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On spiral solutions to generalized crystalline motion with a rotating tip motion
1. | Shibaura Institute of Technology, Fukasaku 309, Minuma-ku, Saitama, 337-8570 |
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: 10.1007/BF01041068. |
[2] |
W. K. Burton, N. Cabrera and F. C. Frank, The Growth of Crystals and the Equilibrium Structure of their Surfaces, Philos. Trans. Roy. Soc. London. Ser. A., 243 (1951), 299-358.
doi: 10.1098/rsta.1951.0006. |
[3] |
B. Fiedler, J.-S. Guo and J.-C. Tsai, Multiplicity of rotating spirals under curvature flows with normal tip motion, J. Differential Equations, 205 (2004), 211-228.
doi: 10.1016/j.jde.2004.02.012. |
[4] |
M.-H. Giga and Y. Giga, Crystalline and level set flow - convergence of a crystalline algorithm for a general anisotropic curvature flow in the plane, Free boundary problems: Theory and applications, I (Chiba, 1999), GAKUTO Internat. Ser. Math. Sci. Appli., Gakkōtosho, Tokyo, 13 (2000), 64-79. |
[5] |
Y. Giga and M. E. Gurtin, A comparison theorem for crystalline evolution in the plane, Quart. J. Appl. Math., 54 (1996), 727-737. |
[6] |
J.-S. Guo, K.-I. Nakamura, T. Ogiwara and J.-C. Tsai, On the Steadily Rotating Spirals, Japan J. Indust. Appl. Math., 23 (2006), 1-19.
doi: 10.1007/BF03167495. |
[7] |
M. E. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane, Oxford, Clarendon Press, 1993. |
[8] |
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: 10.1007/BF03167521. |
[9] |
T. Ishiwata, On the motion of polygonal curves with asymptotic lines by crystalline curvature flow with bulk effect, Discrete Contin. Dyn. Syst., Series S, 4 (2011), 865-873.
doi: 10.3934/dcdss.2011.4.865. |
[10] |
T. Ishiwata, Motion of polygonal curved fronts by crystalline motion: V-shaped solutions and eventual monotonicity, Discrete Contin. Dyn. Syst. Supplement, 1 (2011), 717-726. |
[11] |
T. Ishiwata, Crystalline motion of spiral-shaped polygonal curves with a tip motion, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 53-62.
doi: 10.3934/dcdss.2014.7.53. |
[12] |
J. E. Taylor, Constructions and conjectures in crystalline nondifferential geometry, Proceedings of the Conference on Differential Geometry, Rio de Janeiro, Pitman Monographs Surveys Pure Appl. Math., 52 (1991), 321-336, Pitman London. |
[13] |
S. Yazaki, Point-extinction and geometric expansion of solutions to a crystalline motion, Hokkaido Math. J., 30 (2001), 327-357.
doi: 10.14492/hokmj/1350911957. |
[14] |
H. Imai, N. Ishimura and T. K. Ushijima, A crystalline motion of spiral-shaped curves with symmetry, J. Math. Anal. Appl., 240 (1999), 115-127.
doi: 10.1006/jmaa.1999.6599. |
[15] |
H. Imai, N. Ishimura and T. K. Ushijima, Motion of spirals by crystalline curvature, M2AN Math. Model. Numer. Anal., 33 (1999), 797-806.
doi: 10.1051/m2an:1999164. |
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), 323-391.
doi: 10.1007/BF01041068. |
[2] |
W. K. Burton, N. Cabrera and F. C. Frank, The Growth of Crystals and the Equilibrium Structure of their Surfaces, Philos. Trans. Roy. Soc. London. Ser. A., 243 (1951), 299-358.
doi: 10.1098/rsta.1951.0006. |
[3] |
B. Fiedler, J.-S. Guo and J.-C. Tsai, Multiplicity of rotating spirals under curvature flows with normal tip motion, J. Differential Equations, 205 (2004), 211-228.
doi: 10.1016/j.jde.2004.02.012. |
[4] |
M.-H. Giga and Y. Giga, Crystalline and level set flow - convergence of a crystalline algorithm for a general anisotropic curvature flow in the plane, Free boundary problems: Theory and applications, I (Chiba, 1999), GAKUTO Internat. Ser. Math. Sci. Appli., Gakkōtosho, Tokyo, 13 (2000), 64-79. |
[5] |
Y. Giga and M. E. Gurtin, A comparison theorem for crystalline evolution in the plane, Quart. J. Appl. Math., 54 (1996), 727-737. |
[6] |
J.-S. Guo, K.-I. Nakamura, T. Ogiwara and J.-C. Tsai, On the Steadily Rotating Spirals, Japan J. Indust. Appl. Math., 23 (2006), 1-19.
doi: 10.1007/BF03167495. |
[7] |
M. E. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane, Oxford, Clarendon Press, 1993. |
[8] |
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: 10.1007/BF03167521. |
[9] |
T. Ishiwata, On the motion of polygonal curves with asymptotic lines by crystalline curvature flow with bulk effect, Discrete Contin. Dyn. Syst., Series S, 4 (2011), 865-873.
doi: 10.3934/dcdss.2011.4.865. |
[10] |
T. Ishiwata, Motion of polygonal curved fronts by crystalline motion: V-shaped solutions and eventual monotonicity, Discrete Contin. Dyn. Syst. Supplement, 1 (2011), 717-726. |
[11] |
T. Ishiwata, Crystalline motion of spiral-shaped polygonal curves with a tip motion, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 53-62.
doi: 10.3934/dcdss.2014.7.53. |
[12] |
J. E. Taylor, Constructions and conjectures in crystalline nondifferential geometry, Proceedings of the Conference on Differential Geometry, Rio de Janeiro, Pitman Monographs Surveys Pure Appl. Math., 52 (1991), 321-336, Pitman London. |
[13] |
S. Yazaki, Point-extinction and geometric expansion of solutions to a crystalline motion, Hokkaido Math. J., 30 (2001), 327-357.
doi: 10.14492/hokmj/1350911957. |
[14] |
H. Imai, N. Ishimura and T. K. Ushijima, A crystalline motion of spiral-shaped curves with symmetry, J. Math. Anal. Appl., 240 (1999), 115-127.
doi: 10.1006/jmaa.1999.6599. |
[15] |
H. Imai, N. Ishimura and T. K. Ushijima, Motion of spirals by crystalline curvature, M2AN Math. Model. Numer. Anal., 33 (1999), 797-806.
doi: 10.1051/m2an:1999164. |
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