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Ill-posed Problems of Mathematical Physics and Analysis by M. M. LAVRENT'Ell V. G. ROMANOV S. P. SHISIIAT.SKII Volume 94 r -- "T'RANSLATIONS OF MATHEMATICAL 'MONOGRAPHS J I ,AMerican Mathematical Society I TRANSLATIONS OF MATHEMATICAL MONOGRAPHS VOLUME 64 Iliposed Problems of Mathematical Physics and Analysis by M.M. LAVRENT'EV V. G. ROMANOV S. P. SHISHATSKII American Mathematical Society Providence Rhode Island HEKOPPEKTHbIE 3A)ALIH M A T E M A T H LI E C K O I I I I 3 H K H H A H A J I H 3 A M. M. JIABPEHTbEB B.T. POMAHOB V c. n. III I4IIIATC KI4I 4 AKALEMHSI HAYK CCCP CHBHPCKOE OT)EJIEHHE BbILII4CJII4TEJIbHbIf4 IIEHTP M3)ATEJI16CTBO cHAYKA> MOCKBA 1980 Translated from the Russian by J. R. Schulenberger Translation edited by Lev J. Leifman 1980 Mathematics Subject Classification (1985 Revision). Primary 35R30; Secondary 45 D05. ABSTRACT. A number of applied problems connected with the interpretation of geophysical data and leading to mathematical problems which are ill-posed in the sense of Hadamard are considered. The exposition includes the basic concepts of ill-posed problems, problems of analytic continuation from continua and discrete sets, and analogous problems of continuation of solutions of elliptic and parabolic equations, the main ill-posed boundary value problems for partial differential equations, and results on the theory of Volterra equations of the first kind, in particular, on the theory of operator Volterra equations. A very broad presenta- tion is given of modern results on the problem of uniqueness in integral geometry and on inverse problems for partial differential equations. The monograph is of interest to specialists in applied mathematics, physics, and geophysics. Library of Congress Cataloging-in-Publication Data Lavrent'ev, M. M. (Mikhail MikhaIlovich) 111-posed problems of mathematical physics and analysis. (Translation of mathematical monographs; v. 64) Translation of: Nekorrektnye zadachi matematichesko! fiziki i analiza. Bibliography: p. 1. Boundary value problems-Improperly posed problems. 2. Numerical analysis- Improperly posed problems. 3. Mathematical physics. I. Romanov, V. G. (Vladimir Gavrilovich) II. Shishatskii, S. P. (Serge! Petrovich) III. Title. IV. Series. QC20.7.B6L3813 1986 530.1'55 86-3642 ISBN 0-8218-4517-9 Copyright © 1986 by the American Mathematical Society. All rights reserved. Printed in the United States of America. The American Mathematical Society retains all rights except those granted to the United States government. This book may not be reproduced in any form without permission of the publisher. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Contents Preface v Introduction 1 CHAPTER I. Physical Formulations Leading to Ill-posed Problems 7 1. Continuation of static fields 7 2. Problems for the diffusion equation 10 3. Continuation of fields from discrete sets 11 4. Processing readings of physical instruments 13 5. Inverse problems of geophysics 15 6. Inverse problems of gravimetry 16 7. The inverse kinematic problem of seismology 19 CHAPTER II. Basic Concepts of the Theory of Ill-posed Problems 26 1. Problems well-posed in the Tikhonov sense 26 2. Regularization 32 3. Linear ill-posed problems 39 CHAPTER III. Analytic Continuation 48 1. Formulations of problems and classical results 48 2. Analytic continuation from continua 52 3. Analytic continuation from classes of sets including discrete sets 57 4. Recovery of solutions of elliptic and parabolic equations from their values on sets lying inside the domain of regularity 63 CHAPTER IV. Boundary Value Problems for Differential Equations 72 1. The noncharacteristic Cauchy problem for a parabolic equation. The Cauchy problem for an elliptic equation 73 2. A mixed problem for a parabolic equation with decreasing time 95 3. Cauchy problems with data on a segment of the time axis for degenerate parabolic and pseudoparabolic equations 106 in iv CONTENTS 4. Cauchy problems with data on a timelike surface for hyperbolic and ultrahyperbolic equations 123 CHAPTER V. Volterra Equations 155 1. Regularization of a Volterra equation of the first kind 155 2. Operator Volterra equations of the first kind 159 CHAPTER VI. Integral Geometry 168 1. The problem of finding a function from its spherical means 170 2. Problems of integral geometry on a family of manifolds which is invariant under a group of transformations of the space 175 3. Integral geometry in special classes of functions 183 4. Integral geometry "in the small" 190 5. The problem of integral geometry in plane curves and energy inequalities 202 CHAPTER VII. Multidimensional Inverse Problems for Linear Differential Equations 209 1. Examples of formulations of multidimensional inverse problems. Mathematical problems connected with investigating them 211 2. A general approach to investigating questions of uniqueness and stability of inverse problems 220 3. Inverse problems for hyperbolic equations of second order 227 4. Inverse problems for first-order hyperbolic systems 234 5. Inverse problems for parabolic equations of second order 251 1. The connection between solutions of direct problems for equations of hyperbolic and parabolic types and inverse problems 251 2. The method of descent in inverse problems 252 3. Inverse problems for equations of parabolic type 254 6. An abstract inverse problem and questions of its being well-posed 255 1. Reduction to the investigation of a two-parameter family of linear equations 257 2. The method of linearization in investigating the inverse problem 261 Bibliography 265 Preface The theory of ill-posed problems is a direction of mathematics which has de- veloped intensively in the last two decades and is connected with the most var- ied applied problems: interpretation of readings of many physical instruments and of geophysical, geological, and astronomical observations, optimization of control, management and planning, synthesis of automatic systems, etc. De- velopment of the theory of ill-posed problems was occasioned by the advent of modern computing technology. Various areas of the theory of ill-posed problems can be included in tra- ditional areas of mathematics such as function theory, functional analysis, differential equations, and linear algebra. The concept of a well-posed problem is connected with investigations by the famous French mathematician Hadamard of various boundary value problems for the equations of mathematical physics. Hadamard expressed the opinion that boundary value problems whose solutions do not satisfy certain continuity conditions are not physically meaningful, and he presented examples of such problems. It was subsequently found that Hadamard's opinion was erroneous. It turned out that many problems of mathematical physics which are ill-posed in the sense of Hadamard and, in particular, problems noted by Hadamard himself have real physical content. It also turned out that ill-posed problems arise in many other areas of mathematics which are connected with applications. Such a classical problem of mathematical analysis as the problem of differentiation is ill-posed if it is connected with processing experimental data. In our country several monographs have been published which are devoted to the theory of ill-posed problems and their applications (see [113], [145], [152], [164], [184] and [266]). The present state of the theory of ill-posed problems is reflected most completely in the monograph of A. N. Tikhonov and V. Ya. Arsenin [266] and in the recently published monograph of V. K. Ivanov, V. V. Vasin and V. P. Tanana [113]. However, a number of important V vi PREFACE areas of the theory of ill-posed problems are absent or little reflected in the monograph literature noted. This refers, for example, to ill-posed problems for concrete types of differential equations, problems of analytic continuation, inverse problems for differential equations, and problems of integral geometry. In the present monograph the authors have attempted to fill these gaps in the monograph literature on the theory of ill-posed problems, and also to give a new treatment of some of its areas. The authors express their profound gratitude to A. L. Bukhgeim, who familiarized himself with the manuscript in detail and made a number of useful remarks. Introduction 1. The concept of a well-posed (correct) problem of mathematical physics was formulated at the beginning of this century by the famous French mathematician Hadamard (see [306] and [307]). At the present time this concept is widely presented in textbooks on the equations of mathematical physics or partial differential equations. A problem of mathematical physics or a boundary value problem for a partial differential equation is called well-posed if the following conditions are satisfied: 1) a solution of the problem exists; 2) the solution of the problem is unique; and 3) the solution of the problem depends continuously on the data of the problem (see, for example, [139]). The conditions just formulated require refinement. Namely, both the solution and the data of the problem are considered as elements of some function space, and the conditions for a problem to be well-posed are formulated as follows. A. A solution of the problem exists for all data belonging to some closed subspace in a normed linear space of the type Ck, LP, H,, and belongs to a space of the same type. The subspace is most often either the entire space or a part of the space on which a finite collection of linear functionals vanishes. B. The solution of the problem is unique in some analogous space. C. To infinitesimal variations of the data of the problem in the data space there correspond infinitesimal variations of the solution in the solution space (see [75], [199], and [248]). Having formulated the concept of a well-posed problem, Hadamard presented an example of an ill-posed problem for a differential equation which in his opinion did not correspond to any real physical formulation. This example was the Cauchy problem for the Laplace equation. It can be shown that the solution of the Cauchy problem for the Laplace equation is unique, i.e., this problem satisfies the second condition for being 1 2 INTRODUCTION well-posed. Without dwelling on the question of existence, we shall show that in the Cauchy problem for the Laplace equation continuous dependence of the solution on the data does not hold for any pair of the spaces mentioned. We restrict ourselves to the simplest version of the problem. Let AU=0) Y>01 U = U(X1 Y), u(x, 0) = 0, ayu(x, 0) = a sin nx, x E [0, ir]. (2) The Cauchy problem (1), (2) has the solution a u(x, y) = sinh ny sin nx. n We consider the problem of finding the solution u for some fixed y > 0 on the basis of the Cauchy data at y = 0. It is obvious that for any pair of the function spaces mentioned above and any E > 0, c > 0, and y > 0, it is possible to choose a and n such that liasinnxll < s, sinh ny sin nx > c. lin This example, just as the concept of a well-posed problem itself, is well known. An analogous situation with continuous dependence of the solution on the data occurs also in the Cauchy problem for the heat equation with reverse time: u=u(x,y), au a2 u2, t> 0, (1, ) at ax u(x,0) = f(x). (2') The opinion was expressed that the Cauchy problem for the Laplace equation and for elliptic equations in general is ill-posed, since the Cauchy problem is natural for the description of processes developing in time, while elliptic equations describe steady-state processes and physical fields. The fact that the Cauchy problem for the heat equation with reverse time is ill-posed was associated with the second law of thermodynamics. 2. It turned out that the opinion of Hadamard regarding the Cauchy problem for the Laplace equation and a number of other problems of this same type was erroneous. Problems ill-posed in the classical sense were encountered long ago in the mathematical description of physical phenomena. A number of classical results in mathematical analysis and differential equations of the last century and the beginning of this one can be attributed to the theory of ill-posed problems. However, the systematic study of questions in the theory of ill-posed problems began comparatively recently. In discussing ill-posed problems with representatives of the natural sciences, mathematicians repeatedly asserted that consideration of ill-posed problems