The band-gap structure of the spectrum in a periodic medium of masonry type

Abstract

We consider the spectrum of a class of positive, second-order elliptic systems of partial differential equations defined in the plane R 2 . The coefficients of the equation are assumed to have a special form, namely, they are doubly periodic and of high contrast. More precisely, the plane R 2 is decomposed into an infinite union of the translates of the rectangular periodicity cell Ω 0 , and this in turn is divided into two components, on each of which the coefficients have different, constant values. Moreover, the second component of Ω 0 consist of a neighborhood of the boundary of the cell of the width h and thus has an area comparable to h, where h > 0 is a small parameter.Using the methods of asymptotic analysis we study the position of the spectral bands as h → 0 and in particular show that the spectrum has at least a given, arbitrarily large number of gaps, provided h is small enough.

On the limit spectrum of a degenerate operator in the framework of periodic homogenization or singular perturbation problems

“…Unlike the study addressed in [24] where the geometry considered and the choice of the degenerate operator lead to an unbounded spectrum with gaps, we show here that the limit operator is still of a different nature from the original one despite the absence of gaps at the limit (see also [15] and the references therein); the limit spectrum we get in our case is bounded and described by two coupled equations related respectively to the vibrations in the fibers and in the matrix. Other settings have been studied in [2,11,12,16].…”

“…We now prove that any λ ∈ (µ 0 , µ 1 ) which is an eigenvalue of ( 11) may be attained as a limit of a sequence (λ k ε ) ε ; By this we can conclude that (11) has no other eigenvalues than those obtained as the limits of the eigenvalues λ k ε and thus we can list all its eigenvalues in increasing order. It is then clear that for a fixed k, we cannot have two subsequences ε and ε with two different limits for λ k ε and λ k ε since this would lead to add a new element to the set of eigenvalues of (11); hence (15) holds for the whole sequence ε.…”