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Implicit time discretization for the mean curvature flow equation. (English) Zbl 0821.35003

Summary: We apply the method of implicit time discretization to the mean curvature flow equation including outer forces. In the framework of BV-functions we construct discrete solutions iteratively by minimizing a suitable energy- functional in each time step. Employing geometric and variational arguments we show an energy estimate which assures compactness of the discrete solutions. An additional convergence condition excludes a loss of area in the limit. Thus existence of solutions to the continuous problem can be derived. We append a brief discussion of the related Mullins-Sekerka equation.

MSC:

35A15 Variational methods applied to PDEs
49Q20 Variational problems in a geometric measure-theoretic setting
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References:

[1] Almgren, F., Taylor, J.E., Wang, L.: Curvature driven flows: a variational approach. SIAM Journ. Control and Optimization31, 387–437 (1993) · Zbl 0783.35002 · doi:10.1137/0331020
[2] Brakke, K.: The motion of a surface by its mean curvature. Princeton University Press 1978 · Zbl 0386.53047
[3] Chen, Y-G., Giga, Y., Goto, S.: Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differ. Geom.33, 749–786 (1991) · Zbl 0696.35087
[4] Evans, L.C., Spruck, J.: Motion of level sets by mean curvature I. J. Differ. Geom.33, 635–681 (1991) · Zbl 0726.53029
[5] Giusti, E.: Minimal surfaces and functions of bounded variation. Basel Boston Stuttgart: Birkhäuser Verlag 1984 · Zbl 0545.49018
[6] Luckhaus, S.: The Stefan problem with the Gibbs-Thomson law. Preprint No. 591 Universita di Pisa (1991)
[7] Simon, L.: Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis, Australian National University, Volume3, 1983 · Zbl 0546.49019
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