Anisotropic Calder\'{o}n Problem for a Non-Local Second Order Elliptic Operator

TL;DR

This paper addresses the inverse problem of determining the geometry of a closed Riemannian manifold from boundary measurements for a non-local second order elliptic operator, extending Calderón's problem to a non-local setting.

Contribution

It introduces a method to recover the manifold's geometry from Cauchy data for a non-local elliptic operator, advancing inverse problems in geometric analysis.

Findings

Geometry can be recovered up to gauge from Cauchy data.

The approach extends Calderón's problem to non-local operators.

Provides new insights into inverse boundary value problems on manifolds.

Abstract

This paper investigates the anisotropic Calder\'{o}n problem for a non-local elliptic operator of order 2, on closed Riemannian manifolds. We demonstrate that using the Cauchy data set, we can recover the geometry of a closed Riemannian manifold up to standard gauge.