Calculus of Variations and Geometric Measure Theory

H. T. Nguyen - S. Wang

Brakke Regularity for the Allen-Cahn Flow

created by wang on 13 Nov 2020



Inserted: 13 nov 2020

Year: 2020

ArXiv: 2010.12378 PDF


In this paper we prove an analogue of the Brakke's $\varepsilon$-regularity theorem for the parabolic Allen-Cahn equation. In particular, we show uniform $C^{2,\alpha}$ regularity for the transition layers converging to smooth mean curvature flows as $\varepsilon\rightarrow0$. The proof utilises Allen-Cahn versions of the monotonicity formula, parabolic Lipschitz approximation and blowups. A corresponding gap theorem for entire eternal solutions of the parabolic Allen-Cahn is also obtained. As an application of the regularity theorem, we give an affirmative answer to a question of Ilmanen that there is no cancellation in $\mathbf {BV}$ convergence in the mean convex setting.