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We carry out an asymptotic analysis of a thin nematic liquid crystal in whichone 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 asum of a bulk term penalizing divergence and an Aviles-Giga type wall energyinvolving the cube of the jump in the tangential component of the$\mathbb{S}^1$-valued order parameter. We then derive criticality conditionsfor $E_0$ and analyze minimization of $E_0$ both rigorously and numerically forvarious domains $\Omega$ and a variety of Dirichlet boundary conditions.

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