{
  "id": "2405.04388",
  "version": "v1",
  "published": "2024-05-07T15:13:27.000Z",
  "updated": "2024-05-07T15:13:27.000Z",
  "title": "Boundary unique continuation in planar domains by conformal mapping",
  "authors": [
    "Stefano Vita"
  ],
  "comment": "11 pages, comments are welcome",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $\\Omega\\subset\\mathbb R^2$ be a chord arc domain with small constant. We show that a nontrivial harmonic function which vanishes continuously on a relatively open set of the boundary cannot have the norm of the gradient which vanishes on a subset of positive surface measure (arc length). This result was previously known to be true, and conjectured in higher dimensions by Lin, in Lipschitz domains. Let now $\\Omega\\subset\\mathbb R^2$ be a $C^1$ domain with Dini mean oscillations. We prove that a nontrivial harmonic function which vanishes continuously on a relatively open subset of the boundary $\\partial\\Omega\\cap B_1$ has a finite number of critical points in $\\overline\\Omega\\cap B_{1/2}$. The latter improves some recent results by Kenig and Zhao. Our technique involves a conformal mapping which moves the boundary where the harmonic function vanishes into an interior nodal line of a new harmonic function, after a further reflection. Then, size estimates of the critical set - up to the boundary - of the original harmonic function can be understood in terms of estimates of the interior critical set of the new harmonic function and of the critical set - up to the boundary - of the conformal mapping.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-07T15:13:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30C20",
      "31A05",
      "35J25",
      "42B37"
    ],
    "keywords": [
      "boundary unique continuation",
      "conformal mapping",
      "planar domains",
      "nontrivial harmonic function",
      "critical set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}