{
  "id": "1806.11491",
  "version": "v1",
  "published": "2018-06-29T15:47:43.000Z",
  "updated": "2018-06-29T15:47:43.000Z",
  "title": "On the reverse Faber-Krahn inequalities",
  "authors": [
    "T. V. Anoop",
    "K. Ashok Kumar"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "For $N\\geq 2$, let $\\Omega\\subset \\mathbb{R}^N$ be a $C^1$ smooth multiply connected bounded domain. Let $\\Gamma_1$ be the outer boundary of $\\Omega$ and $\\Gamma_0, \\gamma_1,\\ldots \\gamma_n$ be the boundaries of the interior holes. For $p\\in (1,\\infty)$, we consider the first eigenvalue $\\nu_1(\\Omega)$ of the $p$-Laplacian with mixed boundary conditions: $$-\\Delta_p u= \\nu |u|^{p-2}u \\quad \\text{in } \\, \\Omega;\\qquad u=0 \\text{ on } \\Gamma_1;\\qquad \\frac{\\partial{u}}{\\partial{\\eta}}=0 \\text{ on } \\Gamma_0\\cup\\left(\\cup_{i=1}^n\\gamma_i\\right).$$ Let $\\Omega^\\#$ be the annulus centred at the origin having the same volume as $\\Omega$ and the same outer surface measure as $\\Gamma_1$. For $N=2,$ we show that $\\nu_1$ satisfies the following reverse Faber-Krahn inequality: \\begin{equation}\\label{R-RFK} \\nu_1(\\Omega) \\le \\nu_1(\\Omega^\\#) . \\end{equation} For $N\\ge 3,$ we obtain the above inequality, provided $\\Gamma_1$ is a sphere. In this case, we also show that if the equality holds, then $\\Omega$ is a translation of $\\Omega^\\#$. An analogous result is proved for the case when the Dirichlet condition on $\\Gamma_0$ and the Neumann condition on the rest of the boundaries.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-29T15:47:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P30",
      "35M12",
      "35J92",
      "35J25",
      "35P15"
    ],
    "keywords": [
      "reverse faber-krahn inequality",
      "outer surface measure",
      "mixed boundary conditions",
      "neumann condition",
      "outer boundary"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}