Fractional Dirichlet problems with an overdetermined non‐local 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 . Specifically, we show that if has positive reach and the non‐local normal derivative introduced in Dipierro, Ros‐Oton and Valdinoci [Rev. Mat. Iberoam. 33 (2017), no. 2, 377–416] is constant on an external surface parallel and sufficiently close to , then must be a ball. Remarkably, this conclusion remains valid under the sole assumption that is convex. Moreover, we analyze the quantitative stability of this result under two distinct sets of assumptions on . Finally, we extend our analysis to a broader class of overdetermined Dirichlet problems involving the fractional Laplacian.