Bloch wave homogenization and spectral asymptotic analysis (Q1128297)

The authors consider a second order elliptic equation in a bounded periodically heterogeneous medium and the asymptotic behavior of its eigenvalues \((\lambda^k_{\varepsilon})^{-1}\) is studied as the structure period \(\varepsilon\) tends to zero. More precisely, the asymptotic of renormalized eigenvalues \(\varepsilon^{-2} \lambda^k_{\varepsilon}\), i.e. high frequency asymptotic, is considered. It is proved that this renormalized limit spectrum is devided into two parts: the so-called Bloch spectrum and the so-called boundary layer spectrum. The first one is defined explicitly as the spectrum of a family of new auxiliary problems, while the second corresponds to eigenfunctions concentrating at the boundary of the domain. In the case of a rectangular domain made of an entire periodicity cell, it is given an explicit description of the boundary layer spectrum.