{
  "id": "2005.04608",
  "version": "v1",
  "published": "2020-05-10T09:25:32.000Z",
  "updated": "2020-05-10T09:25:32.000Z",
  "title": "Holder continuity of weak solutions of p-Laplacian PDE's with VMO coefficients",
  "authors": [
    "C. S. Goodtich",
    "m. A. Ragusa"
  ],
  "comment": "Nonlinear Analysis",
  "doi": "10.1016/j.na.2019.03.015",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider solutions $u\\in W^{1,p}\\big(\\Omega;\\mathbb{R}^{N}\\big)$ of the $p$-Laplacian PDE \\begin{equation} \\nabla\\cdot\\big(a(x)|Du|^{p-2}Du\\big)=0,\\notag \\end{equation} for $x\\in\\Omega\\subseteq\\mathbb{R}^{n}$, where $\\Omega$ is open and bounded. More generally, we consider solutions of the elliptic system \\begin{equation} \\nabla\\cdot\\left(a(x)g'\\big(a(x)|Du|\\big)\\frac{Du}{|Du|}\\right)=0\\text{, }x\\in\\Omega\\notag \\end{equation} as well as minimizers of the functional \\begin{equation} \\int_{\\Omega}g\\big(a(x)|Du|\\big)\\ dx.\\notag \\end{equation} In each case, the coefficient map $a\\ : \\ \\Omega\\rightarrow\\mathbb{R}$ is only assumed to be of class $VMO(\\Omega)\\cap L^{\\infty}(\\Omega)$, which means that it may be discontinuous. Without assuming that $x\\mapsto a(x)$ has any weak differentiability, we show that $u\\in\\mathscr{C}_{\\text{loc}}^{0,\\alpha}(\\Omega)$ for each $0<\\alpha<1$. The preceding results are, in fact, a corollary of a much more general result, which applies to the functional \\begin{equation} \\int_{\\Omega}f\\big(x,u,Du\\big)\\ dx\\notag \\end{equation} in case $f$ is only asymptotically convex.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-10T09:25:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B65",
      "49N60",
      "46E35"
    ],
    "keywords": [
      "p-laplacian pdes",
      "holder continuity",
      "weak solutions",
      "vmo coefficients",
      "functional"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}