Semiregular and strongly irregular boundary points for nonlocal Dirichlet problems

TL;DR

This paper classifies boundary points for nonlocal fractional p-Laplacian problems into semiregular and strongly irregular types, revealing their distinct behaviors and dependencies on parameters s and p.

Contribution

It introduces a new classification of boundary points for nonlocal equations and provides a removability result for solutions in Sobolev spaces, expanding understanding of boundary regularity.

Findings

Boundary points divided into semiregular and strongly irregular classes.

A new removability theorem for solutions in Sobolev spaces.

Semiregularity depends on parameters s and p.

Abstract

In this paper we study nonlocal nonlinear equations of fractional $(s,p)$-Laplacian type on $\mathbf{R}^n$. We show that the irregular boundary points for the Dirichlet problem can be divided into two disjoint classes: semiregular and strongly irregular boundary points, with very different behaviour. Two fundamental tools needed to show this are the Kellogg property (from our previous paper) and a new removability result for solutions in the $V^{s,p}$ Sobolev type space, which we deduce more generally also for supersolutions of equations with a right-hand side. Semiregular and strongly irregular points are also characterized in various ways. Finally, it is explained how semiregularity depends on $s$ and $p$.