On Vibrating Thin Membranes with Mass Concentrated Near the Boundary: An Asymptotic Analysis

In a smooth bounded domain $\Omega$ of $\mathbb R^2$ we consider the spectral problem $-\Delta u_{\varepsilon}= \lambda(\varepsilon)\rho_{\varepsilon}u_{\varepsilon}$ with boundary condition $\frac{\partial u_{\varepsilon}}{\partial\nu}=0$. The factor $\rho_{\varepsilon}$ 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.