{
  "id": "1901.08261",
  "version": "v1",
  "published": "2019-01-24T07:29:49.000Z",
  "updated": "2019-01-24T07:29:49.000Z",
  "title": "Perturbation of elliptic operators in 1-sided NTA domains satisfying the capacity density condition",
  "authors": [
    "Murat Akman",
    "Steve Hofmann",
    "José María Martell",
    "Tatiana Toro"
  ],
  "categories": [
    "math.CA",
    "math.AP"
  ],
  "abstract": "Let $\\Omega\\subset\\mathbb{R}^{n+1}$, $n\\ge 2$, be a 1-sided non-tangentially accessible domain (aka uniform domain), that is, a set which satisfies the interior corkscrew and Harnack chain conditions, respectively scale-invariant/quantitative versions of openness and path-connectedness. Assume that $\\Omega$ satisfies the so-called capacity density condition. Let $L_0u=-{\\rm div\\,}(A_0\\nabla u)$, $Lu=-{\\rm div\\,}(A\\nabla u)$ be two real symmetric uniformly elliptic operators, and write $\\omega_{L_0}$, $\\omega_L$ for the associated elliptic measures. The goal of this paper is to find sufficient conditions guaranteeing that $\\omega_L$ satisfies $A_\\infty$ or $RH_q$ conditions with respect to $\\omega_{L_0}$. We show that if the discrepancy of the two matrices satisfies a natural Carleson measure condition with respect to $\\omega_0$ then $\\omega_L\\in A_\\infty(\\omega_0)$. Moreover, $\\omega_L\\in RH_q(\\omega_0)$ for any given $1<q<\\infty$ if the Carleson measure condition holds with a sufficiently small constant. This latter case extends previous work of Fefferman-Kenig-Pipher and Milakis-Pipher-Toro who considered Lipschitz and chord-arc domains respectively. Here we go beyond those settings, the capacity density condition is much weaker than the existence of exterior corkscrew balls. Moreover, the boundaries of our domains need not to be Ahlfors regular and the restriction of the $n$-dimensional Hausdorff measure to the boundary could be even locally infinite. The \"large constant\" case, where discrepancy of the matrices satisfies a Carleson measure condition, is new even for nice domains such as the unit ball, the upper-half space, or chord-arc domains. We emphasize that our domains do not have a nice surface measure: all the analysis is done with the underlying measure $\\omega_{L_0}$, which behaves well in the settings we are considering.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-24T07:29:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31B05",
      "35J08",
      "35J25",
      "42B99",
      "42B25",
      "42B37"
    ],
    "keywords": [
      "capacity density condition",
      "nta domains satisfying",
      "symmetric uniformly elliptic operators",
      "matrices satisfies",
      "chord-arc domains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}