A Ginzburg--Landau-Type Problem for Highly Anisotropic Nematic Liquid Crystals

Abstract

We carry out an asymptotic analysis of a variational problem relevant in the studies of nematic liquid crystalline films when one elastic constant dominates over the others, namelyHere u : Ω → R 2 is a vector field, 0 < ε 1 is a small parameter, and L > 0 is a fixed constant, independent of ε. We identify a candidate for the Γ-limit E 0 , which is a sum of a bulk term penalizing divergence and an Aviles-Giga type wall energy involving the cube of the jump in the tangential component of the S 1 -valued nematic director. We establish the lower bound and provide the recovery sequence for this candidate within a restricted class. Then we consider a set of variational problems for E 0 arising for a choice of domain geometries and boundary conditions. We demonstrate that the criticality conditions for E 0 can be expressed as a pair of scalar conservation laws that share characteristics. We use the method of characteristics to analytically construct critical points of E 0 that we observe numerically.

A Model Problem for Nematic-Isotropic Transitions with Highly Disparate Elastic Constants

“…In Section 3.4, we turn to the derivation of criticality conditions for the proposed Γ-limit, E 0 . As in [17], we find that in the S 1 -valued phase, away from walls, we can phrase criticality in terms of a system of conservation laws sharing characteristics, cf. Corollary 3.11.…”

“…The role played by this tangency condition on the formation of interfacial singularities is investigated through several examples: each of these examples involves analytically rigorous reasoning motivated by numerical experiments. We argue that generically, "wall" singularities between S 1 -valued states of the kind analyzed in [17] are expected near the defects along the phase boundary. *…”

“…In [17], our first endeavor in this direction, we propose a model problem coupling the Ginzburg-Landau potential to an elastic energy density with large elastic disparity, namely inf u∈H 1 (Ω;R 2 )…”

“…The natural space for competitors for this limit should be some subset of H div (Ω; S 1 ), the Hilbert space of L 2 vector fields having L 2 divergence. In order to make sense of the jump set we make the additional assumption in [17] that u ∈ BV(Ω; S 1 ), though this is surely not optimal. As a simple consequence of the Divergence Theorem, it follows that allowable jumps for an H div vector field must satisfy continuity of the normal component u + · ν = u − · ν along J u , (1.4) where ν denotes the normal to J u .…”

“…

Continuing the program initiated in [17], we analyze a model problem based on highly disparate elastic constants that we propose in order to understand corners and cusps that form on the boundary between the nematic and isotropic phases in a liquid crystal. For a bounded planar domain Ω we investigate the ε → 0 asymptotics of the variational problemwithin various parameter regimes for L ε > 0.

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