We consider a 2D smectics model

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 \(\{\partial _zu_{n}\}\) in \(L^{2}\) and \(\{\partial _xu_n\}\) in \(L^q\) for any \(1\le q<p\) under the additional assumption \(\Vert \partial _xu_{n}\Vert _{L^{p }}\le C\) 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.

我们考虑一个2D模型

对于\（\ varepsilon _ {N} \ RIGHTARROW 0 \）和一个序列\（\左\ {U_ {N} \右\} \）具有有界能量\（E _ {\ varepsilon _ {N}} \左（ U_ {N} \右）\） ，我们证明的紧凑\（\ {\局部_zu_ {N} \} \）在\（L ^ {2} \）和\（\ {\局部_xu_n \} \）在附加假设\（\ Vert \ partial _xu_ {n} \ Vert _ {L ^ {p}} \ le C \）下的任何\（1 \ le q <p \）的\（L ^ q \）中一些\（p> 6 \）。当\（\ varepsilon \ rightarrow 0. \）时，我们还证明了\（E _ {\ varepsilon} \）的尖锐下界。 锐界对应于过渡区域中1D ansatz的能量。