Mathematical Surveys and Monographs Volume 183 Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems Gershon Kresin Vladimir Maz'ya American Mathematical Society http://dx.doi.org/10.1090/surv/183 Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems Mathematical Surveys and Monographs Volume 183 Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems Gershon Kresin Vladimir Maz'ya American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair Benjamin Sudakov MichaelA.Singer MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 35A23, 35B50, 35J47, 35K40; Secondary 31B10, 35J30, 35Q35, 35Q74. The text was translated from Russian by Tatiana O. Shaposhnikova. For additional information and updates on this book, visit www.ams.org/bookpages/surv-183 Library of Congress Cataloging-in-Publication Data Kresin, Gershon, author. Maximum principles and sharp constants for solutions of elliptic and parabolic systems / Gershon Kresin, Vladimir Mazya p. cm. — (Mathematical surveys and monographs ; volume 183) Includes bibliographical references and index. ISBN 978-0-8218-8981-7 (alk. paper) 1. Inequalities (Mathematics) 2. Maximum principles (Mathematics) I. Mazya, V. G., author. II. Title. QA295.K85 2012 515.983–dc23 2012020950 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2012 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 171615141312 Contents Introduction 1 Part 1. Elliptic Equations and Systems 7 Chapter 1. Prerequisites on Operators Acting into Finite Dimensional Spaces 9 1.1. Introduction 9 1.2. Linear bounded operators defined on spaces of continuous vector- valued functions and acting into Rm or Cm 10 1.3. Linear bounded operators defined on Lebesgue spaces of vector-valued functions and acting into Rm or Cm 17 1.4. Comments to Chapter 1 20 Chapter 2. Maximum Modulus Principle for Second Order Strongly Elliptic Systems 21 2.1. Introduction 21 2.2. Systems with constant coefficients without lower order terms 23 2.3. General second order strongly elliptic systems 33 2.4. Comments to Chapter 2 52 Chapter 3. Sharp Constants in the Miranda-Agmon Inequalities for Solutions of Certain Systems of Mathematical Physics 55 3.1. Introduction 55 3.2. Best constants in the Miranda-Agmon inequalities for solutions of strongly elliptic systems in a half-space 58 3.3. The Lam´e and Stokes systems in a half-space 64 3.4. Planar deformed state 69 3.5. The system of quasistatic viscoelasticity 71 3.6. Comments to Chapter 3 75 Chapter 4. Sharp Pointwise Estimates for Solutions of Elliptic Systems with Boundary Data from Lp 77 4.1. Introduction 77 4.2. Best constants in pointwise estimates for solutions of strongly elliptic systems with boundary data from Lp 79 4.3. The Stokes system in a half-space 83 4.4. The Stokes system in a ball 85 4.5. The Lam´e system in a half-space 87 4.6. The Lam´esysteminaball 91 4.7. Comments to Chapter 4 92 v vi CONTENTS Chapter 5. Sharp Constant in the Miranda-Agmon Type Inequality for Derivatives of Solutions to Higher Order Elliptic Equations 93 5.1. Introduction 93 5.2. Weak form of the Miranda-Agmon inequality with the sharp constant 94 5.3. Sharp constants for biharmonic functions 98 5.4. Comments to Chapter 5 104 Chapter 6. Sharp Pointwise Estimates for Directional Derivatives and Khavinson’s Type Extremal Problems for Harmonic Functions 105 6.1. Introduction 105 6.2. Khavinson’s type extremal problem for bounded or semibounded harmonic functions in a ball and a half-space 110 6.3. Sharp estimates for directional derivatives and Khavinson’s type extremal problem in a half-space with boundary data from Lp 117 6.4. Sharp estimates for directional derivatives and Khavinson’s type extremal problem in a ball with boundary data from Lp 131 6.5. Sharp estimates for the gradient of a solution of the Neumann problem in a half-space 145 6.6. Comments to Chapter 6 148 Chapter 7. The Norm and the Essential Norm for Double Layer Vector-Valued Potentials 151 7.1. Introduction 151 7.2. Definition and certain properties of a solid angle 154 7.3. Matrix-valued integral operators of the double layer potential type 161 7.4. Boundary integral operators of elasticity and hydrodynamics 173 7.5. Comments to Chapter 7 197 Part 2. Parabolic Systems 201 Chapter 8. Maximum Modulus Principle for Parabolic Systems 203 8.1. Introduction 203 8.2. The Cauchy problem for systems of order 2 205 8.3. Second order systems 217 8.4. The parabolic Lam´e system 230 8.5. Comments to Chapter 8 235 Chapter 9. Maximum Modulus Principle for Parabolic Systems with Zero Boundary Data 237 9.1. Introduction 237 9.2. The case of real coefficients 238 9.3. The case of complex coefficients 246 9.4. Comments to Chapter 9 249 Chapter 10. Maximum Norm Principle for Parabolic Systems without Lower Order Terms 251 10.1. Introduction 251 10.2. Some notation 255 CONTENTS vii 10.3. Representation of the constant K(Rn,T) 256 10.4. Necessary condition for validity of the maximum norm principle for the system ∂u/∂t − A0(x, t, Dx)u = 0 259 10.5. Sufficient condition for validity of the maximum norm principle for the system ∂u/∂t − A0(x, t, Dx)u = 0 262 10.6. Necessary and sufficient condition for validity of the maximum norm principle for the system ∂u/∂t − A0(x, Dx)u = 0 264 10.7. Certain particular cases and examples 269 10.8. Comments to Chapter 10 275 Chapter 11. Maximum Norm Principle with Respect to Smooth Norms for Parabolic Systems 277 11.1. Introduction 277 11.2. Representation for the constant K(Rn,T) 280 11.3. Necessary condition for validity of the maximum norm principle for the system ∂u/∂t − A(x, t, Dx)u = 0 284 11.4. Sufficient condition for validity of the maximum norm principle for the system ∂u/∂t − A(x, t, Dx)u = 0 with scalar principal part 288 11.5. Criteria for validity of the maximum norm principle for the system ∂u/∂t − A(x, Dx)u = 0. Certain particular cases 291 11.6. Example: criterion for validity of the maximum p-norm principle, 2 <p<∞ 294 11.7. Comments to Chapter 11 296 Bibliography 297 List of Symbols 307 Index 313 Bibliography [Ag1] S. Agmon, Multiple layer potentials and the Dirichlet problem for higher order elliptic equations in the plane, I, Comm. Pure Appl. Math., 10 (1957), 179–239. [Ag2] S. Agmon, Maximum theorems for solutions of higher order elliptic equations, Bull. Amer. Math. Soc., 66 (1960), 77–80. [ADN1] S. Agmon, A. Douglis, and L. 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