Fractional differentiability for solutions of the inhomogenous $p$-Laplace system

Abstract

It is shown that if $p\ge 3$ and $u\in W^{1,p}(\Omega ,{\mathbb{R}}^N)$ solves the inhomogenous $p$-Laplace system \[ \operatorname{div} (|\nabla u|^{p-2} \nabla u) = f, \qquad f \in W^{1,p'}(Ω,\mathbb{R}^N), \] then locally the gradient $\nabla u$ lies in the fractional Nikol'skii space ${\mathcal{N}}^{\theta ,2/\theta }$ with any $\theta \in [\frac{2}{p},\frac{2}{p-1})$. To the author's knowledge, this result is new even in the case of $p$-harmonic functions, slightly improving known ${\mathcal{N}}^{2/p,p}$ estimates. The method used here is an extension of the one used by A. Cellina in the case $2\le p<3$ to show $W^{1,2}$ regularity.

10 pages

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