# American Institute of Mathematical Sciences

March  2011, 6(1): 1-35. doi: 10.3934/nhm.2011.6.1

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

Received  June 2010 Revised  December 2010 Published  March 2011

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.
Citation: Delfina Gómez, Sergey A. Nazarov, Eugenia Pérez. Spectral stiff problems in domains surrounded by thin stiff and heavy bands: Local effects for eigenfunctions. Networks & Heterogeneous Media, 2011, 6 (1) : 1-35. doi: 10.3934/nhm.2011.6.1
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