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Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics

Book 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.

Table of Contents

  1. Cover
  2. Half title
  3. Title
  4. Copyright
  5. Contents
  6. Preface
  7. Part 1. Modelling and Mathematical Preliminaries
    1. Chapter 1. Complex Media
    2. Chapter 2. The Maxwell Equations and Constitutive Relations
      1. 2.1 Introduction
      2. 2.2 Fundamentals
      3. 2.3 Constitutive relations
      4. 2.4 The Maxwell equations in complex media: A variety of problems
    3. Chapter 3. Spaces and Operators
      1. 3.1 Introduction
      2. 3.2 Function spaces
      3. 3.3 Standard differential and trace operators
      4. 3.4 Function spaces for electromagnetics
      5. 3.5 Traces
      6. 3.6 Various decompositions
      7. 3.7 Compact embeddings
      8. 3.8 The operators of vector analysis revisited
      9. 3.9 The Maxwell operator
  8. Part 2. Time-Harmonic Deterministic Problems
    1. Chapter 4. Well Posedness
      1. 4.1 Introduction
      2. 4.2 Solvability of the interior problem
      3. 4.3 The eigenvalue problem
      4. 4.4 Low chirality behaviour
      5. 4.5 Comments on exterior domain problems
      6. 4.6 Towards numerics
    2. Chapter 5. Scattering Problems: Beltrami Fields and Solvability
      1. 5.1 Introduction
      2. 5.2 Elliptic, circular and linear polarisation of waves
      3. 5.3 Beltrami fields - The Bohren decomposition
      4. 5.4 Scattering problems: Formulation
      5. 5.5 An introduction to BIEs
      6. 5.6 Properties of Beltrami fields
      7. 5.7 Solvability
      8. 5.8 Generalised Müller’s BIEs
      9. 5.9 Low chirality approximations
      10. 5.10 Miscellanea
    3. Chapter 6. Scattering Problems: A Variety of Topics
      1. 6.1 Introduction
      2. 6.2 Important concepts of scattering theory
      3. 6.3 Back to chiral media: Scattering relations and the far-field operator
      4. 6.4 Using dyadics
      5. 6.5 Herglotz wave functions
      6. 6.6 Domain derivative
      7. 6.7 Miscellanea
  9. Part 3. Time-Dependent Deterministic Problems
    1. Chapter 7. Well Posedness
      1. 7.1 Introduction
      2. 7.2 The Maxwell equations in the time domain
      3. 7.3 Functional framework and assumptions
      4. 7.4 Solvability
      5. 7.5 Other possible approaches to solvability
      6. 7.6 Miscellanea
    2. Chapter 8. Controllability
      1. 8.1 Introduction
      2. 8.2 Formulation
      3. 8.3 Controllability of achiral media: The Hilbert Uniqueness method
      4. 8.4 The forward and backward problems
      5. 8.5 Controllability: Complex media
      6. 8.6 Miscellanea
    3. Chapter 9. Homogenisation
      1. 9.1 Introduction
      2. 9.2 Formulation
      3. 9.3 A formal two-scale expansion
      4. 9.4 The optical response region
      5. 9.5 General bianisotropic media
      6. 9.6 Miscellanea
    4. Chapter 10. Towards a Scattering Theory
      1. 10.1 Introduction
      2. 10.2 Formulation
      3. 10.3 Some basic strategies
      4. 10.4 On the construction of solutions
      5. 10.5 Wave operators and their construction
      6. 10.6 Complex media electromagnetics
      7. 10.7 Miscellanea
    5. Chapter 11. Nonlinear Problems
      1. 11.1 Introduction
      2. 11.2 Formulation
      3. 11.3 Well posedness of the model
      4. 11.4 Miscellanea
  10. Part 4. Stochastic Problems
    1. Chapter 12. Well Posedness
      1. 12.1 Introduction
      2. 12.2 Maxwell equations for random media
      3. 12.3 Functional setting
      4. 12.4 Well posedness
      5. 12.5 Other possible approaches to solvability
      6. 12.6 Miscellanea
    2. Chapter 13. Controllability
      1. 13.1 Introduction
      2. 13.2 Formulation
      3. 13.3 Subtleties of stochastic controllability
      4. 13.4 Approximate controllability I: Random PDEs
      5. 13.5 Approximate controllability II: BSPDEs
      6. 13.6 Miscellanea
    3. Chapter 14. Homogenisation
      1. 14.1 Introduction
      2. 14.2 Ergodic media
      3. 14.3 Formulation
      4. 14.4 A formal two-scale expansion
      5. 14.5 Homogenisation of the Maxwell system
      6. 14.6 Miscellanea
  11. Part 5. Appendices
    1. Appendix A. Some Facts from Functional Analysis
      1. A.1 Duality
      2. A.2 Strong, weak and weak-* convergence
      3. A.3 Calculus in Banach spaces
      4. A.4 Basic elements of spectral theory
      5. A.5 Compactness criteria
      6. A.6 Compact operators
      7. A.7 The Banach-Steinhaus theorem
      8. A.8 Semigroups and the Cauchy problem
      9. A.9 Some fixed point theorems
      10. A.10 The Lax-Milgram lemma
      11. A.11 Gronwall’s inequality
      12. A.12 Nonlinear operators
    2. Appendix B. Some Facts from Stochastic Analysis
      1. B.1 Probability in Hilbert spaces
      2. B.2 Stochastic processes and random fields
      3. B.3 Gaussian measures
      4. B.4 The Q- and the cylindrical Wiener process
      5. B.5 The Itō integral
      6. B.6 Itō formula
      7. B.7 Stochastic convolution
      8. B.8 SDEs in Hilbert spaces
      9. B.9 Martingale representation theorem
    3. Appendix C. Some Facts from Elliptic Homogenisation Theory
      1. C.1 Spaces of periodic functions
      2. C.2 Compensated compactness
      3. C.3 Homogenisation of elliptic equations
      4. C.4 Random elliptic homogenisation theory
    4. Appendix D. Some Facts from Dyadic Analysis
    5. Appendix E. Notation and abbreviations
  12. Bibliography
  13. Index