Neumann筛问题和降维：多尺度方法
The Neumann sieve problem and dimensional reduction: a multiscale approach
摘要：
我们进行多尺度分析的弹性能量的$n$ -维双层薄膜的厚度$2δ$，其层连接通过$ε$ -周期性分布的接触区。将接触区描述为半径为$r\ll ε$（筛孔）的$(n-1)$ -维球的联合体，并假设$δ\ll ε$，我们证明了筛的渐近记忆（如$ε\to 0$）是由一个额外的界面能项的存在见证。此外，我们发现三个不同的极限行为（或制度）依赖于相互消失率的$δ$和$r$。我们还给出了一个明确的非线性电容型公式的界面能密度在每个政权。
英文摘要：
We perform a multiscale analysis for the elastic energy of a $n$-dimensional bilayer thin film of thickness $2δ$ whose layers are connected through an $ε$-periodically distributed contact zone. Describing the contact zone as a union of $(n-1)$-dimensional balls of radius $r\ll ε$ (the holes of the sieve) and assuming that $δ\ll ε$, we show that the asymptotic memory of the sieve (as $ε\to 0$) is witnessed by the presence of an extra interfacial energy term. Moreover we find three different limit behaviors (or regimes) depending on the mutual vanishing rate of $δ$ and $r$. We also give an explicit nonlinear capacitary-type formula for the interfacial energy density in each regime.
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