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 $-\mathrm{\Delta }$ 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.

Keywords:

• Spectral problem
• spectral gap
• Bessel functions
• periodic domain
• cuspidal domain

AMS Classifications:

• 35J05
• 35P10
• 47A10
• 33C10

Acknowledgments

The authors wish to thank V. Chiadò Piat for helping us during the preparation of the paper and for her fruitful discussion, her careful reading and relevant remarks. The author L. D'Elia is a member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Disclosure statement

No potential conflict of interest was reported by the author(s).