On vibrating thin membranes with mass concentrated near the boundary: an asymptotic analysis

Abstract

We consider the spectral problem \begin{equation*} \left\{\begin{array}{ll} -Δu_{\varepsilon}=λ(\varepsilon)ρ_{\varepsilon}u_{\varepsilon} & {\rm in}\ Ω\\ \frac{\partial u_{\varepsilon}}{\partialν}=0 & {\rm on}\ \partialΩ\end{array}\right. \end{equation*} in a smooth bounded domain $\Omega$ of ${\mathbb{R}}^2$. The factor ${\rho }_{\epsilon }$ which appears in the first equation plays the role of a mass density and it is equal to a constant of order ${\epsilon }^{-1}$ in an $\epsilon$-neighborhood of the boundary and to a constant of order $\epsilon$ in the rest of $\Omega$. We study the asymptotic behavior of the eigenvalues $\lambda (\epsilon )$ and the eigenfunctions $u_{\epsilon }$ as $\epsilon$ tends to zero. We obtain explicit formulas for the first and second terms of the corresponding asymptotic expansions by exploiting the solutions of certain auxiliary boundary value problems.