Compactness and sharp lower bound for a 2D smectics model

Abstract: 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.