Abstract. 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.
Stability of Crystalline Evolutions / M. NOVAGA; E. PAOLINI. - In: MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES. - ISSN 0218-2025. - STAMPA. - 15:(2005), pp. 1-17. [10.1142/S0218202505000571]
Stability of Crystalline Evolutions
PAOLINI, EMANUELE
2005
Abstract
Abstract. 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.File | Dimensione | Formato | |
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