Analysis of defects in minimizers for a planar Frank energy

Duration: 55 mins 16 secs

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Description:	Phillips, D (Purdue University) Friday 12 April 2013, 11:30-12:30


Created:	2013-04-26 12:59
Collection:	The Mathematics of Liquid Crystals
Publisher:	Isaac Newton Institute
Copyright:	Phillips, D
Language:	eng (English)


Abstract:	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.

Abstract:

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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