{
  "id": "1708.00900",
  "version": "v1",
  "published": "2017-08-02T19:11:19.000Z",
  "updated": "2017-08-02T19:11:19.000Z",
  "title": "Fractional differentiability for solutions of the inhomogenous $p$-Laplace system",
  "authors": [
    "Michał Miśkiewicz"
  ],
  "comment": "10 pages",
  "categories": [
    "math.AP"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-02T19:11:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B65",
      "35J92"
    ],
    "keywords": [
      "laplace system",
      "fractional differentiability",
      "inhomogenous",
      "fractional nikolkii space",
      "harmonic functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}