zbMATH — the first resource for mathematics

Two examples of nonconvex self-similar solution curves for a crystalline curvature flow. (English) Zbl 1077.53054
The main goal of the present paper consists in the construction of examples of nonconvex self-similar solutions for a crystalline curvature flow with an interfacial energy of which the Wulff shape is a regular triangle or a square.
Let \(\mathcal{P}\) a \(K\)-admissible curve. For each edge \(S_k\), a crystalline curvature \(H_k\) is defined in terms of the length of \(S_k\) and the length of the \(n\)-th edge of the Wulff shape. Under a crystalline curvature flow, each edge \(S_k\) keeps the same direction but moves in the outward normal direction with the velocity \(V_k\) determined by a homogeneous function of some degree \(\alpha>0\) in the crystalline curvature \(H_k\): \[ V_k=\sigma(\theta_k)| H_k| ^{\alpha-1}H_k, \] where \(\sigma\) defines the interfacial energy density and \(\theta_k\) is the exterior normal angle of the \(k\)-th edge \(S_k\).
Among other results, the authors show that, for \(\alpha\in(0,1)\), a nonconvex self-similar solution exists even in the case of free-orientation motions, i.e., \(\sigma(\theta+\pi)=\sigma(\theta)\). This result is in contrast with that obtained in the case of motion by smooth interfacial density, where the curve becomes convex in a finite time.

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
34A34 Nonlinear ordinary differential equations and systems, general theory
74N05 Crystals in solids
Full Text: DOI
[1] Andrews, B.: Evolving convex curves. Calc. Var. Partial Differential Equations, 7 , 315-371 (1998). · Zbl 0931.53030 · doi:10.1007/s005260050111
[2] Andrews, B.: Singularities in crystalline curvature flows. Asian J. Math., 6 , 101-122 (2002). · Zbl 1025.53038
[3] Angenent, S., and Gurtin, M. E.: Multiphase thermomechanics with interfacial structure. II. Evolution of an isothermal interface, Arch. Rational Mech. Anal., 108 , 323-391 (1989). · Zbl 0723.73017 · doi:10.1007/BF01041068
[4] Chou, K.-S., and Zhu, X.-P.: A convexity theorem for a class of anisotropic flows of plane curves. Indiana Univ. Math. J., 48 , 139-154 (1999). · Zbl 0979.53074 · doi:10.1512/iumj.1999.48.1273 · www.iumj.indiana.edu
[5] Giga, M.-H., and Giga, Y.: Crystalline and level set flow - convergence of a crystalline algorithm for a general anisotropic curvature flow in the plane. GAKUTO Internat. Ser. Math. Sci. Appli., 13 , 64-79 (2000). · Zbl 0957.35122
[6] Ishiwata, T., and Yazaki, S.: On the blow-up rate for fast blow-up solutions arising in an anisotropic crystalline motion. J. Comput. Appl. Math., 159 , 55-64 (2003). · Zbl 1033.65055 · doi:10.1016/S0377-0427(03)00556-9
[7] Ishiwata, T., and Yazaki, S.: A fast blow-up solution and degenerate pinching arising in an anisotropic crystalline motion. (Preprint). · Zbl 1282.34042 · doi:10.3934/dcds.2014.34.2069
[8] Stancu, A.: Uniqueness of self-similar solutions for a crystalline flow. Indiana Univ. Math. J., 45 , 1157-1174 (1996). · Zbl 0873.35034 · doi:10.1512/iumj.1996.45.1159
[9] Taylor, J. E.: Constructions and conjectures in crystalline nondifferential geometry. Pitman Monogr. Surveys Pure Appl. Math., 52, Longman Sci. Tech., Harlow, pp. 321-336 (1991). · Zbl 0725.53011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.