{
  "id": "1711.07256",
  "version": "v1",
  "published": "2017-11-20T11:13:38.000Z",
  "updated": "2017-11-20T11:13:38.000Z",
  "title": "Reverse approximation of gradient flows as Minimizing Movements: a conjecture by De Giorgi",
  "authors": [
    "Florentine Fleißner",
    "Giuseppe Savaré"
  ],
  "categories": [
    "math.CA",
    "math.AP"
  ],
  "abstract": "We consider the Cauchy problem for the gradient flow \\begin{equation} \\label{eq:81} \\tag{$\\star$} u'(t)=-\\nabla\\phi(u(t)),\\quad t\\ge 0;\\quad u(0)=u_0, \\end{equation} generated by a continuously differentiable function $\\phi:\\mathbb H \\to \\mathbb R$ in a Hilbert space $\\mathbb H$ and study the reverse approximation of solutions to ($\\star$) by the De Giorgi Minimizing Movement approach. We prove that if $\\mathbb H$ has finite dimension and $\\phi$ is quadratically bounded from below (in particular if $\\phi$ is Lipschitz) then for every solution $u$ to ($\\star$) (which may have an infinite number of solutions) there exist perturbations $\\phi_\\tau:\\mathbb H \\to \\mathbb R \\ (\\tau>0)$ converging to $\\phi$ in the Lipschitz norm such that $u$ can be approximated by the Minimizing Movement scheme generated by the recursive minimization of $\\Phi(\\tau,U,V):=\\frac 1{2\\tau}|V-U|^2+ \\phi_\\tau(V)$: \\begin{equation} \\label{eq:abstract} \\tag{$\\star\\star$} U_\\tau^n\\in \\operatorname{argmin}_{V\\in \\mathbb H} \\Phi(\\tau,U_\\tau^{n-1},V)\\quad n\\in\\mathbb N, \\quad U_\\tau^0:=u_0. \\end{equation} We show that the piecewise constant interpolations with time step $\\tau > 0$ of all possible selections of solutions $(U_\\tau^n)_{n\\in\\mathbb N}$ to ($\\star\\star$) will converge to $u$ as $\\tau\\downarrow 0$. This result solves a question raised by Ennio De Giorgi. We also show that even if $\\mathbb H$ has infinite dimension the above approximation holds for the distinguished class of minimal solutions to ($\\star$), that generate all the other solutions to ($\\star$) by time reparametrization.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-20T11:13:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49M25",
      "34G20",
      "47J25",
      "47J30"
    ],
    "keywords": [
      "gradient flow",
      "reverse approximation",
      "conjecture",
      "giorgi minimizing movement approach",
      "infinite dimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}