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

Abstract

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

Modifications made according to referee's comments. Details added for some estimates. Statement and proof of Corollary 1.5 reformulated

References in corpus (2)

• Multiplicity one and strictly stable Allen-Cahn minimal hypersurfaces
• Brakke Regularity for the Allen-Cahn Flow

Cited by in corpus (1)

• Brakke Regularity for the Allen-Cahn Flow