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.

Allen-Cahn approximation of mean curvature flow in riemannian manifolds II, Brakke’s flows / A. Pisante, F. Punzo. - In: COMMUNICATIONS IN CONTEMPORARY MATHEMATICS. - ISSN 0219-1997. - 17:5(2015), pp. 1450041.1-1450041.35. [10.1142/S0219199714500412]

Allen-Cahn approximation of mean curvature flow in riemannian manifolds II, Brakke’s flows

F. Punzo
Ultimo
2015

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.
Allen-Cahn equation; Riemannian manifold; Huisken's monotonicity formula; mean curvature flow; Brakke's inequality; varifolds
Settore MAT/05 - Analisi Matematica
4-ago-2014
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/254482
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