A non-local inverse problem with boundary response

Tuhin Ghosh

doi:10.4171/rmi/1323

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 $$ \begin{cases} ((-\Delta)^a + q )u = 0 &\operatorname{in} \Omega,\ \operatorname{supp}, u\subseteq \overline{\Omega}\cup \overline{W} ,\ \overline{W} \cap \overline{\Omega}=\emptyset. \end{cases} $$ We show that, by making an exterior to boundary measurement as $\big(u|{W}, \frac{u(x)}{d(x)^a}\big|{\Sigma}\big)$, 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.

Highlights

Introduction & main resultIn this paper, we address the so-called fractional Calderon problem through fractional Schrodinger equation and study the global identifiability of the potential based on the boundary response
We address the so-called fractional Calderon problem
[GSU20]) through fractional Schrodinger equation and study the global identifiability of the potential based on the boundary response

Summary

Introduction & main result

We would like to address the inverse problem through introducing the exterior to boundary response map, and based on that we look for a new global uniqueness result of recovering the potential. The Theorem 1.1 is a global uniqueness result in the inverse problem for the fractional Schrodinger equation with both partial exterior (W ⊂ Ωe) and boundary data (Σ ⊂ ∂Ω). Let U1, U2 ∈ H1(Rn++1, y1−2a) be the solutions of the Robin boundary value problem (2.5) for two different potentials q1, q2 ∈ Cc∞(Ω) respectively, with the same partial Dirichlet data U1(0, x) W = U2(0, x) W = f ∈ Cc∞(W ). We add one Appendix containing local characterization of the large a-harmonic functions in ball and its application which includes boundary unique continuation and local density result

Preliminaries

Fractional Schrodinger equation

Inverse problem

Local characterization of the large a-harmonic functions in ball