Allen-Cahn Approximation of Mean Curvature Flow in Riemannian manifolds II, Brakke's flows

2013/08/02 by Pisante, Adriano, Punzo, Fabio
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1308.0570

Abstract

We prove convergence of solutions to the parabolic Allen-Cahn equation to Brakke's motion by mean curvature in space forms, generalizing previous results from [15] in Euclidean space. We show that a sequence of measures, associated to energy density of solutions of the parabolic Allen-Cahn equation, converges in the limit to a family of rectifiable Radon measures, which evolves by mean curvature flow in the sense of Brakke. A key role is played by a local almost monotonicity formula (a weak counterpart of Huisken's monotonicity formula) proved in [22], to get various density bounds for the limiting measures.

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