A non-local inverse problem with boundary response

Abstract

The problem of interest in this article is to study the (non-local) inverse problem of recovering a potential based on the boundary measurement associated with the fractional Schrödinger equation. Let $0<a<1$, and let $u$ solve

We show that, by making an exterior to boundary measurement as $(u{|}_W,\frac{u(x)}{d(x{)}^a}{|}_{\Sigma })$, it is possible to determine $q$ uniquely in $\Omega$, where $\Sigma \subseteq \partial \Omega$ is a non-empty open subset and $d(x)=d(x,\partial \Omega )$ denotes the boundary distance function.

We also discuss a local characterization of large $a$-harmonic functions in a ball or in the half space and its applications, which include boundary unique continuation and a local density result.