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 \(-\text{ div }(\sigma \nabla u)+qu = 0\) 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 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.