scholarly journals Gradient-flow techniques for the analysis of numerical schemes for multi-phase mean-curvature flow

2018 ◽  
Vol 3 (1) ◽  
pp. 76-89
Author(s):  
Tim Laux
Keyword(s):  
Mean Curvature ◽  
General Framework ◽  
Mean Curvature Flow ◽  
Gradient Flow ◽  
Curvature Flow ◽  
Numerical Schemes ◽  
Two Phase ◽  
Guiding Principle ◽  
Convergence Results ◽  
Multi Phase

Abstract Several recent convergence results for numerical schemes for mean-curvature flow in particular in the multi-phase case with arbitrary surface tensions are discussed. The guiding principle of all these works is the gradient-flow structure of multi-phase mean-curvature flow which is explained in the general framework. For simplicity, the convergence results are presented in the simpler two-phase case.

scholarly journals Two-phase flow problem coupled with mean curvature flow

2012 ◽  
Vol 14 (2) ◽  
pp. 185-203 ◽  
Author(s):  
Chun Liu ◽  
Norifumi Sato ◽  
Yoshihiro Tonegawa
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Two Phase Flow ◽  
Flow Problem ◽  
Curvature Flow ◽  
Phase Flow ◽  
Two Phase

scholarly journals Minimizing movements for forced anisotropic mean curvature flow of partitions with mobilities

2020 ◽  
pp. 1-36
Author(s):  
Giovanni Bellettini ◽  
Antonin Chambolle ◽  
Shokhrukh Kholmatov
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Driving Forces ◽  
Curvature Flow ◽  
Comparison Result ◽  
Two Phase ◽  
Hölder Continuous ◽  
The Family ◽  
Minimizing Movements ◽  
Anisotropic Mean Curvature

Under suitable assumptions on the family of anisotropies, we prove the existence of a weak global 1/(n+1)-Hölder continuous in time mean curvature flow with mobilities of a bounded anisotropic partition in any dimension using the method of minimizing movements. The result is extended to the case when suitable driving forces are present. We improve the Hölder exponent to 1/2 in the case of partitions with the same anisotropy and the same mobility and provide a weak comparison result in this setting for a weak anisotropic mean curvature flow of a partition and an anisotropic mean curvature two-phase flow.

The gradient structure of the mean curvature flow

2015 ◽  
Vol 8 (3) ◽  
Author(s):  
Martijn M. Zaal
Keyword(s):  
Metric Space ◽  
Mean Curvature ◽  
Metric Spaces ◽  
Mean Curvature Flow ◽  
Gradient Flow ◽  
Curvature Flow ◽  
Gradient Structure ◽  
Length Space ◽  
Curve Length ◽  
The Mean

AbstractThe concept of curve of maximal slope, a generalized notion of gradient flow, is extended to the setting of length spaces, which includes all metric spaces. This extended definition is used to show that curves of maximal slope in a metric space do not depend on the full metric, but only on the concept of curve length generated by it. Subsequently, it is shown that a length space can be constructed to describe

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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