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

Continuing the program initiated in Golovaty et al. (SIAM J Math Anal 51(1):276-320, 2018), 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\asymptotics of the variational problem inf12 integral omega mml:mfenced close=) open=(1 epsilon W(u)+epsilon| backward difference u|2+L epsilon(divu)2dx$$\begin{aligned} \inf \displaystyle \frac{1}{2}\int _\Omega \left( \frac{1}{\varepsilon } W(u)+\varepsilon |\nabla u|<^>2 + L_\varepsilon (\mathrm {div}\,u)<^>2 \right) \,\hbox {d}x \end{aligned}$$\end{document}within various parameter regimes for L epsilon 0.$$L_\varepsilon 0.$$\end{document} Here u:omega -> R2$$u:\Omega \rightarrow \mathbb {R}<^>2$$\end{document} and W is a potential vanishing on the unit circle and at the origin. When epsilon MUCH LESS-THANL epsilon -> 0$$\varepsilon \ll L_\varepsilon \rightarrow 0$$\end{document}, 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 epsilon equivalent to L 0we 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 the compactness of energy bounded sequences in L2 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 S1-valued states of the kind analyzed in Golovaty et al. (SIAM J Math Anal 51(1):276-320, 2018) are expected near the defects along the phase boundary.