# American Institute of Mathematical Sciences

October  2006, 14(4): 689-706. doi: 10.3934/dcds.2006.14.689

## Stability of facets of crystals growing from vapor

 1 Graduate School of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Tokyo 153-8914, Japan 2 Institute of Applied Mathematics and Mechanics, Warsaw University, ul. Banacha 2, 07-097 Warsaw, Poland

Received  December 2004 Revised  October 2005 Published  January 2006

Consider a Stefan-like problem with Gibbs-Thomson and kinetic effects as a model of crystal growth from vapor. The equilibrium shape is assumed to be a regular circular cylinder. Our main concern is a problem whether or not a surface of cylindrical crystals (called a facet) is stable under evolution in the sense that its normal velocity is constant over the facet. If a facet is unstable, then it breaks or bends. A typical result we establish is that all facets are stable if the evolving crystal is near the equilibrium. The stability criterion we use is a variational principle for selecting the correct Cahn-Hoffman vector. The analysis of the phase plane of an evolving cylinder (identified with points in the plane) near the unique equilibrium provides a bound for ratio of velocities of top and lateral facets of the cylinders.
Citation: Yoshikazu Giga, Piotr Rybka. Stability of facets of crystals growing from vapor. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 689-706. doi: 10.3934/dcds.2006.14.689
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