{
  "id": "1509.07068",
  "version": "v1",
  "published": "2015-09-23T17:21:04.000Z",
  "updated": "2015-09-23T17:21:04.000Z",
  "title": "$σ-$finiteness of a certain Borel measure associated with a positive weak solution to a quasilinear elliptic PDE in space",
  "authors": [
    "Murat Akman",
    "John Lewis",
    "Andrew Vogel"
  ],
  "comment": "33 pages, 3 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we study the Hausdorff dimension of a certain Borel measure $\\mu_{f}$ in space associated to a positive weak solution to a certain quasilinear elliptic PDE in an open subset and vanishing on a portion of the boundary of that open set. We show that this measure is concentrated on a set of $\\sigma-$finite $n-1$ dimensional Hausdorff measure for $p>n$ and the same result holds for $p=n$ with an assumption on the boundary. We also construct an example of a domain in space for which the corresponding measure has Hausdorff dimension $\\leq n-1-\\delta$ for $p\\geq n$ for some $\\delta$ which depends on various constants including $p$. The first result generalizes the authors previous work when the PDE is the $p-$Laplacian and the second result generalizes the well known theorem of Wolff when $p=2$ and $n=2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-23T17:21:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J25",
      "35J70",
      "37F35",
      "28A78"
    ],
    "keywords": [
      "quasilinear elliptic pde",
      "positive weak solution",
      "borel measure",
      "finiteness",
      "hausdorff dimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}