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

Sergei A. Nazarov; Rafael Orive-Illera; María-Eugenia Pérez-Martínez; Sergei A. Nazarov; Rafael Orive-Illera; María-Eugenia Pérez-Martínez

doi:10.3934/nhm.2019029

Networks and Heterogeneous Media

2019, Volume 14, Issue 4: 733-757. doi: 10.3934/nhm.2019029

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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

Received: 01 October 2018 Revised: 01 May 2019
Primary: 35B27, 35P05, 47A55, 35J25, 47A10; Secondary: 35P10, 35P15, 47A75

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) $.
- Band-gap structure,
- spectral perturbations,
- homogenization,
- perforated media,
- double periodicity,
- Dirichlet-Laplace operator
Citation: Sergei A. Nazarov, Rafael Orive-Illera, María-Eugenia Pérez-Martínez. Asymptotic structure of the spectrum in a Dirichlet-strip with double periodic perforations[J]. Networks and Heterogeneous Media, 2019, 14(4): 733-757. doi: 10.3934/nhm.2019029

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Abstract

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) $.

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