{
  "id": "math/0501071",
  "version": "v1",
  "published": "2005-01-05T19:29:07.000Z",
  "updated": "2005-01-05T19:29:07.000Z",
  "title": "The topology of critical sets of some ordinary differential operators",
  "authors": [
    "Nicolau C. Saldanha",
    "Carlos Tomei"
  ],
  "comment": "15 pages, 4 figures",
  "journal": "Progress in Nonlinear Differential Equations and Their Applications, Vol. 66, 491-504, 2005.",
  "categories": [
    "math.FA",
    "math.CA"
  ],
  "abstract": "We survey recent work of Burghelea, Malta and both authors on the topology of critical sets of nonlinear ordinary differential operators. For a generic nonlinearity $f$, the critical set of the first order nonlinear operator $F_1(u)(t) = u'(t) + f(u(t))$ acting on the Sobolev space $H^1_p$ of periodic functions is either empty or ambient diffeomorphic to a hyperplane. For the second order operator $F_2(u)(t) = -u''(t) + f(u(t))$ on $H^2_D$ (Dirichlet boundary conditions), the critical set is ambient diffeomorphic to a union of isolated parallel hyperplanes. For second order operators on $H^2_p$, the critical set is not a Hilbert manifold but is still contractible and admits a normal form. The third order case is topologically far more complicated.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-01-05T19:29:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34L30",
      "58B05",
      "34B15",
      "46T05"
    ],
    "keywords": [
      "critical set",
      "second order operator",
      "nonlinear ordinary differential operators",
      "first order nonlinear operator",
      "ambient diffeomorphic"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......1071S"
    }
  }
}