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Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics

G. F. Roach, I. G. Stratis, and A. N. Yannacopoulos
Publisher: 
Princeton University Press
Publication Date: 
2012
Number of Pages: 
382
Format: 
Hardcover
Series: 
Princeton Series in Applied Mathematics
Price: 
99.50
ISBN: 
9780691142173
Category: 
Monograph
We do not plan to review this book.

Preface xi

PART 1. MODELLING AND MATHEMATICAL PRELIMINARIES 1

Chapter 1. Complex Media 3
Chapter 2. The Maxwell Equations and Constitutive Relations 9
2.1 Introduction 9
2.2 Fundamentals 9
2.3 Constitutive relations 13
2.4 The Maxwell equations in complex media: A variety of problems 23

Chapter 3. Spaces and Operators 38
3.1 Introduction 38
3.2 Function spaces 38
3.3 Standard difierential and trace operators 45
3.4 Function spaces for electromagnetics 48
3.5 Traces 51
3.6 Various decompositions 52
3.7 Compact embeddings 53
3.8 The operators of vector analysis revisited 54
3.9 The Maxwell operator 56

PART 2. TIME-HARMONIC DETERMINISTIC PROBLEMS 59

Chapter 4. Well Posedness 61
4.1 Introduction 61
4.2 Solvability of the interior problem 62
4.3 The eigenvalue problem 68
4.4 Low chirality behaviour 70
4.5 Comments on exterior domain problems 74
4.6 Towards numerics 77

Chapter 5. Scattering Problems: Beltrami Fields and Solvability 83
5.1 Introduction 83
5.2 Elliptic, circular and linear polarisation of waves 84
5.3 Beltrami fields - The Bohren decomposition 86
5.4 Scattering problems: Formulation 88
5.5 An introduction to BIEs 91
5.6 Properties of Beltrami fields 96
5.7 Solvability 99
5.8 Generalised Muller's BIEs 106
5.9 Low chirality approximations 108
5.10 Miscellanea 109

Chapter 6. Scattering Problems: A Variety of Topics 112
6.1 Introduction 112
6.2 Important concepts of scattering theory 113
6.3 Back to chiral media: Scattering relations and the far-field operator 118
6.4 Using dyadics 124
6.5 Herglotz wave functions 129
6.6 Domain derivative 136
6.7 Miscellanea 140

PART 3. TIME-DEPENDENT DETERMINISTIC PROBLEMS 149

Chapter 7. Well Posedness 151
7.1 Introduction 151
7.2 The Maxwell equations in the time domain 151
7.3 Functional framework and assumptions 152
7.4 Solvability 153
7.5 Other possible approaches to solvability 158
7.6 Miscellanea 162

Chapter 8. Controllability 163
8.1 Introduction 163
8.2 Formulation 163
8.3 Controllability of achiral media: The Hilbert Uniqueness method 165
8.4 The forward and backward problems 167
8.5 Controllability: Complex media 174
8.6 Miscellanea 176

Chapter 9. Homogenisation 180
9.1 Introduction 180
9.2 Formulation 181
9.3 A formal two-scale expansion 184
9.4 The optical response region 188
9.5 General bianisotropic media 199
9.6 Miscellanea 207

Chapter 10. Towards a Scattering Theory 212
10.1 Introduction 212
10.2 Formulation 213
10.3 Some basic strategies 214
10.4 On the construction of solutions 217
10.5 Wave operators and their construction 220
10.6 Complex media electromagnetics 225
10.7 Miscellanea 229

Chapter 11. Nonlinear Problems 231
11.1 Introduction 231
11.2 Formulation 231
11.3 Well posedness of the model 232
11.4 Miscellanea 241

PART 4. STOCHASTIC PROBLEMS 245

Chapter 12. Well Posedness 247
12.1 Introduction 247
12.2 Maxwell equations for random media 248
12.3 Functional setting 249
12.4 Well posedness 250
12.5 Other possible approaches to solvability 255
12.6 Miscellanea 261

Chapter 13. Controllability 263
13.1 Introduction 263
13.2 Formulation 263
13.3 Subtleties of stochastic controllability 264
13.4 Approximate controllability I: Random PDEs 266
13.5 Approximate controllability II: BSPDEs 269
13.6 Miscellanea 272

Chapter 14. Homogenisation 275
14.1 Introduction 275
14.2 Ergodic media 276
14.3 Formulation 279
14.4 A formal two-scale expansion 282
14.5 Homogenisation of the Maxwell system 284
14.6 Miscellanea 288

PART 5. APPENDICES 291

Appendix A. Some Facts from Functional Analysis 293
A.1 Duality 293
A.2 Strong, weak and weak-* convergence 295
A.3 Calculus in Banach spaces 297
A.4 Basic elements of spectral theory 300
A.5 Compactness criteria 303
A.6 Compact operators 304
A.7 The Banach-Steinhaus theorem 308
A.8 Semigroups and the Cauchy problem 308
A.9 Some fixed point theorems 312
A.10 The Lax-Milgram lemma 313
A.11 Gronwall's inequality 314
A.12 Nonlinear operators 315

Appendix B. Some Facts from Stochastic Analysis 316
B.1 Probability in Hilbert spaces 316
B.2 Stochastic processes and random fields 318
B.3 Gaussian measures 319
B.4 The Q- and the cylindrical Wiener process 320
B.5 The Ito integral 321
B.6 Ito formula 324
B.7 Stochastic convolution 325
B.8 SDEs in Hilbert spaces 325
B.9 Martingale representation theorem 326

Appendix C. Some Facts from Elliptic Homogenisation Theory 327
C.1 Spaces of periodic functions 327
C.2 Compensated compactness 329
C.3 Homogenisation of elliptic equations 329
C.4 Random elliptic homogenisation theory 332
Appendix D. Some Facts from Dyadic Analysis (by George Dassios) 334
Appendix E. Notation and abbreviations 341

Bibliography 343
Index 377

Dummy View - NOT TO BE DELETED