Maxwell's Equations with Scalar Impedance: Inverse Problems with data given on a part of the boundary

Abstract

We study Maxwell's equations in time domain in an anisotropic medium. The goal of the paper is to solve an inverse boundary value problem for anisotropies characterized by scalar impedance $α$. This means that the material is conformal, i.e., the electric permittivity $ε$ and magnetic permeability $μ$ are tensors satisfying $μ=α^2ε$. This condition is equivalent to a single propagation speed of waves with different polarizations which uniquely defines an underlying Riemannian structure. 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 with an additional structure of the wave impedance, $(M,g,α)$, where $g$ is the travel-time metric. We study the properties of this system in the first part of the paper. In the second part we consider the inverse problem, that is, the determination of $(M,g,α)$ from measurements done only on an open part of the boundary and on a finite time interval. As an application, in the isotropic case with $M\subset \R^3$, we prove that the boundary data given only on an open part of the boundary determine uniquely the domain $M$ and the coefficients $ε$ and $μ$.