{
  "id": "2305.07802",
  "version": "v1",
  "published": "2023-05-12T23:43:30.000Z",
  "updated": "2023-05-12T23:43:30.000Z",
  "title": "Exceptional domains in higher dimensions",
  "authors": [
    "Ignace Aristide Minlend",
    "Tobias Weth",
    "Jing Wu"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove the existence of nontrivial unbounded exceptional domains in the Euclidean space $\\R^N$, $N\\geq4$. These domains arise as perturbations of complements of straight cylinders in $\\R^N$, and by definition they support a positive harmonic function with vanishing Dirichlet boundary values and constant Neumann boundary values, the so-called roof function. While the domains have a similar shape as those constructed in the recent work \\cite{Fall-MinlendI-Weth3} for the case $N=3$, there is a striking constrast with regard to the shape of corresponding roof functions which are bounded for $N \\ge 4$. Moreover, while the analysis in \\cite{Fall-MinlendI-Weth3} does not extend to higher dimensions, the approach of the present paper depends heavily on the assumption $N \\ge 4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-12T23:43:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "higher dimensions",
      "roof function",
      "constant neumann boundary values",
      "nontrivial unbounded exceptional domains",
      "vanishing dirichlet boundary values"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}