The Neumann Sieve Problem and Dimensional Reduction: a Multiscale Approach

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 ≪ ε (the holes of the sieve) and assuming that δ ≪ ε, we show that the asymptotic memory of the sieve (as ε → 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.