  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 $\Omega$ we investigate the $\varepsilon \to 0$ asymptotics of
the variational problem \[\inf \frac{1}{2}\int_\Omega \left(
\frac{1}{\varepsilon} W(u)+\varepsilon |\nabla u|^2 +
L_\varepsilon(\mathrm{div}\, u)^2 \right) \,dx\] within various parameter
regimes for $L_\varepsilon > 0.$ Here $u:\Omega\to\mathbb{R}^2$ and $W$ is a
potential vanishing on the unit circle and at the origin. When $\varepsilon\ll
L_\varepsilon\to 0$, we show that these functionals $\Gamma-$converge to a
constant multiple of the perimeter of the phase boundary and the divergence
penalty is not felt. However, when $L_\varepsilon \equiv L > 0$, we find that a
tangency requirement along the phase boundary for competitors in the
conjectured $\Gamma$-limit becomes a mechanism for development of
singularities. We establish criticality conditions for this limit and under a
non-degeneracy assumption on the potential we prove compactness of energy
bounded sequences in $L^2$. 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 $\mathbb{S}^1$-valued states are expected near the
defects along the phase boundary.
