A Hopf lemma for the regional fractional Laplacian

We provide a Hopf boundary lemma for the regional fractional Laplacian $(-\mathrm{\Delta }{)}_{\mathrm{\Omega }}^s$, with $\mathrm{\Omega }\subset {\mathbb{R}}^N$ a bounded open set. More precisely, given $u$ a pointwise or weak super-solution of the equation $(-\mathrm{\Delta }{)}_{\mathrm{\Omega }}^{s}u=c(x)u$ in $\mathrm{\Omega }$, we show that the ratio $u(x)\mathrm{/}(\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x,\mathrm{\partial }\mathrm{\Omega }){)}^{2s-1}$ is strictly positive as $x$ approaches the boundary $\mathrm{\partial }\mathrm{\Omega }$ of $\mathrm{\Omega }$. We also prove a strong maximum principle for distributional super-solutions.