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Allen–Cahn approximation of mean curvature flow in Riemannian manifolds, II: Brakke's flows

Adriano Pisante

Dipartimento di Matematica "G. Castelnuovo", Università di Roma "La Sapienza", P.le A. Moro 5, I-00185 Rome, Italy

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and

Fabio Punzo

Dipartimento di Matematica "F. Enriques", Università degli Studi di Milano, via C. Saldini 50, 20133 Milan, Italy

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https://doi.org/10.1142/S0219199714500412Cited by:5 (Source: Crossref)

Abstract

We prove convergence of solutions to the parabolic Allen–Cahn equation to Brakke's motion by mean curvature in Riemannian manifolds with Ricci curvature bounded from below. Our results hold for a general class of initial conditions and extend previous results from [T. Ilmanen, Convergence of the Allen–Cahn equation to the Brakke's motion by mean curvature, J. Differential Geom. 31 (1993) 417–461] even 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 nonpositivity of the limiting energy discrepancy and a local almost monotonicity formula (a weak counterpart of Huisken's monotonicity formula) proved in [Allen–Cahn approximation of mean curvature flow in Riemannian manifolds, I, uniform estimates, to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci.; arXiv:1308.0569], to get various density bounds for the limiting measures.

Keywords:

AMSC: 53C44, 58J35, 35K57, 49Q20

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