{
  "id": "1902.10587",
  "version": "v1",
  "published": "2019-02-27T15:27:57.000Z",
  "updated": "2019-02-27T15:27:57.000Z",
  "title": "Nontrivial solutions to Serrin's problem in annular domains",
  "authors": [
    "Nikola Kamburov",
    "Luciano Sciaraffia"
  ],
  "comment": "21 pages, 1 figure",
  "categories": [
    "math.AP"
  ],
  "abstract": "We construct nontrivial smooth bounded domains $\\Omega \\subseteq \\mathbb{R}^n$ of the form $\\Omega_0 \\setminus \\overline{\\Omega}_1$, bifurcating from annuli, for which there exists a positive solution to the overdetermined boundary value problem \\[ -\\Delta u = 1, \\; u>0 \\quad \\text{in } \\Omega, \\qquad u = 0 ,\\; \\partial_\\nu u = \\text{const} \\quad \\text{on } \\partial\\Omega_0, \\qquad u = \\text{const} ,\\; \\partial_\\nu u = \\text{const} \\quad \\text{on } \\partial \\Omega_1, \\] where $\\nu$ stands for the inner unit normal to $\\partial\\Omega$. From results by Reichel and later by Sirakov, it was known that the condition $\\partial_\\nu u \\leq 0$ on $\\partial\\Omega_1$ is sufficient for rigidity to hold, namely, the only domains which admit such a solution are annuli and solutions are radially symmetric. Our construction shows that the condition is also necessary.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-27T15:27:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35N25",
      "37G25",
      "47A75"
    ],
    "keywords": [
      "serrins problem",
      "nontrivial solutions",
      "annular domains",
      "construct nontrivial smooth bounded domains",
      "overdetermined boundary value problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}