{
  "id": "1206.6641",
  "version": "v2",
  "published": "2012-06-28T11:49:55.000Z",
  "updated": "2014-01-27T15:51:02.000Z",
  "title": "On the Regularity of the Free Boundary for Quasilinear Obstacle Problems",
  "authors": [
    "S. Challal",
    "A. Lyaghfouri",
    "J. F. Rodrigues",
    "R. Teymurazyan"
  ],
  "comment": "40 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We extend basic regularity of the free boundary of the obstacle problem to some classes of heterogeneous quasilinear elliptic operators with variable growth that includes, in particular, the $p(x)$-Laplacian. Under the assumption of Lipschitz continuity of the order of the power growth $p(x)>1$, we use the growth rate of the solution near the free boundary to obtain its porosity, which implies that the free boundary is of Lebesgue measure zero for $p(x)$-Laplacian type heterogeneous obstacle problems. Under additional assumptions on the operator heterogeneities and on data we show, in two different cases, that up to a negligible singular set of null perimeter the free boundary is the union of at most a countable family of $C^1$ hypersurfaces: i) by extending directly the finiteness of the $(n-1)$-dimensional Hausdorff measure of the free boundary to the case of heterogeneous $p$-Laplacian type operators with constant $p$; $1<p<\\infty$; ii) by proving the characteristic function of the coincidence set is of bounded variation in the case of non degenerate or non singular operators with variable power growth $p(x)>1$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-01-27T15:51:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "free boundary",
      "quasilinear obstacle problems",
      "regularity",
      "laplacian type heterogeneous obstacle problems",
      "power growth"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.6641C"
    }
  }
}