Hopf's lemma and constrained radial symmetry for the fractional Laplacian

In this paper we give a new proof of Hopf's boundary point lemma for the fractional Laplacian. With respect to the classical formulation, in the non-local framework the normal derivative of the involved function~$u$ at~$z \in \partial \Omega$ is replaced with the limit of the ratio $u(x)/(\delta_R(x))^s$, where $\delta_R(x)=\mathop{\rm dist}(x, \partial B_R)$ and $B_R \subset \Omega$ is a ball such that $z \in \partial B_R$. Also we consider an overdetermined problem and we prove the that it admits a solution only in a suitable ball centered at the origin. The proof is based on a comparison principle proved along the paper, and on the boundary point lemma mentioned before.