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

Adriano Pisante; Fabio Punzo

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arXiv · 1308.0570

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

Adriano Pisante·Fabio Punzo

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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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Adriano Pisante, Fabio Punzo. 2013-11-18. Allen-Cahn Approximation of Mean Curvature Flow in Riemannian manifolds II, Brakke's flows. https://arxiv.org/abs/1308.0570

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