{
  "id": "math/0607585",
  "version": "v1",
  "published": "2006-07-24T06:28:42.000Z",
  "updated": "2006-07-24T06:28:42.000Z",
  "title": "A Faber-Krahn inequality with drift",
  "authors": [
    "Francois Hamel",
    "Nikolai Nadirashvili",
    "Emmanuel Russ"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $\\Omega$ be a bounded $C^{2,\\alpha}$ domain in $\\R^n$ ($n\\geq 1$, $0<\\alpha<1$), $\\Omega^{\\ast}$ be the open Euclidean ball centered at 0 having the same Lebesgue measure as $\\Omega$, $\\tau\\geq 0$ and $v\\in L^{\\infty}(\\Omega,\\R^n)$ with $\\left\\Vert v\\right\\Vert\\_{\\infty}\\leq \\tau$. If $\\lambda\\_{1}(\\Omega,\\tau)$ denotes the principal eigenvalue of the operator $-\\Delta+v\\cdot\\nabla$ in $\\Omega$ with Dirichlet boundary condition, we establish that $\\lambda\\_{1}(\\Omega,v)\\geq \\lambda\\_{1}(\\Omega^{\\ast},\\tau e\\_{r})$ where $e\\_{r}(x)=x/| x|$. Moreover, equality holds only when, up to translation, $\\Omega=\\Omega^{\\ast}$ and $v=\\tau e\\_{r}$. This result can be viewed as an isoperimetric inequality for the first eigenvalue of the Dirichlet Laplacian with drift. It generalizes the celebrated Rayleigh-Faber-Krahn inequality for the first eigenvalue of the Dirichlet Laplacian.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-07-24T06:28:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P15",
      "47A75",
      "49R50",
      "35J20"
    ],
    "keywords": [
      "dirichlet laplacian",
      "first eigenvalue",
      "dirichlet boundary condition",
      "celebrated rayleigh-faber-krahn inequality",
      "lebesgue measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......7585H"
    }
  }
}