{
  "id": "2109.04233",
  "version": "v1",
  "published": "2021-09-09T12:52:06.000Z",
  "updated": "2021-09-09T12:52:06.000Z",
  "title": "A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness",
  "authors": [
    "Sebastian Hensel",
    "Tim Laux"
  ],
  "comment": "37 pages",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "We propose a new weak solution concept for (two-phase) mean curvature flow which enjoys both (unconditional) existence and (weak-strong) uniqueness properties. These solutions are evolving varifolds, just as in Brakke's formulation, but are coupled to the phase volumes by a simple transport equation. First, we show that, in the exact same setup as in Ilmanen's proof [J. Differential Geom. 38, 417-461, (1993)], any limit point of solutions to the Allen-Cahn equation is a varifold solution in our sense. Second, we prove that any calibrated flow in the sense of Fischer et al. [arXiv:2003.05478] - and hence any classical solution to mean curvature flow - is unique in the class of our new varifold solutions. This is in sharp contrast to the case of Brakke flows, which a priori may disappear at any given time and are therefore fatally non-unique. Finally, we propose an extension of the solution concept to the multi-phase case which is at least guaranteed to satisfy a weak-strong uniqueness principle.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-09T12:52:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53E10",
      "49Q20",
      "35K57",
      "35Q49",
      "28A75"
    ],
    "keywords": [
      "mean curvature flow",
      "varifold solution concept",
      "allen-cahn equation",
      "convergence",
      "weak solution concept"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}