Asymptotic behavior of solutions to a crystalline flow

On the role of kinetic and interfacial anisotropy in the crystal growth theory

“…A similar result is proved for crystalline flow by [St1], [St2]. So it is likely that the value of M (⃗ n)/γ(−⃗ n) on N plays an important role to determine the fully faceted shape.…”

Existence of self-similar evolution of crystals grown from supersaturated vapor

“…In fact there exists a self-similar shrinking solution for smooth strictly convex γ: [Ga], [GaL], [DG], [DGM]. It is extended to a planar crystalline flow by [St1], [St2] and to a crystalline flow in R 3 by [PP]. Note that if βγ = const, it is clear that the Wulff shape of γ always shrinks self-similarly.…”

The Discrete Planar L0-Minkowski Problem

“…While a polygon will shrink to a point by such a deformation, a sequence of its properly normalized shapes will converge in the Hausdorff metric to the solution of the polygonal L 0 -Minkowski problem. This asymptotic behavior has been studied in a different context [13], where our former argument works in the case (i) of Theorem 1.1. The method presented here is different and simpler.…”

“…It was also, independently, defined and studied by Taylor [14], for a particular set C. The flow is well defined as all the sides of the evolving polygon become of zero length simultaneously, at finite time T [14]. It should also be noted that the shape of evolving polygons is independent on the choice of the origin [13], thus we may assume, without any loss of generality, that the origin is the shrinking point K(T). An immediate consequence is the strict positivity of the support numbers to the sides of the boundary "K(t) for t < T. For any other choice of the origin, the support numbers, h i (t), will become signed distances according to the definition…”