{
  "id": "2001.05985",
  "version": "v1",
  "published": "2020-01-16T18:35:23.000Z",
  "updated": "2020-01-16T18:35:23.000Z",
  "title": "A System of Local/Nonlocal $p$-Laplacians: The Eigenvalue Problem and Its Asymptotic Limit as $p\\to1$",
  "authors": [
    "S. Buccheri",
    "J. V. da Silva",
    "L. H. de Miranda"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this work, given $p\\in (1,\\infty)$, we prove the existence and simplicity of the first eigenvalue $\\lambda_p$ and its corresponding eigenvector $(u_p,v_p)$, for the following local/nonlocal PDE system \\begin{equation}\\label{Eq0} \\left\\{ \\begin{array}{rclcl} -\\Delta_p u + (-\\Delta)^r_p u & = & \\frac{2\\alpha}{\\alpha+\\beta}\\lambda |u|^{\\alpha-2}|v|^{\\beta}u & \\mbox{in} & \\Omega \\\\ -\\Delta_p v + (-\\Delta)^s_p v& = & \\frac{2\\beta}{\\alpha+\\beta}\\lambda |u|^{\\alpha}|v|^{\\beta-2}v & \\mbox{in} & \\Omega u& =& 0&\\text{ on } & \\mathbb{R}^N \\setminus \\Omega v& =& 0&\\text{ on } & \\mathbb{R}^N \\setminus \\Omega, \\end{array} \\right. \\end{equation} where $\\Omega$$\\subset$ $\\mathbb{R}^N$ is a bounded open domain, $0<r, s<1$ and $\\alpha(p)+\\beta(p) = p$. Moreover, we address the asymptotic limit as $p \\to \\infty$, proving the explicit geometric characterization of the corresponding first $\\infty-$eigenvalue, namely $\\lambda_{\\infty}$, and the uniformly convergence of the pair $(u_p,v_p)$ to the $\\infty-$eigenvector $(u_{\\infty},v_{\\infty})$. Finally, the triple $(u_{\\infty},v_{\\infty},\\lambda_{\\infty})$ verifies, in the viscosity sense, a limiting PDE system. \\end{abstract}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-16T18:35:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "eigenvalue problem",
      "asymptotic limit",
      "laplacians",
      "local/nonlocal pde system",
      "viscosity sense"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}