NUMERICAL SIMULATIONS OF MEAN CURVATURE FLOW IN THE PRESENCE OF A NONCONVEX ANISOTROPY

1998 ◽  
Vol 08 (04) ◽  
pp. 573-601 ◽  
Author(s):  
FRANCESCA FIERRO ◽  
ROBERTA GOGLIONE ◽  
MAURIZIO PAOLINI
Keyword(s):  
Numerical Simulations ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Finsler Geometry ◽  
Vertical Motion ◽  
Reaction Diffusion ◽  
Curvature Flow ◽  
Reaction Diffusion Equation ◽  
Curvature Driven ◽  
Anisotropic Mean Curvature

In this paper we present and discuss the results of some numerical simulations in order to investigate the mean curvature flow problem in the presence of a nonconvex anisotropy. Mathematically, nonconvexity of the anisotropy leads to the ill-posedness of the evolution problem, which becomes forward–backward parabolic. Simulations presented here refer to two different settings: curvature driven vertical motion of graphs (nonparametric setting) and motion in the normal direction by anisotropic mean curvature of surfaces (parametric setting). In the latter we first relax the problem via an Allen–Cahn type reaction-diffusion equation, in the context of Finsler geometry (diffused interface approximation). Our results suggest three main points. A nonconvex anisotropy and its convexification give rise, for both settings and the discretizations considered, to different evolutions. Wrinkled regions seem to appear only in correspondence to locally concave parts of the anisotropy. Moreover, locally convex regions (interior to the convexification of the so-called Frank diagram) seem to play an important role.

NUMERICAL EVIDENCE OF FATTENING FOR THE MEAN CURVATURE FLOW

1996 ◽  
Vol 06 (06) ◽  
pp. 793-813 ◽  
Author(s):  
FRANCESCA FIERRO ◽  
MAURIZIO PAOLINI
Keyword(s):  
Numerical Simulations ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Normal Direction ◽  
Compact Surface ◽  
Principal Curvatures ◽  
Forcing Term ◽  
The Mean ◽  
Motion By Mean Curvature

In this paper we describe some numerical simulations in the context of mean curvature flow. We recover a few different approaches in modeling the evolution of an interface Σ which evolves according to the law: V=κ+g where V is the velocity in the inward normal direction, κ is the sum of the principal curvatures and g is a given forcing term. We will discuss about the phenomenon of fattening or nonuniqueness of the solution, recalling what is known about this subject. Finally we show some interesting numerical simulations that suggest evidence of fattening starting from different initial interfaces. Of particular interest is the result obtained for a torus in ℝ4 which would be a first example of a regular and compact surface showing evidence of fattening in the case of pure motion by mean curvature (no forcing term).

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 Motion by anisotropic mean curvature as sharp interface limit of an inhomogeneous and anisotropic Allen–Cahn equation

2010 ◽  
Vol 140 (4) ◽  
pp. 673-706 ◽  
Author(s):  
Matthieu Alfaro ◽  
Harald Garcke ◽  
Danielle Hilhorst ◽  
Hiroshi Matano ◽  
Reiner Schätzle
Keyword(s):  
Transition Layer ◽  
Time Scale ◽  
Mean Curvature ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Finsler Metric ◽  
Diffusion Term ◽  
Almost Everywhere ◽  
Anisotropic Reaction Diffusion ◽  
Anisotropic Mean Curvature

We consider the spatially inhomogeneous and anisotropic reaction–diffusion equation ut = m(x)−1 div[m(x)ap(x,∇u)] + ε−2f(u), involving a small parameter ε > 0 and a bistable nonlinear term whose stable equilibria are 0 and 1. We use a Finsler metric related to the anisotropic diffusion term and work in relative geometry. We prove a weak comparison principle and perform an analysis of both the generation and the motion of interfaces. More precisely, we show that, within the time-scale of order ε2|ln ε|, the unique weak solution uε develops a steep transition layer that separates the regions {uε ≈ 0} and {uε | 1}. Then, on a much slower time-scale, the layer starts to propagate. Consequently, as ε → 0, the solution uε converges almost everywhere (a.e.) to 0 in Ω−t and 1 in Ω+t , where Ω−t and Ω+t are sub-domains of Ω separated by an interface Гt, whose motion is driven by its anisotropic mean curvature. We also prove that the thickness of the transition layer is of order ε.

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