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 }\left( 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 a sequence $\left\{ u_{n}\right\}$ with bounded energies $E_{\varepsilon _{n}}\left(u_{n}\right) ,$ we prove compactness of $\{\partial_z u_{n}\}$ in $L^{2}$ and $\{\partial_x u_n\}$ in $L^q$ for any $1\leq q<p$ under the additional assumption $\| \partial_x u_{n}\| _{L^{p }}\leq C$ for some $p>6$. We also prove a sharp lower bound on $E_{\varepsilon }$ when $\varepsilon\rightarrow 0.$ The sharp bound corresponds to the energy of a 1D ansatz in the transition region.