{
  "id": "2205.12717",
  "version": "v1",
  "published": "2022-05-25T12:34:23.000Z",
  "updated": "2022-05-25T12:34:23.000Z",
  "title": "Reverse Faber-Krahn inequalities for Zaremba problems",
  "authors": [
    "T. V. Anoop",
    "Mrityunjoy Ghosh"
  ],
  "comment": "17 pages",
  "categories": [
    "math.AP",
    "math.OC"
  ],
  "abstract": "Let $\\Omega$ be a multiply-connected domain in $\\mathbb{R}^n$ ($n\\geq 2$) of the form $\\Omega=\\Omega_{\\text{out}}\\setminus \\bar{\\Omega_{\\text{in}}}.$ Set $\\Omega_D$ to be either $\\Omega_{\\text{out}}$ or $\\Omega_{\\text{in}}$. For $p\\in (1,\\infty),$ and $q\\in [1,p],$ let $\\tau_{1,q}(\\Omega)$ be the first eigenvalue of \\begin{equation*} -\\Delta_p u =\\tau \\left(\\int_{\\Omega}|u|^q \\text{d}x \\right)^{\\frac{p-q}{q}} |u|^{q-2}u\\;\\text{in} \\;\\Omega,\\; u =0\\;\\text{on}\\;\\partial\\Omega_D, \\frac{\\partial u}{\\partial \\eta}=0\\;\\text{on}\\; \\partial \\Omega\\setminus \\partial \\Omega_D. \\end{equation*} Under the assumption that $\\Omega_D$ is convex, we establish the following reverse Faber-Krahn inequality $$\\tau_{1,q}(\\Omega)\\leq \\tau_{1,q}({\\Omega}^\\bigstar),$$ where ${\\Omega}^\\bigstar=B_R\\setminus \\bar{B_r}$ is a concentric annular region in $\\mathbb{R}^n$ having the same Lebesgue measure as $\\Omega$ and such that (i) (when $\\Omega_D=\\Omega_{\\text{out}}$) $W_1(\\Omega_D)= \\omega_n R^{n-1}$, and $(\\Omega^\\bigstar)_D=B_R$, (ii) (when $\\Omega_D=\\Omega_{\\text{in}}$) $W_{n-1}(\\Omega_D)=\\omega_nr$, and $(\\Omega^\\bigstar)_D=B_r$. Here $W_{i}(\\Omega_D)$ is the $i^{\\text{th}}$ $quermassintegral$ of $\\Omega_D.$ We also establish Sz. Nagy's type inequalities for parallel sets of a convex domain in $\\mathbb{R}^n$ ($n\\geq 3$) for our proof.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-25T12:34:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P15",
      "35P30",
      "49R05",
      "49Q10"
    ],
    "keywords": [
      "reverse faber-krahn inequality",
      "zaremba problems",
      "concentric annular region",
      "nagys type inequalities",
      "first eigenvalue"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}