Symmetry and quantitative stability for the parallel surface fractional torsion problem

We study symmetry and quantitative approximate for an overdetermined problem involving the fractional torsion in a bounded open set ? ? mathvariant="double-struck">R n \Omega \subset \mathbb R^n . More precisely, we prove that if function has alttext="upper C 1"> C 1 encoding="application/x-tex">C^1 level surface which is parallel to boundary alttext="partial-differential normal Omega"> mathvariant="normal">?<!-- ? encoding="application/x-tex">\partial \Omega then encoding="application/x-tex">\Omega ball. If instead assume solution close constant on boundary, quantitatively Our results use techniques are peculiar of nonlocal case as, instance, versions Hopf point lemma Harnack estimates antisymmetric functions. also provide application rural-urban fringes population settlements.