Fractional Dirichlet problems with an overdetermined nonlocal Neumann condition

TL;DR

This paper proves that certain overdetermined fractional torsion problems imply the domain is a ball, with results valid for convex sets and including stability analysis and broader fractional Laplacian problems.

Contribution

It establishes symmetry results for fractional overdetermined problems, including convex domains and stability estimates, extending to a wider class of fractional Laplacian problems.

Findings

Domains with constant nonlocal normal derivative are balls.

Convexity alone suffices for symmetry conclusions.

Quantitative stability results are provided.

Abstract

We investigate symmetry and quantitative approximate symmetry for an overdetermined problem related to the fractional torsion equation in a regular open, bounded set $\Omega \subseteq \mathbb{R}^n$. Specifically, we show that if $\overline{\Omega}$ has positive reach and the nonlocal normal derivative introduced in (Dipierro, Ros-Oton, Valdinoci, Rev. Mat. Iberoam. 33 (2017), no. 2, 377-416) is constant on an external surface parallel and sufficiently close to $\partial \Omega$, then $\Omega$ must be a ball. Remarkably, this conclusion remains valid under the sole assumption that $\Omega$ is convex. Moreover, we analyze the quantitative stability of this result under two distinct sets of assumptions on $\Omega$. Finally, we extend our analysis to a broader class of overdetermined Dirichlet problems involving the fractional Laplacian.