Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term
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ORIGINAL PAPER
Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term Yoshikazu Giga1 • Norbert Pozˇa´r2 Ó The Author(s) 2020
Abstract 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. Mathematics Subject Classification 35K67 35D40 35K55 35B51 35K93
1 Introduction In our previous works [21, 22], we constructed a unique global-in-time level set flow for the crystalline mean curvature flow of the form V ¼ gðm; jr Þ: Here V is the normal velocity of an evolving hypersurface in Rn , n 2, in the direction of a unit normal vector field m and jr is a (purely) crystalline mean curvature of the hypersurface. The anisotropy r is assumed to be crystalline, that is, r : Rn ! R is a positively one-homogeneous function such that fr\1g is a bounded convex polytope. The function
This article is part of the topical collection Viscosity solutions—Dedicated to Hitoshi Ishii on the award of the 1st Kodaira Kunihiko Prize, edited by Kazuhiro Ishige, Shigeaki Koike, Tohru Ozawa, Senjo Shimizu. & Yoshikazu Giga [email protected] Norbert Pozˇa´r [email protected] 1
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo 153-8914, Japan
2
Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University, Kakuma town, Kanazawa, Ishikawa 920-1192, Japan SN Partial Differential Equations and Applications
39 Page 2 of 26
SN Partial Differ. Equ. Appl. (2020)1:39
g 2 CðS n1 RÞ is a given function that is non-decreasing in the second variable so that the problem is at least formally degenerate parabolic; here, S n1 denotes the unit sphere in Rn . We are using the convention that V ¼ jr is the usual mean curvature flow when r is isotropic so that jr is the usual mean curvature. Our goal is to extend the result in [21] to the problem ð1:1Þ V ¼ g m; jr þ f ðx; tÞ ; where f ¼ f ðx; tÞ is a continuous function that is Lipschitz continuous in space variable x uniformly in time t. Namely, we consider a crystalline mean curvature flow with a nonuniform driving force term. We introduce a suitable notion of viscosity solutions to the level set equation for (1.1), which looks slightly weaker than those in [21, 22]. Our main result reads: Theorem 1.1 Assume that g 2 CðS n1 RÞ i
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