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Inverse Problem for Fractional-Laplacian with Lower Order Non-local Perturbations. (arXiv:1810.03567v1 [math.AP])
来源于:arXiv
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 \[(-\D)^t + (-\D)_{\Omega}^{{s}/{2}} \ b
(-\D)_{\Omega}^{{s}/{2}} + q, \quad 0<s<t<1\] on a bounded Lipschitz domain
$\Omega \subset \R^n$. 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 $\R^n \s 查看全文>>