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 $-Delta u_arepsilon=lambda (arepsilon ) ho_arepsilon u_arepsilon$ with boundary condition $rac{partial u_arepsilon}{partial_ u}=0$. The factor $ ho_arepsilon$ plays the role of a mass density, and it is equal to a constant of order $arepsilon^{-1}$ in an $arepsilon$-neighborhood of the boundary and to a constant of order $arepsilon$ in the rest of $Omega$. We study the asymptotic behavior of the eigenvalues $lambda (arepsilon)$ and the eigenfunctions $u_arepsilon$ as $arepsilon$ 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.