Inverse Problem for Fractional-Laplacian with Lower Order Non-local Perturbations

Abstract

In this article, we study a model problem featuring a L\'evy process in a domain with semi-transparent boundary by considering the following perturbed fractional Laplacian operator \[\mathscr{L}_{b,q} := (-\Delta)^t + (-\Delta)_{\Omega}^{s/2} \ b (-\Delta)_\Omega^{s/2} + q, \quad 0<s<t<1\] on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^n$. While the non-locality of the fraction Laplacian $(-\Delta)^t$ depends on entire $\mathbb{R}^n$, in its non-local perturbation the non-locality depends on the domain $\Omega$ through the regional fractional Laplacian term $(-\Delta)^{s/2}_{\Omega}$ and $b$ exhibits the semi-transparency of the process. We analyze the well-posedness of the model and certain qualitative property like unique continuation property, Runge approximation scheme considering its regional non-local perturbation. Then we move into studying the inverse problem and find that by knowing the corresponding Dirichlet to Neumann map (D-N map) of $\mathscr{L}_{b,c}$ on the exterior domain $\mathbb{R}^n \setminus \Omega$, it is possible to determine the lower order perturbations `$b$',`$q$' in $\Omega$. We also discuss the recovery of `$b$', `$q$' from a single measurement and its limitations.