Higher Sobolev regularity for the fractional $p-$Laplace equation in the superquadratic case

TL;DR

This paper demonstrates that solutions to the fractional p-Laplacian equation with p ≥ 2 exhibit enhanced regularity in fractional Sobolev spaces, with stability as the fractional order approaches 1.

Contribution

It establishes higher Sobolev regularity for solutions of the fractional p-Laplacian in the superquadratic case, including conditions for local gradient regularity.

Findings

Solutions gain regularity in fractional Sobolev spaces.

Under certain conditions, solutions are in W^{1,p}_{loc} and gradients are fractional Sobolev.

Estimates remain stable as fractional order s approaches 1.

Abstract

We prove that for $p\ge 2$ solutions of equations modeled by the fractional $p$-Laplacian improve their regularity on the scale of fractional Sobolev spaces. Moreover, under certain precise conditions, they are in $W^{1,p}_{loc}$ and their gradients are in a fractional Sobolev space as well. The relevant estimates are stable as the fractional order of differentiation $s$ reaches $1$.