {x ∈ B(0, R) : xd = 0}. We recall here that B+ (0, R) = B(0, R) ∩ Rd+ . Given f ∈ W j,p (Ω; K) (for some j = 0, . . . , k, where W 0,p = Lp ), we set Ek f = PN j,d (Rd ; K) (h = 1, . . . , N ) are defined as follows. h=1 ηh gh , where the functions gh ∈ W −1 If h < N , then function fh = (ηh f ) ◦ ψh belongs to W j,p (B+ (0, R); K) and is supported in B+ (0, r) for some r < R. Hence, its trivial extension by zero to the whole Rd+ , say fh , belongs to W j,p (Rd+ ; K) and ||fh ||W j,p (Rd+ ;K) ≤ Ch ||f ||W j,p (Ω;K) for some positive constant Ch , independent of f . Denote by gbh the function obtained extending the previous formula to the whole Rd , as explained in the first part of the proof. Finally, set gh = gbh ◦ ψh . Clearly,

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459

bh ||f ||W j,p (Ω;K) , the constant C bh being ingh belongs to W j,p (Rd ; K) and ||gh ||W 1,p (Rd ;K) ≤ C d dependent of f . Finally, we denote by gN the trivial extension to R of the function ηN f . bN ||f ||W j,p (Ω;K) , since ηN f is compactly It belongs to W j,p (Rd ; K) and ||gN ||W 1,p (Rd ;K) ≤ C Pk 2 supported in Ω. Observing that f = h=1 ηh f on Ω, it is easy to check that Ek f extends the function f to Rd . Moreover, it enjoys properties (i) and (ii) in the statement of the proposition.  Remark B.4.6 Extending a function u ∈ W0k,p (Ω; K) (k ∈ N and p ∈ [1, ∞) ∪ {∞}) with a function v ∈ W k,p (Rd ; K) is easier and we do not need to require any smoothness assumption on the open set Ω. Indeed, it suffices to extend u by zero to the whole Rd . Indeed, since u ∈ W0k,p (Ω; K), then there exists a sequence (ψn ) ⊂ Cc∞ (Ω; K) which converges to u in W k,p (Ω; K). Therefore, for every ϕ ∈ Cc∞ (Rd ; K) and every multi-index β with length at most k, we can write Z Z Z vDβ ϕ dx = uDβ ϕ dx = lim ψn Dβ ϕ dx n→∞ d R Ω Ω Z Z |β| β =(−1) lim D ψn ϕ dx = (−1)|β| Dβ uϕ dx n→∞ Ω Ω Z |β| =(−1) Dβ vϕ dx, Rd

so that the distributional derivative Dβ v is the trivial extension to Rd of the function Dβ u. It belongs to Lp (Rd ; K) and ||Dβ v||Lp (Rd ;K) = ||Dβ u||Lp (Ω;K) . Thus, we conclude that v ∈ W k,p (Rd ; K) and ||v||W k,p (Rd ;K) = ||u||W k,p (Ω;K) . Remark B.4.7 If u ∈ W 2,p (Rd+ ; K) ∩ W01,p (Rd+ ; K), then we can extend u to Rd with a function v ∈ W 2,p (Rd ; K) in a simpler way than we did in the proof of Theorem B.4.5: it suffices to consider the function uo , which is the odd extension of u with respect to the last variable. Indeed, let (ψn ) ⊂ Cc∞ (Rd ; K) be a sequence converging to u in W 2,p (Rd+ ; K) (see eo : Rd+ → K the function, Theorem 1.3.4). Moreover, for every ϕ ∈ Cc∞ (Rd ; K), denote by ϕ 0 d defined by ϕo (x) = ϕ(x) − ϕ(x , −xd ) for every x ∈ R+ . Then, for every j ∈ {1, . . . , d − 1} we can write Z Z Z uo Dj ϕ dx = uDj ϕ eo dx = lim ψn Dj ϕ eo dx Rd

n→∞

Rd +

Rd +

Z

Z

= − lim

n→∞

Rd +

Dj ψn ϕ eo dx = −

Rd +

Dj uϕ eo dx,

so that the distributional derivative Dj uo is the odd extension with respect to the variable xd of the derivative Dj u. Hence, it belongs to Lp (Rd ; K). Similarly, we can prove that the distributional derivative Dij uo is the odd extension with respect to the last variable of the function Dij u for every i, j = 1, . . . , d − 1. Therefore, Dij uo belongs to Lp (Rd ; K). To prove that also the distributional derivatives Dd uo and Djd uo belong to Lp (Rd ; K), we argue as follows. We fix a sequence (ψbn ) ⊂ Cc∞ (Rd+ ; K), which converges to u in W 1,p (Rd ; K), and a function ϕ ∈ Cc∞ (Rd ; K). Denoting by ϕ ee : Rd+ → K the function defined by ϕ ee (x) = 0 d ϕ(x) + ϕ(x , −xd ) for every x ∈ R+ and integrating by parts, we get Z Z Z uo (x)Dd ϕ(x) dx = lim ψbn Dd ϕ ee dx = − Dd uϕe dx, Rd

n→∞

Rd +

Rd +

so that Dd uo is the even extension with respect to xd of the function Dd u and it belongs to

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Smooth Domains and Extension Operators

Lp (Rd ; K). Next, we fix a function η ∈ C ∞ ([0, ∞)) such that χ[1,∞) ≤ η ≤ χ[0,∞) and set ηn (xd ) = η(nxd ) for every xd ∈ [0, ∞). Then, we observe that Z Z Z Djd uo ηn ϕ ee dx Dd uDj (ηn ϕ ee ) dx = − lim Dd uo Dj ϕ dx = lim n→∞

Rd

n→∞

Rd +

Rd +

Z =− Rd +

Djd uo ϕ ee dx

for every j = 1, . . . , d − 1, so that Djd uo is the even extension with respect to the last variable of the function Dd u and it belongs to Lp (Rd ; K). Finally, we observe that Z Z Dd uo ηn Dd ϕo dx Dd uo Dd ϕdx = lim n→∞

Rd

Rd +

Z

Z Dd uo Dd (ηn ϕo ) dx − lim

= lim

n→∞

n→∞

Rd +

Z = − lim

n→∞

Rd +

Dd uo ϕo Dd ηn dx Rd +

Z Ddd uηn ϕ eo dx − lim

n→∞

Dd uo ϕo Dd ηn dx Rd +

Z = Rd +

Ddd uϕ eo dx,

so that Ddd uo is the odd extension with respect to xd of the function Ddd u. We have so proved that uo ∈ W 2,p (Rd ; K). Remark B.4.8 Arguing as in Remark B.4.7, it can be easily proved that, if u ∈ W 2,p (Rd+ ; C) for some p ∈ [1, ∞) has null normal derivative on ∂Rd+ , then it can be extended to Rd , even with respect to the last variable. The so obtained function ue belongs to W 2,p (Rd ; C) and ||ue ||W 2,p (Rd ;C) ≤ 21/p ||u||W 2,p (Rd+ ;C) .

B.4.2

Extending functions defined on the boundary of a set

We begin this subsection extending to Rd functions which are H¨older continuous on the boundary of a sufficiently smooth and bounded subset of Rd . Proposition B.4.9 Let Ω be a bounded open set of class C α for some α > 0. Then, there exists an extension operator Eα0 which is bounded from C(∂Ω; K) into C(Ω; K). Moreover, for every β ∈ [0, α], its restriction to C β (∂Ω; K) is also bounded from C β (∂Ω; K) into Cbβ (Rd ; K). Proof Let {Uk : k = 1, . . . , N } be a finite covering of ∂Ω consisting of open sets and denote by ψk : Uk → B(0, r) the associated diffeomorphism of class C α , which is invertible with inverse function of class C α . We consider a partition of the unity {η1 , . . . , ηN } PN subordinated to the previous covering and split u = k=1 uk , where uk = ηk u for every k = 1, . . . , N . Fix a function u ∈ C β (∂Ω; K) for some β ∈ [0, α] and, for every k = 1, . . . , N , let wk : B 0 (0, r) → K be the function defined by wk (y) = uk (ψk−1 (y, 0)) for every y ∈ B 0 (0, r), where B 0 (0, r) is the ball of Rd−1 centered at the origin, with radius r. Function wk belongs to C β (B 0 (0, r); K) and it is easy to extend it to B(0, r), by setting w ek (y) = wk (y1 , . . . , yd−1 , 0) for every y ∈ B(0, r), and this new function belongs to C β (B(0, r); K) and satisfies the estimate ||w ek ||C β (B(0,r);K) = ||wk ||C β (B 0 (0,r);K) ≤ C1 ||u||C β (∂Ω;K) for some positive constant C1 , independent of u. We now multiply w ek by function ϑk ∈ Cc∞ (B(0, r)) which is equal

Semigroups of Bounded Operators and Second-Order PDE’s

461

to one in ψk (supp(ηk )). As a byproduct, the function (ϑk w ek ) ◦ ψk belongs to C β (Uk ; K), is compactly supported in Uk and equals the function uk on Uk ∩ ∂Ω. We extend this function in the trivial way to the whole Rd and we denote by u ek the so obtained function. Then, it follows that ||e uk ||C β (Rd ;K) ≤ C2 ||u||C β (∂Ω;K) , where the constant C2 is independent of u. The PN b function u e = k=1 u ek belongs to Cbβ (Rd ; K) and equals u on ∂Ω. Moreover, ||e u||C β (Rd ;K) ≤

N X

b

||e uk ||C β (Rd ;K) ≤ C2 N ||u||C β (∂Ω;K) . b

k=1

We set Eα0 u := u e. Clearly, the operator Eα0 is bounded from C β (∂Ω; K) into Cbβ (Rd ; K) for every β ∈ [0, α].  Corollary B.4.10 Let Ω be a bounded open set of class C 2+α for some α ∈ (0, 1). Then, there exists a linear bounded operator Eα0 : C([0, T ] × ∂Ω; K) → C([0, T ] × Ω; K) such that its restriction to C 1+α/2,2+α ((0, T )×∂Ω; K) is bounded from this space into C 1+α/2,2+α ((0, T )× Ω; K). Proof The proof is analogous to that of Proposition B.4.9 and in fact the operator Eα0 is defined in the same way. Hence, we skip some details. We fix a finite covering {Uk : k = 1, . . . , N } of ∂Ω and denote by ψk : Uk → B(0, r) the diffeomorphism of class C 2+α associated with the open set Uk , which admits inverse function of class C 2+α as well. Moreover, we consider a partition of the unity {η1 , . . . , ηN } subordinated to this covering PN of ∂Ω and split u = k=1 uk , where uk = ηk u for every k = 1, . . . , N . Fix a function u ∈ C([0, T ] × ∂Ω; K) and, for every k = 1, . . . , N , let vk : [0, T ] × B 0 (0, r) → K be the function defined by vk (t, y) = uk (t, ψk−1 (y, 0)) for every t ∈ [0, T ] and y ∈ B 0 (0, r), where B 0 (0, r) is the ball of Rd−1 centered at the origin, with radius r. Function vk belongs to C([0, T ] × B 0 (0, r); K) and we extend it to [0, T ] × B(0, r) by setting vek (t, y) = ϑk (y)vk (t, y1 , . . . , yd−1 , 0) for every t ∈ [0, T ] and y ∈ B(0, r), where ϑk ∈ Cc∞ (B(0, r)) is equal to one on ψk (supp(ηk )). Function vek belongs to C([0, T ] × B(0, r); K) and ||e vk ||C([0,T ]×B(0,r);K) ≤ C1 ||u||C([0,T ]×∂Ω;K) for some positive constant C1 , independent of u. Hence, the function (t, x) 7→ u0k (t, x) := vek (t, ψk (x)) belongs to C([0, T ] × Uk ; K), is compactly supported in [0, T ]×Uk and equals the function uk on Uk ∩∂Ω. We can thus extend PN this function in the trivial way to [0, T ]×Ω. Now, we can set Eα0 u = k=1 u0k , where u0k is the trivial extension of function u0k to [0, T ] × Ω. Operator Eα0 is bounded from C([0, T ] × ∂Ω; K) into C([0, T ] × Ω; K) and the function Eα0 u equals the function u on ∂Ω. Finally, if u ∈ C 1+α/2,2+α ([0, T ] × ∂Ω; K), then the function vk belongs to C 1+α/2,2+α ((0, T ) × B 0 (0, r); K) and ||vk ||C 1+α/2,2+α ((0,T )×B 0 (0,r);K) ≤ C||u||C 1+α/2,2+α ([0,T ]×∂Ω;K) for some positive constant C2 , independent of u. Hence, repeating the above arguments, we conclude that the operator Eα0 is bounded from C 1+α/2,2+α ([0, T ] × ∂Ω; K) into C 1+α/2,2+α ((0, T ) × Ω; K).  Remark B.4.11 Without any smoothness assumption on the bounded domain Ω, for every α ∈ (0, 1) we can define a nonlinear operator mapping C α (∂Ω; K) into C α (Ω; K). Given a function u ∈ C α (∂Ω; K), we can extend it to Ω with the function u e, defined by u e(x) = inf{ψ(y) + [ψ]C α (∂Ω;K) |x − y|α : y ∈ ∂Ω},

x ∈ Ω.

Clearly, u e(x) = u(x) for every x ∈ ∂Ω. To prove that u e belongs to C α (Ω; K), we fix x1 , x2 ∈ Ω α and observe that u(y) + [u]C α (∂Ω;K) |x2 − y| ≤ u(y) + [u]C α (∂Ω;K) (|x1 − y|α + |x2 − x1 |α ) for every y ∈ ∂Ω. Therefore, u e(x2 ) ≤ u(y) + [u]C α (∂Ω;K) |x1 − y| + [u]C α (∂Ω;K) |x2 − x1 |α for every y ∈ ∂Ω. Taking the minimum with respect to y ∈ ∂Ω, we deduce that u e(x2 ) ≤ u e(x1 ) + [u]C α (∂Ω;K) |x2 − x1 |α . Interchanging x1 and x2 , we conclude that |e u(x2 ) − u e(x1 )| ≤

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Smooth Domains and Extension Operators

[u]C α (∂Ω;K) |x2 −x1 |α . Thus, [e u]C α (Ω;K) ≤ [u]C α (∂Ω;K) . Moreover, if x ∈ Ω and y ∈ ∂Ω, we can estimate |e u(x)| ≤ |u(y)|+|e u(x)−e u(y)| ≤ ||u||∞ +[u]C α (∂Ω;K) diam(Ω), where diam(Ω) denotes the diameter of Ω. Summing up, we have proved that u e ∈ C α (Ω; K) and ||e u||C α (Ω;K) ≤ 2||u||C α (∂Ω;K) . The same definition can be used to extend u ∈ Lip(∂Ω; K) with a function u ∈ Lip(Ω; K). Note that the above definition cannot be used when Ω is replaced by Rd+ . Indeed, in this case the function u e belongs to C α (Rd+ ; K) but it is not bounded as it follows easily from observing that u(y) + [u]Cbα (∂Rd+ ;K) |x − y|α ≥ [u]Cbα (∂Rd+ ;K) |x − y|α − ||u||∞ ≥ [u]Cbα (∂Rd+ ;K) xα d − ||u||∞ d for every y ∈ ∂Rd+ , so that u e(x) ≥ [u]Cbα (∂Rd+ ) xα d − ||u||∞ for every x ∈ R+ .

The following propositions are used in Chapters 8, 11 and 12. Proposition B.4.12 The following properties are satisfied. (i) Let η ∈ Cbα (Rd−1 ; Rd ) be a function such that |η(x0 )| = 1 for every x0 ∈ Rd−1 and supRd−1 ηd < 0, where ηd denotes the last component of the function η. Then, there exists a bounded linear operator E mapping Cb (Rd−1 ; K) into Cb1 (Rd+ ; K) such that ∂Eψ Eψ ≡ 0 and ≡ ψ on ∂Rd+ for every ψ ∈ Cb (Rd−1 ; K). Moreover, E is also ∂η bounded from Cbβ (Rd−1 ; K) into Cbβ+1 (Rd+ ; K) for every β ∈ [1, 2). (ii) Let A = Tr(QD2 ) + hb, ∇i + c be a differential operator, whose coefficients belong to Cbα (Rd+ ) for some α ∈ (0, 1) and hQ(x)ξ, ξi ≥ µ|ξ|2 for every x ∈ Rd+ , ξ ∈ Rd and some positive constant µ. Then, there exists a bounded linear operator S mapping Cbα (Rd−1 ; K) into Cb2+α (Rd+ ; K) such that Sψ = 0 on ∂Rd+ and ASψ = ψ on ∂Rd+ for every ψ ∈ Cbα (Rd+ ; K). Proof (i) We fix a function ϑ ∈ Cc∞ (Rd−1 ), with 0 ≤ ϑ ≤ 1 and ||ϑ||L1 (Rd−1 ) = 1, and a function ζ ∈ C ∞ ([0, ∞)) such that ζ(s) = s for every s ∈ [0, 1/2] and ζ(s) = 0 for every s ≥ 1. Then, we define the operator T by setting Z (T ψ)(x) = ζ(xd ) ψ(x0 − xd y)ϑ(y) dy, x ∈ Rd+ , Rd−1

for every ψ ∈ Cb (Rd−1 ; K). Clearly, T ψ vanishes on ∂Rd+ . Moreover, writing  0  Z x −z (T ψ)(x) = x1−d ζ(x ) ψ(z)ϑ dz d d xd Rd−1 we can easily show that the function T ψ is continuously differentiable in Rd+ and Z −1 (Di T ψ)(x) = ζ(xd )xd ψ(x0 − xd y)Di ϑ(y) dy, Rd−1

Z (Dd T ψ)(x) = [ζ 0 (xd ) − (d − 1)x−1 ζ(x )] ψ(x0 − xd y)ϑ(y) dy d d Rd−1 Z −1 − ζ(xd )xd ψ(x0 − xd y)h∇ϑ(y), yi dy Rd−1

Rd+

for every x ∈ and i = 1, . . . , d − 1. Using the definition of ζ, we can extend the previous functions by continuity to Rd+ . Moreover, we can estimate ||T ψ||C 1 (Rd ;K) ≤ C1 ||ζ||Cb1 ([0,∞)) ||ψ||∞ b

+

(B.4.3)

Semigroups of Bounded Operators and Second-Order PDE’s for some constant C1 , independent of ψ. In particular,   Z 0 (Dd T ψ)(x , 0) = 2 − d − h∇ϑ(y), yi dy ψ(x0 ) = ψ(x0 ),

463

x0 ∈ Rd−1 .

Rd−1

We now suppose that ψ ∈ Cbβ (Rd−1 ; K) for some β ∈ [1, 2). As it is easily seen, Z −1 (Dij T ψ)(x) = xd ζ(xd ) Di ψ(x0 − yxd )Dj ϑ(y) dy, Rd−1

Z (Did T ψ)(x) = [(1 − d)xd−1 ζ(xd ) + ζ 0 (xd )] Di ψ(x0 − xd y)ϑ(y) dy Rd−1 Z −1 − xd ζ(xd ) Di ψ(x0 − xd y)h∇ϑ(y), yi dy Rd−1

for every x ∈

Rd+

and i, j = 1, . . . , d − 1. Similarly, writing   0 Z Z x −z ζ(xd ) 0 0 0 dz h∇ψ(z), x − ziϑ (Dd T ψ)(x) = ζ (xd ) ψ(x − xd y)ϑ(y) dy − xd xdd Rd−1 Rd−1

for every x ∈ Rd+ , we easily see that Dd T ψ is continuously differentiable with respect to xd in Rd+ and Z (Ddd T ψ)(x) = ζ 00 (xd ) ψ(x0 − xd y)ϑ(y) dy Rd−1 Z −1 0 − [2ζ (xd ) − dxd ζ(xd )] h∇ψ(x0 − xd y), yiϑ(y) dy Rd−1 Z −1 + xd ζ(xd ) h∇ψ(x − xd y), yih∇ϑ(y), yi dy. Rd−1

From the previous formulas, it follows that the second-order derivatives of T ψ are actually continuous on Rd+ and ||Dij T ψ||∞ ≤ ||ζ 0 ||∞ ||Di ψ||∞ ||Dj ϑ||L1 (Rd−1 ) , ||Did T ψ||∞ ≤ C2 ||ζ 0 ||∞ ||Di ψ||∞ , ||Ddd T ψ||∞ ≤ C3 (||ζ 00 ||∞ ||ψ||∞ + ||ζ 0 ||∞ |||∇ψ|||∞ ) for some positive constants C2 and C3 , independent of ψ. The assertion follows if β = 1. If β ∈ (1, 2), then we can estimate [Dij T ψ]C β−1 (Rd ;K) ≤[ζ 0 ]C β−1 ([0,∞)) ||Di ψ||∞ ||Dj ϑ||L1 (Rd−1 ) b

+

b

+ ||ζ 0 ||∞ [Di ψ]C β−1 (Rd−1 ;K) ||Dj ϑ||L1 (Rd−1 ) b Z 0 + ||ζ ||∞ [Di ψ]C β−1 (Rd−1 ;K) |y|β−1 |Dj ϑ(y)| dy b

Rd−1

0

≤C4 ||ζ ||C β−1 ([0,∞)) ||Di ψ||C β−1 (Rd−1 ;K) b

b

so that ||Dij T ψ||C β−1 (Rd ;K) ≤ C5 ||ζ 0 ||C 1+β−1 ([0,∞)) ||Di ψ||C β−1 (Rd−1 ;K) +

b

b

(B.4.4)

b

for every i, j = 1, . . . , d − 1. Here, C4 and C5 are positive constants, independent of ψ. Arguing in the same way, we can show that ||Did T ψ||C β−1 (Rd ;K) ≤ C6 ||ζ 0 ||C β−1 ([0,∞)) ||Di ψ||C β−1 (Rd−1 ;K) b

+

b

b

(B.4.5)

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Smooth Domains and Extension Operators

for every i = 1, . . . , d − 1 and ||Ddd T ψ||C β−1 (Rd ;K) ≤ C6 ||ζ||C β+1 ([0,∞)) ||ψ||C β (Rd−1 ;K) , +

b

b

(B.4.6)

b

where the constant C6 is independent of ψ. Estimates (B.4.3)–(B.4.6) show that T is bounded from Cbβ (Rd−1 ; K) into Cbβ+1 (Rd+ ; K). ψ Finally, we set Eψ = T for every ψ ∈ Cb (Rd−1 ; C). Since the supremum over Rd−1 of ηd the function ηd is negative and ηd ∈ Cb1+α (Rd−1 ), the function ψ/ηd belongs to Cbγ (Rd−1 ; K) if ψ ∈ Cbγ (Rd−1 ; K) for every γ ∈ [0, 1 + α]. Note that   ψ ψ ed = ed ∇Eψ = Dd T ηd ηd ∂ Eψ = ψ on ∂Rd+ . The assertion follows. ∂η (ii) The proof is similar to that of property (i). We define the operator S by setting Z ψ(x0 − xd y) ϑ(y) dy, x ∈ Rd+ , (Sψ)(x) = %(xd ) 0 Rd−1 qdd (x − xd y, 0)

on ∂Rd+ . Hence,

for every ψ ∈ Cbα (Rd−1 ; K), where ϑ ∈ C ∞ (Rd−1 ) is a nonnegative even function with integral over Rd−1 equal to one, % ∈ Cb2+α ([0, ∞)) satisfies the conditions %(0) = %0 (0) = 0, %00 (0) = 2(1 + 2d − d2 )−1 . To simplify the notation, we set h = (qdd (·, 0))−1 ψ. Clearly, Sα ψ vanishes on Rd+ . Moreover, Z (Di Sψ)(x) = x−1 %(x ) h(x0 − xd y)Di ϑ(y) dy, d d Rd−1

Z (Dd Sψ)(x) = [%0 (xd ) − (d − 1)%(xd )x−1 ] h(x0 − yxd )ϑ(y) dy, d d−1 R Z −1 0 − xd %(xd ) h(x − xd y)h∇ϑ(y), yi dy, Rd−1

(Dij Sψ)(x) = x−2 d %(xd ) (Did Sψ)(x) =

Z

h(x0 − xd y)Dij ϑ(y) dy,

Rd−1

0 [x−1 d % (xd )

1)x−1 d %(xd )

x−2 d %(xd )]

Z

− (d − − h(x0 − xd y)Di ϑ(y) dy Rd−1 Z − x−2 %(x ) h(x0 − xd y)h∇Di ϑ(y), yi dy, d d Rd−1

(Ddd Sψ)(x) = [%

00

−2 0 (xd )−2(d−1)x−1 d % (xd )−(d−1)(d−2)xd %(xd )]

Z

h(x0 − xd y)ϑ(y) dy

Rd−1

Z −2 0 − [x−1 % (x ) − (d − 1)x %(x )] h(x0 − xd y)h∇ϑ(y), yi dy d d d d d−1 R Z −2 0 2 + xd %(xd ) h(x − xd y)hD ϑ(y)y, yi dy Rd−1

Rd+

for every x ∈ and i, j = 1, . . . , d − 1. As it is easily seen, the previous functions can be extended by continuity to the boundary of Rd+ since % and %0 vanish at zero. Moreover, all the derivatives of the function Sψ, but the function Ddd Sψ, vanish on ∂Rd+ . On the other hand, it can be easily shown that Z Z h∇ϑ(z), zi dz = 1 − d, hD2 ϑ(z)z, zi dz = d(d − 1). (B.4.7) Rd−1

Rd−1

Semigroups of Bounded Operators and Second-Order PDE’s

465

Hence,   Z Z d d2 3−d 1 2 (Ddd Sψ)(·, 0) =% (0) 2 − − − h∇ϑ(y), yi dy + hD ϑ(y)y, yi dy h 2 2 2 2 Rd−1 Rd−1   1 d2 00 = +d− % (0)h = h 2 2 00

on ∂Rd+ , so that (ASψ)(·, 0) = qdd (·, 0)Ddd Sψ(·, 0) = ψ on ∂Rd+ . Finally, note that the second-order derivatives of Sψ belong to Cbα (Rd+ ; K). Indeed, since qdd ≥ µ on Rd+ and it belong to Cbα (Rd+ ; K), the function h belongs to Cbα (Rd−1 ; K). Similarly, we can show that the function η1 : [0, ∞) → R, defined by η1 (r) = r−2 %(r) for every r > 0 and η1 (0) = %00 (0)/2, belongs to Cbα ([0, ∞)). Indeed, Z 1 Z 1 00 00 % (r1 σ)(1 − σ) dσ % (r2 σ)(1 − σ) dσ − |η1 (r2 ) − η1 (r1 )| = 0

0

≤|r2 − r1 |α [%00 ]Cbα ([0,∞))

Z

1

σ α (1 − σ) dσ

0

for every r1 , r2 ∈ [0, ∞). Moreover, ||η1 ||∞ ≤ 2−1 ||%00 ||∞ . Therefore, we can estimate Z |(Dij Sψ)(x) − (Dij Sψ)(z)| ≤|η1 (xd ) − η1 (zd )| |h(x0 − xd y)||Dij ϑ(y)| dy d−1 R Z + |η1 (zd )| |h(x0 − xd y) − h(z 0 − zd y)||Dij ϑ(y)| dy Rd−1

≤||Dij ϑ||L1 (Rd−1 ;K) [η1 ]Cbα ([0,∞)) ||h||∞ |xd − zd |α Z + ||η1 ||∞ [h]Cbα (Rd+ ;K) (|x0 −z 0 |+|y||xd −zd |)α |Dij ϑ(y)| dy Rd−1

≤||Dij ϑ||L1 (Rd−1 ;K) [η1 ]Cbα ([0,∞)) ||h||∞ |xd − zd |α + ||η1 ||∞ [h]Cbα (Rd+ ;K) ||Dij ϑ||L1 (Rd−1 ;K) |x0 − z 0 |α Z + ||η1 ||∞ [h]Cbα (Rd+ ;K) |xd − zd |α |y|α |Dij ϑ(y)| dy Rd−1

e C α (Rd ;K) |x − z|α ≤C[ψ] + b e independent of ψ. Writing for every x, z ∈ Rd+ and some positive constant C, x−1 d %(xd ) =

Z

1

%0 (xd σ) dσ,

0

0 x−1 d % (xd ) =

Z

1

%00 (xd σ) dσ

0

for every xd > 0 and arguing as above, we can deal with the other second-order derivative of Sψ. This completes the proof.  We can now prove Proposition B.2.13. Proof of Proposition B.2.13 Even if it is not difficult, the proof is rather technical. For this purpose we split it in some steps. Step 1: preliminary tools. Let ψi : Ui → B(0, r) (i = 1, . . . , N − 1) be the function in Definition B.2.1. Associated to the covering {Ui : i = 1, . . . , N − 1} of ∂Ω, we consider a partition of the unity {ϑ2i : i = 1, . . . , N − 1} such that each function ϑ]i = ϑi ◦ ψi−1 belongs to Cc∞ (B(0, r)). Indeed, the proof of Corollary B.1.3 shows that, for every i = 1, . . . , N − 1, SN there exists an open set Ui0 b Ui such that ∂Ω ⊂ i=1 Ui0 . Then, ψi (Ui0 ) is a compact subset

466

Smooth Domains and Extension Operators

of B(0, r). Hence, there exists r1 < r such that ψi (Ui0 ) ⊂ B(0, r0 ) (clearly, we can take r1 independent of i). We introduce a function ϑe]i ∈ Cc∞ (B(0, r)) such that χB(0,r0 ) ≤ ϑe]i ≤ 1. The function ϑei = ϑe]i ◦ψi belongs to C 2+α (Ui ) and χUi0 ≤ ϑei ≤ 1. We can define the functions Q ϑbi (i = 1, . . . , N −1) by setting ϑb1 = ϑe1 , ϑbi = ϑbi k 0 such that i=1 ϑi ζi ≥ 1/2 on Ωδ = {x ∈ Rd : dist(x, ∂Ω) < δ}. Since {Ωδ , Ω0 }, where Ω0 = {x ∈ Ω : dist(x, ∂Ω) > δ/2} is an open covering of Λδ = {x ∈ d R : dist(x, Ω) ≤ δ/2}, Corollary B.1.3 shows that there exist two functions χ1 ∈ Cc∞ (Ωδ ) and χ2 ∈ Cc∞ (Ω0 ) such that 0 ≤ χj ≤ 1 on Rd (j = 1, 2) and χ1 + χ22 = 1 on Λδ . Then, the PN −1 function χ = i=1 χ1 ζi ϑi + χ22 is smooth on Rd and χ≥

1 1 1 χ1 + χ22 ≥ (χ1 + χ22 ) = 2 2 2

on Λδ . Hence, we can define the functions φi (i = 1, . . . , N ) and ζN by setting φi =

ϑi χ1 ϕ2 , χ

i = 1, . . . , N − 1,

φN =

χ2 ϕ2 , χ

ζN = χ2 ,

where ϕ2 ∈ Cc∞ (Λδ ) is a smooth function such that ϕ2 ≡ 1 on Ω. It is immediate to check

Semigroups of Bounded Operators and Second-Order PDE’s

467

that the functions φi and ζi (i = 1, . . . , N ) have all the properties in the statement of the proposition.  Proposition B.4.13 Let η ∈ Cbα (Rd−1 ; Rd ) be a function such that |η(x0 )| = 1 for every x0 ∈ Rd−1 and supRd−1 ηd < 0, where ηd denotes the last component of the function η. Then, there exists a bounded linear operator E mapping Lp (Rd−1 ; K) into W 1,p (Rd+ ; K) such that Eψ ≡ 0. Moreover, its restriction to B 1−1/p,p (Rd−1 ; K) is a bounded operator mapping this ∂Eψ ≡ ψ on ∂Rd+ for every ψ ∈ B 1−1/p,p (Rd−1 ; K). space into W 2,p (Rd+ ; K). Moreover, ∂η Proof We will prove that the operator E defined in the proof of Proposition B.4.12 can be extended with a bounded operator mapping Lp (Rd−1 ; K) into W 1,p (Rd+ ; K). The main step consists in showing that the operator T extends with a bounded operator mapping Lp (Rd−1 ; K) (resp. B 1−1/p,p (Rd−1 ; K)) into W 1,p (Rd+ ; K) (resp. W 2,p (Rd+ ; K)). For this purpose, we observe that, for every ψ ∈ Lp (Rd−1 ; K) ∩ Cb (Rd−1 ; K), we can estimate Z p Z ∞ Z p p 0 |ζ(xd )| dxd ||T ψ||Lp (Rd ;K) = ψ(x − xd y)ϑ(y) dy dx0 + d−1 Rd−1 Z0 ∞ ZR Z p 0 ≤ |ζ(xd )| dxd dx |ψ(x0 − xd y)|p |ϑ(y)| dy 0 Rd−1 Rd−1 Z ∞ Z Z p = |ζ(xd )| dxd |ϑ(y)| dy |ψ(x0 − xd y)|p dx0 Rd−1 p p =||ζ||Lp ((0,∞)) ||ψ||Lp (Rd−1 ;K) . 0

Rd−1

Arguing in the same way, it can be shown that |||∇T ψ|||Lp (Rd+ ;K) ≤ C||ψ||Lp−1 (Rd−1 ;K) for some positive constant C, which, as all the forthcoming constants, is independent of ψ. Since Lp (Rd−1 ; K) ∩ Cb (Rd−1 ; K) is dense in Lp (Rd−1 ; K) (see Theorem 1.4.2), we conclude that the operator T extends with a bounded operator mapping Lp (Rd−1 ; K) into W 1,p (Rd+ ; K). Next, we observe that, since the trace operator is bounded from W 1,p (Rd+ ; K) into B 1−1/p,p (Rd−1 ; K) and Ef = 0 on ∂Rd+ for every f ∈ Cb (Rd−1 ; K), function Ef vanishes on ∂Rd+ for every f ∈ Lp (Rd−1 ; K). To prove that T is bounded from B 1−1/p,p (Rd−1 ; K) into W 2,p (Rd+ ; K), we observe that, if ψ ∈ Cc∞ (Rd−1 ; K), then Z (Dij T ψ)(x) = ζ(xd )x−2 ψ(x0 − xd z)Dij ϑ(z) dz d Rd−1

(Did T ψ)(x) = (ζ

0

(xd )x−1 d

− Z

− ζ(xd )x−2 d (Ddd T ψ)(x) = [ζ

00

dζ(xd )x−2 d )

Z

ψ(x0 − xd z)Di ϑ(z) dz

Rd−1

ψ(x0 − xd z)h∇Di ϑ(z), zi dz

Rd−1

(xd )+2(1−d)xd−1 ζ 0 (xd )

+ d(d −

1)x−2 d ζ(xd )]

Z

ψ(x0 − xd z)ϑ(z) dz

Rd−1

Z 0 + 2[dx−1 ζ(x ) − ζ (x )] ψ(x0 − xd z)h∇ϑ(z), zi dz d d d d−1 R Z −2 0 + xd ζ(xd ) ψ(x − xd y)hD2 ϑ(y)y, yi dy Rd−1

for every i, j = 1, . . . , d − 1. Since the integral over Rd−1 of the function Dij ϑ is zero, we can write Z (Dij T ψ)(x) = ζ(xd )x−2 [ψ(x0 − xd z) − ψ(x0 )]Dij ϑ(z) dz d Rd−1

468

Smooth Domains and Extension Operators

and estimate |(Dij T ψ)(x)| ≤Cx−1 d

Z

|ψ(x0 − xd z) − ψ(x0 )||Dij ϑ(z)| dz

Rd−1 1

p0 ≤Cx−1 d ||Dij ϑ||L1 (Rd−1 )

Z

0

|ψ(x − xd z) − ψ(x )| |Dij ϑ(z)| dz

Rd−1

0

Z

|ψ(x0 − xd z) − ψ(x0 )|p |Dij ϑ(z)| dz

Rd−1

Z |Dij ϑ(z)| dz

=Cp

 p1

p

Rd−1

for every x ∈ Rd+ and i, j = 1, . . . , d − 1, so that Z Z ∞ Z Z 0 |(Dij T ψ)(x)|p dx ≤Cp x−p dx dx d d Rd +

0

Rd−1

0

∞

x−p d dxd

Z

|ψ(x0 − xd z) − ψ(x0 )|p dx0 .

Rd−1

Next, we observe that |ψ(x0 − xd z) − ψ(x0 )| ≤|ψ(x0 − xd z) − ψ(x0 − xd z + xd zd−1 ed−1 )| + |ψ(x0 − xd z + xd zd−1 ed−1 ) − ψ(x0 − xd z + xd zd−1 ed−1 + xd zd−2 ed−2 )| + . . . + |ψ 0 (x0 − xd z1 e1 ) − ψ(x0 )| for every x0 ∈ Rd−1 , xd ∈ (0, ∞), so that Z

∞

0

Z

x−p d

Z dxd R

∞

x−p d

      p d−1 d−1 X X 0 ψ x0 − xd z − dx0 zj ej − ψ x − xd z − zj ej d−1 j=h+1

j=h

Z

|ψ(y 0 + xd zh eh ) − ψ(y 0 )|p dy 0 Z ∞ Z =|zh |p−1 σ −p dσ |ψ(y 0 + σeh ) − ψ(y 0 )|p dy 0

=

0

dxd

Rd−1

Rd−1

0 p−1

= : |zh |

[ψ]pp,h

for h = 1, . . . , d − 2, and Z ∞ Z x−p dx |ψ(x0 − xd z) − ψ(x0 − xd z + xd zd−1 ed−1 )|p dx0 d d 0 Rd−1 Z ∞ Z = x−p dx |ψ(y 0 ) − ψ(y 0 + xd zd−1 ed−1 )|p dy 0 d d 0 Rd−1 Z ∞ Z p−1 −p =|zd−1 | σ dσ |ψ(y 0 + σed−1 ) − ψ(y 0 )|p dy 0 0 p−1

= : |zd−1 |

Rd−1

[ψ]pp,d−1 .

Consequently, Z Rd +

|(Dij T ψ)(x)|p dx ≤ Cp

d−1 X

[ψ]pp,h

Z

h=1

|zh |p−1 |Dij ϑ(z)| dz.

Rd−1

Hence, ||Dij T ψ||Lp (Rd+ ;K) ≤ C||ψ||B 1−1/p,p (Rd−1 ;K) . Arguing in the same way, taking into account that Z −2 (Did T ψ)(x) =(ζ 0 (xd )x−1 − dζ(x )x ) [ψ(x0 − xd z) − ψ(x0 )]Di ϑ(z) dz d d d Rd−1

Semigroups of Bounded Operators and Second-Order PDE’s Z − ζ(xd )x−2 [ψ(x0 − xd z) − ψ(x0 )]h∇Di ϑ(z), zi dz, d

469

Rd−1

it can be proved that ||Did T ψ||Lp (Rd+ ;K) ≤ C||ψ||B 1−1/p,p (Rd−1 ;K) for every i = 1, . . . , d − 1 and some positive constant Cp,2 . The second-order derivative Ddd T ψ is trickier to be analyzed. We split it as follows Z −1 0 (Ddd T ψ)(x) ={ζ 00 (xd ) + (d − 1)x−1 [dx ζ(x ) − 2ζ (x ) + 2 − d]} ψ(x0 − xd z)ϑ(z) dz d d d d Rd−1 Z −1 −1 0 + 2xd [dxd ζ(xd ) − ζ (xd ) + 1 − d] ψ(x − xd z)h∇ϑ(z), zi dz Rd−1 Z −1 + x−1 ψ(x0 − xd z)hD2 ϑ(z)z, zi dz d (xd ζ(xd ) − 1) d−1 Z R −1 + xd (d − 1)(d − 2) (ψ(x0 − xd z) − ψ(x0 ))ϑ(z) dz Rd−1 Z + 2(d − 1) (ψ(x0 − xd z) − ψ(x0 ))h∇ϑ(z), zi dz Rd−1 Z + (ψ(x0 − xd z) − ψ(x0 ))hD2 ϑ(z)z, zi dz d−1 R   Z Z 2 2 + x−1 d −3d+2+2(d−1) h∇ϑ(z), zi dz+ hD ϑ(z)z, zi dz ψ(x0 ) d Rd−1

Rd−1

(B.4.8) for every x ∈ Rd+ . Note that the coefficients in front of the three integral terms in the righthand side of (B.4.8) can be extended by continuity at zero, so that they are bounded and continuous functions over (0, ∞). Hence, arguing as in the first part of the proof, we can easily show that the three terms in the right hand side of (B.4.8) belong to Lp (Rd+ ; K) and their Lp -norm can be bounded from above by a positive constant times the Lp (Rd−1 ; K) norm of ψ. The forthcoming three terms can be analyzed as we did for the second-order derivative Dij T ψ: they belong to Lp (Rd+ ; K) and their Lp -norm can be estimated from above by the B 1−1/p,p (Rd−1 ; K) norm of ψ. Finally, by (B.4.7), the last term in the right-hand side of (B.4.8) vanishes. By density, from the above results, it follows that the operator T is bounded from B 1−1/p,p (Rd−1 ; K) into W 2,p (Rd+ ; K). To complete the proof, we observe that, since ηd belongs to C 1+α (Rd−1 ; K), the function ψ/ηd belongs to B 1−1/p,p (Rd−1 ; K) for every ψ ∈ B 1−1/p,p (Rd−1 ; K). Moreover, ||ψ/ηd ||B 1−1/p,p (Rd−1 ;K) ≤ C||ψ||B 1−1/p/p (Rd−1 ;K) . Hence, the function Eψ = T (ψ/ηd ) belongs ∂Eψ = ψ on ∂Rd+ . By density we to W 2,p (Rd+ ; K). Moreover, if ψ ∈ Cc∞ (Rd−1 ; K), then ∂η ∂Eψ deduce that = ψ for every ψ ∈ B 1−1/p,p (Rd−1 ; K). The proof is complete.  ∂η We partly extend Proposition B.4.12 to the case when Rd+ is replaced by a smooth bounded open set. Proposition B.4.14 Assume that η ∈ C 1+α (∂Ω; Rd ), |η(y)| = 1 for each y ∈ ∂Ω and inf ∂Ω hη, νi > 0, where ν(y) denotes the unit outer normal vector to ∂Ω at y. Then the following properties are satisfied. (i) Let Ω be a bounded open set of class C 1+α for some α ∈ (0, 1). Then, there exists a ∂Eα g ≡g bounded linear operator Eα : C(∂Ω; K) → C 1 (Ω; K) such that Eα g ≡ 0 and ∂η

470

Smooth Domains and Extension Operators on ∂Ω for every g ∈ C(∂Ω; K). Moreover, for every β ∈ [1, 1 + α], its restriction to C β (∂Ω; K) belongs to L(C β (∂Ω; K), C β+1 (Ω; K)).

(ii) Let Ω be a bounded open set of class C 2 . Then, for every p ∈ (1, ∞), there exists a bounded linear operator Ep : Lp (∂Ω; K) → W 1,p (Ω; K) such that Ep g vanishes on ∂Ω for every g ∈ Lp (∂Ω; Hd−1 ; K). Moreover, the restriction of the operator Ep to ∂Ep g ≡ g on ∂Ω for B 1−1/p,p (∂Ω; K) belongs to L(B 1−1/p,p (∂Ω; K), W 2,p (Ω; K)), and ∂η every g ∈ B 1−1/p,p (∂Ω; K). Proof We introduce some needed tools. We fix a finite covering {Uj : j = 1, . . . , N } of ∂Ω and associated functions ψj : Uj → B(0, r) which are of class C 1+α , with inverse functions which belong to C 1+α as well. We also fix a partition of the unity {ηj : j = 1, . . . , N } associated with the previous covering of ∂Ω. Moreover, for every i = 1, . . . , N and x ∈ Ui we introduce the vector ξi (x) = (Jac ψi )(x)η(x) and denote by ψi,d the last component of the function ψi . Since ν(x) = −|∇ψi,d (x)|−1 ∇ψi,d (x) for every x ∈ Ui ∩ ∂Ω (see Remark B.2.7(i)), it follows that hξi (x), ed i = −|∇ψi,d (x)|hη(x), ν(x)i for every x ∈ Ui ∩ ∂Ω. Next, we introduce a cut-off function ϕ1 ∈ Cc∞ (B(0, r)) which is equal to one in B(0, r1 ) for some r1 ∈ (0, r). Then, the function ηbi = ϕ1 ξi] + ϕ1 − 1, where ξ ] is the trivial extension of ξ ] = ξ ◦ ψi−1 (·, 0) outside B 0 (0, r), belongs to Cb1+α (Rd−1 ). Here, B 0 (0, σ) denotes the ball of Rd−1 centered at 0 and with radius σ. Clearly, supx∈Rd−1 ηbi,d (x) < 0, where ηbi,d (x) = hb ηi (x), ed i. Moreover, ηbi = ξi on B(0, r1 ) ⊃ supp(ϑ]i ). PN (i) Fix f ∈ C(∂Ω; K) and split it into the sum f = j=1 fj , where fj = ηj f for j = 1, . . . , N . The function fj ◦ ψj−1 (·, 0) is continuous on the open ball B 0 (0, r). Moreover, its support is contained in the ball B 0 (0, r1 ) for some r1 < r. Hence, its trivial extension fj ◦ ψj−1 (·, 0) belongs to BU C(Rd−1 ; K). Fix r2 ∈ (r1 , r) and let Ej be the extension operator in Proposition B.4.12(i) associated with the vector ηbj , where now we assume that the support of the function ϑ is contained in B(0, 1) and the function ζ ∈ C ∞ ([0, ∞)) is such that ζ(s) = s for every s ∈ [0, ε/2] and ζ(s) = 0 if s ≥ ε. Here, ε is chosen such that p ε < (r2 −r1 )∧ r2 − r22 . Then, the function Ej (fj ◦ ψj−1 (·, 0)) belongs to Cb (Rd+ ; K), vanishes ∂ on ∂Rd+ and Ej (fj ◦ ψj−1 (·, 0)) ≡ fj ◦ ψj−1 (·, 0) on ∂Rd+ . Moreover, due to the choice of ∂b ηj ε, the function Ej (fj ◦ ψj−1 (·, 0)) is compactly supported in B(0, r). As a byproduct, the function Eα,j f = (Ej (fj ◦ ψk−1 (·, 0))) ◦ ψj−1 belongs to C 1 (Uj ; K) and vanishes on ∂Ω ∩ Uj . Since it is compactly supported in Uj , we can extend it in the trivial way to Rd and the so PN obtained function Eα,j f belongs to Cb1 (Rd ; K). We set Eα f = j=1 Eα,j f . It is clear that the ∂Eα f function Eα f belongs to C 1 (Ω; K), Eα f = 0 and ≡ f on ∂Ω. Moreover, the operator ∂η 1 Eα is bounded from C(∂Ω; K) into C (Ω; K). We now suppose that f ∈ C β (∂Ω; K) for some β ∈ [0, 1 + α]. By Proposition B.4.12, the function Ej (fj ◦ ψj−1 (·, 0)) belongs to C β+1 (Uj ; K), with C β+1 (Ω; K)-norm can be bounded from above by a positive constant, independent of f , times the C β (∂Ω; K) norm of u. Moreover, since it is compactly supported in Uj , the trivial extension Eα,j f of function Eα,j f to Rd+ belongs to Cbk+1+α (Rd+ ; K). As a byproduct, the operator Eα is bounded from C k+α (∂Ω; K) into C k+1+α (Ω; K). PN (ii) Fix f ∈ Lp (∂Ω; K) and split f = j=1 fj as in the proof of property (i). For every j = 1, . . . , N , it holds that Z Z Z p d−1 p d−1 |fi (σ)| dH (σ) = |fi (σ)| dH (σ) = |fj ◦ ψj−1 (y 0 , 0)|p gi (y 0 ) dy 0 , ∂Ω

∂Ω∩Uxi

B 0 (0,r)

Semigroups of Bounded Operators and Second-Order PDE’s

471

where vi = η ◦ ψi−1 (·, 0) and gi (y 0 ) is the square root of the determinant of the matrix A∗i (x0 )Ai (x0 ), Ai (y 0 ) being the Jacobian matrix of the function ψx−1 (·, 0) evaluated at y 0 ∈ i 0 0 B (0, r). Note that g nowhere vanishes on B (0, r). Hence, there exist two positive constants K1 and K2 such that K1 ≤ g(y 0 ) ≤ K2 for every y 0 ∈ supp(vi (·, 0)), which is a compact subset of B 0 (0, r). Then, the trivial extension of the function vi to Rd−1 belongs to Lp (Rd−1 ; K). Due to Proposition B.4.13, the function Ej (fj ◦ ψj−1 (·, 0)) belongs to W 1,p (Rd+ ; K) and it vanishes on ∂Rd+ . Hence, the function Eα,j f = (Ej (fj ◦ ψk−1 (·, 0))) ◦ ψj−1 , where the operator Ej is as in the proof of property (i), belongs to W 1,p (Ω; K), it vanishes on ∂Ω and its support is contained in B(0, r). Moreover, ||Eα,j f ||W 1,p (Rd+ ;K) ≤ K3 ||f ||Lp (∂Ω,Hd−1 ;K) for some positive PNconstant K3 , independent of f . Hence, we can define the operator Eα by setting Eα = j=1 Eα,j . To prove the last part of the statement, we assume that g belongs to B 1−1/p (∂Ω; K). Then, fj ◦ ψj−1 (·, 0) ∈ B 1−1/p,p (Rd−1 ; K) so that the function Ej (fj ◦ ψj−1 (·, 0)) belongs to W 2,p (Rd+ ; K), ||Ej (fj ◦ ψj−1 (·, 0))||W 2,p (Rd+ ;K) ≤ C4 ||f ||B 1−1/p,p (Rd−1 ;K) , the constant C4 being ∂ independent of f , and Ej (fj ◦ ψj−1 (·, 0)) = fj ◦ ψj−1 (·, 0) on ∂Rd+ . Therefore, the function ∂b ηj ∂ Eα f belongs to W 2,p (Rd+ ; K) and Eα f = f on ∂Ω. This completes the proof.  ∂η We conclude this subsection with the following result. Proposition B.4.15 Let Ω be Rd+ or a bounded open subset of Rd of class C k for some k ∈ N. Then, for every p ∈ (1, ∞), there exists a bounded linear operator Ep0 : B 1−1/p,p (∂Ω; K) → W 1,p (Ω; K) such that (i) the trace of the function Ep0 g on ∂Ω coincides with g for every g ∈ B 1−1/p,p (∂Ω; K); (ii) the restriction of Ep0 to B m−1/p,p (∂Ω; K) is a bounded linear operator which maps B m−1/p,p (∂Ω; K) into W m,p (Ω; K) for every m = 2, . . . , k. Here, we identify ∂Rd+ with Rd−1 . Proof We split the proof into three steps. Step 1. Here, we consider the case Ω = Rd+ and prove property (i). For this purpose, we fix p ∈ (1, ∞), h ∈ {1, . . . , d − 1} and set Z (Rh g)(x) := x−d |g(x0 − Th y) − g(x0 − Th+1 y)| dy, x = (x0 , xd ) ∈ Rd+ , d B

Pd−1 for every g ∈ Cc∞ (Rd−1 ; K), where Th z = j=h zj ej for every z ∈ Rd−1 . Further, we set zbh = (z1 , . . . , zh−1 , zh+1 , zd−1 ) for every z ∈ Rd−1 , B = B(0, xd ) ⊂ Rd−1 , B 0 = B(0, xd ) ⊂ Rd−2 and apply H¨ older’s inequality to get |(Rh g)(x)|

≤x−d d

Z

0

0

p

 p1  Z

|g(x − Th y) − g(x − Th+1 y)| dy B 1−d−p p

≤Cp xd

1− p1 dy

B

Z

xd

Z dyh

−xd

|g(x0 − Th y) − g(x0 − Th+1 y)|p db yh

 p1

B0

for x ∈ Rd+ and some positive constant Cp , independent of x and g (as all the other constants

472

Smooth Domains and Extension Operators

in this proof). Integrating the first and last side of the previous chain of inequalities over Rd+ and changing the order of integration, we can write Z |(Rh g)(x)|p dx Rd +

Z

∞

x1−d−p dxd d

Z

dx0 d−1 0 R Z ∞ Z Z 1−d−p p =Cp xd dxd db yh

≤Cpp

B0

0

Z

xd

Z

dyh −xd Z xd dyh

|g(x0 − Th y) − g(x0 − Th+1 y)|p db yh

B0

|g(x0 − Th y) − g(x0 − Th+1 y)|p dx0 .

(B.4.9)

Rd−1

−xd

Performing the change of unknowns zj = x0j (j = 1, . . . , h), zj = x0j −yj (j = h+1, . . . , d−1) in the last integral, we can write Z ∞ Z Z xd Z 1−p−d xd dxd db yh dyh |g(x0 − Th y) − g(x0 − Th+1 y)|p dx0 0 B0 −xd Rd−1 Z ∞ Z Z xd Z 1−p−d = xd dxd db yh dyh |g(z − yh eh ) − g(z)|p dz 0 B0 −xd Rd−1 Z ∞ Z xd Z −p−1 0 =C xd dxd dyh |g(z − yh eh ) − g(z)|p dz (B.4.10) Rd−1

−xd

0

for some positive constant C 0 . Note that the function σ 7→

Z

|g(z − σeh ) − g(z)|p dz is

Rd−1

even in R and therefore, from (B.4.9) and (B.4.10) it follows that Z Z ∞ Z xd Z ep |(Rh g)(x)|p dx ≤C x−p−1 dx dy |g(z + yh eh ) − g(z)|p dz d h d Rd +

0

Z ep =C

0

∞

Z

∞

dyh 0

yh

xd−p−1 dxd

Rd−1

Z Rd−1

ep [g]pp |g(z + yh eh ) − g(z)|p dz ≤ p−1 C

ep . for some positive constant C Next, we fix i ∈ {2, . . . , d − 1} and denote by Θi the operator with interchanges the components x1 and xi of every vector x ∈ Rd . Applying the result so far proved to the function g ◦ Θi and choosing h = 1, we get ||Si g||Lp (Rd+ ) ≤ Cp0 [g]p for some constant Cp > 0, where Z (Si g)(x) = x−d |g(x0 − y) − g(x0 − y + yi ei )| dy, x ∈ Rd+ , d Rd−1

for every g ∈ B 1−1/p,p (Rd−1 ; K). To unify the notation, we also set S1 = R1 . Now, we are almost done. We introduce the operator S defined by Z (Sg)(x0 , xd ) = ϑ(y)g(x0 − xd y) dy, x ∈ Rd+ , Rd−1

for every g ∈ B 1−1/p,p (Rd−1 ; K), where ϑ ∈ Cc∞ (Rd−1 ) is a nonnegative function with L1 norm equal to one. Function Sg is smooth in Rd+ and it equals the function g on ∂Rd+ . Moreover, Z −d 0 (Di Sg)(x) =xd Di ϑ(x−1 d y)g(x − y) dy d−1 ZR −d 0 0 =xd Di ϑ(x−1 d y)(g(x − y) − g(x − y + yi ei )) dy Rd−1

Semigroups of Bounded Operators and Second-Order PDE’s

473

for every x ∈ Rd+ and i = 1, . . . , d − 1. Therefore, |(Di Sg)(x)| ≤ ||Di ϑ||∞ Si (g) for x ∈ Rd+ , so that Di Sg belongs to Lp (Rd+ ; K) and ||Di Sg||Lp (Rd+ ;K) ≤ Cp00 [g]p for every i = 1, . . . , d − 1 and a positive constant Cp00 . To estimate the Lp -norm of the derivative Dd Sg, we observe that Z 1−d 0 0 0 (Sg)(x) = xd ϑ(x−1 x ∈ Rd+ , d (x − y))(g(y) − g(x )) dy + g(x ), Rd−1

so that |(Dd Sg)(x)|

Z

=x−d d

[(1 −

0 d)ϑ(x−1 d (x

− y)) + Z   −d ≤ d − 1 + sup |z||∇ϑ(z)| xd Rd−1

z∈Rd−1

−1 0 h∇ϑ(x−1 d y), xd yi](g(x

− y) − g(x )) dy 0

|g(x0 − y) − g(x0 )| dy

Rd−1

d−1  X ≤ d − 1 + sup |z||∇ϑ(z)| (Rk g)(x) z∈Rd−1

k=1

for x ∈ Rd+ . Thus, Dd Sg belongs to Lp (Rd+ ; K) and its norm can be bounded from above by a positive constant times the seminorm [g]p . The function Sg(·, xd ) belongs to Lp (Rd−1 ; K) and ||(Sg)(·, xd )||Lp (Rd−1 ;K) ≤ ||g||p . Fix a function ψ ∈ Cc∞ ([0, ∞)) such that ψ(0) = 1 and define the operator Ep0 by setting (Ep0 g)(xd ) = ψ(xd )(Sg)(x),

x ∈ Rd+ ,

for every g ∈ Cc∞ (Rd−1 ; K) and then extending it to B 1−1/p,p (Rd−1 ; K) by density (see Theorem 1.4.2). Indeed, the above results show that Ep0 g belongs to W 1,p (Rd+ ; K) and ||Ep0 g||W 1,p (Rd+ ;K) ≤ Cp∗∗ ||g||B 1−1/p,p (Rd+ ;K) for every g ∈ Cc∞ (Rd−1 ; K) and some positive constant Cp∗∗ . Since Ep0 g = g on ∂Rd+ for every g ∈ B 1−1/p,p (Rd−1 ; K) ∩ C ∞ (Rd−1 ; K), again by density we conclude that Ep0 g = g on ∂Rd+ for every g ∈ B 1−1/p (Rd−1 ; K). Step 2. Here, we complete the proof in the case Ω = Rd+ , proving property (ii). For this purpose, we observe that for every m ∈ N, every multi-index β ∈ (N ∪ {0})d−1 and g ∈ Cc∞ (Rd−1 ; K), it holds that Dxβ0 Ep0 g = Ep0 (Dxβ0 g) and (Dxmd Sg)(x) =

d−1 X

m Y

yij ϑ(y)Di1 ,...,im g(x0 − xd y),

x ∈ Rd+ .

i1 ,...,im =1 j=1

Qm ∞ d−1 Since the function y 7→ ) for every i1 , . . . , im ∈ j=1 yij ϑ(y) belongs to Cc (R {1, . . . , m}, from these two formulas and the results proved so far, we easily conclude that Ep0 g ∈ W m,p (Rd+ ; K) and ||Ep0 g||W m,p (Rd+ ;K) ≤ Ch,p ||g||B m−1/p,p (Rd−1 ;K) for some positive constant Ch,p . Again, the density of Cc∞ (Rd−1 ; K) into B m−1/p,p (Rd−1 ; K) allows us to conclude that the restriction of Ep0 to B m−1/p,p (Rd−1 ; K) is a bounded operator mapping B m−1/p,p (Rd−1 ; K) into W m,p (Rd+ ; K). Step 3. Here, we complete the proof considering the case when Ω is a bounded domain of class C 1 . Taking advantage of Remark B.2.4, we introduce a finite open covering {Uxi : i = 1, . . . , N } of ∂Ω which is a covering also of the set Ωδ := {x ∈ Ω : dist(x, ∂Ω) ≤ δ} for some δ > 0. Fix g ∈ C k (∂Ω; K). By Proposition B.1.1, we can determine N smooth functions PN η1 , . . . , ηN such that supp(ηi ) ⊂ Ωi (i = 1, . . . , N ) and i=1 ηi ≡ 1 on Ω. Clearly, PN g ≡ g , where g = gη for every i = 1, . . . , N . Further, let ψxi : Uxi → B(0, r) i i i=1 i

474

Smooth Domains and Extension Operators

(i = 1, . . . , N ) be a bijective function of class C k , whose inverse function is itself of class C k . For each i ∈ {1, . . . , N }, the function vi , which is the trivial extension outside B 0 (0, r) of the function gi ◦ψx−1 (·, 0), belongs to B k−1/p,p (Rd−1 ; K). Here, B 0 (0, r) is the ball of Rd−1 , i centered at zero and with radius r.P By Remark 1.4.6, for every m = 1, . . . , k there exists N ∗ ∗ a positive constant Cm,p such that i=1 ||vi ||B m−1/p,p (Rd−1 ;K) ≤ Cm,p ||g||B m−1/p,p (∂Ω;K) . For 0 k,p d every i ∈ {1, . . . , N }, the function Ep vi belongs to W (R+ ; K) and ||Ep0 vi ||W m,p (Rd−1 ;K) ≤ em,p ||g||B m−1/p,p (∂Ω;K) for every m = 1, . . . , k. C N X We can now define the operator Ep0 by setting Ep0 g = %i (Ep0 vi ) ◦ ψi , where (Ep0 vi ) ◦ ψi i=1

denotes the trivial extension of the function (Ep0 vi ) ◦ ψi outside Uxi ∩ Ω and %i ∈ Cc∞ (Ωi ) is equal to one in a neighborhood of supp(ηi ). It is easy to check that Ep u belongs to W m,p (Ω; K) for every m = 1, . . . , k and, using the above estimates, we infer that bp,m ||g||B m−1/p,p (∂Ω;K) . ||Ep0 g||W m,p (Ω;K) ≤ C We claim that Ep0 g ≡ g on ∂Ω. To check the claim, we fix x ∈ ∂Ω and denote by Ix the PN set P of all indexes i ∈ {1, . . . , N } such that x ∈ Uxi . Since d i=1 ηi ≡ 1 on ∂Ω, it follows that i∈Ix ηi (x) = 1. Finally, we observe that %i ηi = ηi on R . Based on all these remarks, we can write X X X (Ep0 g)(x) = %i (x)(Ep0 vi )(ψi (x)) = %i (xi )gi (x) = g(x) ηi (x) = g(x). i∈Ix

i∈Ix

i∈Ix

Since C m (∂Ω; K) is dense in B m−1/p,p (∂Ω; K) (see Theorem 1.4.2) and the trace operator is bounded from W m,p (Ω; K) into B m−1/p,p (∂Ω; K) for every m ∈ {1, . . . , k} (see Theorem B.3.1), by density we can complete the proof.  Remark B.4.16 Differently from the case p ∈ (1, ∞) there exist no linear bounded operators mapping L1 (Rd−1 ; K) into W 1,1 (Rd+ ; K). This is an interesting result proved by Peetre (see [30]).

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Index

BU C(Ω; K), 1 BU C k (Ω; K), 2 B k−1/p,p (Rd ; K), 26 B k−1/p,p (∂Ω; K), 30 C (1+α)/2,1+α ([0, T ] × ∂Ω; K), 14 (k+α)/2,k+α Cb (I × Ω; K), 9 Cb0,β (I × Ω; K), 8 Cb0,k (I × Ω; K), 9 Cb1+α,2+β (I × Ω; K), 9 C 1+α/2,2+α ([0, T ] × ∂Ω; K), 14 Cb1,2 (I × Ω; K), 9 Cbα,0 (I × Ω; K), 8 Cbα,1+β (I × Ω; K), 9 Cbα,β (I × Ω; K), 8 α,β Cloc (I × Ω; K), 10 C α (∂Ω; K), 7 C0α (Ω; K), 2 Cbα (Ω; K), 1, 2 α (Ω; K), 2 Cloc k,0 Cb (I × Ω; K), 9 C0 -semigroup, 38 C0 (Ω; K), 1 Cb (Ω; K), 1 DA (1 + α, ∞), 69 DA (α, ∞), 66 L(X, Y ), 417 L∞ -contractive semigroup, 404 L∞ -quasi contractive semigroup, 404 Lp (Ω; K), 15 Lploc (Ω, K), 15 L∞ (Ω, K), 15 L∞ loc (Ω, K), 15 S(ω, θ0 , M ), 57 W k,p (Ω; K), 16 W0k,p (Ω, K), 16 k,p Wloc (Ω, K), 16 ΓT , 85 Σω,θ0 , 57 G(M, ω), 38 Lip(Ω; K), 2

adjoint operator, 47 analytic semigroup, 62 Beurling-Deny criteria, 405 calculus fundamental theorem, 420 Calder´on-Zygmund inequality, 256 Cauchy theorem, 423 classical solution to a boundary value problem on Rd+ with Dirichlet boundary conditions, 291 to an abstract Cauchy problem, 70 to an elliptic equation on Rd , 249 to the Cauchy problem in Ω with Dirichlet boundary conditions, 230 general boundary conditions, 230 to the Cauchy problem on Rd , 87 to the Cauchy problem on Rd+ with Dirichlet boundary conditions, 175 general boundary conditions, 199 closable operator, 417 closed operator, 417 closure of a linear operator, 417 comparison principle, 88 dissipative operator, 47 Dunford integral, 58, 70 Dunford-Pettis criterion, 398 Fr´echet derivative, 420 Fr´echet differentiable function, 420 Gagliardo-Nirenberg inequality, 17 Gaussian upper bounds, 409 graph norm, 417 graph of a linear operator, 417 growth bound, 39 holomorphic, 422 Hopf’s lemma elliptic, 101 parabolic, 88 479

480 improper complex integrals, 421 integral, 419 infinitesimal generator, 40 interior sphere condition, 448

Index strongly continuous semigroup, 38 time regularity results for nonhomogeneous Cauchy problems, 77 trace-method, 429 Trotter-Kato theorem, 48

Laurent expansion, 423 uniformly continuous semigroups, 37 mild solution, 51 Von Neumann series, 425 Nash inequality, 400 Neumann series, 56 Newtonian potential, 256 operators of weak type (p, p), 438 Optimal Schauder estimates the elliptic case, 253, 254, 292, 294, 317, 319, 342, 349 the parabolic case global regularity, 126, 145, 175, 199, 230 spatial regularity, 126, 159, 192, 224, 243 parabolic distance, 10 Poisson kernel, 256 Post-Widder inverse formula, 48, 49 power series expansion, 423 quasi-contractive sesquilinear form, 401 reiteration theorem, 432 resolvent identity, 425 Riemann integral, 418 sectorial operator, 57 semigroup consistent, 399 interpolates, 399 of bounded operators, 37 of contractions, 47 of left-translations on BU C(R), 38 on Lp (R), 40 property, 37 ultracontractive, 398 sign function, 401 Sobolev embedding theorems, 17 strict solution to an abstract Cauchy problem, 50 strong maximum principle elliptic, 104 parabolic, 92

weak holomorphic, 422 weak maximum principle elliptic, 99 parabolic, 85 Yosida approximation, 45