Higher H\"older regularity for the fractional $p$-Laplace equation in the subquadratic case

TL;DR

This paper establishes explicit Hölder regularity estimates for solutions to the fractional p-Laplace equation in the subquadratic case, extending known results and providing near-sharp exponents through Moser iteration and perturbation techniques.

Contribution

It provides the first explicit Hölder estimates for the subquadratic fractional p-Laplace equation, complementing existing results for the superquadratic case.

Findings

Derived explicit Hölder exponents for solutions in the subquadratic case

Extended regularity results to inhomogeneous equations with near-sharp exponents

Utilized Moser iteration and perturbation methods for analysis

Abstract

We study the fractional $p$-Laplace equation $$ (-\Delta_p)^s u = 0 $$ for $0<s<1$ and in the subquadratic case $1<p<2$. We provide H\"older estimates with an explicit H\"older exponent. The inhomogeneous equation is also treated and there the exponent obtained is almost sharp. Our results complement the previous results for the superquadratic case when $p\geq 2$. The arguments are based on a careful Moser-type iteration and a perturbation argument.