Article ID Journal Published Year Pages File Type
4644764 Journal de Mathématiques Pures et Appliquées 2006 34 Pages PDF
Abstract

We study Maxwell's equations in time domain for an anisotropic medium of a special type, characterized by the polarization independent velocity of the wave propagation. In particular, this property is satisfied by all isotropic media. The analysis is based on an invariant formulation of the system of electrodynamics as a Dirac type first order system on a Riemannian 3-manifold. We study the properties of this system in the first part of the paper. The second part is devoted to the inverse problem of the identification of the Riemannian manifold M and the corresponding system of equations from the dynamic boundary data. These data are the boundary ∂M and the admittance map ZT. Physically, this map corresponds to the measurements of the tangential components of the electric and magnetic fields on the boundary at a finite time interval [0,T]. It is shown that, for sufficiently large T>0, ZT determines the Riemannian manifold and the underlying electromagnetic parameters. Similar results are proven in the case when the boundary data are given only on an open part of the boundary. In domains of R3, we describe the group of transformations which preserve the admittance map ZT, providing a complete characterization of the non-uniqueness of the underlying physical problem. In the isotropic case with M⊂R3, we prove that the boundary data given on an open part of the boundary determine the domain M, the permittivity ε and the permeability μ uniquely.

RésuméNous étudions les équations de Maxwell dans un domaine temporel pour un milieu d'un type particulier, où la vitesse de la propagation des ondes est indépendante de la polarisation. En particulier, tout milieu isotrope a cette propriété. L'analyse utilise une formulation des équations de Maxwell en termes de système du premier ordre de type Dirac. Dans la première partie de cet article, nous étudions les propriétés de ce système ; la deuxième partie traite le problème inverse de l'identification de la variété riemannienne M et du système d'équations correspondant pour des données aux limites dynamiques sur une partie ouverte, non vide, du bord.

Related Topics
Physical Sciences and Engineering Mathematics Applied Mathematics