Calculus of Variations and Geometric Measure Theory

M. Novaga - E. Paolini

Stability of Crystalline Evolutions

created on 20 May 2004
modified by novaga on 17 Apr 2016


Published Paper

Inserted: 20 may 2004
Last Updated: 17 apr 2016

Journal: Math. Mod. Meth. Appl. Sc.
Volume: 15
Number: 6
Pages: 1-17
Year: 2005


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.

Keywords: stability, crystal, evolution, wulff