The fractional anisotropic Calder\'{o}n problem for a nonlocal parabolic equation on closed Riemannian manifolds

TL;DR

This paper proves that the geometry of a closed Riemannian manifold can be uniquely determined by boundary measurements of a nonlocal parabolic equation, extending Calderón's inverse problem to a nonlocal, parabolic context.

Contribution

It establishes a uniqueness result for the fractional anisotropic Calderón problem on closed Riemannian manifolds using local source-to-solution data.

Findings

Unique determination of manifold geometry from local data

Analysis of nonlocal parabolic operators via spectrum and semigroup theory

Extension of Calderón problem to nonlocal parabolic equations on manifolds

Abstract

We consider the fractional anisotropic Calder\'on problem for the nonlocal parabolic equation $(\partial_t -\Delta_g)^s u=f$ ($0<s<1$) on closed Riemannian manifolds. More concretely, we can determine the Riemannian manifold $(M,g)$ up to isometry by using the local source-to-solution map in an arbitrarily small open cylinder in the spacetime domain. This can be regarded as a nonlocal analog of the anisotropic Calder\'on problem in the parabolic setting. We also study several useful properties for nonlocal parabolic operators by using comprehensive spectrum analysis with semigroup theory.