Mathematical analysis of deterministic and stochastic problems in complex media electromagnetics / G. F. Roach, I. G. Stratis, A. N. Yannacopoulos.

Format:

Book

Author/Creator:

Roach, G. F. (Gary Francis)

Contributor:

Stratis, I. G. (Ioannis G.), 1955-

Yannacopoulos, A. N. (Athanasios N.), 1968-

Series:

Princeton series in applied mathematics

Language:

English

Subjects (All):

Electromagnetism--Mathematics.

Electromagnetism.

Stochastic control theory.

Mathematical analysis.

Physical Description:

xv, 382 pages : illustrations ; 25 cm.

Place of Publication:

Princeton : Princeton University Press, 2012.

Summary:

Electromagnetic complex media are artificial materials that affect the propagation of electromagnetic waves in surprising ways not usually seen in nature. Because of their wide range of important applications, these materials have been intensely studied over the past twenty-five years, mainly from the perspectives of physics and engineering. But a body of rigorous mathematical theory has also gradually developed, and this is the first book to present that theory.

Designed for researchers and advanced graduate students in applied mathematics, electrical engineering, and physics, this book introduces the electromagnetics of complex media through a systematic, state-of-the-art account of their mathematical theory. The book combines the study of well posedness, homogenization, and controllability of Maxwell equations complemented with constitutive relations describing complex media. The book treats deterministic and stochastic problems both in the frequency and time domains. It also covers computational aspects and scattering problems, among other important topics. Detailed appendices make the book self-contained in terms of mathematical prerequisites, and accessible to engineers and physicists as well as mathematicians. Book jacket.

Contents:

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 differential 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 Müller'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.

Notes:

Includes bibliographical references and index.

ISBN:

9780691142173

0691142173

OCLC:

758646782