Cheeger cuts and Robin spectral minimal partitions of metric graphs

TL;DR

This paper investigates the relationship between Cheeger cuts and Robin spectral minimal partitions on metric graphs, establishing existence, convergence, and asymptotic behaviors of these partitions as Robin parameters vary.

Contribution

It proves the existence of minimising partitions for both Cheeger and Robin spectral functionals and demonstrates their convergence and asymptotic relations as Robin parameters tend to zero or infinity.

Findings

Robin spectral minimal energies converge to Cheeger constants as Robin parameter approaches zero.

Robin spectral minimal partitions converge to Cheeger cuts as Robin parameter approaches zero.

As Robin parameter approaches infinity, partitions converge to Dirichlet minimal energy partitions.

Abstract

We study partition problems based on two ostensibly different kinds of energy functionals defined on $k$-partitions of metric graphs: Cheeger-type functionals whose minimisers are the $k$-Cheeger cuts of the graph, and the corresponding values are the $k$-Cheeger constants of the graph; and functionals built using the first eigenvalue of the Laplacian with positive, i.e. absorbing, Robin (delta) vertex conditions at the boundary of the partition elements. We prove existence of minimising $k$-partitions, $k \geq 2$, for both these functionals. We also show that, for each $k \geq 2$, as the Robin parameter $\alpha \to 0$, up to a renormalisation the spectral minimal Robin energy converges to the $k$-Cheeger constant. Moreover, up to a subsequence, the Robin spectral minimal $k$-partitions converge in a natural sense to a $k$-Cheeger cut of the graph. Finally, we show that as $\alpha \to \infty$ there is convergence in a similar sense to the corresponding Dirichlet minimal energy and partitions. It is strongly expected that similar results hold on general (smooth, bounded) Euclidean domains and manifolds.