Springer Tracts in Natural Philosophy Volume 11 Edited by C. Truesdell Co-Editors: R.Aris· L.Collatz· G.Fichera· P.Germain J. Keller· M. M. Schiffer· A. Seeger M. M. Lavrentiev Some Improperly Posed Problems of Mathematical Physics Translation revised by Robert J. Sacker Springer-Verlag New York Inc. 1967 Professor Dr. M. M. LAVRENTIEV Academy of Science, Siberian Department Novosibirsk - U.d.S.S.R. Dr. Robert J. SACKER New York University Courant Institute of Mathematical Sciences ISBN-13: 978-3-642-88212-8 e-ISBN-13: 978-3-642-88210-4 DOl: 10.1007/978-3-642-88210-4 All rights reserved, especially that of translation into foreign languages. It is also forbidden to reproduce this book, either whole or in part, by photomechanical means (photostat, microfilm and/or microcard) or by other procedure without written permission from the PUblishers. © by Springer-Verlag Berlin' Heidelberg 1967. Library of Congress Catalog Card-Number 67-13673 Sof'tcover reprint of the hardcover 1st edition 1967 Title-No. 6739 Preface This monograph deals with the problems of mathematical physics which are improperly posed in the sense of Hadamard. The first part covers various approaches to the formulation of improperly posed problems. These approaches are illustrated by the example of the classical improperly posed Cauchy problem for the Laplace equation. The second part deals with a number of problems of analytic continuations of analytic and harmonic functions. The third part is concerned with the investigation of the so-called inverse problems for differential equations in which it is required to determine a dif ferential equation from a certain family of its solutions. Novosibirsk June, 1967 M. M. LAVRENTIEV Table of Contents Chapter I Formu1ation of some Improperly Posed Problems of Mathematic:al Physics § 1 Improperly Posed Problems in Metric Spaces. . . . . . . . . § 2 A Probability Approach to Improperly Posed Problems. .. 8 Chapter II Analytic Continuation § 1 Analytic Continuation of a Function of One Complex Variable from a Part of the Boundary of the Region of Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13 § 2 The Cauchy Problem for the Laplace Equation . . . . . .. 18 § 3 Determination of an Analytic Function from its Values on a Set Inside the Domain of Regularity. . . . . . . . . . . .. 22 § 4 Analytic Continuation of a Function of Two Real Variables 32 § 5 Analytic Continuation of Harmonic Functions from a Circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 § 6 Analytic Continuation of Harmonic Function with Cylin drical Symmetry . . . . . . . . . . . . . . . . . . . . . . . .. 42 Chapter III Inverse Problems for Differential Equations § 1 The Inverse Problem for a Newtonian Potential . . . . . .. 45 § 2 A Class of Nonlinear Integral Equations . .. ....... 55 § 3 Inverse Problems for Some Non-Newtonian Potentials . .. 62 § 4 An Inverse Problem for the Wave Equation. . . . . . . . .. 63 § 5 On a Class of Inverse Problems for Differential Equations. 65 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . " 70 Chapter I Formulation of some Improperly Posed Problems of Mathematical Physics § 1. Improperly Posed Problems in Metric Spaces The notion of correctness· introduced at the beginning of our century by the French mathematician HADAMARD plays an important role in the investiga tion of the problems of mathematical physics. One often says that a problem is solved if its correctness is established. Various authors present notions of correctness which coincide in their essence but differ in details. We give one of the possible definitions of correct ness which is convenient for our aims. Let $, Fbe some complete metric spaces, and let A<p be a function with the domain of definition $ and the range of values F. Consider the equation A<p=J (1.1) Let us point out that most problems of mathematical physics can be reduced to the investigation of the solution of equation (1.1) with a given function A and right-hand side f We say that the problem of solving (1.1) is properly posed if the following conditions are satisfied: 1) The solution of (1.1) exists for any IE F. 2) The solution of (1.1) is unique in $. 3) The solution of (1.1) depends continuously on the right-hand sidef In other words, the problem of solving (1.1) is properly posed ifthere exists a function Bf defined and continuous over all of F, which is inverse to the function A<p. Linear problems are most often considered in mathematical physics. In this case $, F are BANACH spaces, and A is a linear operator. The BANACH Hsp:a,c es $, Fin concrete problems are the known functional spaces Cl, Lp' W;, S p' ••• with the carriers in some n-dimensional space of the independent variables or on any part of the spaces of independent variables. * Translator's note: Hereafter we shall refer to the correctness or incorrectness of problems which are respectively properly posed or improperly posed. I Springer Tracts, Vol. II, Lavrentiev 2 Formulation of some Improperly Posed Problems of Mathematical Physics The first requirement of correctness is that the problem should not be overdetermined, and superfluous conditions should not be imposed. The second requirement is that the solution be unique. The third requirement of correctness, continuity of the inverse function Bf, arises from the fact that in the real problems of mathematical physics the right hand side of equation (1.1) is obtained from measurements made with the aid of actual instruments and is therefore known only approximately. Therefore, it has been felt for a long time that if at any point f the function Bf is dis continuous, then the solution q> cannot be uniquely recovered from the right hand side/. HADAMARD introduced the notion of correctness by giving an example of an improperly posed problem which became classical and was included in most text-books on mathematical physics. The example is the CAUCHY problem for the LAPLACE equation. It is weIl-known that the solution of this linear problem does not depend continuously on the data obtained from any of the functional spaces mentioned above. On the basis of this HADAMARD concluded that CAUCHY'S problem for the LAPLACE equation and, in general, all problems exhibiting a similar dependence of the solution on the right-hand side, do not correspond to any real formulations, i. e., they are not problems of mathematical physics. It was discovered later that HADAMARD'S conclusion was erroneous and many real problems of mathematical physics lead to problems which are improperly posed in the sense of HADAMARD. In particular a number of im portant problems of geophysics lead to the CAUCHY problem for the LAPLACE equation. There are stilI further examples of linear as weIl as nonlinear improperly posed problems which are important in the applications, namely the solution of the heat equation for negative time and CAUCHY data on the boundary, the nonhyperbolic CAUCHY problem for the wave equation, inverse problems of potential, and a number of inverse problems for differential equations. At present there exists a number of approaches to the investigation of improperly posed problems. We will now explain them using the above classical CAUCHY problem for the LAPLACE equation as an illustration. We consider one of the simplest versions of the CAUCHY problem for the LAPLACE equation. Let u (x, y) be a twice continuously differentiable function of the variables x, y in the rectangle O~x~n, O~y~H (D) satisfying the foIlowing conditions Au=O, (x,y)eD (1.2) o oy u (x, O)=u (0, y)=u (n, y)=O. § 1. Improperly Posed Problems in Metric Spaces 3 It is required to determine its values on the segment y = h, 0 ::s; x ::s; n by its values on the segment y = 0, 0 ::s; x ::s; n. The problem stated is equivalent to the solution of equation (1.1) where ({J, f stand for the functions of the variable x ((J=U(X, h) J=u(x,O). A is a linear integral operator with the kernel The functional spaces CP, F considered in the given example are the HILBERT functional spaces with the scalar products of the form 00 (({J 1, ({J2) = L Ak({J~' ({J~ o (1.3) 00 L (f"J2)= J1.df·Jf o i/ where ({J~ and are the FOURIER coefficient of the functions ({Jj and fj, and }'k and,uk> 0 are some sequences. Wi We note that spaces of such form are, in particular, the spaces with the norm It may be easily seen that the solution of (1.1) in the ease under considera tion is unique, but for its existence and continuity it is insufficient to let (/) and wI F be some with finite I. To make the problem properly posed one can introduce a sufficiently "strong" norm in the data space F or a sufficiently "weak" one in the space (/). It is quite evident that the solution of (Ll) is correct in the sense of our defini tion, if the sequences Ak,,uk in (1.3) satisfy the inequality Ak ekh-<C rIIk (1.4) where C is a constant. The inequalities (1.4), in particular, are satisfied by the following sequences )..k,,uk corresponding to the above possibilities _e-kh Al'k - , J1.~=1 (1.5) A;= 1, 1I"_ekh rk- • 4 Formulation of some Improperly Posed Problems of Mathematical Physics However, the above approaches have the following defects. In the first case the statement does not cover the range of problems in which the errors in the data may be considered to be small only in the "ordinary" functional spaces; in the second case solutions of the problem may be objects which are not functions in the usual sense and do not correspond to physical reality. We now formulate some approaches to the question of correctness of problems of the type under consideration which are free of the above defects and, in our opinion, quite natural from the standpoint of applications. The first approach consists of changing the notion of correctness, namely, to one having requirements different from 1), 2), and 3). We now state the requirements. In addition to the spaces if> and F and the operator A, let there be given some closed set Me if>. We call the problem for the solution of (1.1) properly posed according to TYKHONOV if the following conditions are fulfilled. 1) It is a-priori known that the solution<p exists for some class of data and belongs to the given set M, <p E M. 2) The solution is unique in a class of functions belonging to M. 3) Arbitrarily small changes of the right-hand side of [which do not carry the solution <p out of M correspond to arbitrarily small changes in the solu tion <po We denote by MA the image of M after the application to the space if> of the operator A. Requirement 3) can be restated in the following manner, 3) The solution of equation (1.1) depends continuously on the right-hand side [on the set MA• If M is a compact set the following statement holds (see [17]). If equation (1.1) satisfies the requirements 1), 2) of correctness due to TYKHONOV, then there exists a function IX (r) such that a) IX (r) is a continuous nondecreasing function with IX (0) = O. b) for any <Pl' <P2 E M satisfying the inequality the following holds Thus, the requirement of continuous dependence 3) is satisfied if 1) and 2) are satisfied. We note that, if a problem is properly posed according to TYCHONOV and we replace the metric spaces if>, Fby their supspaces M, MA then the problem becomes properly posed in the usual sense. § 1. Improperly Posed Problems in Metric Spaces 5 The necessity of examining spaces ifJ, Ftogether with M, MA is due to the fact that in real problems the errors committed in the determination of the right-hand side f usually lead to f outside of MA• The consideration of the problem according to TYKHONOV'S formulation gives the possibility of construct ing an approximate solution with a certain guaranteed degree of accuracy in spite of the fact that an exact solution of (1.1) with approximate data either does not exist at all or may strongly deviate from the "true" solution. In the CAUCHY problem for the LAPLACE equation we consider as the set M the sets defined in the following manner: we denote by A hi an integral operator with the kernel f ~)= ~ sinkx·sink~ Khl (x, cosh-1 khl 1t 0 and let if We now give evidence that our formulation is natural for the case in which the CAUCHY problem for the LAPLACE equation is used for the solution of the geophysical problem of interpreting the gravitational or magnetic anomalies. The problem of interpretation of geophysical data is as follows: the characteris tics of some physical field interacting with the inner layers of the earth are measured on the earth's surface. Certain characteristics of the structure of these layers are required to be determined. In interpreting the constant magnetic and gravitational fields there often arises the following situation. An upper layer of the earth's crust consists of uniform sedimentary rocks. The thickness of the sedimentary rock layer may be estimated from below by some constant M. The anomalies observed on the earth surface, i.e., the CAUCHY data for a harmonic function, are generated by some bodies lying at a depth exceeding H and the ultimate aim of an observer is to determine these bodies. This problem is rather complicated; its solution is not unique if one proceeds only from measurements of a field on the surface. However, by proper evaluation of the general geological situation and with well chosen supplementary hypotheses, the observer often succeeds in finding solutions with satisfactory accuracy. It is important in this problem to make good estimates on the number of the bodies, their approximate position, and their size. For the case in which the average depth of embedding of the bodies is substantially less than their horizontal dimensions and the spacing between them, the distinct extrema of the anomaly correspond to distinct bodies. However, if the average depth of imbedding roughly coincides with or exceeds the characteristics horizontal