Initial Boundary Value Problems in Mathematical Physics

Transcription

1 Initial Boundary Value Problems in Mathematical Physics

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

3 C o p y r i g h t 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 ISBN (ebook) DOI / 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.L ' ISBN 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.

4 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

5 Contents 1. Introduction 2. Linear operators Function spaces Linear operators Discussing the Laplace operator Boundary value problems The wave equation Formulation Existence and uniqueness The spectrum of A and boundary value problems The spectrum of A for bounded domains The spectrum of A for exterior domains The unique continuation principle Exterior boundary value problems Representation theorems and a priori estimates The limiting absorption principle Integral transformations The free space problem for the wave equation A Fourier transform representation of the solution Distributional solutions Fundamental solutions and the Huygens principle Asymptotic wave profiles The wave equation continued: time-asymptotic behaviour of the solutions Solutions in bounded domains The behaviour of solutions in exterior domains The existence of wave operators Proof of Theorem Linear acoustics Formulation Existence and uniqueness 131 vii

6 viii 7.3 The free space problem The spectrum of A Maxwell's equations Formulation Existence and uniqueness The free space problem Boundary value problems for bounded domains Proof of the selection theorem Exterior boundary value problems Linear acoustics and Maxwell's equations continued Formulation of a unified approach The free space problem The spectrum of A Exterior boundary value problems The existence of wave operators A Schrodinger equation Formulation The spectrum of A The free space problem The existence of wave operators Linear elasticity Formulation Existence and uniqueness The free space problem Exterior boundary value problems The plate equation Formulation Existence and uniqueness The free space problem Exterior boundary value problems Linear thermoelasticity Formulation Existence and uniqueness The free space problem Boundary value problems 244 Appendix 247 A.1 Proof of Theorem A.2 Proof of Korn's inequality 251 References 256 Notation 262 Index 263