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E-raamat: Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics

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Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media ElectromagneticsSuurem pilt
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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.

Arvustused

"This monograph is of a very high standard, allowing the reader to learn many facets of the rapidly growing field of complex media and to get up-to-date information on a number of open research problems."--Vilmos Komornik, Mathematical Reviews

Preface xi
PART 1 MODELLING AND MATHEMATICAL PRELIMINARIES
1(58)
Chapter 1 Complex Media
3(6)
Chapter 2 The Maxwell Equations and Constitutive Relations
9(29)
2.1 Introduction
9(1)
2.2 Fundamentals
9(4)
2.3 Constitutive relations
13(10)
2.4 The Maxwell equations in complex media: A variety of problems
23(15)
Chapter 3 Spaces and Operators
38(21)
3.1 Introduction
38(1)
3.2 Function spaces
38(7)
3.3 Standard differential and trace operators
45(3)
3.4 Function spaces for electromagnetics
48(3)
3.5 Traces
51(1)
3.6 Various decompositions
52(1)
3.7 Compact embeddings
53(1)
3.8 The operators of vector analysis revisited
54(2)
3.9 The Maxwell operator
56(3)
PART 2 TIME-HARMONIC DETERMINISTIC PROBLEMS
59(90)
Chapter 4 Well Posedness
61(22)
4.1 Introduction
61(1)
4.2 Solvability of the interior problem
62(6)
4.3 The eigenvalue problem
68(2)
4.4 Low chirality behaviour
70(4)
4.5 Comments on exterior domain problems
74(3)
4.6 Towards numerics
77(6)
Chapter 5 Scattering Problems: Beltrami Fields and Solvability
83(29)
5.1 Introduction
83(1)
5.2 Elliptic, circular and linear polarisation of waves
84(2)
5.3 Beltrami fields - The Bohren decomposition
86(2)
5.4 Scattering problems: Formulation
88(3)
5.5 An introduction to BIEs
91(5)
5.6 Properties of Beltrami fields
96(3)
5.7 Solvability
99(7)
5.8 Generalised Muller's BIEs
106(2)
5.9 Low chirality approximations
108(1)
5.10 Miscellanea
109(3)
Chapter 6 Scattering Problems: A Variety of Topics
112(37)
6.1 Introduction
112(1)
6.2 Important concepts of scattering theory
113(5)
6.3 Back to chiral media: Scattering relations and the far-field operator
118(6)
6.4 Using dyadics
124(5)
6.5 Herglotz wave functions
129(7)
6.6 Domain derivative
136(4)
6.7 Miscellanea
140(9)
PART 3 TIME-DEPENDENT DETERMINISTIC PROBLEMS
149(96)
Chapter 7 Well Posedness
151(12)
7.1 Introduction
151(1)
7.2 The Maxwell equations in the time domain
151(1)
7.3 Functional framework and assumptions
152(1)
7.4 Solvability
153(5)
7.5 Other possible approaches to solvability
158(4)
7.6 Miscellanea
162(1)
Chapter 8 Controllability
163(17)
8.1 Introduction
163(1)
8.2 Formulation
163(2)
8.3 Controllability of achiral media: The Hilbert Uniqueness method
165(2)
8.4 The forward and backward problems
167(7)
8.5 Controllability: Complex media
174(2)
8.6 Miscellanea
176(4)
Chapter 9 Homogenisation
180(32)
9.1 Introduction
180(1)
9.2 Formulation
181(3)
9.3 A formal two-scale expansion
184(4)
9.4 The optical response region
188(11)
9.5 General bianisotropic media
199(8)
9.6 Miscellanea
207(5)
Chapter 10 Towards a Scattering Theory
212(19)
10.1 Introduction
212(1)
10.2 Formulation
213(1)
10.3 Some basic strategies
214(3)
10.4 On the construction of solutions
217(3)
10.5 Wave operators and their construction
220(5)
10.6 Complex media electromagnetics
225(4)
10.7 Miscellanea
229(2)
Chapter 11 Nonlinear Problems
231(14)
11.1 Introduction
231(1)
11.2 Formulation
231(1)
11.3 Well posedness of the model
232(9)
11.4 Miscellanea
241(4)
PART 4 STOCHASTIC PROBLEMS
245(46)
Chapter 12 Well Posedness
247(16)
12.1 Introduction
247(1)
12.2 Maxwell equations for random media
248(1)
12.3 Functional setting
249(1)
12.4 Well posedness
250(5)
12.5 Other possible approaches to solvability
255(6)
12.6 Miscellanea
261(2)
Chapter 13 Controllability
263(12)
13.1 Introduction
263(1)
13.2 Formulation
263(1)
13.3 Subtleties of stochastic controllability
264(2)
13.4 Approximate controllability I: Random PDEs
266(3)
13.5 Approximate controllability II: BSPDEs
269(3)
13.6 Miscellanea
272(3)
Chapter 14 Homogenisation
275(16)
14.1 Introduction
275(1)
14.2 Ergodic media
276(3)
14.3 Formulation
279(3)
14.4 A formal two-scale expansion
282(2)
14.5 Homogenisation of the Maxwell system
284(4)
14.6 Miscellanea
288(3)
PART 5 APPENDICES
291(2)
Appendix A Some Facts from Functional Analysis
293(23)
A.1 Duality
293(2)
A.2 Strong, weak and weak-* convergence
295(2)
A.3 Calculus in Banach spaces
297(3)
A.4 Basic elements of spectral theory
300(3)
A.5 Compactness criteria
303(1)
A.6 Compact operators
304(4)
A.7 The Banach-Steinhaus theorem
308(1)
A.8 Semigroups and the Cauchy problem
308(4)
A.9 Some fixed point theorems
312(1)
A.10 The Lax-Milgram lemma
313(1)
A.11 Gronwall's inequality
314(1)
A.12 Nonlinear operators
315(1)
Appendix B Some Facts from Stochastic Analysis
316(11)
B.1 Probability in Hilbert spaces
316(2)
B.2 Stochastic processes and random fields
318(1)
B.3 Gaussian measures
319(1)
B.4 The Q- and the cylindrical Wiener process
320(1)
B.5 The Ito integral
321(3)
B.6 Ito formula
324(1)
B.7 Stochastic convolution
325(1)
B.8 SDEs in Hilbert spaces
325(1)
B.9 Martingale representation theorem
326(1)
Appendix C Some Facts from Elliptic Homogenisation Theory
327(7)
C.1 Spaces of periodic functions
327(2)
C.2 Compensated compactness
329(1)
C.3 Homogenisation of elliptic equations
329(3)
C.4 Random elliptic homogenisation theory
332(2)
Appendix D Some Facts from Dyadic Analysis (by George Dassios)
334(7)
Appendix E Notation and abbreviations
341(2)
Bibliography 343(34)
Index 377
G. F. Roach is professor emeritus in the Department of Mathematics and Statistics at the University of Strathclyde. I. G. Stratis is professor in the Department of Mathematics at the National and Kapodistrian University, Athens. A. N. Yannacopoulos is professor in the Department of Statistics at the Athens University of Economics and Business.
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