Calculus of Variations and Geometric Measure Theory

S. Stuvard - Y. Tonegawa

End-time regularity theorem for Brakke flows

created by stuvard on 16 Dec 2022
modified on 22 Mar 2024


Published Paper

Inserted: 16 dec 2022
Last Updated: 22 mar 2024

Journal: Math. Annalen
Year: 2024
Doi: 10.1007/s00208-024-02826-8

ArXiv: 2212.07727 PDF
Links: Published version


For a general $k$-dimensional Brakke flow in $\mathbb{R}^n$ locally close to a $k$-dimensional plane in the sense of measure, it is proved that the flow is represented locally as a smooth graph over the plane with estimates on all the derivatives up to the end-time. Moreover, at any point in space-time where the Gaussian density is close to $1$, the flow can be extended smoothly as a mean curvature flow up to that time in a neighborhood: this extends White's local regularity theorem to general Brakke flows. The regularity result is in fact obtained for more general Brakke-like flows, driven by the mean curvature plus an additional forcing term in a dimensionally sharp integrability class or in a Hölder class.