Stability estimates for the inverse fractional conductivity problem

TL;DR

This paper establishes a logarithmic stability estimate for the inverse fractional conductivity problem on smooth domains, advancing understanding of the problem's stability and optimality in the context of nonlocal equations.

Contribution

It provides the first resolution of technical challenges in deriving stability estimates for the fractional conductivity inverse problem, extending classical results to a nonlocal setting.

Findings

Logarithmic stability estimate for the inverse fractional conductivity problem

Proof of the optimality of the stability estimates

Extension of classical stability results to fractional and nonlocal equations

Abstract

We study the stability of an inverse problem for the fractional conductivity equation on bounded smooth domains. We obtain a logarithmic stability estimate for the inverse problem under suitable a priori bounds on the globally defined conductivities. The argument has three main ingredients: 1. the logarithmic stability of the related inverse problem for the fractional Schr\"odinger equation by R\"uland and Salo; 2. the Lipschitz stability of the exterior determination problem; 3. utilizing and identifying nonlocal analogies of Alessandrini's work on the stability of the classical Calder\'on problem. The main contribution of the article is the resolution of the technical difficulties related to the last mentioned step. Furthermore, we show the optimality of the logarithmic stability estimates, following the earlier works by Mandache on the instability of the inverse conductivity problem, and by R\"uland and Salo on the analogous problem for the fractional Schr\"odinger equation.