{
  "id": "1205.2027",
  "version": "v1",
  "published": "2012-05-09T16:32:13.000Z",
  "updated": "2012-05-09T16:32:13.000Z",
  "title": "Stability estimates in $H^1_0$ for solutions of elliptic equations in varying domains",
  "authors": [
    "José M. Arrieta",
    "Gerassimos Barbatis"
  ],
  "comment": "10 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider second-order uniformly elliptic operators subject to Dirichlet boundary conditions. Such operators are considered on a bounded domain $\\Omega$ and on the domain $\\phi(\\Omega)$ resulting from $\\Omega$ by means of a bi-Lipschitz map $\\phi$. We consider the solutions $u$ and $\\tilde u$ of the corresponding elliptic equations with the same right-hand side $f\\in L^2(\\Omega\\cup\\phi(\\Omega))$. Under certain assumptions we estimate the difference $\\|\\nabla\\tilde u-\\nabla u\\|_{L^2(\\Omega\\cup\\phi(\\Omega))}$ in terms of certain measure of vicinity of $\\phi$ to the identity map. For domains within a certain class this provides estimates in terms of the Lebesgue measure of the symmetric difference of $\\phi(\\Omega)$ and $\\Omega$, that is $|\\phi(\\Omega)\\triangle \\Omega|$. We provide an example which shows that the estimates obtained are in a certain sense sharp.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-05-09T16:32:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stability estimates",
      "varying domains",
      "second-order uniformly elliptic operators subject",
      "dirichlet boundary conditions",
      "bi-lipschitz map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1205.2027A"
    }
  }
}