Fractional Dirichlet problems with an overdetermined nonlocal Neumann condition

Abstract

We investigate symmetry and quantitative approximate symmetry for an overdetermined problem related to the fractional torsion equation in a regular open, bounded set $\hat{I}\text{c◯}\subseteq {\mathbb{R}}^n$. Specifically, we show that if $\overline{\hat{I}}\text{c◯}$ 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 $\partial \hat{I}\text{c◯}$, then $\hat{I}\text{c◯}$ must be a ball. Remarkably, this conclusion remains valid under the sole assumption that $\hat{I}\text{c◯}$ is convex. Moreover, we analyze the quantitative stability of this result under two distinct sets of assumptions on $\hat{I}\text{c◯}$. Finally, we extend our analysis to a broader class of overdetermined Dirichlet problems involving the fractional Laplacian.

2 figures. Comments are welcome. Final version