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).

INTERFACE ESTIMATES FOR THE FULLY ANISOTROPIC ALLEN-CAHN EQUATION AND ANISOTROPIC MEAN-CURVATURE FLOW

1996 ◽  
Vol 06 (08) ◽  
pp. 1103-1118 ◽  
Author(s):  
CHARLES M. ELLIOTT ◽  
MAURIZIO PAOLINI ◽  
REINER SCHÄTZLE
Keyword(s):  
Convex Function ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
Kinetic Term ◽  
Normal Velocity ◽  
Cahn Equation ◽  
The Second Fundamental Form ◽  
Anisotropic Mean Curvature

In this paper, we prove that solutions of the anisotropic Allen-Cahn equation in double-obstacle form with kinetic term [Formula: see text] in {|φ|<1}, where A is a convex function, homogeneous of degree two, and β depends only on the direction of ∇φ, converge to an anisotropic mean-curvature flow [Formula: see text] Here VN and R denote respectively the normal velocity and the second fundamental form of the interface, and [Formula: see text]. We prove this in the case when the above flow admits a smooth solution, and we establish that the Hausdorff-distance between the zero-level set of φ and the interface of the flow is of order O(ε2).

scholarly journals Self-Expanders of the Mean Curvature Flow

2021 ◽  
Author(s):  
Knut Smoczyk
Keyword(s):  
Scalar Curvature ◽  
Fundamental Form ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Vector ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
The Second Fundamental Form ◽  
The Mean ◽  
Mean Convex

AbstractWe study self-expanding solutions $M^{m}\subset \mathbb {R}^{n}$ M m ⊂ ℝ n of the mean curvature flow. One of our main results is, that complete mean convex self-expanding hypersurfaces are products of self-expanding curves and flat subspaces, if and only if the function |A|2/|H|2 attains a local maximum, where A denotes the second fundamental form and H the mean curvature vector of M. If the principal normal ξ = H/|H| is parallel in the normal bundle, then a similar result holds in higher codimension for the function |Aξ|2/|H|2, where Aξ is the second fundamental form with respect to ξ. As a corollary we obtain that complete mean convex self-expanders attain strictly positive scalar curvature, if they are smoothly asymptotic to cones of non-negative scalar curvature. In particular, in dimension 2 any mean convex self-expander that is asymptotic to a cone must be strictly convex.

ANCIENT SOLUTIONS OF CODIMENSION TWO SURFACES WITH CURVATURE PINCHING

2020 ◽  
Vol 102 (1) ◽  
pp. 162-171
Author(s):  
ZHENGCHAO JI
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
Geometric Flows ◽  
Codimension Two ◽  
The Second Fundamental Form ◽  
Ancient Solutions ◽  
The Mean ◽  
Curvature Pinching

We prove rigidity theorems for ancient solutions of geometric flows of immersed submanifolds. Specifically, we find conditions on the second fundamental form that characterise the shrinking sphere among compact ancient solutions for the mean curvature flow in codimension two surfaces.

scholarly journals Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow

2020 ◽  
Vol 18 (1) ◽  
pp. 1518-1530
Author(s):  
Xuesen Qi ◽  
Ximin Liu
Keyword(s):  
Laplace Operator ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
Coefficient Function ◽  
First Eigenvalue ◽  
Initial Hypersurface ◽  
The Mean ◽  
The Laplace Operator

Abstract In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the p-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao’s work. Moreover, we give an example to specify applications of conclusions obtained above.

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 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.

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