Polarization and Moment Tensors (eBook)

Preface 6
Contents 7
1 Introduction 11
2 Layer Potentials and Transmission Problems 17
Introduction 17
2.1 Notation and Preliminaries 17
2.2 Layer Potentials on Smooth Domains 24
2.3 Layer Potentials on Lipschitz Domains 34
2.4 Neumann and Dirichlet Functions 49
2.5 Representation Formula 55
2.6 Energy Identities 60
2.7 Anisotropic Transmission Problem 61
2.8 Periodic Isotropic Transmission Problem 62
2.9 Periodic Anisotropic Transmission Problem 69
2.10 Further Results and Open Problems 76
3 Uniqueness for Inverse Conductivity Problems 77
Introduction 77
3.1 Uniqueness With Many Measurements 78
3.2 Uniqueness With One Measurement 81
3.3 Further Results and Open Problems 84
4 Generalized Isotropic and Anisotropic Polarization Tensors 85
Introduction 85
4.1 Definition 86
4.2 Explicit Formulae 91
4.3 Extreme Conductivity Cases 98
4.4 Uniqueness Result 100
4.5 Symmetry and Positivity of GPTs 101
4.6 Estimates of the Harmonic Moments 104
4.7 Optimal Bounds for the Polarization Tensor 107
4.8 Monotonocity 113
4.9 Estimates of the Center of Mass 114
4.10 Polarization Tensors of Multiple Inclusions 116
4.11 Explicit Formulae for the Polarization Tensor of Multiple Disks 122
4.12 Anisotropic Polarization Tensors 129
4.13 Further Results and Open Problems 137
5 Full Asymptotic Formula for the Potentials 138
Introduction 138
5.1 Energy Estimates 140
5.2 Asymptotic Expansion 144
5.3 Derivation of the Asymptotic Formula for Closely Spaced Small Inclusions 149
5.4 Derivation of the Asymptotic Formula for Anisotropic Inclusions 151
5.5 Further Results and Open Problems 152
6 Near-Boundary Conductivity Inclusions 154
Introduction 154
6.1 Optimal Gradient Estimates 155
6.2 Asymptotic Expansions 159
6.3 Further Results and Open Problems 169
7 Impedance Imaging of Conductivity Inclusions 170
Introduction 170
7.1 Preliminary 171
7.2 Projection Algorithm — Reconstruction of a Single Inclusion 172
7.3 Quadratic Algorithm — Detection of Closely Spaced Inclusions 178
7.4 Simple Pole Method 181
7.5 Least-Squares Algorithm 182
7.6 Variational Algorithm 183
7.7 Linear Sampling Method 185
7.8 Lipschitz-Continuous Dependence and Moment Estimations 191
7.9 Detection of Anisotropic Inclusions 195
7.10 Further Results and Open Problems 201
8 Effective Properties of Electrical Composites 204
Introduction 204
8.1 Computation of Effective Conductivity 206
8.2 Anisotropic Composites 214
8.3 Further Results and Open Problems 219
9 Transmission Problem for Elastostatics 220
Introduction 220
9.1 Layer Potentials for the Lame System 220
9.2 Kelvin Matrix Under Unitary Transformations 224
9.3 Transmission Problem 227
9.4 Complex Representation of the Displacement Field 234
9.5 Periodic Green’s Function 239
9.6 Further Results and Open Problems 244
10 Elastic Moment Tensor 245
Introduction 245
10.1 Asymptotic Expansion in Free Space 245
10.2 Properties of EMTs 249
10.3 EMTs Under Linear Transformations 256
10.4 EMTs for Ellipses 259
10.5 EMTs for Elliptic Holes and Hard Ellipses 264
10.6 Further Results and Open Problems 267
11 Full Asymptotic Expansions of the Displacement Field 268
Introduction 268
11.1 Full Asymptotic Expansions 268
11.2 Further Results and Open Problems 274
12 Imaging of Elastic Inclusions 276
Introduction 276
12.1 Detection of EMTs 276
12.2 Representation of the EMTs by Ellipses 280
12.3 Detection of the Location 282
12.4 Numerical Results 284
12.5 Further Results and Open Problems 290
13 Effective Properties of Elastic Composites 292
Introduction 292
13.1 Derivation of the Effective Elastic Properties 293
13.2 Further Results and Open Problems 296
A Appendices 297
Introduction 297
A.1 Compact Operators 297
A.2 Theorem of Coifman, McIntosh, and Meyer 298
A.3 Continuity Method 299
References 301
Index 317