Fractional Calderón problem on a closed Riemannian manifold

Feizmohammadi, Ali

doi:10.1090/tran/9106

Fractional Calderón problem on a closed Riemannian manifold
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by Ali Feizmohammadi

Trans. Amer. Math. Soc. 377 (2024), 2991-3013

DOI: https://doi.org/10.1090/tran/9106

Published electronically: February 8, 2024

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Abstract:

Given a fixed $\alpha \in (0,1)$, we study the inverse problem of recovering the isometry class of a smooth closed and connected Riemannian manifold $(M,g)$, given the knowledge of a source-to-solution map for the fractional Laplace equation $(-\Delta _g)^\alpha u=f$ on the manifold subject to an arbitrarily small observation region $\mathcal O$ where sources can be placed and solutions can be measured. This can be viewed as a non-local analogue of the well known anisotropic Calderón problem that is concerned with the limiting case $\alpha =1$. In this paper, we solve the non-local problem under the assumption that the a priori known observation region $\mathcal O$ belongs to some Gevrey class while making no geometric assumptions on the inaccessible region of the manifold, namely $M\setminus \mathcal O$. Our proof is based on discovering a connection to a variant of Carlson’s theorem in complex analysis that reduces the inverse problem to Gel’fand inverse spectral problem.

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