The Calderón problem for variable coefficients nonlocal elliptic operators

Abstract

In this paper, we introduce an inverse problem of a Schrödinger 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 $n\ge 2$. Our results generalize the recent initiative [16] of introducing and solving inverse problem for fractional Schrödinger 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.

41 pages