Abstract
AbstractIn this paper, we construct a family of integral varifolds, which is a global weak solution to the volume preserving mean curvature flow in the sense of$$L^2$$L2-flow. This flow is also a distributional BV-solution for a short time, when the perimeter of the initial data is sufficiently close to that of a ball with the same volume. To construct the flow, we use the Allen–Cahn equation with a non-local term motivated by studies of Mugnai, Seis, and Spadaro, and Kim and Kwon. We prove the convergence of the solution for the Allen–Cahn equation to the family of integral varifolds with only natural assumptions for the initial data.
Funder
Japan Society for the Promotion of Science
Publisher
Springer Science and Business Media LLC
Subject
Mechanical Engineering,Mathematics (miscellaneous),Analysis
Reference48 articles.
1. Almgren, F., Taylor, J.E., Wang, L.: Curvature-driven flows: a variational approach. SIAM J. Control. Optim. 31, 387–438, 1993
2. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. The Clarendon Press, Oxford University Press, New York (2000)
3. Antonopoulou, D., Karali, G., Sigal, I.M.: Stability of spheres under volume-preserving mean curvature flow. Dyn. Partial Differ. Equ. 7, 327–344, 2010
4. Athanassenas, M.: Volume-preserving mean curvature flow of rotationally symmetric surfaces. Comment. Math. Helv. 72, 52–66, 1997
5. Brakke, K.A.: The Motion of a Surface by Its Mean Curvature. Princeton University Press, Princeton (1978)
Cited by
2 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献