The Transition to a Point Constraint in a Mixed Biharmonic Eigenvalue Problem
Alan E. Lindsay, Michael J. Ward, Théodore Kolokolnikov · SIAM Journal on Applied Mathematics · 2015
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 of $\delta\ll 1$: $\delta\ll {\mathcal O}(\varepsilon^2)$, $\delta={\mathcal O}(\varepsilon^2)$, and ${\mathcal O}(\varepsilon^2)\ll \delta\ll 1$. In the first two of these regimes, the limiting behavior depends of the radius of the hole $\varepsilon$, while in the regime ${\mathcal O}(\varepsilon^2)\ll \delta\ll 1$ the eigenvalue is asymptotically independent of $\varepsilon$. Therefore, it is this regime that provides a transition to the point constraint behavior characteristic of the range $\delta={\mathcal O}(1)$. The asymptotic results for the eigenvalues are validated by full numerical simulations of the PDE.