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

Abstract:

It is shown that if $p \geqslant 3$ and $u \in W^{1,p}(\Omega ,\mathbb {R}^N)$ solves the inhomogeneous $p$-Laplace system \[ \operatorname {div} (|\nabla u|^{p-2} \nabla u) = f, \qquad f \in W^{1,p’}(\Omega ,\mathbb {R}^N), \] then locally the gradient $\nabla u$ lies in the fractional Nikol’skiĭ space $\mathcal {N}^{\theta ,2/\theta }$ with any $\theta \in [ \tfrac {2}{p}, \tfrac {2}{p-1} )$. 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 $2 \leqslant p < 3$ to show $W^{1,2}$ regularity.