Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics ：Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics ( Princeton Series in Applied Mathematics )

Publication subTitle ：Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics

Publication series ：Princeton Series in Applied Mathematics

Author： Roach G. F.;Stratis I. G.;Yannacopoulos A. N.;

Publisher： Princeton University Press‎

Publication year： 2012

E-ISBN: 9781400842650

P-ISBN(Paperback): 9780691142173

Subject： O441 electromagnetics

Keyword：物理学,数理科学和化学,数学

Language： ENG

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Description

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.

Chapter

PART 1: MODELLING AND MATHEMATICAL PRELIMINARIES

Chapter 1 Complex Media

Chapter 2 The Maxwell Equations and Constitutive Relations

2.1 Introduction

2.2 Fundamentals

2.3 Constitutive relations

2.4 The Maxwell equations in complex media: A variety of problems

Chapter 3 Spaces and Operators

3.1 Introduction

3.2 Function spaces

3.3 Standard differential and trace operators

3.4 Function spaces for electromagnetics

3.5 Traces

3.6 Various decompositions

3.7 Compact embeddings

3.8 The operators of vector analysis revisited

3.9 The Maxwell operator

PART 2: TIME-HARMONIC DETERMINISTIC PROBLEMS

Chapter 4 Well Posedness

4.1 Introduction

4.2 Solvability of the interior problem

4.3 The eigenvalue problem

4.4 Low chirality behaviour

4.5 Comments on exterior domain problems

4.6 Towards numerics

Chapter 5 Scattering Problems: Beltrami Fields and Solvability

5.1 Introduction

5.2 Elliptic, circular and linear polarisation of waves

5.3 Beltrami fields - The Bohren decomposition

5.4 Scattering problems: Formulation

5.5 An introduction to BIEs

5.6 Properties of Beltrami fields

5.7 Solvability

5.8 Generalised Müller's BIEs

5.9 Low chirality approximations

5.10 Miscellanea

Chapter 6 Scattering Problems: A Variety of Topics

6.1 Introduction

6.2 Important concepts of scattering theory

6.3 Back to chiral media: Scattering relations and the far-field operator

6.4 Using dyadics

6.5 Herglotz wave functions

6.6 Domain derivative

6.7 Miscellanea

PART 3: TIME-DEPENDENT DETERMINISTIC PROBLEMS

Chapter 7 Well Posedness

7.1 Introduction

7.2 The Maxwell equations in the time domain

7.3 Functional framework and assumptions

7.4 Solvability

7.5 Other possible approaches to solvability

7.6 Miscellanea

Chapter 8 Controllability

8.1 Introduction

8.2 Formulation

8.3 Controllability of achiral media: The Hilbert Uniqueness method

8.4 The forward and backward problems

8.5 Controllability: Complex media

8.6 Miscellanea

Chapter 9 Homogenisation

9.1 Introduction

9.2 Formulation

9.3 A formal two-scale expansion

9.4 The optical response region

9.5 General bianisotropic media

9.6 Miscellanea

Chapter 10. Towards a Scattering Theory

10.1 Introduction

10.2 Formulation

10.3 Some basic strategies

10.4 On the construction of solutions

10.5 Wave operators and their construction

10.6 Complex media electromagnetics

10.7 Miscellanea

Chapter 11 Nonlinear Problems

11.1 Introduction

11.2 Formulation

11.3 Well posedness of the model

11.4 Miscellanea

PART 4: STOCHASTIC PROBLEMS

Chapter 12 Well Posedness

12.1 Introduction

12.2 Maxwell equations for random media

12.3 Functional setting

12.4 Well posedness

12.5 Other possible approaches to solvability

12.6 Miscellanea

Chapter 13. Controllability

13.1 Introduction

13.2 Formulation

13.3 Subtleties of stochastic controllability

13.4 Approximate controllability I: Random PDEs

13.5 Approximate controllability II: BSPDEs

13.6 Miscellanea

Chapter 14 Homogenisation

14.1 Introduction

14.2 Ergodic media

14.3 Formulation

14.4 A formal two-scale expansion

14.5 Homogenisation of the Maxwell system

14.6 Miscellanea

PART 5: APPENDICES

Appendix A Some Facts from Functional Analysis

A.1 Duality

A.2 Strong, weak and weak-* convergence

A.3 Calculus in Banach spaces

A.4 Basic elements of spectral theory

A.5 Compactness criteria

A.6 Compact operators

A.7 The Banach-Steinhaus theorem

A.8 Semigroups and the Cauchy problem

A.9 Some fixed point theorems

A.10 The Lax-Milgram lemma

A.11 Gronwall's inequality

A.12 Nonlinear operators

Appendix B Some Facts from Stochastic Analysis

B.1 Probability in Hilbert spaces

B.2 Stochastic processes and random fields

B.3 Gaussian measures

B.4 The Q- and the cylindrical Wiener process

B.5 The Itō integral

B.6 Itō formula

B.7 Stochastic convolution

B.8 SDEs in Hilbert spaces

B.9 Martingale representation theorem

Appendix C Some Facts from Elliptic Homogenisation Theory

C.1 Spaces of periodic functions

C.2 Compensated compactness

C.3 Homogenisation of elliptic equations

C.4 Random elliptic homogenisation theory

Appendix D Some Facts from Dyadic Analysis

Appendix E Notation and abbreviations

Bibliography

Index

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