Asymptotics beyond All Orders in a Model of Crystal Growth

In its simplest form, the geometric model of crystal growth is a third‐order, nonlinear, ordinary differential equation for θ(s, ε): urn:x-wiley:-wiley:00222526:media:sapm1991852129-math-0001 A needle crystal is a solution that satisfies boundary conditions urn:x-wiley:-wiley:00222526:media:sapm1991852129-math-0002 The geometric model admits a needle‐crystal solution for ε = 0; for small ε, it admits an asymptotic expansion that is valid to all orders for such a solution. Even so, we prove that the geometric model in this form admits no needle crystal for any small, nonzero ε, a fact that lies beyond all orders of the asymptotic expansion. A more complicated version of the geometric model is urn:x-wiley:-wiley:00222526:media:sapm1991852129-math-0003 where α represents crystalline anisitropy. We show that for 0 < α < 1, the geometric model admits needle crystals for a discrete set of values of α. The number of such values of α increases like ε−1 as ε → 0.