Allen-Cahn Approximation of Mean Curvature Flow in Riemannian manifolds I, uniform estimates

2013/08/02 by Adriano Pisante, Pisante, Adriano, Fabio Punzo +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1308.0569

openalex publication_date 2013/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We are concerned with solutions to the parabolic Allen-Cahn equation in Riemannian manifolds. For a general class of initial condition we show non positivity of the limiting energy discrepancy. This in turn allows to prove almost monotonicity formula (a weak counterpart of Huisken's monotonicity formula) which gives a local uniform control of the energy densities at small scales. Such results will be used in [40] to extend previous important results from [31] in Euclidean space, showing convergence of solutions to the parabolic Allen-Cahn equations to Brakke's motion by mean curvature in space forms.

Related

• Allen-Cahn Approximation of Mean Curvature Flow in Riemannian manifolds II, Brakke's flows
• Brakke Regularity for the Allen-Cahn Flow
• Convergence of the Allen-Cahn equation with constraint to Brakke's mean curvature flow
• Second order estimates for transition layers and a curvature estimate for the parabolic Allen-Cahn
• Convergence rates of the Allen-Cahn equation to mean curvature flow: A\n short proof based on relative entropies