Asymptotics and estimates for spectral minimal partitions of metric graphs

TL;DR

This paper investigates the asymptotic behavior and estimates of spectral minimal partitions on metric graphs, establishing bounds, asymptotic laws, and exploring the structure and behavior of these partitions.

Contribution

It provides sharp bounds and Weyl-type asymptotics for spectral minimal energies on metric graphs, extending understanding of their structure and asymptotic properties.

Findings

Spectral minimal energies satisfy a Weyl-type asymptotic law.

No second term generally exists in the asymptotic expansion.

Various behaviors are possible for minimal partitions asymptotically.

Abstract

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.