A Hopf lemma for the regional fractional Laplacian

Abstract

We provide a Hopf boundary lemma for the regional fractional Laplacian (−∆) s Ω , with Ω ⊂ R N a bounded open set. More precisely, given u a pointwise or weak super-solution of the equation (−∆) s Ω u = c(x)u in Ω, we show that the ratio u(x)/(dist(x, ∂Ω)) 2s−1 is strictly positive as x approaches the boundary ∂Ω of Ω. We also prove a strong maximum principle for distributional super-solutions.