Homogenization of periodically perforate media

We consider Bloch wave decomposition to study the homogenization of periodi cally heterogeneous and perforate media in a bounded domain Omega of R-n. A ssuming that epsilon is the size of the periodicity of the structure and of the holes, we study the asymptotic behavior, as epsilon --> 0, of the solu tion of an elliptic boundary value problem with strongly oscillating coeffi cients posed in Omega(epsilon) (Omega(epsilon) being Omega minus the holes) with a homogeneous Dirichlet condition on the boundary of Omega and a Neum ann condition on the boundary of the holes. We first consider the case when Omega is R-n and then localize the problem for a bounded domain Omega. A n ew characterization of the homogenized coefficients is given via the Hessia n at the origin of the first Bloch eigenvalue.