On the transfer of stability from the local to the fractional anisotropic Calder\'on problem with exterior measurements

TL;DR

This paper establishes the first stability estimates for the fractional Calderón problem with exterior data, transferring uniqueness results from the classical case using advanced quantitative techniques.

Contribution

It introduces a novel approach to transfer stability from the classical to the fractional Calderón problem without Liouville transforms, addressing unique challenges in unbounded geometries.

Findings

First stability estimates for the fractional Calderón problem with exterior data.

Successful transfer of uniqueness results using quantitative methods.

Overcoming geometric and dimensionality mismatches in measurement domains.

Abstract

We study the quantitative transfer of uniqueness from the classical to the fractional Calder\'on problem with exterior data. This allows us to deduce the first stability estimates for the principal part of the isotropic fractional Calder\'on problem with exterior data in the absence of Liouville transforms. Our argument relies on careful quantitative unique continuation and Runge approximation estimates. Due to the unbounded geometry and the mismatch of the dimensionalities of the measurement domains (exterior data on an open set vs boundary data on a co-dimension one manifold) novel challenges arise compared to the setting of source-to-solution measurements on closed manifolds.