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Stability of facets of self-similar motion of a crystal

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 an aid of a convex variational
problem with a convex obstacle type constraint.

We derive
necessary and
sufficient conditions for the non-breaking of facets in terms of the
size and the supersaturation at space infinity when the motion is
self-similar.