{
  "id": "1302.5546",
  "version": "v3",
  "published": "2013-02-22T10:50:35.000Z",
  "updated": "2013-10-28T18:22:29.000Z",
  "title": "Existence of critical points with semi-stiff boundary conditions for singular perturbation problems in simply connected planar domains",
  "authors": [
    "Xavier Lamy",
    "Petru Mironescu"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $\\Omega$ be a smooth bounded simply connected domain in $\\mathbb{R}^2$. We investigate the existence of critical points of the energy $E_\\varepsilon (u)=1/2\\int_\\Omega |\\nabla u|^2+1/(4\\varepsilon^2)\\int_\\Omega (1-|u|^2)^2$, where the complex map $u$ has modulus one and prescribed degree $d$ on the boundary. Under suitable nondegeneracy assumptions on $\\Omega$, we prove existence of critical points for small $\\varepsilon$. More can be said when the prescribed degree equals one. First, we obtain existence of critical points in domains close to a disc. Next, we prove that critical points exist in \"most\" of the domains.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-10-28T18:22:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J57",
      "35J61"
    ],
    "keywords": [
      "critical points",
      "simply connected planar domains",
      "semi-stiff boundary conditions",
      "singular perturbation problems",
      "bounded simply connected domain"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1302.5546L"
    }
  }
}