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.

scholarly journals Minimizing Movements for Mean Curvature Flow of Partitions

2018 ◽  
Vol 50 (4) ◽  
pp. 4117-4148 ◽  
Author(s):  
Giovanni Bellettini ◽  
Shokhrukh Yu. Kholmatov
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Minimizing Movements

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.

Translating Solitons of the Mean Curvature Flow Asymptotic to Hyperplanes in ℝn+1

2018 ◽  
Vol 2020 (24) ◽  
pp. 10114-10153 ◽  
Author(s):  
Eddygledson S Gama ◽  
Francisco Martín
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Normal Component ◽  
Vertical Cylinder ◽  
Curvature Flow ◽  
Grim Reaper ◽  
The Family ◽  
The Mean ◽  
Translating Solitons

Abstract A translating soliton is a hypersurface $M$ in ${\mathbb{R}}^{n+1}$ such that the family $M_t= M- t \,\mathbf e_{n+1}$ is a mean curvature flow, that is, such that normal component of the velocity at each point is equal to the mean curvature at that point $\mathbf{H}=\mathbf e_{n+1}^{\perp }.$ In this paper we obtain a characterization of hyperplanes that are parallel to the velocity and the family of tilted grim reaper cylinders as the only translating solitons in $\mathbb{R}^{n+1}$ that are $C^1$-asymptotic to two half-hyperplanes outside a non-vertical cylinder. This result was proven for translators in $\mathbb{R}^3$ by the 2nd author, Perez-Garcia, Savas-Halilaj, and Smoczyk under the additional hypotheses that the genus of the surface was locally bounded and the cylinder was perpendicular to the translating velocity.

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.

scholarly journals De Giorgi’s inequality for the thresholding scheme with arbitrary mobilities and surface tensions

2022 ◽  
Vol 61 (1) ◽  
Author(s):  
Tim Laux ◽  
Jona Lelmi
Keyword(s):  
Energy Dissipation ◽  
General Theory ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Gradient Flows ◽  
Surface Tensions ◽  
New Variant ◽  
Minimizing Movements ◽  
Energy Convergence

AbstractWe provide a new convergence proof of the celebrated Merriman–Bence–Osher scheme for multiphase mean curvature flow. Our proof applies to the new variant incorporating a general class of surface tensions and mobilities, including typical choices for modeling grain growth. The basis of the proof are the minimizing movements interpretation of Esedoḡlu and Otto and De Giorgi’s general theory of gradient flows. Under a typical energy convergence assumption we show that the limit satisfies a sharp energy-dissipation relation.

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