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

n 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 ∈∂Oω is replaced with the limit of the ratio u(x)/(δR(x))s, where δR(x) = dist(x, ∂BR) and BR ⊂Oω is a ball such that z ∈∂BR. More precisely, we show that lim inf B∋x→z u(x)/(δR(x))s > 0. Also we consider the overdetermined problem { (-Δ)s u = 1 in Oω u = 0 in ℝN \ Oω lim Oω∋x→z u(x)/(δOω(x))s = q(|z|) for every z ∈∂Oω. Here Oω is a bounded open set in ℝN, N ≥ 1, containing the origin and satisfying the interior ball condition, δOω(x) = dist(x, ∂Oω), and (-Δ)s, s ∈(0, 1), is the fractional Laplace operator defined, up to normalization factors, as (-Δ)s u(x) = P.V. ∫ ℝN u(x)-u(y) |x-y|N+2s dy. We show that if the function q(r) grows fast enough with respect to r, then the problem 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