{
  "id": "2401.14568",
  "version": "v1",
  "published": "2024-01-25T23:44:57.000Z",
  "updated": "2024-01-25T23:44:57.000Z",
  "title": "A counterexample regarding a two-phase problem for harmonic measure in VMO",
  "authors": [
    "Xavier Tolsa"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:2209.14346",
  "categories": [
    "math.AP",
    "math.CA"
  ],
  "abstract": "Let $\\Omega^+\\subset\\mathbb R^{n+1}$ be a vanishing Reifenberg flat domain such that $\\Omega^+$ and $\\Omega^-=\\mathbb R^{n+1}\\setminus\\overline {\\Omega^+}$ have joint big pieces of chord-arc subdomains and the outer unit normal to $\\Omega^+$ belongs to $VMO(\\omega^+)$, where $\\omega^\\pm$ is the harmonic measure of $\\Omega^\\pm$. Up to now it was an open question if these conditions imply that $\\log\\dfrac{d\\omega^-}{d\\omega^+} \\in VMO(\\omega^+)$. In this paper we answer this question in the negative by constructing an appropriate counterexample in $\\mathbb R^2$, with the additional property that the outer unit normal to $\\Omega^+$ is constant $\\omega^+$-a.e. in $\\partial\\Omega^+$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-25T23:44:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31B15",
      "31B20",
      "28A75"
    ],
    "keywords": [
      "harmonic measure",
      "two-phase problem",
      "outer unit normal",
      "counterexample regarding",
      "joint big pieces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}