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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

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

[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

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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