Boundary layers in the homogenization of a spectral problem in fluid-solid structures

This paper is devoted to the asymptotic analysis of the spectrum of a mathematical model that describes the vibrations of a coupled fluid-solid periodic structure. In a previous work [Arch. Rational Mech. Anal., 135 (1996), pp. 197-257] we proved by means of a Bloch wave homogenization method that, in the limit as the period goes to zero, the spectrum is made of three parts: the macroscopic or homogenized spectrum, the microscopic or Bloch spectrum, and a third component, the so-called boundary layer spectrum. While the two first parts were completely described as the spectrum of some limit problem, the latter was merely defined as the set of limit eigenvalues corresponding to sequences of eigenvectors concentrating on the boundary. It is the purpose of this paper to characterize explicitly this boundary layer spectrum with the help of a family of limit problems revealing the intimate connection between the periodic microstructure and the boundary of the domain. We therefore obtain a "completeness" result, i.e., a precise description of all possible asymptotic behaviors of sequences of eigenvalues, at least for a special class of polygonal domains.