On the motion of polygonal curves with asymptotic lines by crystalline curvature flow with bulk effect

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August 2011, 4(4): 865-873. doi: 10.3934/dcdss.2011.4.865

On the motion of polygonal curves with asymptotic lines by crystalline curvature flow with bulk effect

Tetsuya Ishiwata ^1,

1.	Shibaura Institute of Technology, Fukasaku 309, Minuma-ku, Saitama, 337-8570, Japan

Received September 2009 Revised November 2009 Published November 2010

Abstract
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The behavior of polygonal curves with asymptotic lines to crystalline motion with the bulk effect is discussed. We show sufficient conditions for global existence of the solutions and characterize facet-extinction patterns. We also show the eventual monotonicity of shape of the solution curves, that is, the solutions become V-shaped in finite time.

Keywords: Eventual monotonicity., Crystalline curvature, Crystalline motion.

Mathematics Subject Classification: Primary: 34A34, 39A12, 74N05; Secondary: 53A04, 34A2.

Citation: Tetsuya Ishiwata. On the motion of polygonal curves with asymptotic lines by crystalline curvature flow with bulk effect. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 865-873. doi: 10.3934/dcdss.2011.4.865

References:

[1]	S. Angenent and M. E. Gurtin, Multiphase thermomechanics with interfacial structure, 2. Evolution of an isothermal interface, Arch. Rational Mech. Anal., 108 (1989), 323-391. doi: doi:10.1007/BF01041068.
[2]	Y. Giga and M. E. Gurtin, A comparison theorem for crystalline evolution in the plane, Quart. J. Appl. Math., LIV (1996), 727-737.
[3]	M. E. Gurtin, "Thermomechanics of Evolving Phase Boundaries in the Plane," Oxford, Clarendon Press, 1993.
[4]	T. Ishiwata, Motion of non-convex polygons by crystalline curvature and almost convexity phenomena, Japan Journal of Industrial and Applied Mathematics, 25 (2008), 233-253. doi: doi:10.1007/BF03167521.
[5]	Y. Marutani, H. Ninomiya and R. Weidenfeld, Traveling curved fronts of anisotropic curvature flows, Japan Journal of Industrial and Applied Mathematics, 23 (2006), 83-104. doi: doi:10.1007/BF03167500.
[6]	S. Yazaki, Point-extinction and geometric expansion of solutions to a crystalline motion, Hokkaido Math. J., 30 (2001), 327-357.

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References:

[1]	S. Angenent and M. E. Gurtin, Multiphase thermomechanics with interfacial structure, 2. Evolution of an isothermal interface, Arch. Rational Mech. Anal., 108 (1989), 323-391. doi: doi:10.1007/BF01041068.
[2]	Y. Giga and M. E. Gurtin, A comparison theorem for crystalline evolution in the plane, Quart. J. Appl. Math., LIV (1996), 727-737.
[3]	M. E. Gurtin, "Thermomechanics of Evolving Phase Boundaries in the Plane," Oxford, Clarendon Press, 1993.
[4]	T. Ishiwata, Motion of non-convex polygons by crystalline curvature and almost convexity phenomena, Japan Journal of Industrial and Applied Mathematics, 25 (2008), 233-253. doi: doi:10.1007/BF03167521.
[5]	Y. Marutani, H. Ninomiya and R. Weidenfeld, Traveling curved fronts of anisotropic curvature flows, Japan Journal of Industrial and Applied Mathematics, 23 (2006), 83-104. doi: doi:10.1007/BF03167500.
[6]	S. Yazaki, Point-extinction and geometric expansion of solutions to a crystalline motion, Hokkaido Math. J., 30 (2001), 327-357.

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