Overdetermined problems for the fractional Laplacian in exterior and annular sets

We consider a fractional elliptic equation in an unbounded set with both Dirichlet and fractional normal derivative datum prescribed. We prove that the domain and the solution are necessarily radially symmetric. We also study the extension of the result in bounded non-convex regions, as well as the radial symmetry of the solution when the set is assumed a priori to be rotationally symmetric.