{
  "id": "math/0610494",
  "version": "v1",
  "published": "2006-10-16T17:27:53.000Z",
  "updated": "2006-10-16T17:27:53.000Z",
  "title": "Maximal monotone operators are selfdual vector fields and vice-versa",
  "authors": [
    "Nassif Ghoussoub"
  ],
  "comment": "8 pages. Updated versions --if any-- of this author's papers can be downloaded at http://www.pims.math.ca/~nassif/",
  "categories": [
    "math.AP"
  ],
  "abstract": "If $L$ is a selfdual Lagrangian $L$ on a reflexive phase space $X\\times X^*$, then the vector field $x\\to \\bar\\partial L(x):=\\{p\\in X^*; (p,x)\\in \\partial L(x,p)\\}$ is maximal monotone. Conversely, any maximal monotone operator $T$ on $X$ is derived from such a potential on phase space, that is there exists a selfdual Lagrangian $L$ on $X\\times X^*$ (i.e, $L^*(p, x) =L(x, p)$) such that $T=\\bar\\partial L$. This solution to problems raised by Fitzpatrick can be seen as an extension of a celebrated result of Rockafellar stating that maximal cyclically monotone operators are actually of the form $T=\\partial \\phi$ for some convex lower semi-continuous function on $X$. This representation allows for the application of the selfdual variational theory --recently developed by the author-- to the equations driven by maximal monotone vector fields. Consequently, solutions to equations of the form $\\Lambda x\\in Tx$ for a given map $\\Lambda: D(\\Lambda)\\subset X\\to X^*$, can now be obtained by minimizing functionals of the form $I(x)=L(x,\\Lambda x)-< x, \\Lambda x>$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-10-16T17:27:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "maximal monotone operator",
      "selfdual vector fields",
      "selfdual lagrangian",
      "phase space",
      "vice-versa"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....10494G"
    }
  }
}