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

We consider the spectral problem \begin{equation*} \left{\begin{array}{ll} -\Delta u_{\varepsilon}=\lambda(\varepsilon)\rho_{\varepsilon}u_{\varepsilon} & {\rm in}\ \Omega\ \frac{\partial u_{\varepsilon}}{\partial\nu}=0 & {\rm on}
\partial\Omega \end{array}\right. \end{equation*} in a smooth bounded domain $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$. 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.