We consider a partially hinged rectangular plate and its normal modes. The dynamical properties of the plate are influenced by the spectrum of the associated eigenvalue problem. In order to improve the stability of the plate, we place a certain amount of denser material in appropriate regions. If we look at the partial differential equation appearing in the model, this corresponds to insert a suitable weight coefficient inside the equation. A possible way to locate such regions is to study the eigenvalue problem associated to the aforementioned weighted equation. In this paper, we focus our attention essentially on the first eigenvalue and on its minimization in terms of the weight. We prove the existence of minimizing weights inside special classes and we try to describe them together with the corresponding eigenfunctions.

中文翻译：

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关于加筋部分铰接板的第一频率

我们考虑一个部分铰接的矩形板及其正常模式。板的动力学特性受相关特征值问题的谱的影响。为了提高板的稳定性，我们在适当的区域放置了一定数量的密度较大的材料。如果我们看一下模型中出现的偏微分方程，这对应于在方程中插入一个合适的权重系数。定位此类区域的一种可能方法是研究与上述加权方程相关的特征值问题。在本文中，我们主要关注第一个特征值及其权重最小化。我们证明了在特殊类中最小化权重的存在，并尝试将它们与相应的特征函数一起描述。