Determining both leading coefficient and source in a nonlocal elliptic equation

TL;DR

This paper proves the unique determination of both the leading coefficient and source term in a nonlocal elliptic equation using exterior Dirichlet-to-Neumann data, highlighting differences from the local case.

Contribution

It establishes the first uniqueness result for simultaneous recovery of coefficient and source in a nonlocal elliptic inverse problem using exterior data.

Findings

Unique determination of $\sigma$ and $F$ from exterior DN map.

The nonlocal case differs fundamentally from the local case ($s=1$).

Uses unique continuation and extension techniques for proof.

Abstract

In this short note, we investigate an inverse source problem associated with a nonlocal elliptic equation $\left( -\nabla \cdot \sigma \nabla \right)^s u =F$ that is given in a bounded open set $\Omega\subset \mathbb{R}^n$, for $n\geq 3$ and $0<s<1$. We demonstrate both $\sigma$ and $F$ can be determined uniquely by using the exterior Dirichlet-to-Neumann (DN) map in $\Omega_e:=\mathbb{R}^n\setminus \overline{\Omega}$. The result is intriguing in that analogous theory cannot be true for the local case generally, that is, $s=1$. The key ingredients to prove the uniqueness is based on the unique continuation principle for nonlocal elliptic operators and the reduction from the nonlocal to the local via the Stinga-Torrea extension problem.