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Motion of nonadmissible convex polygons by crystalline curvature. (English) Zbl 1132.53036
An evolution of a closed plane convexe polygon \(P(t)\) governed by the crystalline motion equation is analyzed, see S. Angenent and M. E. Gurtin [Arch. Ration. Mech. Anal. 108, No. 4, 323–391 (1989; Zbl 0723.73017)], J. E. Taylor [Differential geometry. A symposium in honour of Manfredo do Carmo, Proc. Int. Conf., Rio de Janeiro/Bras. 1988, Pitman Monogr. Surv. Pure Appl. Math. 52, 321–336 (1991; Zbl 0725.53011)], S. B. Angenent and M. E. Gurtin [J. Reine Angew. Math. 446, 1–47 (1994; Zbl 0784.35124)], M. E. Gurtin [Thermomechanics of evolving phase boundaries in the plane. Oxford Mathematical Monographs. Oxford: Clarendon Press (1993; Zbl 0787.73004)], J. E. Taylor [Differential geometry. Part 1: Partial differential equations on manifolds. Proceedings of a summer research institute, held at the University of California, Los Angeles, CA, USA, July 8–28, 1990. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 54, Part 1, 417–438 (1993; Zbl 0823.49028)], F. Almgren and J. E. Taylor [J. Differ. Geom. 42, No. 1, 1–22 (1995; Zbl 0867.58020)]. It is known that if \(P(0)\) is an admissible polygon, i.e., the set of its normal angles equals that of the Wulff shape, than \(P(t)\) evolves, while its admissibility is preserved, and eventually it shrinks to a single point or collapses to a line segment in a finite time [cf. B. Andrews, Asian J. Math. 6, No. 1, 101–121 (2002; Zbl 1025.53038), M.-H. Giga and Y. Giga, Proceedings of the international conference on free boundary problems: theory and applications, Chiba, Japan, November 7–13, 1999. I. Tokyo: Gakkotosho. GAKUTO Int. Ser., Math. Sci. Appl. 13, 64–79 (2000; Zbl 0957.35122)], T. Ishiwata and S. Yazaki [J. Comput. Appl. Math. 159, No. 1, 55–64 (2003; Zbl 1033.65055)].
The author discusses the case of a nonadmissible initial data \(P(0)\) and proves that either \(P(t)\) shrinks to a point in a finite time, or some edges disappearing occurs at most finitely many times so that \(P(t)\) becomes admissible. As corollary, if \(P(0)\) is not admissible than \(P(t)\) does not collapse to a line segment without becoming an admissible polygon. The discussion is illustrated by five typical examples.

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53A04 Curves in Euclidean and related spaces
74N05 Crystals in solids
34A34 Nonlinear ordinary differential equations and systems, general theory
Full Text: DOI
[1] B. Andrews, Singularities in crystalline curvature flows, Asian J. Math. 6 (2002), no. 1, 101-121. · Zbl 1025.53038
[2] S. Angenent and M. E. Gurtin, Multiphase thermomechanics with interfacial structure. II. Evolution of an isothermal interface, Arch. Rational Mech. Anal. 108 (1989), no. 4, 323-391. · Zbl 0723.73017 · doi:10.1007/BF01041068
[3] M.-H. Giga and Y. Giga, Crystalline and level set flow-convergence of a crystalline al- gorithm for a general anisotropic curvature flow in the plane, in Free boundary problems: theory and applications, I (Chiba, 1999), 64-79, Gakk\? otosho, Tokyo, 2000. · Zbl 0957.35122
[4] M.-H. Giga, Y. Giga, and H. Hontani, Self-similar expanding solutions in a sector for a crystalline flow, SIAM J. Math. Anal. 37 (2005), no. 4, 1207-1226 (electronic). · Zbl 1093.74048 · doi:10.1137/040614372
[5] M. E. Gurtin, Thermomechanics of evolving phase boundaries in the plane, Oxford Univ. Press, New York, 1993. · Zbl 0787.73004
[6] C. Hirota, T. Ishiwata and S. Yazaki, Some results on singularities of solutions to an anisotropic crystalline curvature flow, in Mathematical approach to nonlinear phenom- ena: modelling, analysis and simulations, 119-128, Gakk\? otosho, Tokyo, 2005. · Zbl 1163.34340
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[8] , Numerical study and examples on singularities of solutions to anisotropic crys- talline curvature flows of nonconvex polygonal curves, Proceedings of MSJ-IRI 2005 “Asymptotic Analysis and Singularity”, (Sendai, 2005), Advanced Studies in Pure Math- ematics, to appear.
[9] H. Hontani, M.-H. Giga, Y. Giga and K. Deguchi, Expanding selfsimilar solutions of a crystalline flow with applications to contour figure analysis, Discrete Appl. Math. 147 (2005), no. 2-3, 265-285. · Zbl 1117.65036 · doi:10.1016/j.dam.2004.09.015
[10] K. Ishii and H. M. Soner, Regularity and convergence of crystalline motion, SIAM J. Math. Anal. 30 (1999), no. 1, 19-37 (electronic). · Zbl 0963.35082 · doi:10.1137/S0036141097317347
[11] T. Ishiwata, T. K. Ushijima, H. Yagisita and S. Yazaki, Two examples of nonconvex self-similar solution curves for a crystalline curvature flow, Proc. Japan Acad. Ser. A Math. Sci. 80 (2004), no. 8, 151-154. i i i i i i i 170 · Zbl 1077.53054 · doi:10.3792/pjaa.80.151
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