{
  "id": "1109.6765",
  "version": "v2",
  "published": "2011-09-30T09:16:16.000Z",
  "updated": "2011-10-11T12:42:59.000Z",
  "title": "On the gradient flow of a one-homogeneous functional",
  "authors": [
    "Ariela Briani",
    "Antonin Chambolle",
    "Matteo Novaga",
    "Giandomenico Orlandi"
  ],
  "journal": "Confluentes Mathematici (CM) 03, 04 (2012) 617-635",
  "doi": "10.1142/S1793744211000461",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the gradient flow of a one-homogeneous functional, whose dual involves the derivative of a constrained scalar function. We show in this case that the gradient flow is related to a weak, generalized formulation of the Hele-Shaw flow. The equivalence follows from a variational representation, which is a variant of well-known variational representations for the Hele-Shaw problem. As a consequence we get existence and uniqueness of a weak solution to the Hele-Shaw flow. We also obtain an explicit representation for the Total Variation flow in one dimension and easily deduce basic qualitative properties, concerning in particular the \"staircasing effect\".",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-10-11T12:42:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gradient flow",
      "one-homogeneous functional",
      "hele-shaw flow",
      "total variation flow",
      "well-known variational representations"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1109.6765B"
    }
  }
}