Contents
Preface ix
Foreword xi
Chapter 1. Preliminaries 1
1.1. Sets and functions. Differential operators 1
1.2. Uniqueness of the continuation 4
1.3. Elliptic boundary value problems for second-order equations 10
1.4. Fundamental solutions 14
1.5. Measures and their potentials 15
1.6. Properties of volume and simple layer potentials 18
1.7. Stability of the Dirichlet problem 22
1.8. The sweeping out method of Poincare 27
Chapter 2. Results Overlook 33
2.1. Problem formulation 33
2.2. Uniqueness results 36
2.3. Stability and the numerical solution 38
2.4. Existence problem 43
2.5. Use of one complex variable 46
2.6. Nonstationary problems 48
Chapter 3. Uniqueness Theorems 55
3.1. Unknown domain 55
3.2. Unknown domain and density 62
3.3. Unknown density 64
3.4. Examples of nonuniqueness 68
3.5. Methods of the Fourier transform 76
3.6. Stability estimates 80
3.7. Commentary 84
Chapter 4. Singular Points of the Exterior Potential 89
4.1. Regularity and analyticity of domains and their potentials 89
4.2. Singularities of the corner potentials 94
4.3. Singularities and the Fourier transform 97
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