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, namely, $ \inf E_\varepsilon(u)$, where $E_\varepsilon(u) := \frac{1}{2}\int_\Omega \left\{\varepsilon\,|\nabla u|^2 + \frac{1}{\varepsilon} \,(|u|^2 - 1)^2 + L \,(\mathrm{div}\, u)^2\right\} \,dx. $ Here $u: \Omega \to \mathbb{R}^2$ is a vector field, $0 < \varepsilon \ll 1 $ is a small parameter, and $L > 0$ is a fixed constant, independent of $\varepsilon$. We identify a candidate for the $\Gamma$-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 $\mathbb{S}^1$-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 from various choices of domain geometry 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.