{
  "id": "2105.11424",
  "version": "v1",
  "published": "2021-05-24T17:30:45.000Z",
  "updated": "2021-05-24T17:30:45.000Z",
  "title": "The Neumann and Dirichlet problems for the total variation flow in metric measure spaces",
  "authors": [
    "Wojciech Górny",
    "José M. Mazón"
  ],
  "comment": "This article continues the work in arXiv:2103.13373 and arXiv:2105.00432",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the Neumann and Dirichlet problems for the total variation flow in metric measure spaces. We prove existence and uniqueness of weak solutions and study their asymptotic behaviour. Furthermore, in the Neumann problem we provide a notion of solutions which is valid for $L^1$ initial data, as well as prove their existence and uniqueness. Our main tools are the first-order linear differential structure due to Gigli and a version of the Gauss-Green formula.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-24T17:30:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49J52",
      "58J35",
      "35K90",
      "35K92"
    ],
    "keywords": [
      "total variation flow",
      "metric measure spaces",
      "dirichlet problems",
      "first-order linear differential structure",
      "weak solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}