Gaps in the spectrum of two-dimensional square packing of stiff disks

Abstract

In this paper we investigate via an asymptotic method the opening of gaps in the spectrum of a stiff problem for the Laplace operator $-\Delta$ in ${\mathbb{R}}^2$ perforated by contiguous circular holes. The density and the stiffness constants are of order ${\epsilon }^{-2m}$ and ${\epsilon }^{-1}$ in the holes with $m\in (0,1/2)$. We provide an explicit expression of the leading terms of the eigenvalues and the corresponding eigenfunctions which are related to the Bessel functions of the first kind.