Smectic C* liquid crystal films are modeled with a relaxed Frank energy,
∫Ω(ks(divu)2+kb(curlu)2+12ϵ2(1−|u|2)2)dx.
Here ks and kb represent the two dimensional splay and bend moduli for the film respectively with ks,kb>0, Ω is a planar domain, and u is an R2-valued vector field with fixed boundary data having degree d>0. We study the limiting pattern for a sequence of minimizers {uϵ} as ϵ→0. We prove that the pattern contains d, degree one defects and that it has a either a radial or circular asymptotic form near each defect depending on the relative values of ks and kb. We further characterize a renormalized energy for the problem and show that it is minimized by the limit. This is joint work with Sean Colbert-Kelly.
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