scholarly journals Nonparametric Mean Curvature Flow with Nearly Vertical Contact Angle Condition

2021 ◽  
Vol 54 (1) ◽  
pp. 28-55
Author(s):  
Zhenghuan Gao
Keyword(s):  
Contact Angle ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Angle Condition ◽  
Vertical Contact

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.

Translating spacelike graphs by mean curvature flow with prescribed contact angle

2014 ◽  
Vol 103 (6) ◽  
pp. 499-508 ◽  
Author(s):  
Shanze Gao ◽  
Guanghan Li ◽  
Chuanxi Wu
Keyword(s):  
Contact Angle ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow

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 Mean curvature flow with free boundary outside a hypersphere

2017 ◽  
Vol 369 (12) ◽  
pp. 8319-8342 ◽  
Author(s):  
Glen Wheeler ◽  
Valentina-Mira Wheeler
Keyword(s):  
Free Boundary ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow

scholarly journals Ancient solutions for Andrews’ hypersurface flow

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Peng Lu ◽  
Jiuru Zhou
Keyword(s):  
Ricci Flow ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Euclidean Spaces ◽  
Round Cylinder ◽  
Ancient Solutions ◽  
Singular Time

AbstractWe construct the ancient solutions of the hypersurface flows in Euclidean spaces studied by B. Andrews in 1994.As time {t\rightarrow 0^{-}} the solutions collapse to a round point where 0 is the singular time. But as {t\rightarrow-\infty} the solutions become more and more oval. Near the center the appropriately-rescaled pointed Cheeger–Gromov limits are round cylinder solutions {S^{J}\times\mathbb{R}^{n-J}}, {1\leq J\leq n-1}. These results are the analog of the corresponding results in Ricci flow ({J=n-1}) and mean curvature flow.

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