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Epsilon-regularity for the Brakke flow with boundary

Carlo Gasparetto
Vol. 18 (2025), No. 4, 857–894
DOI: 10.2140/apde.2025.18.857
Abstract

We prove that, if a Brakke flow with boundary is close enough to a stationary half-plane with density 1, then it is C1,α . Our approach is based on viscosity techniques introduced by Savin in the context of elliptic equations. The same techniques can be used to give an alternative proof of Brakke’s (interior) regularity theorem.

Keywords
mean curvature flows, Brakke flows, Brakke's theorem, $\varepsilon$-regularity, varifolds, Allard's theorem, boundary regularity, viscosity, small perturbation solutions
Mathematical Subject Classification
Primary: 53E10
Secondary: 35B65, 35D40
Milestones
Received: 22 June 2022
Revised: 4 September 2023
Accepted: 3 January 2024
Published: 27 March 2025
Authors
Carlo Gasparetto
Dipartimento di Matematica
Università di Pisa
Pisa
Italy
© 2025 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY).

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