A Hopf lemma for the regional fractional Laplacian

TL;DR

This paper establishes a boundary behavior lemma and a maximum principle for the regional fractional Laplacian, enhancing understanding of boundary conditions for nonlocal operators in bounded domains.

Contribution

It introduces a Hopf boundary lemma and a strong maximum principle specifically for the regional fractional Laplacian, extending classical results to nonlocal operators.

Findings

The ratio u(x)/dist(x,∂Ω)^{2s-1} is positive near the boundary.

A strong maximum principle holds for distributional super-solutions.

Provides tools for boundary analysis of nonlocal PDEs.

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.