Spectral stiff problems in domains surrounded
by thin stiff and heavy bands: Local effects for  eigenfunctions

Delfina Gómez; Sergey A. Nazarov; Eugenia Pérez; Delfina Gómez; Sergey A. Nazarov; Eugenia Pérez

doi:10.3934/nhm.2011.6.1

Networks and Heterogeneous Media

2011, Volume 6, Issue 1: 1-35. doi: 10.3934/nhm.2011.6.1

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Spectral stiff problems in domains surrounded by thin stiff and heavy bands: Local effects for eigenfunctions

1.
Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Avenida de los Castros s/n., Santander, 39005
2.
Institute of Mechanical Engineering Problems, RAN V.O.Bol'shoi pr., 61, StPetersburg, 199178
3.
Departamento de Matemática Aplicada y Ciencias de la Computación, Universided de Cantabria, Avenida de los Castros s/n, 39005 Santander

Received: 01 June 2010 Revised: 01 December 2010
Primary: 35P05, 35P20; Secondary: 35B25, 73D30, 47A55, 47A75, 49R05.

We consider the Neumann spectral problem for a second order differential operator, with piecewise constants coefficients, in a domain $\Omega_\varepsilon$ of $R^2$. Here $\Omega_\varepsilon$ is $\Omega \cup \omega_\varepsilon \cup \Gamma$, where $\Omega$ is a fixed bounded domain with boundary $\Gamma$, $\omega_\varepsilon$ is a curvilinear band of variable width $O(\varepsilon)$, and $\Gamma=\overline{\Omega}\cap \overline {\omega_\varepsilon}$. The density and stiffness constants are of order $O(\varepsilon^{-m-1})$ and $O(\varepsilon^{-1})$ respectively in this band, while they are of order $O(1)$ in $\Omega$; $m$ is a positive parameter and $\varepsilon \in (0,1)$, $\varepsilon\to 0$. Considering the range of the low, middle and high frequencies, we provide asymptotics for the eigenvalues and the corresponding eigenfunctions. For $m>2$, we highlight the middle frequencies for which the corresponding eigenfunctions may be localized asymptotically in small neighborhoods of certain points of the boundary.
- Stiff problems,
- asymptotic analysis,
- spectral analysis,
- localized eigenfunctions
Citation: Delfina Gómez, Sergey A. Nazarov, Eugenia Pérez. Spectral stiff problems in domains surroundedby thin stiff and heavy bands: Local effects for eigenfunctions[J]. Networks and Heterogeneous Media, 2011, 6(1): 1-35. doi: 10.3934/nhm.2011.6.1

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Abstract

We consider the Neumann spectral problem for a second order differential operator, with piecewise constants coefficients, in a domain $\Omega_\varepsilon$ of $R^2$. Here $\Omega_\varepsilon$ is $\Omega \cup \omega_\varepsilon \cup \Gamma$, where $\Omega$ is a fixed bounded domain with boundary $\Gamma$, $\omega_\varepsilon$ is a curvilinear band of variable width $O(\varepsilon)$, and $\Gamma=\overline{\Omega}\cap \overline {\omega_\varepsilon}$. The density and stiffness constants are of order $O(\varepsilon^{-m-1})$ and $O(\varepsilon^{-1})$ respectively in this band, while they are of order $O(1)$ in $\Omega$; $m$ is a positive parameter and $\varepsilon \in (0,1)$, $\varepsilon\to 0$. Considering the range of the low, middle and high frequencies, we provide asymptotics for the eigenvalues and the corresponding eigenfunctions. For $m>2$, we highlight the middle frequencies for which the corresponding eigenfunctions may be localized asymptotically in small neighborhoods of certain points of the boundary.

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