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

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...

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Bibliographic Details
Published in:SN partial differential equations and applications 2020-12, Vol.1 (6), Article 39
Main Authors: Giga, Yoshikazu, Požár, Norbert
Format: Article
Language:English
Subjects:
Analysis
Mathematical and Computational Biology
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Numerical Analysis
Original Paper
Partial Differential Equations
Theoretical
Viscosity solutions - Dedicated to Hitoshi Ishii on the award of the 1st Kodaira Kunihiko Prize
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Summary: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.
ISSN:2662-2963
2662-2971
DOI:10.1007/s42985-020-00040-0