论文标题
2D元素模型的紧凑性和锋利的下限
Compactness and sharp lower bound for a 2D smectics model
论文作者
论文摘要
我们考虑一个2D晶状体模型\ begin {qore*} e_ {ε} \ left（u \ right）= \ frac {1} {2} {2} {2} \int_Ω\ frac {1} {\ varepsilon} {\ varepsilon} } {2} u_ {x} ^{2} \ right） ^{2}+\ varepsilon \ left（u_ {xx} \ right） ^{2} dx \，dz。 \ end {qore*}对于$ \ varepsilon _ {n} \ rightarrow 0 $和一个序列$ \ left \ {u_ {n} \ right \} $，带有有界的能量$ e _ { $ \ {\ partial_z u_ {n} \} $ in $ l^{2} $和$ \ {\ partial_x u_n \} $ in $ l^q $ in $ 1 \ leq q <p <p <p <p <p <p $， \ partial_x u_ {n} \ | _ {l^{p}} \ leq c $对于某些$ p> 6 $。当$ \ varepsilon \ rightarrow 0时，我们还证明了$ e _ {\ varepsilon} $的尖锐下限。
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 $\varepsilon _{n}\rightarrow 0$ 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.
