A Ginzburg-Landau type problem for highly anisotropic nematic liquid crystals

TL;DR

This paper analyzes the asymptotic behavior of a thin nematic liquid crystal model with dominant elastic constant, deriving a limit energy that combines divergence penalization and wall energy, with rigorous and numerical minimization results.

Contribution

It derives the Gamma-limit of the energy for highly anisotropic nematic liquid crystals, revealing a combined divergence and wall energy structure, and analyzes minimizers both rigorously and numerically.

Findings

Gamma-limit energy combines divergence penalty and wall energy.

Criticality conditions for the limit energy are established.

Numerical minimization explores various domains and boundary conditions.

Abstract

We carry out an asymptotic analysis of a thin nematic liquid crystal in which one elastic constant dominates over the others, namely \begin{align} \label{energyab} \inf E_\varepsilon(u)\quad\mbox{where}\quad 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. \end{align} 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 derive 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 order parameter. We then derive criticality conditions for $E_0$ and analyze minimization of $E_0$ both rigorously and numerically for various domains $\Omega$ and a variety of Dirichlet boundary conditions.