Recovering stable kernels from exterior measurements

Abstract

We study an inverse problem for translation-invariant symmetric stable operators of the form \begin{equation*} L_a u(x)=\mathrm{P.V.}\int_{\mathbb R^n}(u(x)-u(y))\frac{a((x-y)/|x-y|)}{|x-y|^{n+2s}}\,dy, \quad 0<s<1, \end{equation*} where the unknown is the even angular density $a$ on $\mathbb{S}n$. For a bounded open set $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$, with $\hat{I}{\text{c◯}}_e={\mathbb{R}}^n\setminus \overline{\hat{I}}\text{c◯}$, we consider restricted exterior Dirichlet-to-Neumann maps $\hat{I}{›}_a^{W_1,W_2}$, where exterior data are supported in $W_1⋐\hat{I}{\text{c◯}}_e$ and the nonlocal Neumann data are observed on $W_2⋐\hat{I}{\text{c◯}}_e$. We prove three recovery results for the leading angular density. In the overlapping regime $W_1\cap W_2\ne \varnothing$, the exterior diagonal singularity determines every smooth elliptic angular density. In the separated regime ${\overline{W}}_1\cap {\overline{W}}_2=\varnothing$, where this singularity is absent, we prove uniqueness in the finite harmonic angular class by an exact factorization of the stable symbol. We also prove separated-data uniqueness for real-analytic angular densities when the source and observation sets lie in the unbounded exterior component, using analytic continuation of the off-diagonal Dirichlet-to-Neumann kernel and a far-field asymptotic argument.

22 pages. All comments are welcome