A Hopf's Lemma and the Boundary Regularity for the Fractional P-Laplacian

TL;DR

This paper establishes a Hopf's lemma for the fractional p-Laplacian on a half-space and proves boundary Holder continuity of positive solutions, advancing understanding of boundary regularity in nonlocal PDEs.

Contribution

It introduces a Hopf's lemma for the fractional p-Laplacian and demonstrates boundary Holder regularity for positive solutions, filling gaps in boundary behavior analysis.

Findings

Derivative along outward normal is strictly positive on boundary

Positive solutions are Holder continuous up to the boundary

Provides tools for boundary regularity in nonlocal PDEs

Abstract

We begin the paper with a Hopf's lemma for a fractional p-Laplacian problem on a half-space. Specifically speaking, we show that the derivative of the solution along the outward normal vector is strictly positive on the boundary of the half-space. Next we show that positive solutions to a fractional p-Laplacian equation possess certain Holder continuity up to the boundary.