Fractional Dirichlet problems with an overdetermined nonlocal Neumann condition

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