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

Abstract

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 $\partial \Omega$.