非均匀板的谱optimization
Spectral optimization of inhomogeneous plates
摘要：
本文研究非均匀平板的光谱优化问题。特别是，我们优化的第一特征值的振动板相对于其厚度和/或密度。我们的结果是三重的。首先，我们证明了存在一个最佳的厚度，使用精细的工具铰链上的重排类的拓扑性质。其次，在圆板的情况下，我们提供了一个表征的最佳厚度的Talenti不等式。最后，我们证明了一个稳定的结果时，假设板的厚度和密度线性相关。该证明依赖于应用于双调和算子的H收敛工具。
英文摘要：
This article is devoted to the study of spectral optimisation for inhomogeneous plates. In particular, we optimise the first eigenvalue of a vibrating plate with respect to its thickness and/or density. Our result is threefold. First, we prove existence of an optimal thickness, using fine tools hinging on topological properties of rearrangement classes. Second, in the case of a circular plate, we provide a characterisation of this optimal thickness by means of Talenti inequalities. Finally, we prove a stability result when assuming that the thickness and the density of the plate are linearly related. This proof relies on H-convergence tools applied to biharmonic operators.
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