Optimisation of Eigenvalues of the Dirichlet Laplacian with a Surface Area Restriction
Optimisation of Eigenvalues of the Dirichlet Laplacian with a Surface Area Restriction
复制标题
表面积限制的狄利克雷拉普拉斯特征值优化
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
P. Freitas
中科院分区：
文献类型：
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作者：
Pedro R. S. Antunes;P. Freitas
我们对Dirichlet Laplacian的低频进行了数值优化，分别在2维和3维范围内进行了周长和表面积的限制。在前一种情况下，我们处理前50个特征值，并测量相应的优化器接近磁盘的速度，而在后一种情况下，我们优化前20个特征值。我们推导出了一系列优化器必须满足的理论相容条件，并用这些条件检验了我们的数值结果。我们还考虑了具有固定周长的矩形和具有曲面限制的平行四面体的情况，分别计算了前$$10^7$$107和$$10^6$$106的最优特征值。在此背景下，我们证明了在任何维度上收敛到立方体，并将数值结果与我们的收敛速度的理论估计进行了比较。
We perform a numerical optimisation of the low frequencies of the Dirichlet Laplacian with perimeter and surface area restrictions, in two and 3-dimensions, respectively. In the former case, we handle the first 50 eigenvalues and measure the rate at which the corresponding optimisers approach the disk, while in the latter we optimise the first twenty eigenvalues. We derive theoretical compatibility conditions which must be satisfied by a sequence of optimisers and test our numerical results against these. We also consider the cases of rectangles with a fixed perimeter and parallelepipeds with a surface restriction for which we compute the first $$10^7$$107 and $$10^6$$106 optimal eigenvalues, respectively. In this context, we prove convergence to the cube in any dimensions and compare the numerical results with our theoretical estimates for the rate of convergence.