On the Eigenvalues of the Biharmonic Steklov Problem on a Thin Set

Abstract

This paper investigates the asymptotic behavior of the eigenvalues of the biharmonic operator on a thin set with Steklov boundary conditions. The thin set is a tubular neighborhood ${\omega }_{\epsilon }$ of a planar smooth domain $\Omega$. We prove that, as $\epsilon \to 0$, all Steklov eigenvalues satisfy ${\lambda }_{\epsilon ,k}\sim \epsilon {\lambda }_k$, where ${\lambda }_k$ is the $k$th eigenvalue of an explicit one-dimensional limiting problem on $(0,|\partial \Omega \left|\right)$ involving the curvature $\kappa$. The proof relies on the use of Fermi coordinates, a key coercivity estimate for the rescaled bilinear form, and $E$-compact convergence of the associated resolvent operators.