{
  "id": "1902.08662",
  "version": "v1",
  "published": "2019-02-22T20:19:43.000Z",
  "updated": "2019-02-22T20:19:43.000Z",
  "title": "Sharp solvability criteria for Dirichlet problems of mean curvature type in Riemannian manifolds: non-existence results",
  "authors": [
    "Yunelsy N Alvarez",
    "Ricardo Sá Earp"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "It is well known that the \\textit{Serrin condition} is a necessary condition for the solvability of the Dirichlet problem for the prescribed mean curvature equation in bounded domains of $\\mathbb{R}^n$ with certain regularity. In this paper we investigate this fact for the vertical mean curvature equation in the product $ M^n \\times \\mathbb{R}$. Precisely, given a $\\mathscr{C}^2$ bounded domain $\\Omega$ in $M$ and a function $H = H (x, z) $ continuous in $\\overline{\\Omega}\\times\\mathbb{R}$ and non-decreasing in the variable $z$, we prove that the \\textit{strong Serrin condition} $(n-1)\\mathcal{H}_{\\partial\\Omega}(y) \\geq n\\sup\\limits_{z\\in\\mathbb{R}} |H(y,z)| \\ \\forall \\ y\\in\\partial\\Omega$, is a necessary condition for the solvability of the Dirichlet problem in a large class of Riemannian manifolds within which are the Hadamard manifolds and manifolds whose sectional curvatures are bounded above by a positive constant. As a consequence of our results we deduce Jenkins-Serrin and Serrin type sharp solvability criteria.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-22T20:19:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dirichlet problem",
      "mean curvature type",
      "riemannian manifolds",
      "non-existence results",
      "mean curvature equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}