A1 Journal article (refereed)
Consistency of the Flat Flow Solution to the Volume Preserving Mean Curvature Flow (2024)

Julin, V., & Niinikoski, J. (2024). Consistency of the Flat Flow Solution to the Volume Preserving Mean Curvature Flow. Archive for Rational Mechanics and Analysis, 248(1), Article 1. https://doi.org/10.1007/s00205-023-01944-y

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Publication details

All authors or editors: Julin, Vesa; Niinikoski, Joonas

Journal or series: Archive for Rational Mechanics and Analysis

ISSN: 0003-9527

eISSN: 1432-0673

Publication year: 2024

Publication date: 07/12/2023

Volume: 248

Issue number: 1

Article number: 1

Publisher: Springer

Publication country: Germany

Publication language: English

DOI: https://doi.org/10.1007/s00205-023-01944-y

Publication open access: Openly available

Publication channel open access: Partially open access channel

Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/92658

Web address of parallel published publication (pre-print): https://arxiv.org/abs/2206.05002

Abstract

We consider the flat flow solution, obtained via a discrete minimizing movement scheme, to the volume preserving mean curvature flow starting from C1,1-regular set. We prove the consistency principle, which states that (any) flat flow solution agrees with the classical solution as long as the latter exists. In particular the flat flow solution is unique and smooth up to the first singular time. We obtain the result by proving the full regularity for the discrete time approximation of the flat flow such that the regularity estimates are stable with respect to the time discretization. Our method can also be applied in the case of the mean curvature flow and thus it provides an alternative proof, not relying on comparison principle, for the consistency between the flat flow solution and the classical solution for C1,1-regular initial sets.

Keywords: partial differential equations

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Preliminary JUFO rating: 3