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

TL;DR

This paper analyzes the spectral gaps in a two-dimensional Laplace operator with a periodic array of stiff, perforated disks, providing explicit asymptotic formulas for eigenvalues and eigenfunctions related to Bessel functions.

Contribution

It introduces an asymptotic method to explicitly characterize the opening of spectral gaps in a perforated domain with specific stiffness and density scaling.

Findings

Explicit leading-order eigenvalue expressions derived

Eigenfunctions related to Bessel functions of the first kind

Identification of spectral gap openings in the perforated domain

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 $\varepsilon^{-2m}$ and $\varepsilon^{-1}$ 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.