A Hopf lemma and regularity for 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 (−△) s p u ∈ C 1 (R n ) for smooth functions u.

A Hopf lemma for the regional fractional Laplacian

“…For the Hopf boundary lemma for weak super-solutions related to the fractional p-Laplacian, we refer to [9] and references therein. Other references on the Hopf boundary lemma for fractional Laplacian can be found in [1,5,7,12,16,17]. However, to the best of our knowledge, an analogue result for the regional fractional Laplacian has not been investigated before.…”

Sub-solutions and a point-wise Hopf's lemma for fractional p -Laplacian

“…Wenxiong Chen and Congming Li in [7] prove a boundary estimate for (−∆) s p , which is a key part in the moving plane method, and the boundary estimate plays the role of Hopf's lemma to some degree. Leandro M. Del Pezzo and Alexander Quaas in [13], Wenxiong Chen, Congming Li and Shijie Qi in [9] prove a Hopf's lemma for u ∈ W s,p (Ω), where…”

Hopf’s lemma for viscosity solutions to a class of non-local equations with applications

“…A first result in this direction has been obtained in [26,30], where the authors proved it for Dirichlet exterior value problems involving the fractional Laplacian (−∆) α/2 , α ∈ (0, 2). The problem has been studied for the fractional p-Laplacian in [19,44].…”