Motion by mean curvature of curves on surfaces using the Allen–Cahn equation

2015 ◽  
Vol 97 ◽  
pp. 126-132 ◽  
Author(s):  
Yongho Choi ◽  
Darae Jeong ◽  
Seunggyu Lee ◽  
Minhyun Yoo ◽  
Junseok Kim
Keyword(s):  
Mean Curvature ◽  
Cahn Equation ◽  
Motion By Mean Curvature ◽  
Curves On Surfaces

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

2015 ◽  
Vol 17 (05) ◽  
pp. 1450041
Author(s):  
Adriano Pisante ◽  
Fabio Punzo
Keyword(s):  
Riemannian Manifolds ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Initial Conditions ◽  
Curvature Flow ◽  
Monotonicity Formula ◽  
Uniform Estimates ◽  
Cahn Equation ◽  
Motion By Mean Curvature ◽  
Density Of Solutions

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.

Mean curvature flow by the Allen–Cahn equation

2015 ◽  
Vol 26 (4) ◽  
pp. 535-559 ◽  
Author(s):  
D. S. LEE ◽  
J. S. KIM
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Curve Shortening Flow ◽  
Cahn Equation ◽  
Angle Condition ◽  
Motion By Mean Curvature ◽  
Curve Shortening ◽  
Three Space ◽  
Stable Hybrid

In this paper, we investigate motion by mean curvature using the Allen–Cahn (AC) equation in two and three space dimensions. We use an unconditionally stable hybrid numerical scheme to solve the equation. Numerical experiments demonstrate that we can use the AC equation for applications to motion by mean curvature. We also study the curve-shortening flow with a prescribed contact angle condition.

scholarly journals Convergence rate of the Allen-Cahn equation to generalized motion by mean curvature

2011 ◽  
Vol 12 (2) ◽  
pp. 267-294 ◽  
Author(s):  
Matthieu Alfaro ◽  
Jérôme Droniou ◽  
Hiroshi Matano
Keyword(s):  
Convergence Rate ◽  
Mean Curvature ◽  
Cahn Equation ◽  
Motion By Mean Curvature
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