Compactness and sharp lower bound for a 2D smectics model

Abstract

We consider a 2D smectics model \begin{equation*} E_{ε}\left( u\right) =\frac{1}{2}\int_Ω\frac{1}{\varepsilon }\left( u_{z}-\frac{1% }{2}u_{x}^{2}\right) ^{2}+\varepsilon \left( u_{xx}\right) ^{2}dx\,dz. \end{equation*} For ${\epsilon }_n\to 0$ and a sequence $\left\{u_n\right\}$ with bounded energies $E_{{\epsilon }_n}\left(u_n\right),$ we prove compactness of $\{{\partial }_z{u}_n\}$ in $L^2$ and $\{{\partial }_x{u}_n\}$ in $L^q$ for any $1\le q<p$ under the additional assumption $\Vert {\partial }_x{u}_n{\Vert }_{L^p}\le C$ for some $p>6$. We also prove a sharp lower bound on $E_{\epsilon }$ when $\epsilon \to 0.$ The sharp bound corresponds to the energy of a 1D ansatz in the transition region.

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