论文标题
光谱图的光谱最小分区的渐近学和估计值
Asymptotics and estimates for spectral minimal partitions of metric graphs
论文作者
论文摘要
我们研究了最近在[Kennedy等人，Calc，Calc。 var。 60（2021），61]。我们为不同类别的分区中的最小分区能提供了尖锐的下层和上层估计；尽管下限让人联想到经典的公制图形不平等现象，但上限更多地参与其中，并且也反映了公制图的组合结构。结合起来，我们推断出这些光谱最小的能量还满足了与具有各种顶点条件的量子图拉普拉斯人的特征值相似的Weyl型渐近法。在两个示例中，我们表明，在最小分区能量的渐近扩展中，通常没有第二项，但是证明可以表明各种行为是可能的。我们还研究了最小分区本身的渐近行为的某些方面。
We study properties of spectral minimal partitions of metric graphs within the framework recently introduced in [Kennedy et al, Calc. Var. 60 (2021), 61]. We provide sharp lower and upper estimates for minimal partition energies in different classes of partitions; while the lower bounds are reminiscent of the classic isoperimetric inequalities for metric graphs, the upper bounds are more involved and mirror the combinatorial structure of the metric graph as well. Combining them, we deduce that these spectral minimal energies also satisfy a Weyl-type asymptotic law similar to the well-known one for eigenvalues of quantum graph Laplacians with various vertex conditions. Drawing on two examples we show that in general no second term in the asymptotic expansion for minimal partition energies can exist, but show that various kinds of behaviour are possible. We also study certain aspects of the asymptotic behaviour of the minimal partitions themselves.
