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

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 $${\mathcal{L}}_{b,q}:=(-\mathrm{\Delta }{)}^t+(-\mathrm{\Delta }{)}_{\mathrm{\Omega }}^{s\mathrm{/}2}b(-\mathrm{\Delta }{)}_{\mathrm{\Omega }}^{s\mathrm{/}2}+q,0<s<t<1$$ on a bounded Lipschitz domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$. While the non-locality of the fraction Laplacian $(-\mathrm{\Delta }{)}^t$ depends on entire ${\mathbb{R}}^n$, in its non-local perturbation the non-locality depends on the domain $\mathrm{\Omega }$ through the regional fractional Laplacian term $(-\mathrm{\Delta }{)}_{\mathrm{\Omega }}^{s\mathrm{/}2}$ 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 ${\mathbb{R}}^n\setminus \mathrm{\Omega }$, it is possible to determine the lower order perturbations $b$',$q$’ in $\mathrm{\Omega }$. We also discuss the recovery of $b$', $q$’ from a single measurement and its limitations.