Asymptotic Behaviour of Extremal Averages of Laplacian Eigenvalues
Asymptotic Behaviour of Extremal Averages of Laplacian Eigenvalues
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拉普拉斯特征值极值平均值的渐近行为
DOI：
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发表时间：
2017
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通讯作者：
P. Freitas
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作者：
P. Freitas
我们研究了在测度和表面测度限制下 $$mathbb {R}^{n}$$Rn 域上狄利克雷拉普拉斯算子特征值平均值极值的收敛性。在前一种情况下，我们证明 n / 2 次幂的平均值序列是次可加的，并确定其渐近线中高频极限的第一项。在后一种情况下，我们表明随着频率趋向无穷大，最小化器序列收敛到球。类似的结果也适用于诺伊曼边界条件。
We study the convergence of extrema of averages of eigenvalues of the Dirichlet Laplacian on domains in $$mathbb {R}^{n}$$Rn under both measure and surface measure restrictions. In the former case we prove that the sequence of averages to the power n / 2 is sub-additive and determine the first term in its asymptotics in the high-frequency limit. In the latter case, we show that the sequence of minimisers converges to the ball as the frequency goes to infinity. Similar results hold for Neumann boundary conditions.