  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.
