Symmetry and quantitative stability for the parallel surface fractional torsion problem

We study symmetry and quantitative approximate symmetry for an overdetermined problem involving the fractional torsion problem in a bounded open set Ω ⊂ R n \Omega \subset \mathbb R^n . More precisely, we prove that if the fractional torsion function has a C 1 C^1 level surface which is parallel to the boundary ∂ Ω \partial \Omega then Ω \Omega 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 Ω \Omega is close to a ball. Our results use techniques which are peculiar of 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.