A uniqueness result in the inverse problem for the anisotropic Schr\"odinger type equation from local measurements

TL;DR

This paper proves that the coefficients of an anisotropic Schrödinger type equation can be uniquely identified within a domain from local boundary measurements, assuming they are piecewise constant on a known partition.

Contribution

It establishes a uniqueness result for the inverse boundary value problem with local data for anisotropic Schrödinger equations with piecewise constant coefficients.

Findings

Unique determination of coefficients from local boundary data

Applicable to piecewise constant anisotropic media

Advances inverse problem theory for Schrödinger equations

Abstract

We consider the inverse boundary value problem of the simultaneous determination of the coefficients $\sigma$ and $q$ of the equation $-\mbox{div}(\sigma \nabla u)+qu = 0$ 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 $n\geq 3$. 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.