Determining both leading coefficient and source in a nonlocal elliptic equation

In this short note, we investigate an inverse (source) problem associated with a nonlocal elliptic equation (equation presented) that is given in a bounded open set (equation presented) for n ≥ 3 and 0 < s < 1 . We demonstrate both the leading coefficient σ and the source F can be determined uniquely by using the exterior Dirichlet-to-Neumann (DN) map in (equation presented). The result is intriguing in that analogous theory cannot be true for the local case generally, that is, s = 1 {s=1}. The key ingredients to prove the uniqueness are 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.