Some Improperly Posed Problems of Mathematical Physics

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M. M. Lavrentiev

Some Improperly Posed Problems of Mathematical Physics

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. L avrentiev Academy of Science, Siberian Department Novosibirsk - U.d.S.S.R.

Dr. Robert J. Sacker New York University Courant Institute of Mathematical Sciences

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 Printed in Germany 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. L a v r e n t i e v

Table of Contents

Chapter 1

Formulation of some Improperly Posed Problems of Mathematical Physics

§ 1 Improperly Posed Problems in Metric Spaces........................

1

§ 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 R egularity...................................................................................

13

§ 2 The Cauchy Problem for the Laplace E q u a tio n ..................

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 S y m m e try .......................................................................

42

Chapter III Inverse Problems for Differential Equations § 1 The Inverse Problem for a Newtonian P o te n tia l..................

45

§ 2 A Class of Nonlinear Integral E q u a tio n s ..............................

55

§ 3 Inverse Problems for Some Non-Newtonian Potentials . . .

62

§ 4 An Inverse Problem for the WaveEquation............................

63

§ 5 On a Class of Inverse Problems for Differential Equations . 65 B ibliography......................................................................................................

70

Chapter 1

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 H a d a m a r d 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 0 , F be some complete metric spaces, and let Aq> be a function with the domain of definition 0 and the range of values F. Consider the equation

A