A new version of Brakke's local regularity theorem

2016/01/25 by Ananda Lahiri, Lahiri, Ananda
Mathematics · #35B45 #49Q20 #53C44 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Numerical methods in inverse problems #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1601.06710

openalex publication_date 2016/01/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider an integral Brakke flow (μ_t), t∈ [0,T], inside some ball in Euclidean space. If μ₀ has small height, its measure does not deviate too much from that of a plane and if μ_T is non-empty, then Brakke's local regularity theorem yields that (μ_t) is actually smooth and graphical inside a smaller ball for times t∈ (C,T-C) for some constant C. Here we extend this result to times t∈ (C,T). The main idea is to prove that a Brakke flow that is initially locally graphical with small gradient will remain graphical for some time. Moreover we use the new local regularity theorem to generalise White's regularity theorem to Brakke flows.

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