Fractional Calderón problem on a closed Riemannian manifold

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 $\alpha =1$. While the latter problem is widely open in dimensions three and higher, we solve the non-local problem in broad geometric generality, assuming only a local property on the a priori known observation region $\mathcal{O}$ while making no geometric assumptions on the inaccessible region of the manifold, namely $M\setminus \mathcal{O}$. Our proof is based on discovering a hidden connection to a variant of Carlson's theorem in complex analysis that allows us to reduce the non-local inverse problem to the Gel'fand inverse spectral problem.

arXiv commentUpdated the introduction with more references. Comments are welcome

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Cited by in corpus (1)

• The Calderón problem for nonlocal operators