{
  "id": "0903.0984",
  "version": "v1",
  "published": "2009-03-05T13:23:16.000Z",
  "updated": "2009-03-05T13:23:16.000Z",
  "title": "$Γ$-convergence of some super quadratic functionals with singular weights",
  "authors": [
    "Giampiero Palatucci",
    "Yannick Sire"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the $\\Gamma$-convergence of the following functional ($p>2$) $$ F_{\\epsilon}(u):=\\epsilon^{p-2}\\int_{\\Omega}|Du|^p d(x,\\partial \\Omega)^{a}dx+\\frac{1}{\\epsilon^{\\frac{p-2}{p-1}}}\\int_{\\Omega}W(u) d(x,\\partial \\Omega)^{-\\frac{a}{p-1}}dx+\\frac{1}{\\sqrt{\\epsilon}}\\int_{\\partial\\Omega}V(Tu)d\\mathcal{H}^2, $$ where $\\Omega$ is an open bounded set of $\\mathbb{R}^3$ and $W$ and $V$ are two non-negative continuous functions vanishing at $\\alpha, \\beta$ and $\\alpha', \\beta'$, respectively. In the previous functional, we fix $a=2-p$ and $u$ is a scalar density function, $Tu$ denotes its trace on $\\partial\\Omega$, $d(x,\\partial \\Omega)$ stands for the distance function to the boundary $\\partial\\Om$. We show that the singular limit of the energies $F_{\\epsilon}$ leads to a coupled problem of bulk and surface phase transitions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-03-05T13:23:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B26",
      "49J45",
      "49Q20"
    ],
    "keywords": [
      "super quadratic functionals",
      "singular weights",
      "convergence",
      "scalar density function",
      "surface phase transitions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0903.0984P"
    }
  }
}