# Asymptotic behavior of solutions to an area-preserving motion by crystalline curvature

Kybernetika (2007)

- Volume: 43, Issue: 6, page 903-912
- ISSN: 0023-5954

## Access Full Article

top## Abstract

top## How to cite

topYazaki, Shigetoshi. "Asymptotic behavior of solutions to an area-preserving motion by crystalline curvature." Kybernetika 43.6 (2007): 903-912. <http://eudml.org/doc/33906>.

@article{Yazaki2007,

abstract = {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.},

author = {Yazaki, Shigetoshi},

journal = {Kybernetika},

keywords = {essentially admissible polygon; crystalline curvature; the Wulff shape; isoperimetric inequality; essentially admissible polygon; crystalline curvature; the Wulff shape; isoperimetric inequality},

language = {eng},

number = {6},

pages = {903-912},

publisher = {Institute of Information Theory and Automation AS CR},

title = {Asymptotic behavior of solutions to an area-preserving motion by crystalline curvature},

url = {http://eudml.org/doc/33906},

volume = {43},

year = {2007},

}

TY - JOUR

AU - Yazaki, Shigetoshi

TI - Asymptotic behavior of solutions to an area-preserving motion by crystalline curvature

JO - Kybernetika

PY - 2007

PB - Institute of Information Theory and Automation AS CR

VL - 43

IS - 6

SP - 903

EP - 912

AB - 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.

LA - eng

KW - essentially admissible polygon; crystalline curvature; the Wulff shape; isoperimetric inequality; essentially admissible polygon; crystalline curvature; the Wulff shape; isoperimetric inequality

UR - http://eudml.org/doc/33906

ER -

## References

top- Almgren F., Taylor J. E., Flat flow is motion by crystalline curvature for curves with crystalline energies, J. Differential Geom. 42 (1995), 1–22 (1995) Zbl0867.58020MR1350693
- Angenent S., Gurtin M. E., 10.1007/BF01041068, Evolution of an isothermal interface. Arch. Rational Mech. Anal. 108 (1989), 323–391 (1989) MR1013461DOI10.1007/BF01041068
- Gage M., 10.1090/conm/051/848933, Contemp. Math. 51 (1986), 51–62 (1986) MR0848933DOI10.1090/conm/051/848933
- Giga Y., Anisotropic curvature effects in interface dynamics, Sūgaku 52 (2000), 113–127; English transl., Sūgaku Expositions 16 (2003), 135–152
- Gurtin M. E., Thermomechanics of Evolving Phase Boundaries in the Plane, Clarendon Press, Oxford 1993 Zbl0787.73004MR1402243
- Hontani H., Giga M.-H., Giga, Y., Deguchi K., 10.1016/j.dam.2004.09.015, Discrete Appl. Math. 147 (2005), 265–285 Zbl1117.65036MR2127078DOI10.1016/j.dam.2004.09.015
- 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 (1989) MR1244207
- 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 (1991) Zbl0725.53011MR1173051
- 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 (1990) MR1216599
- Taylor J. E., Cahn, J., Handwerker C., 10.1016/0956-7151(92)90090-2, Acta Metall. 40 (1992), 1443–1474 (1992) DOI10.1016/0956-7151(92)90090-2
- Yazaki S., 10.1007/s005260100094, Calc. Var. 14 (2002), 85–105 Zbl1143.37320MR1883601DOI10.1007/s005260100094
- Yazaki S., On an anisotropic area-preserving crystalline motion and motion of nonadmissible polygons by crystalline curvature, Sūrikaisekikenkyūsho Kōkyūroku 1356 (2004), 44–58
- Yazaki S., 10.2977/prims/1199403812, Publ. Res. Inst. Math. Sci. 43 (2007), 155–170 Zbl1132.53036MR2317117DOI10.2977/prims/1199403812

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.