Level Set Method for Motion by Mean Curvature
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rnoti-p1148.pdf
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Author(s)
Colding, Tobias
Minicozzi, William
Date Issued
November 2016
Journal
Notices of the American Mathematical Society
Publisher
American Mathematical Society (AMS)
Citation
Colding, Tobias Holck and William P. Minicozzi. “Level Set Method for Motion by Mean Curvature.” Notices of the American Mathematical Society 63, 10 (November 2016): 1148–1153 © 2016 American Mathematical Society
Version
Final published version
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1090/NOTI1439
