Nonlocal inverse problem with boundary response

Abstract

The problem of interest in this article is to study the (nonlocal) inverse problem of recovering a potential based on the boundary measurement associated with the fractional Schr\"{o}dinger equation. Let $0<a<1$, and $u$ solves \[\begin{cases} \left((-\Delta)^a + q\right)u = 0 \mbox{ in } \Omega\\ supp\, u\subseteq \overline{\Omega}\cup \overline{W}\\ \overline{W} \cap \overline{\Omega}=\emptyset. \end{cases} \] We show that by making the exterior to boundary measurement as $\left(u|_{W}, \frac{u(x)}{d(x)^a}\big|_{\Sigma}\right)$, it is possible to determine $q$ uniquely in $\Omega$, where $\Sigma\subseteq\partial\Omega$ be a non-empty open subset and $d(x)=d(x,\partial\Omega)$ denotes the boundary distance function. We also discuss local characterization of the large $a$-harmonic functions in ball and its application which includes boundary unique continuation and local density result.