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Minimizing movements for forced anisotropic mean curvature flow of partitions with mobilities

Part of: Existence theories Global differential geometry

Published online by Cambridge University Press:  17 August 2020

Giovanni Bellettini ,
Antonin Chambolle  and
Shokhrukh Kholmatov
Giovanni Bellettini
Affiliation:
University of Siena, via Roma 56, 53100, Siena, Italy International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34151, Trieste, Italy ([email protected])
Antonin Chambolle
Affiliation:
CMAP, Ecole Polytechnique, CNRS, 91128, Palaiseau Cedex, France ([email protected])
Shokhrukh Kholmatov
Affiliation:
University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria ([email protected])
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Abstract

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Under suitable assumptions on the family of anisotropies, we prove the existence of a weak global 1/(n+1)-Hölder continuous in time mean curvature flow with mobilities of a bounded anisotropic partition in any dimension using the method of minimizing movements. The result is extended to the case when suitable driving forces are present. We improve the Hölder exponent to 1/2 in the case of partitions with the same anisotropy and the same mobility and provide a weak comparison result in this setting for a weak anisotropic mean curvature flow of a partition and an anisotropic mean curvature two-phase flow.

Keywords

mean curvature flowpartitionsminimizing movementsforcinganisotropymobility

MSC classification

Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
Secondary: 49J45: Methods involving semicontinuity and convergence; relaxation 49J53: Set-valued and variational analysis
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 4 , August 2021 , pp. 1135 - 1170
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © The Author(s) 2020. Published by Cambridge University Press

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