scholarly journals Numerical methods for anisotropic mean curvature flow based on a discrete time variational formulation

2011 ◽  
Vol 9 (3) ◽  
pp. 637-662 ◽  
Author(s):  
Adam Oberman ◽  
Stanley Osher ◽  
Ryo Takei ◽  
Richard Tsai
Keyword(s):  
Numerical Methods ◽  
Discrete Time ◽  
Mean Curvature ◽  
Variational Formulation ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Anisotropic Mean Curvature

scholarly journals Anisotropic mean curvature flow of Lipschitz graphs and convergence to self-similar solutions

2021 ◽  
Author(s):  
Annalisa Cesaroni ◽  
Heiko Kröner ◽  
Matteo Novaga
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Time Behavior ◽  
Long Time ◽  
Self Similar ◽  
The Stability ◽  
Lipschitz Graphs ◽  
Similar Solutions ◽  
Anisotropic Mean Curvature

We consider the anisotropic mean curvature flow of entire Lipschitz graphs. We prove existence and uniqueness of expanding self-similar solutions which are asymptotic to a prescribed cone, and we characterize the long time behavior of solutions, after suitable rescaling, when the initial datum is a sublinear perturbation of a cone. In the case of regular anisotropies, we prove the stability of self-similar solutions asymptotic to strictly mean convex cones, with respect to perturbations vanishing at infinity. We also show the stability of hyperplanes, with a proof which is novel also for the isotropic mean curvature flow.

scholarly journals APPROXIMATION OF THE ANISOTROPIC MEAN CURVATURE FLOW

2007 ◽  
Vol 17 (06) ◽  
pp. 833-844 ◽  
Author(s):  
ANTONIN CHAMBOLLE ◽  
MATTEO NOVAGA
Keyword(s):  
Mean Curvature ◽  
Variational Approach ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Mean Curvature Motion ◽  
Curvature Motion ◽  
Anisotropic Mean Curvature ◽  
Anisotropic Case

In this paper, we provide simple proofs of consistency for two well-known algorithms for mean curvature motion, Almgren–Taylor–Wang's1variational approach, and Merriman–Bence–Osher's algorithm.29Our techniques, based on the same notion of strict sub- and superflows, also work in the (smooth) anisotropic case.

The limit of the anisotropic double-obstacle Allen–Cahn equation

1996 ◽  
Vol 126 (6) ◽  
pp. 1217-1234 ◽  
Author(s):  
Charles M. Elliott ◽  
Reiner Schätzle
Keyword(s):  
Convex Function ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
Normal Velocity ◽  
Cahn Equation ◽  
The Second Fundamental Form ◽  
Image Position ◽  
Anisotropic Mean Curvature

In this paper, we prove that solutions of the anisotropic Allen–Cahn equation in doubleobstacle formwhere A is a strictly convex function, homogeneous of degree two, converge to an anisotropic mean-curvature flowwhen this equation admits a smooth solution in ℝn. Here VN and R respectively denote the normal velocity and the second fundamental form of the interface, and More precisely, we show that the Hausdorff-distance between the zero-level set of φ and the interface of the above anisotropic mean-curvature flow is of order O(ε2).

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