In this post, we will see the book Some Improperly Posed Problems Of Mathematical Physics by M.M. Lavrentiev.

About the book

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 differential equation from a certain family of its solutions.

The book was translated from Russian by was published in 1969.

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You can get the book here.

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Contents

Chapter I 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 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 Cylindrical Symmetry 420

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