A Hopf lemma for the regional fractional Laplacian

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