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

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 & 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. doi: doi:10.1007/BF01041068. Google Scholar
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show all references

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. doi: doi:10.1007/BF01041068. Google Scholar
[2]	Y. Giga and M. E. Gurtin, A comparison theorem for crystalline evolution in the plane,, Quart. J. Appl. Math., LIV (1996), 727. Google Scholar
[3]	M. E. Gurtin, "Thermomechanics of Evolving Phase Boundaries in the Plane,", Oxford, (1993). Google Scholar
[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. doi: doi:10.1007/BF03167521. Google Scholar
[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. doi: doi:10.1007/BF03167500. Google Scholar
[6]	S. Yazaki, Point-extinction and geometric expansion of solutions to a crystalline motion,, Hokkaido Math. J., 30 (2001), 327. Google Scholar

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