A uniqueness result in the inverse problem for the anisotropic Schrödinger type equation from local measurements

Abstract

We consider the inverse boundary value problem of the simultaneous determination of the coefficients $\sigma$ and $q$ of the equation $-\mbox{div}(σ\nabla u)+qu = 0$ from knowledge of the so-called Neumann-to-Dirichlet map, given locally on a non-empty curved portion $\Sigma$ of the boundary $\partial \Omega$ of a domain $\Omega \subset {\mathbb{R}}^n$, with $n\ge 3$. We assume that $\sigma$ and $q$ are \textit{a-priori} known to be a piecewise constant matrix-valued and scalar function, respectively, on a given partition of $\Omega$ with curved interfaces. We prove that $\sigma$ and $q$ can be uniquely determined in $\Omega$ from the knowledge of the local map.