Regularity for the fractional $p$-Laplace equation

TL;DR

This paper investigates higher Sobolev and Hölder regularity for solutions of the fractional p-Laplace equation, providing precise local estimates that unify and extend known regularity results across different regimes of the fractional order.

Contribution

It establishes new local regularity estimates for solutions of the fractional p-Laplace equation, covering various regimes of the fractional order and recovering classical results as special cases.

Findings

Stable estimates as s approaches 1, recovering classical p-Laplace regularity.

Almost W^{1+s,2}_{loc}-regularity for the fractional Laplace equation when p=2.

Precise local regularity estimates depending on the fractional order s.

Abstract

Higher Sobolev and H\"older regularity is studied for local weak solutions of the fractional $p$-Laplace equation of order $s$ in the case $p\ge 2$. Depending on the regime considered, i.e. $$0<s\le\tfrac{p-2}{p}\quad \text{or} \quad\tfrac{p-2}{p}<s<1,$$ precise local estimates are proven. The relevant estimates are stable if the fractional order $s$ reaches $1$; the known Sobolev regularity estimates for the local $p$-Laplace are recovered. The case $p=2$ reproduces the almost $W^{1+s,2}_{\rm loc}$-regularity for the fractional Laplace equation of any order $s\in(0,1)$.