Global H\"older regularity for the fractional $p$-Laplacian

TL;DR

This paper establishes boundary regularity results for solutions to the fractional p-Laplacian equation, using barrier methods to prove Hölder continuity up to the boundary for both singular and degenerate cases.

Contribution

It provides the first boundary regularity proof for fractional p-Laplacian problems with Dirichlet conditions, covering both singular and degenerate regimes.

Findings

Proves $C^eta$ regularity up to the boundary for solutions.

Employs barrier arguments tailored to singular and degenerate cases.

Extends regularity theory for non-local nonlinear equations.

Abstract

By virtue of barrier arguments we prove $C^\alpha$-regularity up to the boundary for the weak solutions of a non-local nonlinear problem driven by the fractional $p$-Laplacian operator. The equation is boundedly inhomogeneous and the boundary conditions are of Dirichlet type. We employ different methods according to the singular ($p<2$) of degenerate ($p>2$) case.