Article
Full entry |
PDF
(0.7 MB)
Feedback
PDF
(0.7 MB)
Feedback
Keywords:
essentially admissible polygon; crystalline curvature; the Wulff shape; isoperimetric inequality
essentially admissible polygon; crystalline curvature; the Wulff shape; isoperimetric inequality
Summary:
Asymptotic behavior of solutions of an area-preserving crystalline curvature flow equation is investigated. In this equation, the area enclosed by the solution polygon is preserved, while its total interfacial crystalline energy keeps on decreasing. In the case where the initial polygon is essentially admissible and convex, if the maximal existence time is finite, then vanishing edges are essentially admissible edges. This is a contrast to the case where the initial polygon is admissible and convex: a solution polygon converges to the boundary of the Wulff shape without vanishing edges as time tends to infinity.
References:
[1] Almgren F., Taylor J. E.: Flat flow is motion by crystalline curvature for curves with crystalline energies. J. Differential Geom. 42 (1995), 1–22 MR 1350693 | Zbl 0867.58020
[2] Angenent S., Gurtin M. E.: Multiphase thermomechanics with interfacial structure, 2. Evolution of an isothermal interface. Arch. Rational Mech. Anal. 108 (1989), 323–391 DOI 10.1007/BF01041068 | MR 1013461
[3] Gage M.: On an area-preserving evolution equations for plane curves. Contemp. Math. 51 (1986), 51–62 DOI 10.1090/conm/051/848933 | MR 0848933
[4] Giga Y.: Anisotropic curvature effects in interface dynamics. Sūgaku 52 (2000), 113–127; English transl., Sūgaku Expositions 16 (2003), 135–152
[5] Gurtin M. E.: Thermomechanics of Evolving Phase Boundaries in the Plane. Clarendon Press, Oxford 1993 MR 1402243 | Zbl 0787.73004
[6] Hontani H., Giga M.-H., Giga, Y., Deguchi K.: Expanding selfsimilar solutions of a crystalline flow with applications to contour figure analysis. Discrete Appl. Math. 147 (2005), 265–285 DOI 10.1016/j.dam.2004.09.015 | MR 2127078 | Zbl 1117.65036
[7] Roberts S.: A line element algorithm for curve flow problems in the plane. CMA Research Report 58 (1989); J. Austral. Math. Soc. Ser. B 35 (1993), 244–261 MR 1244207
[8] Taylor J. E.: Constructions and conjectures in crystalline nondifferential geometry. In: Proc. Conference on Differential Geometry, Rio de Janeiro, Pitman Monographs Surveys Pure Appl. Math. 52 (1991), 321–336, Pitman London MR 1173051 | Zbl 0725.53011
[9] Taylor J. E.: Motion of curves by crystalline curvature, including triple junctions and boundary points. Diff. Geom.: partial diff. eqs. on manifolds (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., 54 (1993), Part I, 417–438, AMS, Providence MR 1216599
[10] Taylor J. E., Cahn, J., Handwerker C.: Geometric models of crystal growth. Acta Metall. 40 (1992), 1443–1474 DOI 10.1016/0956-7151(92)90090-2
[11] Yazaki S.: On an area-preserving crystalline motion. Calc. Var. 14 (2002), 85–105 DOI 10.1007/s005260100094 | MR 1883601 | Zbl 1143.37320
[12] Yazaki S.: On an anisotropic area-preserving crystalline motion and motion of nonadmissible polygons by crystalline curvature. Sūrikaisekikenkyūsho Kōkyūroku 1356 (2004), 44–58
[13] Yazaki S.: Motion of nonadmissible convex polygons by crystalline curvature. Publ. Res. Inst. Math. Sci. 43 (2007), 155–170 DOI 10.2977/prims/1199403812 | MR 2317117 | Zbl 1132.53036
Search
Partner of
