We study the discrete spectra of boundary value problems for the biharmonic operator describing oscillations of a Kirchhoff plate in the form of a locally perturbed strip with rigidly clamped or simply supported edges. The two methods are applied: variational and asymptotic. The first method shows that for a narrowing plate the discrete spectrum is empty in both cases, whereas for a widening plate at least one eigenvalue appears below the continuous spectrum cutoff for rigidly clamped edges. The presence of the discrete spectrum remains an open question for simply supported edges. The asymptotic method works only for small variations of the boundary. While for a small smooth perturbation the construction of asymptotics is generally the same for both types of boundary conditions, the asymptotic formulas for eigenvalues can differ substantially even in the main correction term for a perturbation with corner points.

中文翻译：

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局部微扰带形式的无限基尔霍夫板的离散谱

我们研究了双调和算子的边界值问题的离散谱，该算子描述了基尔霍夫板的振荡，其形式为具有刚性夹紧或简单支撑边缘的局部扰动条带。应用了两种方法：变分方法和渐近方法。第一种方法表明，对于变窄的板，离散光谱在两种情况下都是空的，而对于加宽的板，至少有一个特征值出现在刚性夹紧边缘的连续光谱截止点之下。对于简单支持的边缘，离散频谱的存在仍然是一个悬而未决的问题。渐近方法仅适用于边界的微小变化。而对于小的平滑扰动，渐近线的构造对于两种类型的边界条件通常是相同的，