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-so lid periodic structure. In a previous work [Arch. Rational Mech. Anal. , 135 (1996), pp. 197-257] we proved by means of a Bloch wave homogeni zation method that, in the limit as the period goes to zero, the spect rum is made of three parts: the macroscopic or homogenized spectrum, t he microscopic or Bloch spectrum, and a third component, the so-called boundary layer spectrum. While the two first parts were completely de scribed as the spectrum of some limit problem, the latter was merely d efined as the set of limit eigenvalues corresponding to sequences of e igenvectors concentrating on the boundary. It is the purpose of this p aper to characterize explicitly this boundary layer spectrum with the help of a family of limit problems revealing the intimate connection b etween the periodic microstructure and the boundary of the domain. We therefore obtain a ''completeness'' result, i.e., a precise descriptio n of all possible asymptotic behaviors of sequences of eigenvalues, at least for a special class of polygonal domains.