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arXiv · 2409.02012

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

Verena Bögelein·Frank Duzaar·Naian Liao·Giovanni Molica Bisci·Raffaella Servadei
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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.

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Verena Bögelein, Frank Duzaar, Naian Liao, Giovanni Molica Bisci, Raffaella Servadei. 2024-09-03. Gradient regularity for $(s,p)$-harmonic functions. https://arxiv.org/abs/2409.02012

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