{
  "id": "2207.12076",
  "version": "v2",
  "published": "2022-07-25T11:52:49.000Z",
  "updated": "2022-10-30T16:34:33.000Z",
  "title": "Perturbation Theory for Second Order Elliptic Operators with BMO Antisymmetric Part",
  "authors": [
    "Martin Dindoš",
    "Erik Sätterqvist",
    "Martin Ulmer"
  ],
  "comment": "50 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In the present paper we study perturbation theory for the $L^p$ Dirichlet problem on bounded chord arc domains for elliptic operators in divergence form with potentially unbounded antisymmetric part in BMO. Specifically, given elliptic operators $L_0 = \\mbox{div}(A_0\\nabla)$ and $L_1 = \\mbox{div}(A_1\\nabla)$ such that the $L^p$ Dirichlet problem for $L_0$ is solvable for some $p>1$; we show that if $A_0 - A_1$ satisfies certain Carleson condition, then the $ L^q$ Dirichlet problem for $L_1$ is solvable for some $q \\geq p$. Moreover if the Carleson norm is small then we may take $q=p$. We use the approach first introduced in Fefferman-Kenig-Pipher '91 on the unit ball, and build on Milakis-Pipher-Toro '11 where the large norm case was shown for symmetric matrices on bounded chord arc domains. We then apply this to solve the $L^p$ Dirichlet problem on a bounded Lipschitz domain for an operator $L = \\mbox{div}(A\\nabla)$, where $A$ satisfies a Carleson condition similar to the one assumed in Kenig-Pipher '01 and Dindo\\v{s}-Petermichl-Pipher '07 but with unbounded antisymmetric part.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-10-30T16:34:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J25"
    ],
    "keywords": [
      "second order elliptic operators",
      "bmo antisymmetric part",
      "perturbation theory",
      "dirichlet problem",
      "bounded chord arc domains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}