Quasicrystals and the Wulff-Shape

K. Böröczky; U. Schnell

doi:10.1007/pl00009430

Quasicrystals and the Wulff-Shape

Discrete & Computational Geometry ◽

10.1007/pl00009430 ◽

1999 ◽

Vol 21 (3) ◽

pp. 421-436 ◽

Cited By ~ 4

Author(s):

K. Böröczky ◽

U. Schnell

Keyword(s):

Wulff Shape

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On Uniqueness of the Wulff Shape for Cellular Automata

Fundamenta Informaticae ◽

10.3233/fi-1991-14104 ◽

1991 ◽

Vol 14 (1) ◽

pp. 75-89

Author(s):

Paweł Wlaź

Keyword(s):

Cellular Automata ◽

Large Class ◽

Growth Function ◽

Wulff Shape ◽

Transition Rules

In this paper, ordered transition rules are investigated. Such rules describe an increment of mono-crystals and for every rule one can calculate so called Wulff Shape. It is shown that for some large class of these rules, there exists at most one growth function which generates a given Wulff Shape.

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The Wulff shape as the asymptotic limit of a growing crystalline interface

Asian Journal of Mathematics ◽

10.4310/ajm.1997.v1.n3.a6 ◽

1997 ◽

Vol 1 (3) ◽

pp. 560-571 ◽

Cited By ~ 40

Author(s):

Stanley Osher ◽

Barry Merriman

Keyword(s):

Asymptotic Limit ◽

Wulff Shape

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Wulff shape of strontium titanate nanocuboids

Surface Science ◽

10.1016/j.susc.2014.10.014 ◽

2015 ◽

Vol 632 ◽

pp. L22-L25 ◽

Cited By ~ 14

Author(s):

Lawrence Crosby ◽

James Enterkin ◽

Federico Rabuffetti ◽

Kenneth Poeppelmeier ◽

Laurence Marks

Keyword(s):

Strontium Titanate ◽

Wulff Shape

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The Wulff shape minimizes an anisotropic Willmore functional

Interfaces and Free Boundaries ◽

10.4171/ifb/104 ◽

2004 ◽

pp. 351-359 ◽

Cited By ~ 18

Author(s):

Ulrich Clarenz

Keyword(s):

Wulff Shape ◽

Willmore Functional

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STABILITY OF CRYSTALLINE EVOLUTIONS

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202505000571 ◽

2005 ◽

Vol 15 (06) ◽

pp. 921-937 ◽

Cited By ~ 3

Author(s):

MATTEO NOVAGA ◽

EMANUELE PAOLINI

Keyword(s):

Integral Curve ◽

Jacobian Matrix ◽

Convex Polyhedra ◽

Wulff Shape ◽

Stability Properties ◽

Fixed Volume ◽

The Stability

In this paper we analyze the stability properties of the Wulff-shape in the crystalline flow. It is well known that the Wulff-shape evolves self-similarly, and eventually shrinks to a point. We consider the flow restricted to the set of convex polyhedra, we show that the crystalline evolutions may be viewed, after a proper rescaling, as an integral curve in the space of polyhedra with fixed volume, and we compute the Jacobian matrix of this field. If the eigenvalues of such a matrix have real part different from zero, we can determine if the Wulff-shape is stable or unstable, i.e. if all the evolutions starting close enough to the Wulff-shape converge or not, after rescaling, to the Wulff-shape itself.

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