**Mathematical Surveys and Monographs**

Volume: 34;
1990;
191 pp;
Hardcover

MSC: Primary 35;
Secondary 31; 86

**Print ISBN: 978-0-8218-1532-8
Product Code: SURV/34**

List Price: $102.00

AMS Member Price: $81.60

MAA Member Price: $91.80

**Electronic ISBN: 978-1-4704-1261-6
Product Code: SURV/34.E**

List Price: $96.00

AMS Member Price: $76.80

MAA Member Price: $86.40

# Inverse Source Problems

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*Victor Isakov*

Inverse problems arise in many areas of mathematical physics, and applications
are rapidly expanding to such areas as geophysics, chemistry, medicine, and
engineering. The main theme of this book is uniqueness, stability, and
existence of solutions of inverse problems for partial differential equations.
Focusing primarily on the inverse problem of potential theory and closely
related questions such as coefficient identification problems, this book will
give readers an understanding of the results of a substantial part of the
theory of inverse problems and of some of the new ideas and methods used.

The author provides complete proofs of most general uniqueness theorems for the
inverse problem of gravimetry, a detailed study of regularity properties
(including examples of non-regular domains with regular potentials),
counterexamples to uniqueness and uniqueness theorems, and a treatment of the
theory of non-stationary problems. In addition, the book deals with the
orthogonality method, formulates several important unsolved problems, and
suggests certain technical means appropriate for further study; some numerical
methods are also outlined. Requiring a background in the basics of
differential equations and function theory, this book is directed at
mathematicians specializing in partial differential equations and potential
theory, as well as physicists, geophysicists, and engineers.

#### Table of Contents

# Table of Contents

## Inverse Source Problems

- Contents vii8 free
- Preface ix10 free
- Foreword xi12 free
- Chapter 1. Preliminaries 116 free
- 1.1. Sets and functions. Differential operators 116
- 1.2. Uniqueness of the continuation 419
- 1.3. Elliptic boundary value problems for second-order equations 1025
- 1.4. Fundamental solutions 1429
- 1.5. Measures and their potentials 1530
- 1.6. Properties of volume and simple layer potentials 1833
- 1.7. Stability of the Dirichlet problem 2237
- 1.8. The sweeping out method of Poincaré 2742

- Chapter 2. Results Overlook 3348
- Chapter 3. Uniqueness Theorems 5570
- Chapter 4. Singular Points of the Exterior Potential 89104
- Chapter 5. Existence Theory 105120
- 5.1. Local existence theorems 105120
- 5.2. On global existence of a solution 110125
- 5.3. Proofs of local theorems 112127
- 5.4. An analogy of Korn-Lichtenstein-Giraud's inequality 119134
- 5.5. Proof of Theorem 5.1.7 in the plane case 129144
- 5.6. Existence theorem for the density problem 133148
- 5.7. Commentary 135150

- Chapter 6. Parabolic Problems 137152
- Chapter 7. Hyperbolic Problems 161176
- Bibliography 181196
- Index 193208 free