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Contents
Preface vii
Introduction xix
1. Basic Problems 1
1.1 Statement of Electrostatic Problems 1
1.2 Statement of the Basic Problem for Dielectric Bodies . . . . 4
1.3 Reduction of the Basic Problems 5
1.4 Reduction of the Static Problems 10
2. Iterative Processes for Solving Fredholm's Integral
Equations for Static Problems 13
2.1 An Iterative Process for Solving the Problem of Equilibrium 13
2.2 An Iterative Process for Solving the Electrostatic Problems
for Dielectric Bodies 15
2.3 A Stable Iterative Process 20
2.4 An Iterative Process for Screens 21
3. Calculating Electric Capacitance 25
3.1 Capacitance of Solid Conductors and Screens 25
3.2 Variational Principles 28
3.3 Capacitance of Conductors in an Anisotropic Medium . . . 31
3.4 Physical Analogues of Capacitance 37
3.5 Calculating the Potential Coefficients 37
4. Numerical Examples 43
4.1 Introduction 43
4.2 Capacitance of a Circular Cylinder 44
4.3 Capacitances of Parallelepipeds 45
4.4 Interaction Between Conductors 49
5. Calculating Polarizability Tensors 53
5.1 Calculating Polarizability Tensors 53
5.2 Polarizability Tensors of Screens 57
5.3 Polarizability Tensors of Flaky-Homogeneous Bodies . . . . 58
5.4 Variational Principles for Polarizability Tensors 59
6. Iterative Methods: Mathematical Results 69
6.1 Iterative Methods for Solving Predholm Equations 69
6.2 Iterative Processes for Solving Some Operator Equations . . 76
6.3 Iterative Processes for Solving the Exterior and Interior
Problems 79
6.4 An Iterative Process for Solving the Fredholm Integral Equations
88
7. Wave Scattering by Small Bodies 93
7.1 Introduction 93
7.2 Scalar Wave Scattering: The Single-Body Problem 94
7.3 Scalar Wave Scattering: The Many-Body Problem 101
7.4 Electromagnetic Wave Scattering 106
7.5 Radiation from Small Apertures 115
7.6 An Inverse Problem of Radiation Theory 121
8. Fredholm Alternative and a Characterization of
Fredholm Operators 125
8.1 Fredholm Alternative 125
8.1.1 Introduction 126
8.1.2 Proofs 128
8.2 A Characterization of Unbounded Fredholm Operators . . . 131
8.2.1 Statement of the result 131
8.2.2 Proof 132
8.3 Fredholm Alternative for Analytic Operators 135
9. Boundary-Value Problems in Rough Domains 137
9.1 Introduction 138
9.2 Proofs 141
9.3 Exterior Boundary-Value Problems 144
9.4 Quasiisometrical Mappings 151
9.4.1 Definitions and main properties 151
9.4.2 Interior metric and boundary metrics 152
9.4.3 Boundary behavior of quasiisometrical homeomorphisms
155
9.5 Quasiisometrical Homeomorphisms and Embedding Operators 157
9.5.1 Compact embedding operators for rough domains . . 158
9.5.2 Examples 160
9.6 Conclusions 162
10. Low Frequency Asymptotics 163
10.1 Introduction 163
10.2 Integral Equation Method for the Dirichlet Problem . . . . 165
10.3 Integral Equation Method for the Neumann Problem . . . . 171
10.4 Integral Equation Method for the Robin Problem 173
10.5 The Method based on the Predholm Property 180
10.6 The Method based on the Maximum Principle 186
10.7 Continuity with Respect to a Parameter 188
10.7.1 Introduction 189
10.7.2 Proofs 191
11. Finding Small Inhomogeneities from Scattering Data 193
11.1 Introduction 193
11.2 Basic Equations 194
11.3 Justification of the Proposed Method 196
12. Modified Rayleigh Conjecture and Applications 201
12.1 Modified Rayleigh Conjecture and Applications 201
12.1.1 Introduction 201
12.1.2 Direct scattering problem and MRC 203
12.1.3 Inverse scattering problem and MRC 204
12.1.4 Proofs 206
12.2 Modified Rayleigh Conjecture Method 207
12.2.1 Introduction 207
12.2.2 Numerical Experiments 211
12.2.3 Conclusions 215
12.3 Modified Rayleigh Conjecture for Static Fields 216
12.3.1 Solving boundary-value problems by MRC 217
12.3.2 Proofs 218
Appendix A Optimal with Respect to Accuracy
Algorithms for Calculation of Multidimensional
Weakly Singular Integrals and Applications to
Calculation of Capacitances of Conductors of
Arbitrary Shapes 221
A.I Introduction 221
A.2 Definitions of Optimality 223
A.3 Classes of Functions 224
A.4 Auxiliary Statements 227
A. 5 Optimal Methods for Calculating Integrals of the Form (A.I) 232
A.5.1 Lower bounds for the functionals £nm and £# . . . . 232
A.5.2 Optimal cubature formulas for calculating integrals
(A.I) 243
A.6 Optimal Methods for Calculating Integrals of the Form Tf 255
A.6.1 Lower bounds for the functionals C,mn and CN • • • • 255
A.6.2 Cubature formulas 260
A. 7 Calculation of Weakly Singular Integrals on Non-Smooth
Surfaces 263
A.8 Calculation of Weights of Cubature Formulas 267
A.9 Iterative Methods for Calculating Electrical Capacitancies . 269
A.10 Numerical Examples 272
Problems 275
Bibliographical Notes 277
Bibliography 279
List of Symbols 289
Index 291

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