Fine boundary regularity for the degenerate fractional $p$-Laplacian

TL;DR

This paper proves that solutions to a degenerate fractional p-Laplacian equation are regularly behaved up to the boundary, specifically showing they are Hölder continuous when scaled by the distance to the boundary.

Contribution

It establishes boundary regularity results for solutions to the degenerate fractional p-Laplacian, a problem not previously well-understood in this context.

Findings

Solutions are weighted Hölder continuous up to the boundary.

The regularity is characterized by the solution divided by the boundary distance raised to the power s.

The methods involve barriers, superposition principles, and comparison techniques.

Abstract

We consider a pseudo-differential equation driven by the fractional $p$-Laplacian with $p\ge 2$ (degenerate case), with a bounded reaction $f$ and Dirichlet type conditions in a smooth domain $\Omega$. By means of barriers, a nonlocal superposition principle, and the comparison principle, we prove that any weak solution $u$ of such equation exhibits a weighted H\"older regularity up to the boundary, that is, $u/d^s\in C^\alpha(\overline\Omega)$ for some $\alpha\in(0,1)$, $d$ being the distance from the boundary.