A1 Journal article (refereed)
Flat flow solution to the mean curvature flow with volume constraint (2024)

Julin, V. (2024). Flat flow solution to the mean curvature flow with volume constraint. Advances in Calculus of Variations, Early online. https://doi.org/10.1515/acv-2023-0047

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Publication details

All authors or editors: Julin, Vesa

Journal or series: Advances in Calculus of Variations

ISSN: 1864-8258

eISSN: 1864-8266

Publication year: 2024

Publication date: 24/04/2024

Volume: Early online

Publisher: Walter de Gruyter GmbH

Publication country: Germany

Publication language: English

DOI: https://doi.org/10.1515/acv-2023-0047

Publication open access: Openly available

Publication channel open access: Partially open access channel

Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/94472

Web address of parallel published publication (pre-print): https://arxiv.org/abs/2301.10089

Abstract

In this paper I will revisit the construction of a global weak solution to the volume preserving mean curvature flow via discrete minimizing movement scheme by Mugnai, Seis and Spadaro [L. Mugnai, C. Seis and E. Spadaro, Global solutions to the volume-preserving mean-curvature flow, Calc. Var. Partial Differential Equations 55 2016, 1, Article ID 18]. This method is based on the gradient flow approach due to Almgren, Taylor and Wang [F. Almgren, J. E. Taylor and L. Wang, Curvature-driven flows: a variational approach, SIAM J. Control Optim. 31 1993, 2, 387–438] and Luckhaus and Sturzenhecker [S. Luckhaus and T. Sturzenhecker, Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differential Equations 3 1995, 2, 253–271] and my aim is to replace the volume penalization with the volume constraint directly in the discrete scheme, which from practical point of view is perhaps more natural. A technical novelty is the proof of the density estimate which is based on second variation argument

Keywords: partial differential equations; differential geometry

Free keywords:  mean-curvature flow; volume constraint; gradient flow; time discretization

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