Hokkaido Mathematical Journal

Point-extinction and geometric expansion of solutions to a crystalline motion

Shigetoshi YAZAKI

Full-text: Open access

Abstract

We consider the asymptotic behavior of solutions to a generalized crystalline motion which describes evolution of plane curves driven by nonsmooth interfacial energy. Our main results say that solution polygonal curves expand to infinity or shrink to a single point depending on the size of initial data and the sign of the driving force term. In the expanding case, we show that any rescaled solution polygon converges to the boundary of the Wulff shape for the driving force term and hence if the driving force term is a constant, then any solution polygon approaches to an expanding regular polygon even if the motion is anisotropic. We also give lower and upper bounds of the extinction time for the shrinking case. In the appendix, we shall explain the notion of a discrete curvature and crystalline curvature from a numerical point of view.

Article information

Source
Hokkaido Math. J., Volume 30, Number 2 (2001), 327-357.

Dates
First available in Project Euclid: 22 October 2012

Permanent link to this document
https://projecteuclid.org/euclid.hokmj/1350911957

Digital Object Identifier
doi:10.14492/hokmj/1350911957

Mathematical Reviews number (MathSciNet)
MR1844823

Zentralblatt MATH identifier
1192.82077

Subjects
Primary: 58F25
Secondary: 53A04: Curves in Euclidean space 73B30 34A26: Geometric methods in differential equations 34A34: Nonlinear equations and systems, general 82D25: Crystals {For crystallographic group theory, see 20H15}

Keywords
crystalline motion crystalline curvature discrete curvature motion by curvature curve-shortening point-extinction geometric expansion the Wulff shape estimates of blow-up time entropy estimate comparison principle isoperimetric ratio

Citation

YAZAKI, Shigetoshi. Point-extinction and geometric expansion of solutions to a crystalline motion. Hokkaido Math. J. 30 (2001), no. 2, 327--357. doi:10.14492/hokmj/1350911957. https://projecteuclid.org/euclid.hokmj/1350911957


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