A Hopf lemma for the regional fractional Laplacian

Abstract

We provide a Hopf boundary lemma for the regional fractional Laplacian $(-\Delta {)}_{\Omega }^s$, with $\Omega \subset {\mathbb{R}}^N$ a bounded open set. More precisely, given $u$ a pointwise or weak super-solution of the equation $(-\Delta {)}_{\Omega }^{s}u=c(x)u$ 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.

typos corrected