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
dmitry@uakron.edu
Michael Novack
Department of Mathematics
Indiana University
Bloomington, IN
mrnovack@indiana.edu
Peter Sternberg
Department of Mathematics
Indiana University
Bloomington, IN
sternber@indiana.edu
Raghavendra Venkatraman
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA
rvenkatr@andrew.cmu.edu
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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