Rigorous results in selection of steady needle crystals

Abstract

This paper concerns the existence of solutions to a steady needle crystal growth problem in a one-sided model. We rigorously prove that for small nonzero anisotropy γ, analytic symmetric needle crystal solutions exist in the limit of surface tension ε 2 if only if the stokes constant S for a relatively simple nonlinear differential equation is zero. This Stokes constant S depends on the parameter β=2 9/7 γε −8/7 and earlier numerical calculations by a number of investigators have shown this to be zero for a discrete set of values of β. It is also proven that for γ=0, there can be no symmetric needle crystal solution in the considered space. The methodology consists of two steps. First, the original problem is reduced to a weak half-strip problem for any γ in a compact set of [0,1) by relaxation of the symmetry condition. The weak problem is shown to have a unique solution in the function space considered for any γ∈[0, γ m ] for some γ m >0. When a symmetry is invoked, the weak problem is shown equivalent to the original needle crystal problem. Next, by considering the behavior of the solution in neighborhood of an appropriate complex turning point for γ∈(0, γ m ], we extract an exponentially small term in ε as ε→0 + that generally violates the symmetric condition. We prove that the symmetry condition is satisfied for small ε when the parameter β is constrained appropriately.