论文标题
相对于域的优化steklov-lamé特征值
Optimization of the Steklov-Lamé eigenvalues with respect to the domain
论文作者
论文摘要
这项工作涉及与变量域对steklov-laméeigenvalues的行为相关的理论和数值方面。在建立磁盘的特征结构后，我们证明，对于某些类别的LAMé参数，该磁盘最大化面积下的第一个非零特征值或尺寸二维中的周边约束。这些特征值的上限可根据标量steklov特征值找到，涉及各种几何量。我们证明，对于$ \ varepsilon $ -Cone域的互补的Hausdorff收敛性，steklov-lamé特征值是上半连接的，因此，存在形状，使这些特征值最大化，这些特征值和体积约束。提出了一种基于基本解决方案的数值方法来计算steklov-lamé特征值，从而使数值研究最大化前十个非零特征值的形状。
This work deals with theoretical and numerical aspects related to the behavior of the Steklov-Lamé eigenvalues on variable domains. After establishing the eigenstructure for the disk, we prove that for a certain class of Lamé parameters, the disk maximizes the first non-zero eigenvalue under area or perimeter constraints in dimension two. Upper bounds for these eigenvalues can be found in terms of the scalar Steklov eigenvalues, involving various geometric quantities. We prove that the Steklov-Lamé eigenvalues are upper semicontinuous for the complementary Hausdorff convergence of $\varepsilon$-cone domains and, as a consequence, there exist shapes maximizing these eigenvalues under convexity and volume constraints. A numerical method based on fundamental solutions is proposed for computing the Steklov-Lamé eigenvalues, allowing to study numerically the shapes maximizing the first ten non-zero eigenvalues.
