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ОбложкаColton D. Integral equation methods in scattering theory / D.Colton, R.Kress. - Philadelphia: SIAM, 2013. - xvi, 271 p. - (Classics in applied mathematics; 72). - Bibliogr.: p.261-267. - Ind.: p.269-271. - ISBN 978-1-611973-15-0
Шифр: (И/В16-С71) 02
 

Место хранения: 02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents
 
Preface to the Classics Edition ................................ xi

Preface ........................................................ xv
Symbols ...................................................... xiii

1  The Riesz-Fredholm Theory for Compact Operators ............. 1
   1.1  Compact Operators ....................................... 2
   1.2  The Riesz Theory ........................................ 9
   1.3  The Fredholm Theory .................................... 16
   1.4  A Singular Perturbation Problem ........................ 23
   1.5  Successive Approximations .............................. 26
2  Regularity Properties of Surface Potentials ................. 31
   2.1  Geometry of Surfaces ................................... 32
   2.2  Holder Continuity ...................................... 37
   2.3  Weakly Singular Integral Operators on Surfaces ......... 39
   2.4  Single- and Double-Layer Potentials .................... 46
   2.5  Derivatives of Single- and Double-Layer Potentials ..... 51
   2.6  Vector Potentials ...................................... 58
   2.7  Integral Operators for Boundary-Value Problems ......... 61
3  Boundary-Value Problems for the Scalar Helmholtz Equation ... 65
   3.1  Time-Harmonic Acoustic Scattering ...................... 66
   3.2  Green's Representation Theorem and Sommerfeld's
        Radiation Condition .................................... 68
   3.3  The Dirichlet and Neumann Boundary-Value Problems:
        Uniqueness Theorems .................................... 75
   3.4  The Existence of Solutions to the Dirichlet and
        Neumann Problems ....................................... 79
   3.5  Boundary Integral Equations of the First Kind .......... 87
   3.6  Modified Integral Equations ............................ 90
   3.7  The Impedance Boundary-Value Problem ................... 97
   3.8  The Transmission Boundary-Value Problem ................ 99
   3.9  Integral Equations Based on the Representation
        Theorems .............................................. 102
   3.10 The Two-Dimensional Case .............................. 106
4  Boundary-Value Problems for the Time-Harmonic Maxwell
   Equations and the Vector Helmholtz Equation ................ 108
   4.1  Time-Harmonic Electromagnetic Scattering .............. 109
   4.2  Representation Theorems and Radiation Conditions ...... 110
   4.3  The Boundary-Value Problems for a Perfect Conductor:
        Uniqueness Theorems ................................... 121
   4.4  Existence of Solutions to the Electromagnetic
        Boundary-Value Problems by Integral Equations of the
        Second Kind ........................................... 126
   4.5  Boundary Integral Equations of the First Kind ......... 136
   4.6  Modified Integral Equations ........................... 140
   4.7  The Impedance Boundary-Value Problem .................. 146
   4.8  Integral Equations Based on the Representation
        Theorems .............................................. 147
5  Low Frequency Behavior of Solutions to Boundary-Value
   Problems in Scattering Theory .............................. 150
   5.1  Iterative Methods for Solving the Exterior Dirichlet
        and Neumann Problems .................................. 151
   5.2  Iterative Methods for Electromagnetic Problems ........ 154
   5.3  Low Wave Number Behavior of Solutions to the
        Exterior Electromagnetic Boundary-Value Problems ...... 158
6  The Inverse Scattering Problem: Exact Data ................. 173
   6.1  Entire Functions of Exponential Type .................. 175
   6.2  Far-Field Patterns and Their Classification ........... 182
   6.3  Uniqueness of Solutions to the Inverse
        Scattering Problem .................................... 192
7  Improperly Posed Problems and Compact Families ............. 197
   7.1  A Priori Assumptions and the Solution of Improperly
        Posed Problems ........................................ 198
   7.2  Linearized Improperly Posed Problems in Scattering
        Theory ................................................ 206
   7.3  Normal Families of Univalent Functions ................ 211
8  The Determination of the Shape of an Obstacle from Inexact
   Far-Field Data ............................................. 219
   8.1  A Model Problem ....................................... 221
   8.2  The Determination of the Shape of an Obstacle in fig.12 ... 232
   8.3  The Determination of the Shape of an Obstacle in fig.13 ... 239
9  Optimal Control Problems in Radiation and Scattering
   Theory ..................................................... 244
   9.1  Weak Compactness in Hilbert Space ..................... 245
   9.2  Optimal Control for a Radiation Problem ............... 247
   9.3  Optimal Control for a Scattering Problem .............. 254

References .................................................... 261
Index ......................................................... 269

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