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 ∈ ∂ Ω is replaced with the limit of the ratio u(x)/(δR(x))ˢ, where δR(x)= dist(x, ∂ BR) and BR ⊂ Ω is a ball such that z ∈ ∂ BR. 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.