## Hokkaido Mathematical Journal

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

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

#### 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