The Calder\'on problem for variable coefficients nonlocal elliptic operators

TL;DR

This paper studies an inverse problem for a variable coefficient nonlocal elliptic operator, showing how to determine an unknown potential from exterior measurements, extending previous work on fractional Schr"odinger operators.

Contribution

It generalizes the inverse problem for fractional Schr"odinger operators to variable coefficient nonlocal elliptic operators and establishes regularity results for the related direct problem.

Findings

Unique determination of potential q from exterior data

Extension of inverse problem techniques to variable coefficients

Regularity results for the associated degenerate elliptic operator

Abstract

In this paper, we introduce an inverse problem of a Schr\"odinger type variable nonlocal elliptic operator $(-\nabla\cdot(A(x)\nabla))^{s}+q)$, for $0<s<1$. We determine the unknown bounded potential $q$ from the exterior partial measurements associated with the nonlocal Dirichlet-to-Neumann map for any dimension $n\geq2$. Our results generalize the recent initiative [16] of introducing and solving inverse problem for fractional Schr\"odinger operator $((-\Delta)^{s}+q)$ for $0<s<1$. We also prove some regularity results of the direct problem corresponding to the variable coefficients fractional differential operator and the associated degenerate elliptic operator.