Bloch wave homogenization and spectral asymptotic analysis

We consider a second-order elliptic equation in a bounded periodic heterogeneous medium and study the asymptotic behavior of its spectrum, as the structure period goes to zero. We use a new method of Bloch wave homogenization which, unlike the classical homogenization method, characterizes a renormalized limit of the spectrum, namely sequences of eigenvalues of the order of the square of the medium period. We prove that such a renormalized limit spectrum is made of two parts: the so-called Bloch spectrum, which is explicitly defined as the spectrum of a family of limit problems, and the so-called boundary layer spectrum, which is made of limit eigenvalues corresponding to sequences of eigenvectors concentrating on the boundary of the domain. This analysis relies also on a notion of Bloch measures which can be seen as ad hoc Wigner measures in the context of semi-classical analysis. Finally, for rectangular domains made of entire periodicity cells, a variant of the Bloch wave homogenization method gives an explicit characterization of the boundary layer spectrum too.