Calculus of Variations and Geometric Measure Theory

H. T. Nguyen - S. Wang

Second order estimates for transition layers and a curvature estimate for the parabolic Allen-Cahn

created by wang on 13 Nov 2020



Inserted: 13 nov 2020

Year: 2020

ArXiv: 2003.11886 PDF


The parabolic Allen-Cahn equation is a semilinear partial differential equation linked to the mean curvature flow by a singular perturbation. We show an improved convergence property of the parabolic Allen-Cahn equation to the mean curvature flow, which is the parabolic analogue of the improved convergence property of the elliptic Allen-Cahn to minimal surfaces by Wang-Wei and Chodosh-Mantoulidis. More precisely, we show if the phase-transition level sets are converging in $C^2$, then they converge in $C^{2,\theta}$. As an application, we obtain a curvature estimate for parabolic Allen-Cahn equation, which can be viewed as a diffused version of Brakke's and White's regularity theorem for mean curvature flow