Gradient regularity for $(s,p)$-harmonic functions

TL;DR

This paper investigates the regularity of fractional p-harmonic functions, demonstrating their differentiability, integrability, and Hölder continuity, with results stable as the fractional order approaches 1, recovering classical p-harmonic regularity.

Contribution

It establishes new regularity results for $(s,p)$-harmonic functions, including differentiability and fractional differentiability of the weak gradient, bridging fractional and classical p-harmonic theory.

Findings

$(s,p)$-harmonic functions are weakly differentiable.

The weak gradient is locally integrable to any power $q\,\geq 1$.

Functions are Hölder continuous with arbitrary exponent in $(0,1)$.

Abstract

We study the local regularity properties of $(s,p)$-harmonic functions, i.e. local weak solutions to the fractional $p$-Laplace equation of order $s\in (0,1)$ in the case $p\in (1,2]$. It is shown that $(s,p)$-harmonic functions are weakly differentiable and that the weak gradient is locally integrable to any power $q\geq 1$. As a result, $(s,p)$-harmonic functions are H\"older continuous to arbitrary H\"older exponent in $(0,1)$. In addition, the weak gradient of $(s,p)$-harmonic functions has certain fractional differentiability. All estimates are stable when $s$ reaches $1$, and the known regularity properties of $p$-harmonic functions are formally recovered, in particular the local $W^{2,2}$-estimate.