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Publication 18-CNA-029

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

Dmitry Golovaty
Department of Mathematics
University of Akron
Akron OH

Michael Novack
Department of Mathematics
Indiana University
Bloomington, IN

Peter Sternberg
Department of Mathematics
Indiana University
Bloomington, IN

Raghavendra Venkatraman
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA

Abstract: 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 $\epsilon \to 0$ asymptotics of the variational problem \begin{align*} \inf \frac{1}{2}\int_\Omega \left( \frac{1}{\epsilon} W(u)+\epsilon |\nabla u|^2 + L_\epsilon( div u)^2 \right) \,dx \end{align*} within various parameter regimes for $L_\epsilon > 0.$ Here $u:\Omega\to \mathbb{R}^2$ and $W$ is a potential vanishing on the unit circle and at the origin. When $\epsilon\ll L_\epsilon \to 0$, 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 $L_\epsilon \equiv L > 0$, 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 of the kind analyzed by Golovaty, Sternberg and Venkatraman in the absence of phase transitions are expected near the defects along the phase boundary.

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