{
  "id": "1907.07102",
  "version": "v1",
  "published": "2019-07-04T17:23:54.000Z",
  "updated": "2019-07-04T17:23:54.000Z",
  "title": "Harmonic measure and quantitative connectivity: geometric characterization of the $L^p$-solvability of the Dirichlet problem",
  "authors": [
    "Jonas Azzam",
    "Steve Hofmann",
    "José María Martell",
    "Mihalis Mourgoglou",
    "Xavier Tolsa"
  ],
  "comment": "This paper is a combination of arXiv:1712.03696 and arXiv:1803.07975",
  "categories": [
    "math.CA",
    "math.AP"
  ],
  "abstract": "It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak-$A_\\infty$ property) of harmonic measure with respect to surface measure, on the boundary of an open set $ \\Omega\\subset \\mathbb{R}^{n+1}$ with Ahlfors-David regular boundary, is equivalent to the solvability of the Dirichlet problem in $\\Omega$, with data in $L^p(\\partial\\Omega)$ for some $p<\\infty$. In this paper, we give a geometric characterization of the weak-$A_\\infty$ property, of harmonic measure, and hence of solvability of the $L^p$ Dirichlet problem for some finite $p$. This characterization is obtained under background hypotheses (an interior corkscrew condition, along with Ahlfors-David regularity of the boundary) that are in the nature of best possible: we provide counter-examples in the absence of either of them (or even one of the two, upper or lower, Ahlfors-David bounds).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-04T17:23:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31B05",
      "35J25",
      "42B25",
      "42B37"
    ],
    "keywords": [
      "dirichlet problem",
      "harmonic measure",
      "geometric characterization",
      "quantitative connectivity",
      "solvability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}