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 $\Omega$ of $\mathbb R^2$. The factor $\rho_{\varepsilon}$ which appears in the first equation plays the role of a mass density and it is equal to a constant of order $\varepsilon^{-1}$ in an $\varepsilon$-neighborhood of the boundary and to a constant of order $\varepsilon$ in the rest of $\Omega$. We study the asymptotic behavior of the eigenvalues $\lambda(\varepsilon)$ and the eigenfunctions $u_{\varepsilon}$ as $\varepsilon$ 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.