Abstract
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.
Highlights
This is a continuation of our series of papers, [GR1], [GR2], [GR3], [GR4]
We studied there a one phase Stefan-like problem with Gibbs-Thomson and kinetic effect where the interfacial energy is so singular that its Wulff shape-equilibrium shape is a regular circular cylinder
We assume that the initial shape of crystal is a cylinder which may have not have the aspect ratio of the Wulff shape
Summary
This is a continuation of our series of papers, [GR1], [GR2], [GR3], [GR4]. We studied there a one phase Stefan-like problem with Gibbs-Thomson and kinetic effect where the interfacial energy is so singular that its Wulff shape-equilibrium shape is a regular circular cylinder. It is our intention to consider energy density functions γ which are only Lipschitz continuous and (positively) 1-homogeneous, and surfaces S with edges It follows that γ is differentiable a.e., but the normals to our interface S fall precisely into the set of points on non-differentiability of γ. In this paper we address the question of facet stability of circular straight cylinders evolving by (1.1)–(1.5) for surface energy density γ given by (2.1) below without imposing any additional restrictions on γ nor β. The characterization of region of stability in term of the scale factor a(t) ([GR4, Theorem 4.8] and [GR4, Theorem 4.14]) depended heavily on the transparent structure of solutions and available various estimates especially on velocities of facets They were obtained with the help of another general result namely so-called. An important role in this process will be played by the Cahn-Hoffman vector
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