Nonlinear approximation of 3D smectic liquid crystals: sharp lower bound and compactness

TL;DR

This paper establishes a sharp lower bound and compactness results for a 3D smectic liquid crystal energy model, extending known 2D results and introducing new analytical tools like 3D Jin-Kohn entropies.

Contribution

It introduces 3D analogues of Jin-Kohn entropies to derive a sharp energy lower bound and proves compactness of gradients under specific conditions, advancing the mathematical understanding of smectic liquid crystals.

Findings

Sharp lower bound on the 3D smectic energy as epsilon approaches zero.

Energy equipartition between bending and compression strains.

Compactness of the gradient sequence under sign conditions on Laplacian in xy-plane.

Abstract

We consider the 3D smectic energy $$\mathcal{E}_{\epsilon }\left( u\right) =\frac{1}{2}\int_{\Omega }\frac{1}{\varepsilon } \left( u_z-\frac{( u_x)^{2}+( u_y)^{2}}{2}\right) ^{2}+\varepsilon \left( u_{xx}+u_{yy}\right)^{2}\,dx\,dy\,dz. $$ The model contains as a special case the well-known 2D Aviles-Giga model. We prove a sharp lower bound on $\mathcal{E}_{\varepsilon}$ as $\varepsilon \to 0$ by introducing 3D analogues of the Jin-Kohn entropies. The sharp bound corresponds to an equipartition of energy between the bending and compression strains and was previously demonstrated in the physics literature only when the approximate Gaussian curvature of each smectic layer vanishes. Also, for $\varepsilon_{n}\rightarrow 0$ and an energy-bounded sequence $\{u_n \}$ with $\|\nabla u_n\|_{L^{p}(\Omega)},\,\|\nabla u_n\|_{L^2(\partial \Omega)}\leq C$ for some $p>6$, we obtain compactness of $\nabla u_{n}$ in $L^{2}$ assuming that $\Delta_{xy}u_{n}$ has constant sign for each $n$.