On Level Set Formulations for Anisotropic Mean Curvature Flow and Surface Diffusion

2005 ◽  
pp. 227-237 ◽  
Author(s):  
Ulrich Clarenz ◽  
Frank Haußer ◽  
Martin Rumpf ◽  
Axel Voigt ◽  
Ulrich Weikard
Keyword(s):  
Surface Diffusion ◽  
Level Set ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Anisotropic Mean Curvature

Computation of geometric partial differential equations and mean curvature flow

Acta Numerica ◽  
2005 ◽  
Vol 14 ◽  
pp. 139-232 ◽  
Author(s):  
Klaus Deckelnick ◽  
Gerhard Dziuk ◽  
Charles M. Elliott
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Level Set ◽  
Phase Field ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Partial Differential ◽  
Geometric Partial Differential Equations ◽  
Geometric Pdes

This review concerns the computation of curvature-dependent interface motion governed by geometric partial differential equations. The canonical problem of mean curvature flow is that of finding a surface which evolves so that, at every point on the surface, the normal velocity is given by the mean curvature. In recent years the interest in geometric PDEs involving curvature has burgeoned. Examples of applications are, amongst others, the motion of grain boundaries in alloys, phase transitions and image processing. The methods of analysis, discretization and numerical analysis depend on how the surface is represented. The simplest approach is when the surface is a graph over a base domain. This is an example of a sharp interface approach which, in the general parametric approach, involves seeking a parametrization of the surface over a base surface, such as a sphere. On the other hand an interface can be represented implicitly as a level surface of a function, and this idea gives rise to the so-called level set method. Another implicit approach is the phase field method, which approximates the interface by a zero level set of a phase field satisfying a PDE depending on a new parameter. Each approach has its own advantages and disadvantages. In the article we describe the mathematical formulations of these approaches and their discretizations. Algorithms are set out for each approach, convergence results are given and are supported by computational results and numerous graphical figures. Besides mean curvature flow, the topics of anisotropy and the higher order geometric PDEs for Willmore flow and surface diffusion are covered.

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 On the horizontal mean curvature flow for axisymmetric surfaces in the Heisenberg group

2014 ◽  
Vol 16 (03) ◽  
pp. 1350027 ◽  
Author(s):  
Fausto Ferrari ◽  
Qing Liu ◽  
Juan J. Manfredi
Keyword(s):  
Heisenberg Group ◽  
Level Set Method ◽  
Level Set ◽  
Viscosity Solutions ◽  
Mean Curvature ◽  
Explicit Solution ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Existence And Stability ◽  
The Mean

We study the horizontal mean curvature flow in the Heisenberg group by using the level-set method. We prove the uniqueness, existence and stability of axisymmetric viscosity solutions of the level-set equation. An explicit solution is given for the motion starting from a subelliptic sphere. We also give several properties of the level-set method and the mean curvature flow in the Heisenberg group.

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