Quantitative results for fractional overdetermined problems in exterior and annular sets

TL;DR

This paper studies fractional overdetermined problems in exterior and annular domains, proving symmetry of solutions and providing quantitative stability estimates when conditions are nearly satisfied.

Contribution

It establishes the radial symmetry of solutions and introduces a quantitative stability analysis for fractional overdetermined problems in exterior and annular sets.

Findings

Solutions are radially symmetric under overdetermined conditions.

Quantitative stability estimates measure how domain deviations relate to perturbations.

Results extend classical symmetry results to fractional and unbounded domains.

Abstract

We consider overdetermined problems related to the fractional capacity. In particular we study $s$-harmonic functions defined in unbounded exterior sets or in bounded annular sets, and having a level set parallel to the boundary. We first classify the solutions of the overdetermined problems, by proving that the domain and the solution itself are radially symmetric. Then we prove a quantitative stability counterpart of the symmetry results: we assume that the overdetermined condition is slightly perturbed and we measure, in a quantitative way, how much the domain is close to a symmetric set.