On reconstruction in the inverse conductivity problem with one measurement

Abstract

We consider an inverse problem for electrically conductive material occupying a domain $\Omega$ in $\Bbb R^2$. Let $\gamma$ be the conductivity of $\Omega$, and $D$ a subdomain of $\Omega$. We assume that $\gamma$ is a positive constant $k$ on $D$, $k\not=1$ and is $1$ on $\Omega\setminus D$; both $D$ and $k$ are unknown. The problem is to find a reconstruction formula of $D$ from the Cauchy data on $\partial\Omega$ of a non-constant solution $u$ of the equation $\nabla\cdot\gamma\nabla u=0$ in $\Omega$. We prove that if $D$ is known to be a convex polygon such that $\text{diam}\,D<\text{dist}\,(D,\partial\Omega)$, there are two formulae for calculating the support function of $D$ from the Cauchy data.