Fine Boundary Regularity For The Fractional (p,q)-Laplacian

TL;DR

This paper establishes fine boundary regularity results for solutions to fractional (p,q)-Laplacian equations, demonstrating weighted Hölder regularity up to the boundary using novel barrier methods.

Contribution

It introduces a new barrier construction and boundary Harnack method for analyzing boundary regularity of fractional (p,q)-Laplacian solutions, even without scaling or homogeneity.

Findings

Proves $u/d_ ext{boundary}^s$ is Hölder continuous up to the boundary.

Extends regularity results to sign-changing data.

Develops techniques applicable to a range of fractional operators.

Abstract

In this article, we deal with the fine boundary regularity, a weighted H\"{o}lder regularity of weak solutions to the problem involving the fractional $(p,q)$ Laplacian denoted by $(-\Delta)_{p}^{s} u + (-\Delta)_{q}^{s} u = f(x)$ in $\Omega,$ and $u=0$ in $\mathbb{R}^N\setminus\Omega;$ where $\Omega$ is a $C^{1,1}$ bounded domain and $2 \leq p \leq q <\infty.$ For $0<s<1$ and for non-negative data $f\in L^{\infty}(\Omega),$ we employ the nonlocal analogue of the boundary Harnack method to establish that $u/{d_{\Omega}^{s}} \in C^{\alpha}(\Bar{\Omega})$ for some $\alpha \in (0,1),$ where $d_\Omega(x)$ is the distance of $x$ from the boundary. A novel barrier construction allows us to analyse the regularity theory even in the absence of the scaling or the homogeneity properties of the operator. Additionally, we extend our idea to sign changing bounded $f$ as well and prove a fine boundary regularity for fractional $(p,q)$ Laplacian for some range of $s.$