Second-Order Estimates for Transition Layers and a Curvature Estimate for the Parabolic Allen–Cahn

Abstract

Abstract The parabolic Allen–Cahn equation is a semilinear partial differential equation that is closely linked to the mean curvature flow by a singular perturbation. Motivated by the work of Wang–Wei [ 21] and Chodosh–Mantoulidis [ 3] in the elliptic setting, we initiate the corresponding regularity theory for parabolic Allen–Cahn flows. In particular, we establish an improved convergence property of parabolic Allen–Cahn flows to the mean curvature flow: if the phase-transition level sets converge in $C^{2}$, then they converge in $C^{2,\theta }$ as well. As an application, we obtain a curvature estimate for the parabolic Allen–Cahn equation, which can be seen as a diffused version of Brakke’s [ 1] and White’s [ 24] regularity theorems for mean curvature flow.