# American Institute of Mathematical Sciences

December  2019, 14(4): 733-757. doi: 10.3934/nhm.2019029

## Asymptotic structure of the spectrum in a Dirichlet-strip with double periodic perforations

 1 Saint-Petersburg State University, Universitetskaya nab. 7–9, St. Petersburg, 199034, Russia 2 Institute of Problems of Mechanical Engineering RAS, V.O., Bolshoj pr., 61, St. Petersburg, 199178, Russia 3 Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Nicolás Cabrera 13-15, Campus de Cantoblanco-UAM, Madrid, 28049, Spain 4 Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, Spain 5 Departamento de Matemática Aplicada y Ciencias de la Computación, Universidad de Cantabria, Avenida de las Castros s/n, 39005 Santander, Spain

* Corresponding author: María-Eugenia Pérez-Martínez

Received  October 2018 Revised  May 2019 Published  October 2019

Fund Project: The first author is supported by Russian Foundation on Basic Research, grant 18-01-00325.
The second author is supported by the Spanish MINECO through the "Severo Ochoa Programme for Centres of Excellence in RaD" (SEV-2015-0554) and MTM2017-89976-P.
The third author is supported by the Spanish MINECO grant MTM2013- 44883-P and MICINN grant PGC2018-098178-B-I00.

We address a spectral problem for the Dirichlet-Laplace operator in a waveguide $\Pi^ \varepsilon$. $\Pi^ \varepsilon$ is obtained from repsilon an unbounded two-dimensional strip $\Pi$ which is periodically perforated by a family of holes, which are also periodically distributed along a line, the so-called "perforation string". We assume that the two periods are different, namely, $O(1)$ and $O( \varepsilon)$ respectively, where $0< \varepsilon\ll 1$. We look at the band-gap structure of the spectrum $\sigma^ \varepsilon$ as $\varepsilon\to 0$. We derive asymptotic formulas for the endpoints of the spectral bands and show that $\sigma^ \varepsilon$ has a large number of short bands of length $O( \varepsilon)$ which alternate with wide gaps of width $O(1)$.

Citation: Sergei A. Nazarov, Rafael Orive-Illera, María-Eugenia Pérez-Martínez. Asymptotic structure of the spectrum in a Dirichlet-strip with double periodic perforations. Networks & Heterogeneous Media, 2019, 14 (4) : 733-757. doi: 10.3934/nhm.2019029
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##### References:
a) The perforated strip $\Pi^ \varepsilon$ is obtained by removing the double periodic family of holes $\overline{\omega^ \varepsilon}$ from the strip $\Pi\equiv(-\infty, \infty)\times(0, H)$. The periodicities $1$ and $\varepsilon H$ come from the width of he periodicity cell $\varpi^ \varepsilon$ and the distance between two consecutive holes in the perforation string. b) The periodicity cell $\varpi^ \varepsilon$ is obtained by removing a periodic family of holes of diameter $O( \varepsilon)$ from $\varpi^0\equiv (-1/2, 1/2)\times(0, H)$. It contains one perforation string
The strip $\Xi$ with the hole $\omega$. $\Xi$ is involved with the unit cell for the homogenization problem (13)-(16)