Gel'fand-Calder\'on's inverse problem for anisotropic conductivities on bordered surfaces in $\mathbb{R}^3$

TL;DR

This paper presents an explicit method to reconstruct a Riemann surface and isotropic conductivity from boundary measurements of an anisotropic conductivity on a bordered surface in three-dimensional space, establishing uniqueness up to boundary-preserving diffeomorphisms.

Contribution

It provides a constructive procedure to recover a Riemann surface and isotropic conductivity from Dirichlet-to-Neumann data for anisotropic conductivities on bordered surfaces in $ eal^3$.

Findings

Unique reconstruction of Riemann surface and isotropic conductivity from boundary data.

Explicit procedure to find a biholomorphic surface and boundary values of a quasiconformal map.

Proof of uniqueness of anisotropic conductivities up to boundary-preserving diffeomorphisms.

Abstract

Let $X$ be a smooth bordered surface in $\real^3$ with smooth boundary and $\hat \sigma$ a smooth anisotropic conductivity on $X$. If the genus of $X$ is given, then starting from the Dirichlet-to-Neumann operator $\Lambda_{\hat \sigma}$ on $\partial X$, we give an explicit procedure to find a unique Riemann surface $Y$ (up to a biholomorphism), an isotropic conductivity $\sigma$ on $Y$ and the boundary values of a quasiconformal diffeomorphism $F: X \to Y$ which transforms $\hat \sigma$ into $\sigma$. As a corollary we obtain the following uniqueness result: if $\sigma_1, \sigma_2$ are two smooth anisotropic conductivities on $X$ with $\Lambda_{\sigma_1}= \Lambda_{\sigma_2}$, then there exists a smooth diffeomorphism $\Phi: \bar X \to \bar X$ which transforms $\sigma_1$ into $\sigma_2$.