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 \[ \operatorname{div} (|\nabla u|^{p-2} \nabla u) = f, \qquad f \in W^{1,p'}(\Omega,\mathbb{R}^N), \] then locally the gradient $\nabla u$ lies in the fractional Nikol'skii space $\mathcal{N}^{\theta,2/\theta}$ with any $\theta \in [ \tfrac{2}{p}, \tfrac{2}{p-1} )$. 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 $2 \le p < 3$ to show $W^{1,2}$ regularity.
- Publication:
-
arXiv e-prints
- Pub Date:
- August 2017
- DOI:
- 10.48550/arXiv.1708.00900
- arXiv:
- arXiv:1708.00900
- Bibcode:
- 2017arXiv170800900M
- Keywords:
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- Mathematics - Analysis of PDEs;
- 35B65;
- 35J92
- E-Print:
- 10 pages