---
_id: '10011'
abstract:
- lang: eng
  text: 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.
acknowledgement: This project has received funding from the European Research Council
  (ERC) under the European Union’s Horizon 2020 research and innovation programme
  (grant agreement No 948819), and from the Deutsche Forschungsgemeinschaft (DFG,
  German Research Foundation) under Germany’s Excellence Strategy – EXC-2047/1 – 390685813.
  The content of this paper was developed and parts of it were written during a visit
  of the first author to the Hausdorff Center of Mathematics (HCM), University of
  Bonn. The hospitality and the support of HCM are gratefully acknowledged.
article_number: '2109.04233'
article_processing_charge: No
arxiv: 1
author:
- first_name: Sebastian
  full_name: Hensel, Sebastian
  id: 4D23B7DA-F248-11E8-B48F-1D18A9856A87
  last_name: Hensel
  orcid: 0000-0001-7252-8072
- first_name: Tim
  full_name: Laux, Tim
  last_name: Laux
citation:
  ama: 'Hensel S, Laux T. A new varifold solution concept for mean curvature flow:
    Convergence of  the Allen-Cahn equation and weak-strong uniqueness. <i>arXiv</i>.
    doi:<a href="https://doi.org/10.48550/arXiv.2109.04233">10.48550/arXiv.2109.04233</a>'
  apa: 'Hensel, S., &#38; Laux, T. (n.d.). A new varifold solution concept for mean
    curvature flow: Convergence of  the Allen-Cahn equation and weak-strong uniqueness.
    <i>arXiv</i>. <a href="https://doi.org/10.48550/arXiv.2109.04233">https://doi.org/10.48550/arXiv.2109.04233</a>'
  chicago: 'Hensel, Sebastian, and Tim Laux. “A New Varifold Solution Concept for
    Mean Curvature Flow: Convergence of  the Allen-Cahn Equation and Weak-Strong Uniqueness.”
    <i>ArXiv</i>, n.d. <a href="https://doi.org/10.48550/arXiv.2109.04233">https://doi.org/10.48550/arXiv.2109.04233</a>.'
  ieee: 'S. Hensel and T. Laux, “A new varifold solution concept for mean curvature
    flow: Convergence of  the Allen-Cahn equation and weak-strong uniqueness,” <i>arXiv</i>.
    .'
  ista: 'Hensel S, Laux T. A new varifold solution concept for mean curvature flow:
    Convergence of  the Allen-Cahn equation and weak-strong uniqueness. arXiv, 2109.04233.'
  mla: 'Hensel, Sebastian, and Tim Laux. “A New Varifold Solution Concept for Mean
    Curvature Flow: Convergence of  the Allen-Cahn Equation and Weak-Strong Uniqueness.”
    <i>ArXiv</i>, 2109.04233, doi:<a href="https://doi.org/10.48550/arXiv.2109.04233">10.48550/arXiv.2109.04233</a>.'
  short: S. Hensel, T. Laux, ArXiv (n.d.).
date_created: 2021-09-13T12:17:10Z
date_published: 2021-09-09T00:00:00Z
date_updated: 2023-05-03T10:34:38Z
day: '09'
department:
- _id: JuFi
doi: 10.48550/arXiv.2109.04233
ec_funded: 1
external_id:
  arxiv:
  - '2109.04233'
keyword:
- Mean curvature flow
- gradient flows
- varifolds
- weak solutions
- weak-strong uniqueness
- calibrated geometry
- gradient-flow calibrations
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2109.04233
month: '09'
oa: 1
oa_version: Preprint
project:
- _id: 0aa76401-070f-11eb-9043-b5bb049fa26d
  call_identifier: H2020
  grant_number: '948819'
  name: Bridging Scales in Random Materials
publication: arXiv
publication_status: submitted
status: public
title: 'A new varifold solution concept for mean curvature flow: Convergence of  the
  Allen-Cahn equation and weak-strong uniqueness'
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2021'
...