{
  "id": "1603.06391",
  "version": "v1",
  "published": "2016-03-21T11:32:28.000Z",
  "updated": "2016-03-21T11:32:28.000Z",
  "title": "$C^{1,α}$ regularity for the normalized $p$-Poisson problem",
  "authors": [
    "Amal Attouchi",
    "Mikko Parviainen",
    "Eero Ruosteenoja"
  ],
  "comment": "30 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the normalized $p$-Poisson problem $$-\\Delta^N_p u=f \\qquad \\text{in}\\quad \\Omega.$$ The normalized $p$-Laplacian $\\Delta_p^{N}u:=|D u|^{2-p}\\Delta_p u$ is in non-divergence form and arises for example from stochastic games. We prove $C^{1,\\alpha}_{loc}$ regularity with nearly optimal $\\alpha$ for viscosity solutions of this problem. In the case $f\\in L^{\\infty}\\cap C$ and $p>1$ we use methods both from viscosity and weak theory, whereas in the case $f\\in L^q\\cap C$, $q>\\max(n,\\frac p2,2)$, and $p>2$ we rely on the tools of nonlinear potential theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-21T11:32:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J60",
      "35B65",
      "35J92"
    ],
    "keywords": [
      "poisson problem",
      "regularity",
      "nonlinear potential theory",
      "non-divergence form",
      "weak theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160306391A"
    }
  }
}