The bloch approximation in periodically perforated media

摘要

We consider a periodically heterogeneous and perforated medium filling an open domain Omega of R-N. Assuming that the size of the periodicity of the structure and of the holes is O(epsilon), we study the asymptotic behavior, as epsilon -> 0, of the solution of an elliptic boundary value problem with strongly oscillating coefficients posed in Omega(epsilon) e ( Omega(epsilon) e being Omega 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 Omega is R-N and then localize the problem for a bounded domain Omega, considering a homogeneous Dirichlet condition on the boundary of Omega.