The Bloch Approximation in Periodically Perforated Media

We consider a periodically heterogeneous and perforated medium filling an open domain Ω of ℝ^N. Assuming that the size of the periodicity of the structure and of the holes is O(ε), we study the asymptotic behavior, as ε → 0, of the solution of an elliptic boundary value problem with strongly oscillating coefficients posed in Ω^ε (Ω^ε being Ω minus the holes) with a Neumann condition on the boundary of the holes. We use Bloch wave decomposition to introduce an approximation of the solution in the energy norm which can be computed from the homogenized solution and the first Bloch eigenfunction. We first consider the case where Ωis ℝ^N and then localize the problem for a bounded domain Ω, considering a homogeneous Dirichlet condition on the boundary of Ω.