Homogenization in a Perforated Domain Including a Thin Full Interlayer
Alain P. Bourgeat, Roland Tapiéro · Birkhäuser Basel eBooks · 1993
We study the behavior of the solution of a model elliptic problem, the Poisson equation −Δu ε = f on a domain D ε including many tiny holes. These holes are periodically distributed with a period diameter equal to ɛ except in a thin, unperforated layer of thickness η crossing the domain. We show that, if ɛ, η tend to zero and the ratio η/ɛ tends to zero or a constant, the behavior of the limit solution is just as if there were no unperforated layer and only a whole, periodically perforated domain. If $$\varepsilon {\varepsilon ^{\tfrac{2}{3}}}$$ , the asymptotic behavior of the solution is different everywhere in D ɛ from the classical solution of Lions [10] in perforated domains. Proofs are given using energy estimates in Sobolev spaces and the framework of homogenization theory; limits can be found by the classical techniques using “test functions” and “two-scale convergence”.