{
  "id": "1401.0351",
  "version": "v1",
  "published": "2014-01-02T01:03:16.000Z",
  "updated": "2014-01-02T01:03:16.000Z",
  "title": "On Second Order Elliptic and Parabolic Equations of Mixed Type",
  "authors": [
    "Gong Chen",
    "Mikhail Safonov"
  ],
  "comment": "16 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "It is known that solutions to second order uniformly elliptic and parabolic equations, either in divergence or nondivergence (general) form, are H\\\"{o}lder continuous and satisfy the interior Harnack inequality. We show that even in the one-dimensional case ($x\\in R^1$), these properties are not preserved for equations of mixed divergence-nondivergence structure: for elliptic equations. \\begin{equation*} D_i(a^1_{ij}D_ju)+a^2_{ij}D_{ij}u=0, \\end{equation*} and parabolic equations \\begin{equation*} p\\partial_t u=D_i(a_{ij}D_ju), \\end{equation*} where $p=p(t,x)$ is a bounded strictly positive function. The H\\\"{o}lder continuity and Harnack inequality are known if $p$ does not depend either on $t$ or on $x$. We essentially use homogenization techniques in our construction. Bibliography: 23 titles.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-01-02T01:03:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "parabolic equations",
      "second order elliptic",
      "mixed type",
      "second order uniformly elliptic",
      "interior harnack inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1401.0351C"
    }
  }
}