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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.

MSC:
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
34A34 Nonlinear ordinary differential equations and systems, general theory
74N05 Crystals in solids
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[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
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