{
  "id": "1704.03194",
  "version": "v1",
  "published": "2017-04-11T08:33:58.000Z",
  "updated": "2017-04-11T08:33:58.000Z",
  "title": "On multiplicity of eigenvalues and symmetry of eigenfunctions of the $p$-Laplacian",
  "authors": [
    "Benjamin Audoux",
    "Vladimir Bobkov",
    "Enea Parini"
  ],
  "comment": "14 pages, 1 figure",
  "categories": [
    "math.AP",
    "math.SP"
  ],
  "abstract": "We investigate multiplicity and symmetry properties of higher eigenvalues and eigenfunctions of the $p$-Laplacian under homogeneous Dirichlet boundary conditions on certain symmetric domains $\\Omega \\subset \\mathbb{R}^N$. By means of topological arguments, we show how symmetries of $\\Omega$ help to construct subsets of $W_0^{1,p}(\\Omega)$ with suitably high Krasnosel'ski\\u{\\i} genus. In particular, if $\\Omega$ is a ball $B \\subset \\mathbb{R}^N$, we obtain the following chain of inequalities: $$ \\lambda_2(p;B) \\leq \\dots \\leq \\lambda_{N+1}(p;B) \\leq \\lambda_\\ominus(p;B). $$ Here $\\lambda_i(p;B)$ are variational eigenvalues of the $p$-Laplacian on $B$, and $\\lambda_\\ominus(p;B)$ is the eigenvalue which has an associated eigenfunction whose nodal set is an equatorial section of $B$. If $\\lambda_2(p;B)=\\lambda_\\ominus(p;B)$, as it holds true for $p=2$, the result implies that the multiplicity of the second eigenvalue is at least $N$. In the case $N=2$, we can deduce that any third eigenfunction of the $p$-Laplacian on a disc is nonradial. The case of other symmetric domains and the limit cases $p=1$, $p=\\infty$ are also considered.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-11T08:33:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J92",
      "35P30",
      "35A15",
      "35A16",
      "55M25",
      "35B06"
    ],
    "keywords": [
      "multiplicity",
      "symmetric domains",
      "homogeneous dirichlet boundary conditions",
      "symmetry properties",
      "result implies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}