论文标题
公制图的光谱分区理论
A theory of spectral partitions of metric graphs
论文作者
论文摘要
我们引入了一个抽象框架，以研究公制图中的聚类：在适当地测定图形分区的空间之后，我们将拉普拉西亚人限制在簇上，从而产生并利用其光谱差距来定义分区能量的几个概念；这是众所周知的平面域谱最小分区理论的图形，并在[band \ textIt {et al}，comm。\Math。\phys。\ \ textbf {311}（2012），815----838]中包含设置。我们专注于在此类分区定义的大量功能的优化者的存在，同时也研究其定性特性，包括稳定性，规律性和参数依赖性。我们还详细讨论了他们与节点分区理论的相互作用。与域不同，度量图的一维设置允许显式计算和分析 - 而不是数值结果。我们不仅在[conti \ textit {et al}中所研究的光谱最小分区理论中恢复了主要断言，calc。 helffer \ textit {et al}，ann。\ inst。\ henripoincaré肛门。
We introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several notions of partition energies; this is the graph counterpart of the well-known theory of spectral minimal partitions on planar domains and includes the setting in [Band \textit{et al}, Comm.\ Math.\ Phys.\ \textbf{311} (2012), 815--838] as a special case. We focus on the existence of optimisers for a large class of functionals defined on such partitions, but also study their qualitative properties, including stability, regularity, and parameter dependence. We also discuss in detail their interplay with the theory of nodal partitions. Unlike in the case of domains, the one-dimensional setting of metric graphs allows for explicit computation and analytic -- rather than numerical -- results. Not only do we recover the main assertions in the theory of spectral minimal partitions on domains, as studied in [Conti \textit{et al}, Calc.\ Var.\ \textbf{22} (2005), 45--72; Helffer \textit{et al}, Ann.\ Inst.\ Henri Poincaré Anal.\ Non Linéaire \textbf{26} (2009), 101--138], but we can also generalise some of them and answer (the graph counterparts of) a few open questions.
