Determining anisotropic real-analytic metric from boundary electromagnetic information

TL;DR

This paper demonstrates how to reconstruct a compact, real-analytic Riemannian 3-manifold with boundary from boundary electromagnetic measurements, explicitly characterizing the Dirichlet-to-Neumann map and determining electromagnetic parameters throughout the manifold.

Contribution

It provides an explicit local representation of the electromagnetic Dirichlet-to-Neumann map and proves unique reconstruction of the manifold and parameters from boundary data without topological restrictions.

Findings

Reconstruction of the manifold from boundary electromagnetic data.

Explicit symbol expression for the Dirichlet-to-Neumann map.

Unique determination of electromagnetic parameters in the manifold.

Abstract

For a compact, connected, oriented Riemannian $3$-manifold $(M, g)$ with smooth boundary $\partial M$, we explicitly give a local representation and a full symbol expression for the electromagnetic Dirichlet-to-Neumann map by factorizing Maxwell's equations and using an isometric transform. We prove that one can reconstruct a compact, connected, real-analytic Riemannian $3$-manifold $M$ with boundary from the set of tangential electric fields and tangential magnetic fields, given on a non-empty open subset $\Gamma$ of the boundary, of all electric and magnetic fields with tangential electric data supported in $\Gamma$. We note that for this result we need no assumption on the topology of the manifold other than compactness and connectedness, nor do we need a priori knowledge of all of $\partial M$. In addition, as a by-product of the explicit symbol expression of $\Lambda_{g,\Gamma}$, we show that for a given smooth Riemannian metric $g$, the electromagnetic Dirichlet-to-Neumann map $\Lambda_{g,\Gamma}$ uniquely determines all order tangential and normal derivatives of electromagnetic parameters $\mu$ and $\sigma$ on $\Gamma$. Therefore, $\mu$ and $\sigma$ are completely determined in $M$ by $\Lambda_{g,\Gamma}$ if these two parameter functions and metric $g$ are all real analytic in $M$ up to $\Gamma$.