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 \begin{document}$ \mathbb{R}^2 $\end{document} . The coefficients of the equation are assumed to have a special form, namely, they are doubly periodic and of high contrast. More precisely, the plane \begin{document}$ \mathbb{R}^2 $\end{document} is decomposed into an infinite union of the translates of the rectangular periodicity cell \begin{document}$ \Omega^0 $\end{document} , and this in turn is divided into two components, on each of which the coefficients have different, constant values. Moreover, the second component of \begin{document}$ \Omega^0 $\end{document} consist of a neighborhood of the boundary of the cell of the width \begin{document}$ h $\end{document} and thus has an area comparable to \begin{document}$ h $\end{document} , where \begin{document}$ h>0 $\end{document} is a small parameter. Using the methods of asymptotic analysis we study the position of the spectral bands as \begin{document}$ h \to 0 $\end{document} and in particular show that the spectrum has at least a given, arbitrarily large number of gaps, provided \begin{document}$ h $\end{document} is small enough.