viii CONTENTS

4.4. Computation of some potentials 99

4.5. Commentary 103

Chapter 5. Existence Theory 105

5.1. Local existence theorems 105

5.2. On global existence of a solution 110

5.3. Proofs of local theorems 112

5.4. An analogy of Korn-Lichtenstein-Giraud's inequality 119

5.5. Proof of Theorem 5.1.7 in the plane case 129

5.6. Existence theorem for the density problem 133

5.7. Commentary 135

Chapter 6. Parabolic Problems 137

6.1. Preliminaries 137

6.2. The final overdetermination 142

6.3. A lateral overdetermination: df/dxn = 0 149

6.4. A lateral overdetermination: df/dt = 0 152

6.5. A domain problem 154

6.6. Commentary 159

Chapter 7. Hyperbolic Problems 161

7.1. Preliminaries 161

7.2. On the final overdetermination 164

7.3. The lateral overdetermination: n = 1 166

7.4. The lateral overdetermination: the density problem for 2 n 170

7.5. Recovering a domain 173

7.6. Commentary 180

Bibliography 181

Index 193