Symmetry and quantitative stability for the parallel surface fractional torsion problem

TL;DR

This paper investigates symmetry properties of solutions to the fractional torsion problem, establishing conditions under which the domain must be a ball and quantifying how close a domain is to a ball when the solution is nearly constant on a parallel surface.

Contribution

It proves that certain geometric conditions on the fractional torsion function imply the domain is a ball, and provides quantitative estimates for near-spherical domains, using techniques specific to nonlocal operators.

Findings

Domains with a $C^1$ level surface parallel to the boundary are balls.

Solutions close to constant on a parallel surface imply the domain is close to a ball.

Introduces nonlocal techniques like fractional Hopf lemma and boundary Harnack estimates.

Abstract

We study symmetry and quantitative approximate symmetry for an overdetermined problem involving the fractional torsion problem in a bounded domain $\Omega \subset \mathbb R^n$. More precisely, we prove that if the fractional torsion function has a $C^1$ level surface which is parallel to the boundary $\partial \Omega$ then the domain is a ball. If instead we assume that the solution is close to a constant on a parallel surface to the boundary, then we quantitatively prove that $\Omega$ is close to a ball. Our results use techniques which are peculiar to the nonlocal case as, for instance, quantitative versions of fractional Hopf boundary point lemma and boundary Harnack estimates for antisymmetric functions. We also provide an application to the study of rural-urban fringes in population settlements.