On Instability Properties of the Fractional Calder\'{o}n Problem

TL;DR

This paper establishes exponential instability results for the fractional Calderón problem across various geometries and coefficients, demonstrating the optimality of existing logarithmic stability estimates.

Contribution

It generalizes previous instability results to arbitrary geometries and variable coefficients, including anisotropic metrics, and confirms the optimality of known stability bounds.

Findings

Proves exponential instability for fractional Calderón problems in general geometries.

Extends instability results to variable, low-regularity, and anisotropic metrics.

Shows that existing logarithmic stability estimates are optimal.

Abstract

We prove exponential instability properties for the fractional Calder\'on problem and the conductivity formulation of the fractional Calder\'on problem in the regime of fractional powers $s\in (0,1)$. We particularly focus on two settings: First, we discuss instability properties in general domain geometries with scaling critical $L^{\frac{n}{2s}}$ potentials and constant background metrics. Secondly, we investigate instability properties in general geometries with $L^{\frac{n}{2s}}$ potentials and low regularity, variable coefficient, possibly anisotropic background metrics. In both settings we make use of the methods introduced in \cite{KRS21} and we deduce strong compression estimates for the forward problem. In the first setting this is based on analytic smoothing estimates for a suitable comparison operator while in the second setting involving low regularity metrics this is based on an iterated compression gain. We thus generalize the results from \cite{RS18} to generic geometries and variable coefficients and further also discuss the setting of fractional conductivity equations. In particular, this proves that the logarithmic stability estimates for the fractional Calder\'on problem from \cite{RS20} are optimal.