{
  "id": "2004.09243",
  "version": "v1",
  "published": "2020-04-20T12:49:02.000Z",
  "updated": "2020-04-20T12:49:02.000Z",
  "title": "Existence of parabolic minimizers to the total variation flow on metric measure spaces",
  "authors": [
    "Vito Buffa",
    "Michael Collins",
    "Cintia Pacchiano Camacho"
  ],
  "categories": [
    "math.AP",
    "math.FA",
    "math.MG"
  ],
  "abstract": "We give an existence proof for variational solutions $u$ associated to the total variation flow. Here, the functions being considered are defined on a metric measure space $(\\mathcal{X}, d, \\mu)$ satisfying a doubling condition and supporting a Poincar\\'e inequality. For such parabolic minimizers that coincide with a time-independent Cauchy-Dirichlet datum $u_0$ on the parabolic boundary of a space-time-cylinder $\\Omega \\times (0, T)$ with $\\Omega \\subset \\mathcal{X}$ an open set and $T > 0$, we prove existence in the weak parabolic function space $L^1_w(0, T; \\mathrm{BV}(\\Omega))$. In this paper, we generalize results from a previous work by B\\\"ogelein, Duzaar and Marcellini by introducing a more abstract notion for $\\mathrm{BV}$-valued parabolic function spaces. We argue completely on a variational level.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-20T12:49:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49J27",
      "49J40",
      "49J45",
      "30L99",
      "35A15"
    ],
    "keywords": [
      "total variation flow",
      "metric measure space",
      "parabolic minimizers",
      "weak parabolic function space",
      "time-independent cauchy-dirichlet datum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}