{
  "id": "math/0608136",
  "version": "v1",
  "published": "2006-08-05T07:19:09.000Z",
  "updated": "2006-08-05T07:19:09.000Z",
  "title": "Rearrangement inequalities and applications to isoperimetric problems for eigenvalues",
  "authors": [
    "Francois Hamel",
    "Nikolai Nadirashvili",
    "Emmanuel Russ"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $\\Omega$ be a bounded $C^{2}$ domain in $\\R^n$, and let $\\Omega^{\\ast}$ be the Euclidean ball centered at 0 and having the same Lebesgue measure as $\\Omega$. Consider the operator $L=-\\div(A\\nabla)+v\\cdot \\nabla +V$ on $\\Omega$ with Dirichlet boundary condition. We prove that minimizing the principal eigenvalue of $L$ when the Lebesgue measure of $\\Omega$ is fixed and when $A$, $v$ and $V$ vary under some constraints is the same as minimizing the principal eigenvalue of some operators $L^*$ in the ball $\\Omega^*$ with smooth and radially symmetric coefficients. The constraints which are satisfied by the original coefficients in $\\Omega$ and the new ones in $\\Omega^*$ are expressed in terms of some distribution functions or some integral, pointwise or geometric quantities. Some strict comparisons are also established when $\\Omega$ is not a ball.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-08-05T07:19:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P15",
      "47A75",
      "49R50",
      "35J20"
    ],
    "keywords": [
      "isoperimetric problems",
      "rearrangement inequalities",
      "principal eigenvalue",
      "applications",
      "lebesgue measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......8136H"
    }
  }
}