---
_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
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. arXiv.
doi:10.48550/arXiv.2109.04233'
apa: 'Hensel, S., & Laux, T. (n.d.). A new varifold solution concept for mean
curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness.
arXiv. https://doi.org/10.48550/arXiv.2109.04233'
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.”
ArXiv, n.d. https://doi.org/10.48550/arXiv.2109.04233.'
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,” arXiv.
.'
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.”
ArXiv, 2109.04233, doi:10.48550/arXiv.2109.04233.'
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'
...