Initial Boundary Value Problems in Mathematical Physics

Initial Boundary Value Problems in Mathematical Physics

Initial Boundary Value Problems in Mathematical Physics Rolf leis University of Bonn Federal Republic of Germany Springer Fachmedien Wiesbaden GmbH

C o p y r i g h t 1 9 8 6 b y S p r i n g e r F a c h m e d i e n W i e s b a d e n Copyright~ ' Originally published by B.G. Teubner, Stuttgart in 1986 All rights reserved. No part of this book may be reproduced by any means, nor transmitted, nor translated into a machine language without the written permission of the publisher. CIP-Kurztitelaufnahme der Deutschsn Bibliothsk: Leis, Rolf: Initial boundary value problems in mathematical physics/rolf Leis. ISBN 978-3-519-02102-5 ISBN 978-3-663-10649-4 (ebook) DOI 10.1007/978-3-663-10649-4 Library of Congress Cataloging-in-Publication Data. Leis, Rolf. Initial boundary value problems in mathematical physics "Lectures... given at the University of Bonn in the academic year 1983/4 and in part at the University of Strathclyde, Glasgow"-Pref. Bibliography: p. 000 Includes indexes. 1. Initial value problems--addresses, essays, lectures. 2. Boundary value problems--addresses, essays, lectures. I. Title. QA378.L45 1986 515.3'5 85-12473 ISBN 978-3-519-02102-5 British Library Cataloguing in Publication Data: Leis, R. Initial boundary value problems in mathematical physics 1. Initial value problems I. Title 515.3"5 QA378 Typeset by Macmillan India Ltd., Bangalore 25.

Preface The lectures presented in this book were given at the University of Bonn in the academic year 1983/4 and in part at the University of Strathclyde, Glasgow. Their aim was to introduce graduate students of both mathematics and physics to the timedependent theory of linear equations of mathematical physics and to classical scattering theory. Using Hilbert space methods, the theory was developed so that the asymptotic behaviour of the solutions for large t could be discussed. The presentation of the theory is made using equations that are of particular interest for the mathematician and the physicist. The wave equation is considered, as are firstorder systems of linear acoustics and electromagnetism. This is followed by a discussion of a Schrodinger equation with a Coulomb potential, the equations of linear elasticity, the plate equation, and the equations of thermoelasticity. The reader is assumed to have a basic knowledge of functional analysis. However, for convenience, the required background is presented in the second chapter. Because of the breadth of the subject, restrictions have to be made. For instance, the treatment of boundary value problems is confined to bounded and exterior domains (domains with bounded complement). Half spaces, for example, will not be discussed. Furthermore, when dealing with exterior boundary value problems, we assume the existence of a sufficiently large ball outside of which the medium is homogeneous and usually isotropic. The lectures are primarily concerned with vibrations; damping terms are only occasionally considered. The selection of the material follows personal interests and tastes. Only a few historical remarks are included, and although a long list of references is given it is not intended to be exhaustive. Many colleagues and students helped me to improve the lectures and I would like to thank them all. Special thanks are due to H. D. Alber, Y. Ma, D. F. McGhee, R. Picard, R. Racke, G. F. Roach, and S. Seidler, who prepared the notes of similar lectures given in 1981, and tow. Wendt, K. J. Witsch, and C. H. Wilcox, who spent a sabbatical year at the University of Bonn. I want to thank Mrs R. Muller for her expert typing of the manuscript and, last but not least, the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 72, for generous and continuous support. Bonn October 1985 Rolf Leis v

Contents 1. Introduction 2. Linear operators 8 2.1 Function spaces 8 2.2 Linear operators 16 2.3 Discussing the Laplace operator 23 2.4 Boundary value problems 27 3. The wave equation 33 3.1 Formulation 33 3.2 Existence and uniqueness 36 4. The spectrum of A and boundary value problems 45 4.1 The spectrum of A for bounded domains 45 4.2 The spectrum of A for exterior domains 57 4.3 The unique continuation principle 64 4.4 Exterior boundary value problems 69 4.5 Representation theorems and a priori estimates 75 4.6 The limiting absorption principle 81 4.7 Integral transformations 88 5. The free space problem for the wave equation 97 5.1 A Fourier transform representation of the solution 97 5.2 Distributional solutions 100 5.3 Fundamental solutions and the Huygens principle 104 5.4 Asymptotic wave profiles 106 6. The wave equation continued: time-asymptotic behaviour of the solutions 112 6.1 Solutions in bounded domains 112 6.2 The behaviour of solutions in exterior domains 112 6.3 The existence of wave operators 116 6.4 Proof of Theorem 6.2 119 7. Linear acoustics 129 7.1 Formulation 129 7.2 Existence and uniqueness 131 vii

viii 7.3 The free space problem 133 7.4 The spectrum of A 142 8. Maxwell's equations 146 8.1 Formulation 146 8.2 Existence and uniqueness 147 8.3 The free space problem 149 8.4 Boundary value problems for bounded domains 154 8.5 Proof of the selection theorem 165 8.6 Exterior boundary value problems 167 9. Linear acoustics and Maxwell's equations continued 170 9.1 Formulation of a unified approach 170 9.2 The free space problem 174 9.3 The spectrum of A 181 9.4 Exterior boundary value problems 184 9.5 The existence of wave operators 189 10. A Schrodinger equation 191 10.1 Formulation 191 10.2 The spectrum of A 193 10.3 The free space problem 196 10.4 The existence of wave operators 197 11. Linear elasticity 201 11.1 Formulation 201 11.2 Existence and uniqueness 206 11.3 The free space problem 210 11.4 Exterior boundary value problems 217 12. The plate equation 220 12.1 Formulation 220 12.2 Existence and uniqueness 223 12.3 The free space problem 224 12.4 Exterior boundary value problems 229 13. Linear thermoelasticity 231 13.1 Formulation 231 13.2 Existence and uniqueness 233 13.3 The free space problem 236 13.4 Boundary value problems 244 Appendix 247 A.1 Proof of Theorem 5.6 247 A.2 Proof of Korn's inequality 251 References 256 Notation 262 Index 263