Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics

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