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arXiv · 1810.03567

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

Sombuddha Bhattacharyya·Tuhin Ghosh·Gunther Uhlmann
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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 $Ω\subset \mathbb{R}^n$. While the non-locality of the fraction Laplacian $(-Δ)^t$ depends on entire $\mathbb{R}^n$, in its non-local perturbation the non-locality depends on the domain $Ω$ through the regional fractional Laplacian term $(-Δ)^{s/2}_Ω$ 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 $\mathbb{R}^n \setminus Ω$, it is possible to determine the lower order perturbations `$b$',`$q$' in $Ω$. We also discuss the recovery of `$b$', `$q$' from a single measurement and its limitations.

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Sombuddha Bhattacharyya, Tuhin Ghosh, Gunther Uhlmann. 2020-11-11. Inverse Problem for Fractional-Laplacian with Lower Order Non-local Perturbations. https://doi.org/10.13140/rg.2.2.15643.52000

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