{
  "id": "1110.3295",
  "version": "v3",
  "published": "2011-10-14T18:44:56.000Z",
  "updated": "2012-12-10T22:03:48.000Z",
  "title": "Regularity of solutions to degenerate $p$-Laplacian equations",
  "authors": [
    "David Cruz-Uribe",
    "Kabe Moen",
    "Virginia Naibo"
  ],
  "comment": "v3 several revisions. Final version. To appear in JMAA",
  "categories": [
    "math.AP",
    "math.CA"
  ],
  "abstract": "We prove regularity results for solutions of the equation \\[div(< AXu,X u>^{(p-2)/2} AX u) = 0,\\] $1<p<\\infty$, where $X=(X_1,...,X_m)$ is a family of vector fields satisfying H\\\"ormander's ellipticity condition, $A$ is an $m\\times m$ symmetric matrix that satisfies degenerate ellipticity conditions. If the degeneracy is of the form \\[\\lambda w(x)^{2/p}|\\xi|^2\\leq < A(x)\\xi,\\xi>\\leq \\Lambda w(x)^{2/p}|\\xi|^2,\\] $w \\in A_p$, then we show that solutions are locally H\\\"older continuous. If the degeneracy is of the form \\[ k(x)^{-2/p'}|\\xi|^2\\leq < A(x)\\xi,\\xi>\\leq k(x)^{2/p}|\\xi|^2, \\] $k\\in A_{p'}\\cap RH_\\tau$,where $\\tau$ depends on the homogeneous dimension, then the solutions are continuous almost everywhere, and we give examples to show that this is the best result possible. We give an application to maps of finite distortion.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-12-10T22:03:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "laplacian equations",
      "satisfies degenerate ellipticity conditions",
      "best result",
      "regularity results",
      "degeneracy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1110.3295C"
    }
  }
}