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

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 $\Omega$ isa 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 stiffnessconstants 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, middleand high frequencies, we provide asymptotics for the eigenvaluesand the corresponding eigenfunctions. For $m>2$, we highlight the middlefrequencies for which the corresponding eigenfunctions may belocalized asymptotically in small neighborhoods of certain pointsof the boundary.

Locations

• Networks and Heterogeneous Media - View - PDF