Compactness and Sharp Lower Bound for a 2D Smectics Model

We consider a 2D smectics model \begin{equation*} E_{\epsilon }\left( u\right) =\frac{1}{2}\int_\Omega \frac{1}{\varepsilon u_{z}-\frac{1% }{2}u_{x}^{2}\right) ^{2}+\varepsilon \left( u_{xx}\right) ^{2}dx\,dz. \end{equation*} For $\varepsilon _{n}\rightarrow 0$ and sequence $\left\{ u_{n}\right\} $ with bounded energies $E_{\varepsilon _{n}}\left(u_{n}\right) ,$ we prove compactness of $\{\partial_z u_{n}\}$ in $L^{2}$ $\{\partial_x u_n\}$ $L^q$ for any $1\leq q6$. also sharp lower bound on }$ when $\varepsilon\rightarrow 0.$ The corresponds to energy 1D ansatz transition region.