  Let $\Omega$ be a bounded domain in $\mathbb R^2$ with smooth boundary
$\partial\Omega$, and let $\omega_h$ be the set of points in $\Omega$ whose
distance from the boundary is smaller than $h$. We prove that the eigenvalues
of the biharmonic operator on $\omega_h$ with Neumann boundary conditions
converge to the eigenvalues of a limiting problem in the form of system of
differential equations on $\partial\Omega$.
