Level set method for motion by mean curvature

Document Type : Translation Paper

Authors

Vali-e-Asr University of Rafsanjan, Rafsanjan

Abstract

This paper is a translation of the the following paper into Persian:
[T. H. Colding and W. P. Minicozzi II, Level Set Method For Motion by Mean Curvature, Notices of the AMS, 63 no. 10 (2016) 1148–1153.]
 
 Modeling of a wide class of physical phenomena, such as crystal growth and flame propagation, leads to tracking fronts moving with curvature-dependent speed. When the speed is the curvature this leads to one of the classical degenerate nonlinear second-order differential equations on Euclidean space. One naturally wonders, “What is the regularity of solutions?” A priori solutions are only defined in a weak sense, but it turns out that they are always twice differentiable classical solutions. This result is optimal; their second derivative is continuous only in very rigid situations that have a simple geometric interpretation. The proof weaves together analysis and geometry. Without deeply understanding the underlying geometry, it is impossible to prove fine analytical properties.

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References

[1] T. H. Colding and W. P. Minicozzi II, Uniqueness of blowups and Łojasiewicz inequalities, Annals of Math., 182 (1) (2015) 221–285. MR 3374960
[2] ——, Differentiability of the arrival time, Comm. Pure Appl. Math., DOI: 10.1002/cpa.21635.
[3] ——, Regularity of the level set flow, submitted, https://arxiv.org/abs/1606.05185.
[4] T. H. Colding, W. P. Minicozzi II, and E. K. Pedersen, Mean curvature flow, Bull. Amer. Math. Soc., (N.S.) 52 (2015), no. 2, 297–333. MR 3312634

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History

  • Receive Date: 26 September 2018
  • Revise Date: 26 September 2019
  • Accept Date: 28 September 2019
  • Publish Date: 22 May 2019

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