A Hopf lemma and regularity for fractional $p-$Laplacians

TL;DR

This paper investigates the fractional p-Laplacian, establishing a Hopf lemma for positive solutions and identifying conditions for the operator's regularity, advancing understanding of its qualitative properties.

Contribution

It introduces a Hopf lemma for fractional p-Laplacian equations and determines optimal conditions for the operator's smoothness, filling gaps in regularity theory.

Findings

Established a Hopf lemma for positive weak super-solutions.

Derived an optimal condition for the fractional p-Laplacian to be in C^1.

Enhanced understanding of regularity properties of fractional p-Laplacians.

Abstract

In this paper, we study qualitative properties of the fractional $p$-Laplacian. Specifically, we establish a Hopf type lemma for positive weak super-solutions of the fractional $p-$Laplacian equation with Dirichlet condition. Moreover, an optimal condition is obtained to ensure $(-\triangle)_p^s u\in C^1(\mathbb{R}^n)$ for smooth functions $u$.