Hopf’s Lemma and Constrained Radial Symmetry for the Fractional Laplacian

In this paper we prove 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)) , where δR(x) = dist(x, ∂BR) and BR ⊂ Ω is a ball such that z ∈ ∂BR. More precisely, we show that lim inf B∋x→z u(x) (δR(x)) > 0 . Also we consider the overdetermined problem  (−∆) u = 1 in Ω u = 0 in R \ Ω lim Ω∋x→z u(x) (δΩ(x)) s = q(|z|) for every z ∈ ∂Ω . Here Ω is a bounded open set in R , N ≥ 1, containing the origin and satisfying the interior ball condition, δΩ(x) = dist(x, ∂Ω), and (−∆), s ∈ (0, 1), is the fractional Laplace operator defined, up to normalization factors, as