The Neumann sieve problem and dimensional reduction: a multiscale approach

Abstract

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.