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

TL;DR

The paper formulates Maxwell's equations in time domain on a compact oriented 3-manifold as a Dirac-type first-order system for differential forms, with a travel-time metric $g$ and scalar impedance $\alpha$ encoding a conformal anisotropy via $\mu=\alpha^2\epsilon$. It analyzes the boundary admittance map $\mathcal{Z}^T$ and establishes Holmgren–John type unique continuation and controllability results, showing boundary data can influence and determine interior wave states. For the inverse problem, boundary admittance data determine the travel-time metric from boundary distance functions, enable focusing sequences to recover interior fields, and allow reconstruction of the impedance $\alpha$ (up to a global scalar fixed by energy), with full reconstruction from data on an open boundary subset $\Gamma$; in the isotropic setting with $M\subset \mathbb{R}^3$, the boundary data determine uniquely the domain and the coefficients $\epsilon$ and $\mu$. The work also characterizes non-uniqueness in general bounded domains via diffeomorphisms and outlines future directions including inverse boundary spectral problems and reconstruction algorithms.

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 $\alpha$. This means that the material is conformal, i.e., the electric permittivity $\epsilon$ and magnetic permeability $\mu$ are tensors satisfying $\mu =\alpha^2\epsilon$. 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, $(M,g,\alpha)$, 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 $(M,g,\alpha)$ 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 $\epsilon$ and $\mu$.

Figures (3)

• Figure 1: Left: The double cone of influence. Right: For $T$ large enough, the double cone contains a slice $\{T/2\}\times M$.
• Figure 2: The set $S$ in the case when $J=2$.
• Figure 3: Vector $\vec{v}$ is the right polarization of the electromagnetic wave in the plane $M\times \{t\}$. The reconstructed polarization $\vec{w}$ has wrong direction, if the transformation matrix $K(x)$ is not isotropic.

Theorems & Definitions (39)

• Proposition 1.1
• Lemma 1.2
• Definition 1.3
• Lemma 1.4
• Lemma 1.5
• Theorem 1.6
• Definition 1.7
• Definition 1.8
• Theorem 1.9
• Proposition 1.10