Stability of facets of self-similar motion of a crystal

Abstract

We are concerned with a quasi-steady Stefan type problem with Gibbs-Thomson relation and the mobility term which is a model for a crystal growing from supersaturated vapor. The evolving crystal and the Wulff shape of the interfacial energy are assumed to be (right-circular) cylinders. In pattern formation deciding what are the conditions which guarantee that the speed in the normal direction is constant over each facet, so that the facet does not break, is an important question. We formulate such a condition with the aid of a convex variational problem with a convex obstacle type constraint. We derive necessary and sufficient conditions for the nonbreaking of facets in terms of the size and the supersaturation at space infinity when the motion is self-similar.