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

Abstract

In this article, we study a model problem featuring a Lévy process in a domain with semi-transparent boundary by considering the following perturbed fractional Laplacian operator \[\mathscr{L}_{b,q} := (-Δ)^t + (-Δ)_Ω^{s/2} \ b (-Δ)_Ω^{s/2} + q, \quad 0<s<t<1\] 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 $(-\Delta {)}_{\Omega }^{s/2}$ 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 ${\mathcal{L}}_{b,c}$ 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.