Ill-Posed Problems of Mathematical Physics and Analysis

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IIl-posed Problems of Mathematical Physics and Analysis

Copyright © 1986 by the American Mathematical Society
     ISBN 0-8218-4517-9
     QC20.7.B6L3813 1986 530.1'55
     LCCN 86-3642

Contents

Preface

Introduction

CHAPTER I  Physical Formulations Leading to Ill-posed Problems
     §1. Continuation of static fields
     §2. Problems for the diffusion equation
     §3. Continuation of fields from discrete sets
     §4. Processing readings of physical instruments
     §5. Inverse problems of geophysics
     §6. Inverse problems of gravimetry
     §7. The inverse kinematic problem of seismology

CHAPTER II  Basic Concepts of the Theory of Ill-posed Problems
     §1. Problems well-posed in the Tikhonov sense
     §2. Regularization
     §3. Linear ill-posed problems

CHAPTER III  Analytic Continuation
     §1. Formulations of problems and classical results
     §2. Analytic continuation from continua
     §3. Analytic continuation from classes of sets including discrete sets
     §4. Recovery of solutions of elliptic and parabolic equations from their values on sets lying inside the domain of regularity

CHAPTER IV  Boundary Value Problems for Differential Equations
     §1. The noncharacteristic Cauchy problem for,a parabolic equation. The Cauchy problem for an elliptic equation
     §2. A mixed problem for a parabolic equation with decreasing time
     §3. Cauchy problems with data on a segment of the time axis for degenerate parabolic and pseudoparabolic equations
     §4. Cauchy problems with data on a timelike surface for hyperbolic and ultrahyperbolic equations

CHAPTER V  Volterra Equations
     §1. Regularization of a Volterra equation of the first kind
     §2. Operator Volterra equations of the first kind

CHAPTER VI  Integral Geometry
     §1. The problem of finding a function from its spherical means
     §2. Problems of integral geometry on a family of manifolds which is invariant under a group of transformations of the space
     §3. Integral geometry in special classes of functions
     §4. Integral geometry "in the small"
     §5. The problem of integral geometry on plane curves and energy inequalities

CHAPTER VII  Multidimensional Inverse Problems for Linear Differential Equations
     §1. Examples of formulations of multidimensional inverse problems. Mathematical problems connected with investigating them
     §2. A general approach to investigating questions of uniqueness and stability of inverse problems
     §3. Inverse problems for hyperbolic equations of second order
     §4. Inverse problems for first-order hyperbolic systems
     §5. Inverse problems for parabolic equations of second order
          1. The connection between solutions of direct problems for equations of hyperbolic and parabolic types and inverse problem
          2. The method of descent in inverse problems.
          3. Inverse problems for equations of parabolic type
     §6. An abstract inverse problem and questions of its being well-posed
          1. Reduction to the investigation of a two-parameter family of linear equations.
          2. The method of linearization in investigating the inverse problem

Bibliography*

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