On electromagnetic waves in complex linear media in nonsmooth domains

Abstract

Communicated by I. StratisElectromagnetic wave propagation in more complex linear materials such as bi-anisotropic media have come to a considerable attention within the last 15-20 years. We shall propose a general framework to approach a class of highly complex materials. Such problems have been extensively studied mostly in the time-harmonic case. In this paper, we focus on the time-dependent case. A well-posedness result for a large class of media is obtained. We also analyze Drude-Born-Fedorov type media in nonsmooth domains of arbitrary topological genus within this framework.

On a connection between the Maxwell system, the extended Maxwell system, the Dirac operator and gravito‐electromagnetism

“…such a model situation is not as easily accessible. In contrast, to discuss the degeneracy situation considered in , even ϵ and σ are allowed to vanish in a region, requires more general strategies,which are beyond the current framework of assumptions, see for example the problem class discussed in . It is also noteworthy that we do not assume that M 0 and M 1 are block‐diagonal, as in the classical anisotropic inhomogeneous media case, so general bi‐anisotropic and chiral media are covered by the same description, see , compare for example . Remark (See‐saw regularity and initial‐value problems) With the help of the notion of Sobolev lattices, the equation may also be read line by line. The price one has to pay is that even if for some , we only get that the solution U , and belong to the space .…”

“…In contrast, to discuss the degeneracy situation considered in [21], even and are allowed to vanish in a region, requires more general strategies,which are beyond the current framework of assumptions, see for example the problem class discussed in [20]. (c) It is also noteworthy that we do not assume that M 0 and M 1 are block-diagonal, as in the classical anisotropic inhomogeneous media case, so general bi-anisotropic and chiral media are covered by the same description, see [22], compare for example [23,24].…”

A non‐autonomous model for the evolution of electromagnetic fields in complex media

“…DBF‐like relations, in the time domain, can be formally justified when the α , σ e , σ m are localized functions of their arguments and attention is confined in the optical response region, assuming instantaneous responses in the material (see and for a more detailed analysis ). Well‐posedness results on the DBF‐like model, including a wide class of domains , with nonsmooth boundaries ∂ Ω, can be found in .…”

“…This approximation has been extensively used in the modeling of chiral media, especially in the time harmonic case. However, as we can see in [1,2,5,[7][8][9][10][11][12][13], DBF-like approximations have been used for the study of electromagnetic fields in chiral media in the time domain also. DBF-like relations, in the time domain, can be formally justified when the˛, e , m are localized functions of their arguments and attention is confined in the optical response region, assuming instantaneous responses in the material (see [1,7,8] and for a more detailed analysis [5]).…”

“…DBF-like relations, in the time domain, can be formally justified when the˛, e , m are localized functions of their arguments and attention is confined in the optical response region, assuming instantaneous responses in the material (see [1,7,8] and for a more detailed analysis [5]). Well-posedness results on the DBF-like model, including a wide class of domains  R 3 , with nonsmooth boundaries @ , can be found in [13].As we can see, in the nonlocal in time constitutive relations of chiral media in the time domain, the chiral effects are represented as kernels in convolution terms, whereas in the DBF-like approximation, the chiral effects are represented only by a constantˇ. A reasonable aim for this DBF-like, local in time approximation, is that the chiral effects should depend on t and not just be a constantˇ.…”