The Transition to a Point Constraint in a Mixed Biharmonic Eigenvalue Problem

The mixed-order eigenvalue problem −δ∆u + ∆u + λu = 0 with δ > 0, modeling small amplitude vibrations of a thin plate, is analyzed in a bounded 2-D 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 −δ∆u, 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. Three ranges of δ ≪ 1 are uncovered and analyzed: δ = O(ε), δ ≪ O(ε), and O(ε) ≪ δ ≪ 1. In the regime O(ε) ≪ δ ≪ 1 it is shown that the leading-order asymptotic behavior of an eigenvalue of the mixed-order eigenvalue problem 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.