On the eigenvalues of the biharmonic operator with Neumann boundary conditions on a thin set

Let Ω$\Omega$ be a bounded domain in R2$\mathbb {R}^2$ with smooth boundary ∂Ω$\partial \Omega$, and let ωh$\omega _h$ be the set of points in Ω$\Omega$ whose distance from the boundary is smaller than h$h$. We prove that the eigenvalues of the biharmonic operator on ωh$\omega _h$ with Neumann boundary conditions converge to the eigenvalues of a limiting problem in the form of a system of differential equations on ∂Ω$\partial \Omega$.