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An inverse boundary value problem for the Maxwell's equations with partial data

Jian Zhai

TL;DR

The paper studies the time-domain inverse boundary value problem for Maxwell's equations with partial boundary data, seeking local recovery of $\varepsilon$, $\mu$, and $\sigma$ from the impedance map. Building a geometric optics framework, it reduces the problem to microlocal geometric inverse problems along geodesics of the wave-speed metric $g=c^{-2}ds^2$ and leverages the convex foliation to enable local uniqueness. It first establishes boundary determination of all coefficient jets on the data boundary, then reconstructs interior parameters by recovering the wavespeed via lens relations, the ratio $\sigma/\varepsilon$ through geodesic-ray transforms, and finally $\varepsilon$ via a local transverse-ray transform and a closing fourth-order elliptic equation, invoking unique continuation. The results extend dynamical boundary control methods to electrodynamics with partial data, providing a rigorous pathway to locally recover material parameters from boundary measurements.

Abstract

We consider an inverse boundary problem for the dynamical Maxwell's equations. We show that the electric permittivity, conductivity, and magnetic permeability can be uniquely determined locally if there is a strictly convex foliation with respect to the wave speed.

An inverse boundary value problem for the Maxwell's equations with partial data

TL;DR

The paper studies the time-domain inverse boundary value problem for Maxwell's equations with partial boundary data, seeking local recovery of , , and from the impedance map. Building a geometric optics framework, it reduces the problem to microlocal geometric inverse problems along geodesics of the wave-speed metric and leverages the convex foliation to enable local uniqueness. It first establishes boundary determination of all coefficient jets on the data boundary, then reconstructs interior parameters by recovering the wavespeed via lens relations, the ratio through geodesic-ray transforms, and finally via a local transverse-ray transform and a closing fourth-order elliptic equation, invoking unique continuation. The results extend dynamical boundary control methods to electrodynamics with partial data, providing a rigorous pathway to locally recover material parameters from boundary measurements.

Abstract

We consider an inverse boundary problem for the dynamical Maxwell's equations. We show that the electric permittivity, conductivity, and magnetic permeability can be uniquely determined locally if there is a strictly convex foliation with respect to the wave speed.

Paper Structure

This paper contains 7 sections, 3 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Geometric optics
  3. boundary determination
  4. Interior determination
  5. Recovery of the wavespeed
  6. Recovery of $\frac{\sigma}{\varepsilon}$
  7. Recovery of $\varepsilon$

Key Result

Theorem 1

The three parameters $\varepsilon,\mu,\sigma$ can be uniquely determined in $\kappa^{-1}([0,1])\cap\overline{\Omega}$ by $\Lambda_{\varepsilon,\mu,\sigma}^\Gamma$, if $T$ is greater than the length of all geodesics, in the metric $c^{-2}\mathrm{d}s^2$, completely contained in $\kappa^{-1}([0,1])\cap

Theorems & Definitions (4)

  • Definition 1
  • Theorem 1
  • Lemma 1
  • Theorem 2