Under Consideration for Publication in the Siam Journal of Applied Math the Transition to a Point Constraint in a Mixed Biharmonic Eigenvalue Problem

The mixed-order eigenvalue problem $-\delta \Delta^2 u + \Delta u + \lambda u = 0$ with $\delta>0$, modeling small amplitude vibrations of a thin plate, is analyzed in a bounded two-dimensional domain $\Omega$ that contains a single small hole of radius $\varepsilon$ centered at some $x_0\in \Omega$. Clamped conditions are imposed on the boundary of $\Omega$ and on the boundary of the small hole. In the limit $\varepsilon\to 0$, and for $\delta={\mathcal O}(1)$, the limiting problem for $u$ must satisfy the additional point constraint $u(x_0)=0$. To determine how the eigenvalues of the Laplacian in a domain with a small hole are perturbed by adding the small fourth-order term $-\delta \Delta^2 u$, together with an additional boundary condition on $\partial\Omega$ and on the hole boundary, the asymptotic behavior of the eigenvalues of the mixed-order eigenvalue problem are studied in the dual limit $\varepsilon\to 0$ and $\delta\to 0$. Leading-order behaviors of eigenvalues are determined for three ranges ...