Higher H\"older regularity for the fractional $p-$Laplacian in the superquadratic case

TL;DR

This paper establishes higher H"older regularity for solutions to fractional p-Laplacian equations in the superquadratic case, providing explicit exponents and nearly sharp results for non-homogeneous problems.

Contribution

It introduces new regularity results with explicit H"older exponents for fractional p-Laplacian equations when p≥2 and 0<s<1, including non-homogeneous cases.

Findings

Derived explicit H"older exponents for solutions.

Proved near-sharp regularity results for non-homogeneous equations.

Extended regularity theory to the fractional p-Laplacian in the superquadratic case.

Abstract

We prove higher H\"older regularity for solutions of equations involving the fractional $p-$Laplacian of order $s$, when $p\ge 2$ and $0<s<1$. In particular, we provide an explicit H\"older exponent for solutions of the non-homogeneous equation with data in $L^q$ and $q>N/(s\,p)$, which is almost sharp whenever $s\,p\leq (p-1)+N/q$. The result is new already for the homogeneous equation.