scholarly journals Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term

2020 ◽  
Vol 1 (6) ◽  
Author(s):  
Yoshikazu Giga ◽  
Norbert Požár
Keyword(s):  
Level Set ◽  
Mean Curvature ◽  
Driving Force ◽  
Mean Curvature Flow ◽  
Approximate Solutions ◽  
Curvature Flow ◽  
Flow Equation ◽  
Force Term ◽  
Existence Of A Solution ◽  
Unique Existence

Abstract A general purely crystalline mean curvature flow equation with a nonuniform driving force term is considered. The unique existence of a level set flow is established when the driving force term is continuous and spatially Lipschitz uniformly in time. By introducing a suitable notion of a solution a comparison principle of continuous solutions is established for equations including the level set equations. An existence of a solution is obtained by stability and approximation by smoother problems. A necessary equi-continuity of approximate solutions is established. It should be noted that the value of crystalline curvature may depend not only on the geometry of evolving surfaces but also on the driving force if it is spatially inhomogeneous.

scholarly journals Convergence of the Numerical Scheme for Regularised Riemannian Mean Curvature Flow Equation

2018 ◽  
Vol 72 (1) ◽  
pp. 123-140
Author(s):  
Matúš Tibenský ◽  
Angela Handlovičová
Keyword(s):  
Image Segmentation ◽  
Finite Volume ◽  
Level Set ◽  
Mean Curvature ◽  
Numerical Scheme ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Flow Equation ◽  
Finite Volume Scheme ◽  
Volume Scheme

Abstract The aim of the paper is to study problem of image segmentation and missing boundaries completion introduced in [Mikula, K.—Sarti, A.––Sgallarri, A.: Co-volume method for Riemannian mean curvature flow in subjective surfaces multiscale segmentation, Comput. Vis. Sci. 9 (2006), 23–31], [Mikula, K.—Sarti, A.—Sgallari, F.: Co-volume level set method in subjective surface based medical image segmentation, in: Handbook of Medical Image Analysis: Segmentation and Registration Models (J. Suri et al., eds.), Springer, New York, 583–626, 2005], [Mikula, K.—Ramarosy, N.: Semi-implicit finite volume scheme for solving nonlinear diffusion equations in image processing, Numer. Math. 89 (2001), 561–590] and [Tibenský, M.: VyužitieMetód Založených na Level Set Rovnici v Spracovaní Obrazu, Faculty of mathematics, physics and informatics, Comenius University, Bratislava, 2016]. We generalize approach presented in [Eymard, R.—Handlovičová, A.—Mikula, K.: Study of a finite volume scheme for regularised mean curvature flow level set equation, IMA J. Numer. Anal. 31 (2011), 813–846] and apply it in the field of image segmentation. The so called regularised Riemannian mean curvature flow equation is presented and the construction of the numerical scheme based on the finite volume method approach is explained. The principle of the level set, for the first time given in [Osher, S.—Sethian, J. A.: Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys. 79 (1988), 12–49] is used. Based on the ideas from [Eymard, R.—Handlovičová, A.– –Mikula, K.: Study of a finite volume scheme for regularised mean curvature flow level set equation, IMA J. Numer. Anal. 31 (2011), 813–846] we prove the stability estimates on the numerical solution and the uniqueness of the numerical solution. In the last section, there is a proof of the convergence of the numerical scheme to the weak solution of the regularised Riemannian mean curvature flow equation and the proof of the convergence of the approximation of the numerical gradient is mentioned as well.

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 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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