A smectic liquid crystal model in the periodic setting

TL;DR

This paper analyzes the asymptotic behavior of a 2D smectic liquid crystal model in a periodic setting, establishing bounds, existence of minimizers, and lower bounds as the parameter approaches zero.

Contribution

It proves the existence of minimizers for the smectics energy model and establishes bounds and asymptotic lower bounds as the parameter tends to zero.

Findings

Energy controls $L^p$ and Besov norms of $w$

Existence of minimizers for the energy model

Asymptotic lower bound for energy as epsilon approaches zero

Abstract

We consider the asymptotic behavior as $\varepsilon $ goes to zero of the 2D smectics model in the periodic setting given by \begin{equation*} \mathcal{E}_{\varepsilon }( w) =\frac{1}{2}\int_{\mathbb{T}^{2}}\frac{1}{ \varepsilon }\left( \left\vert \partial_{1}\right\vert ^{-1}\left( \partial_{2}w-\partial_{1}\frac{1}{2}w^{2}\right) \right) ^{2}+\varepsilon \left( \partial_{1}w\right) ^{2}dx . \end{equation*} We show that the energy $\mathcal{E}_\varepsilon(w)$ controls suitable $L^p$ and Besov norms of $w$ and use this to demonstrate the existence of minimizers for $\mathcal{E}_\varepsilon(w)$, which has not been proved for this smectics model before, and compactness in $L^p$ for an energy-bounded sequence. We also prove an asymptotic lower bound for $\mathcal{E}_\varepsilon(w)$ as $\varepsilon \to 0$ by means of an entropy argument.