{
  "id": "1505.07403",
  "version": "v1",
  "published": "2015-05-27T17:18:27.000Z",
  "updated": "2015-05-27T17:18:27.000Z",
  "title": "The first nontrivial eigenvalue for a system of $p-$Laplacians with Neumann and Dirichlet boundary conditions",
  "authors": [
    "Leandro M. Del Pezzo",
    "Julio D. Rossi"
  ],
  "comment": "21 pages, 1 figure",
  "categories": [
    "math.AP"
  ],
  "abstract": "We deal with the first eigenvalue for a system of two $p-$Laplacians with Dirichlet and Neumann boundary conditions. If $\\Delta_{p}w=\\mbox{div}(|\\nabla w|^{p-2}w)$ stands for the $p-$Laplacian and $\\frac{\\alpha}{p}+\\frac{\\beta}{q}=1,$ we consider $$ \\begin{cases} -\\Delta_pu= \\lambda \\alpha |u|^{\\alpha-2} u|v|^{\\beta} &\\text{ in }\\Omega,\\\\ -\\Delta_q v= \\lambda \\beta |u|^{\\alpha}|v|^{\\beta-2}v &\\text{ in }\\Omega,\\\\ \\end{cases} $$ with mixed boundary conditions $$ u=0, \\qquad |\\nabla v|^{q-2}\\dfrac{\\partial v}{\\partial \\nu }=0, \\qquad \\text{on }\\partial \\Omega. $$ We show that there is a first non trivial eigenvalue that can be characterized by the variational minimization problem $$ \\lambda_{p,q}^{\\alpha,\\beta} = \\min \\left\\{\\dfrac{\\displaystyle\\int_{\\Omega}\\dfrac{|\\nabla u|^p}{p}\\, dx +\\int_{\\Omega}\\dfrac{|\\nabla v|^q}{q}\\, dx} {\\displaystyle\\int_{\\Omega} |u|^\\alpha|v|^{\\beta}\\, dx} \\colon (u,v)\\in \\mathcal{A}_{p,q}^{\\alpha,\\beta}\\right\\}, $$ where $$ \\mathcal{A}_{p,q}^{\\alpha,\\beta}=\\left\\{(u,v)\\in W^{1,p}_0(\\Omega)\\times W^{1,q}(\\Omega)\\colon uv\\not\\equiv0\\text{ and }\\int_{\\Omega}|u|^{\\alpha}|v|^{\\beta-2}v \\, dx=0\\right\\}. $$ We also study the limit of $\\lambda_{p,q}^{\\alpha,\\beta} $ as $p,q\\to \\infty$ assuming that $\\frac{\\alpha}{p} \\to \\Gamma \\in (0,1)$, and $ \\frac{q}{p} \\to Q \\in (0,\\infty)$ as $p,q\\to \\infty.$ We find that this limit problem interpolates between the pure Dirichlet and Neumann cases for a single equation when we take $Q=1$ and the limits $\\Gamma \\to 1$ and $\\Gamma \\to 0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-27T17:18:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "first nontrivial eigenvalue",
      "dirichlet boundary conditions",
      "first non trivial eigenvalue",
      "variational minimization problem",
      "limit problem interpolates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150507403D"
    }
  }
}