# Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics

G. F. Roach
I. G. Stratis
A. N. Yannacopoulos
Pages: 400
https://www.jstor.org/stable/j.ctt7t1xz

1. Front Matter
(pp. i-vi)
(pp. vii-x)
3. Preface
(pp. xi-xvi)
Gary F. Roach, Ioannis G. Stratis and Athanasios N. Yannacopoulos
4. ### PART 1. MODELLING AND MATHEMATICAL PRELIMINARIES

• Chapter One Complex Media
(pp. 3-8)

In recent years technology has replaced Hercules as far as the labours are concerned: the progress in theoretical studies, followed by impressive experimental work and achievements, is reaching the everyday lives of ordinary people and is rapidly changing our habits and lives.

A big part of this technological revolution, which emerged in the late twentieth century and is propagating with increasing speed and expanding front, is the result of complex media. Complex media are artificial materials exhibiting properties, based on their structure rather than their composition, superior to those in naturally existing materials. Nevertheless, there certainly do exist materials in...

• Chapter Two The Maxwell Equations and Constitutive Relations
(pp. 9-37)

The aim of this chapter is twofold: first, to introduce the constitutive relations which are commonly used in electromagnetic theory for the mathematical modelling of complex electromagnetic media. In the context of the present work these constitutive relations are to be understood as operators connecting the electric flux density and the magnetic flux density with the electric and the magnetic fields. These relations are considered as formal expressions of the physical laws that govern the electromagnetics of complex media. When they are introduced into the Maxwell equations, we obtain differential equations (PDEs) that govern the evolution of the electromagnetic fields;...

• Chapter Three Spaces and Operators
(pp. 38-58)

Let$\cal{O}$be an open set in$\mathbb{R}^N$such that it is locally on one side of its boundary$\Gamma : = \partial \cal{O}$, which is supposed to be bounded and Lipschitz. We shall be interested mainly in the case ofN= 3, so in the following unless explicitly stated otherwise, we are considering this case. Further, without loss of generality, we suppose that Γ is connected (for otherwise, one could work separately at each connected component). Such a set$\cal{O}$will be referred to as “regular” in what follows. Letndenote the outward unit normal vector to Γ. In addition,...

5. ### PART 2. TIME-HARMONIC DETERMINISTIC PROBLEMS

• Chapter Four Well Posedness
(pp. 61-82)

Time-harmonic problems constitute an important class within the theory of electromagnetics in complex media. In treating such problems we assume that the temporal evolution of these fields is periodic, with a fixed and prescribed period$\varpi$. This class covers a variety of important applications, such as wave guide problems, scattering problems, etc. Under the assumption of temporally periodic fields we may use the constitutive relations introduced in Section 2.3.3, which do not contain the convolution terms. The absence of the convolution terms simplifies the treatment of the Maxwell equations, (PDEs) which in this case become partial differential equations depending...

• Chapter Five Scattering Problems: Beltrami Fields and Solvability
(pp. 83-111)

This chapter deals with the solvability of time-harmonic electromagnetic wave scattering by an obstacle: either the obstacle or the environment in which it is embedded, or both, is (are) occupied by a chiral material. Regarding general references for scattering theory, we refer to the books [103], [106], [218], [260], [303], [358], [410], [411], [412] and the book chapter [258]. The corresponding theory for an achiral obstacle and the surrounding environment is well known and established: see [103], [106] and the references therein.

We assume that the scatterer and its surrounding space are homogeneous: this allows us to use boundary integral...

• Chapter Six Scattering Problems: A Variety of Topics
(pp. 112-148)

In this chapter we continue our study of scattering problems in the case where the considered fields have harmonic time dependence and the involved chiral media are homogeneous. For the reader’s convenience we start with a section (Section 6.2) containing various important concepts of scattering theory; for simplicity, the presentation is done for the relatively simple case of the scalar Helmholtz equation, whose vector analogue is strongly related to the Maxwell equations. In the following sections we present a variety of topics. In particular, Section 6.3 deals with the establishment of the reciprocity principle, the general scattering theorem and the...

6. ### PART 3. TIME-DEPENDENT DETERMINISTIC PROBLEMS

• Chapter Seven Well Posedness
(pp. 151-162)

In this chapter we deal with electromagnetic fields in complex media in the time domain. The most general form for the constitutive relations is assumed to be nonlocal in time, in the form of convolutions. The convolution models dispersive effects in the medium. We address questions related to the well posedness of the Maxwell equations in this setting.

The structure of the chapter is as follows. In Section 7.2 we discuss the Maxwell equations in complex media in the time domain and show that they can be expressed as integrodifferential equations. In Section 7.3 we provide a convenient functional setting...

• Chapter Eight Controllability
(pp. 163-179)

In this chapter we present some issues concerning the controllability of the Maxwell equations for complex media in the time domain. Controllability issues constitute an important class of problems that present considerable interest both from the mathematical as well as from the applications point of view.

The structure of this chapter is as follows. In Section 8.2 we formulate the problem and discuss our main strategy for its treatment, using a fixed point approach. This approach is based on the controllability problem for the Maxwell equations and treats the integrodifferential terms in the constitutive relations for the complex media as...

• Chapter Nine Homogenisation
(pp. 180-211)

Composite materials containing finely mixed constituent parts, possibly exhibiting a well-defined structure, are encountered almost everywhere, either in natural forms or as fabricated materials (e.g., bones, wood, metals, rocks, polycrystalline materials, or concrete, carbon fibres, ceramics, foams, etc., respectively; see, e.g., [322]). They are designed to display desirable properties that may not be exhibited by homogeneous media. Complex electromagnetic media are often composite materials.

The evolution of physical phenomena in composite materials may be modelled using boundary value problems with a periodic structure. This periodic structure leads to complications in both the analytic and the numerical treatment of these problems,...

• Chapter Ten Towards a Scattering Theory
(pp. 212-230)

Scattering theories can provide methods for developing robust approximation methods for solving wave problems. In this chapter we indicate how such theories can be developed when wave motions in chiral media are studied; we show, for the sake of illustration, how a relatively simple scattering theory involving achiral materials can be modified to accommodate problems involving a class of chiral materials (see also [368]).

We begin by remarking that a scattering process describes the effects of a perturbation on a system about which everything is known in the absence of the perturbation. Such a process can be conveniently characterised in...

• Chapter Eleven Nonlinear Problems
(pp. 231-244)

Nonlinearity is inherent in nature and accounts for a number of interesting phenomena. To keep within the context of electromagnetic media, nonlinearity appears in a number of cases in which the dispersion relations or equivalently the coefficients of the constitutive relations change as a function of the field amplitudes. This behaviour is very common in dielectrics, where it has been experimentally verified, theoretically studied, and widely studied mathematically. Furthermore, the interplay between dispersion and nonlinearity in dielectrics has led to the observation of solitary waves, which has important applications to optical communications. Regarding chiral media, although third-order nonlinear effects were...

7. ### PART 4. STOCHASTIC PROBLEMS

• Chapter Twelve Well Posedness
(pp. 247-262)

The aim of this part of the book is to examine the effects of randomness on the evolution and behaviour of electromagnetic fields in chiral media. Since our basic interest is the development of a mathematically rigorous framework for the study of this problem, our approach is to model random effects through the introduction of a general class of random fields¹ into the system of equations that governs the evolution of the electromagnetic fields. We do not take up the question of self-consistent mathematical modelling of the random effects, starting from the starting point of first principles physical theory. Such...

• Chapter Thirteen Controllability
(pp. 263-274)

This chapter addresses the problem of controllability for stochastic complex electromagnetic media. Our starting point is the stochastic integrodifferential equations that model the evolution of the fields, with a control procedure to be selected so that the system is driven to a desired final state. The controllability problem for stochastic media is more complicated than that for deterministic media and includes subtleties that must be addressed to reach a satisfactory answer.

The stucture of this chapter is as follows: In Section 13.2 we set the model, while in Section 13.3 we discuss the subtle issues introduced by the stochasticity in...

• Chapter Fourteen Homogenisation
(pp. 275-290)

In certain classes of materials the spatial structure is not sufficiently regular as to be modelled by periodic functions. Such materials can be modelled as random media having some sort of statistical periodicity. This statistical periodicity is expressed mathematically through the concept ofergodicity. This concept is powerful enough to generalise periodicity and allows us to build a homogenisation theory that bypasses the need for periodic structure. From the applications point of view, this generalisation leads to more realistic models. In nature there is actually no such thing as a deterministic periodic structure; materials are subject to random imperfections that...

8. ### PART 5. APPENDICES

• Appendix A. Some Facts from Functional Analysis
(pp. 293-315)
• Appendix B. Some Facts from Stochastic Analysis
(pp. 316-326)
• Appendix C. Some Facts from Elliptic Homogenisation Theory
(pp. 327-333)
• Appendix D. Some Facts from Dyadic Analysis
(pp. 334-340)
George Dassios
• Appendix E. Notation and abbreviations
(pp. 341-342)
9. Bibliography
(pp. 343-376)
10. Index
(pp. 377-382)
11. Back Matter
(pp. 383-383)