An Approximation Scheme for Motion by Mean Curvature with Right-Angle Boundary Condition

2001 ◽  
Vol 33 (2) ◽  
pp. 369-389 ◽  
Author(s):  
Hitoshi Ishii ◽  
Katsuyuki Ishii
Keyword(s):  
Boundary Condition ◽  
Mean Curvature ◽  
Approximation Scheme ◽  
Motion By Mean Curvature

Equilibrium solutions to generalized motion by mean curvature

1998 ◽  
Vol 8 (5) ◽  
pp. 845-858 ◽  
Author(s):  
Tom Ilmanen ◽  
Peter Sternberg ◽  
William P. Ziemer
Keyword(s):  
Mean Curvature ◽  
Equilibrium Solutions ◽  
Motion By Mean Curvature

Gradient Estimates for Spacelike Mean Curvature Flow with Boundary Conditions

2018 ◽  
Vol 62 (2) ◽  
pp. 459-469
Author(s):  
Ben Lambert
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Mean Curvature ◽  
Neumann Boundary Condition ◽  
Mean Curvature Flow ◽  
Neumann Boundary ◽  
Curvature Flow ◽  
Gradient Estimate ◽  
Gradient Estimates ◽  
The Mean

AbstractWe prove a gradient estimate for graphical spacelike mean curvature flow with a general Neumann boundary condition in dimension n = 2. This then implies that the mean curvature flow exists for all time and converges to a translating solution.

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.

Generalized motion by mean curvature with Neumann conditions and the Allen-Cahn model for phase transitions

1995 ◽  
Vol 5 (2) ◽  
pp. 255-279 ◽  
Author(s):  
Markos Katsoulakis ◽  
Georgios T. Kossioris ◽  
Fernando Reitich
Keyword(s):  
Phase Transitions ◽  
Mean Curvature ◽  
Motion By Mean Curvature
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