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.   The extension of the result in bounded non-convex regions is also studied, as well as the radial symmetry of the solution when the set is a priori supposed to be rotationally symmetric. 
Nicola Soave and Enrico Valdinoci
math.AP