THE TRANSITION TO A POINT CONSTRAINT IN A MIXED BIHARMONIC EIGENVALUE PROBLEM

The mixed-order eigenvalue problem –δΔ2u + Δu + λu = 0 with δ > 0, modeling small amplitude vibrations of a thin plate, is analyzed in a bounded two-dimensional domain Ω that contains a single small hole of radius ε centered at some x0 ∈ Ω. Clamped conditions are imposed on the boundary of Ω and on the boundary of the small hole. In the limit ε → 0, and for δ = O(1), the limiting problem for u must satisfy the additional point constraint u(x0) = 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 –δΔ2u, together with an additional boundary condition on ∂Ω and on the hole boundary, the asymptotic behavior of the eigenvalues of the mixed-order eigenvalue problem are studied in the dual limit ε → 0 and δ → 0. Leading-order behaviors of eigenvalues are determined for three ranges of δ ≪ 1: δ ≪ O(ε2), δ = O(ε2), and O(ε2) ≪ δ ≪ 1. In the first two of these regimes, the limiting behavior depends of the radius of the hole ε, while in the regime O(ε2) ≪ δ ≪ 1 the eigenvalue is asymptotically independent of ε. Therefore, it is this regime that provides a transition to the point constraint behavior characteristic of the range δ = O(1). The asymptotic results for the eigenvalues are validated by full numerical simulations of the PDE.