We consider a partially hinged composite plate problem and we investigate qualitative properties, e.g. symmetry and monotonicity, of the eigenfunction corresponding to the density minimizing the first eigenvalue. The analysis is performed by showing related properties of the Green function of the operator and by applying polarization with respect to a fixed plane. As a by-product of the study, we obtain a Hopf type boundary lemma for the operator having its own theoretical interest. The statements are complemented by numerical results.

中文翻译：

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关于部分铰接复合板的对称性

我们考虑了部分铰接的复合板问题，并研究了对应于最小化第一特征值的密度的特征函数的定性性质，例如对称性和单调性。通过显示操作员的格林函数的相关属性并通过相对于固定平面施加极化来执行分析。作为研究的副产品，我们为具有自己理论兴趣的算子获得了Hopf型边界引理。这些陈述由数值结果补充。