Asymptotics and Estimates for Spectral Minimal Partitions of Metric Graphs

Abstract

We study properties of spectral minimal partitions of metric graphs within the framework recently introduced in Kennedy et al. (Calc Var 60:6, 2021). 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.

Phase Separation, Optimal Partitions, and Nodal Solutions to the Yamabe Equation on the Sphere

“…Particular interest has been shown when the cost involves Dirichlet eigenvalues (leading to spectral optimal partitions) both in Euclidean spaces (see for instance the survey [6,33] or the recent [1,42,57] and references therein), and in the context of metric graphs (see e.g. [34,36] and references).…”

Standing waves on quantum graphs

“…The functions V and A are called respectively electric and magnetic potentials and they act on the domain of H by multiplication. In what follows, we will not consider the magnetic Schrödinger operator because it played only a very minor role until now (the only references known to the authors in nonlinear problems are [44,83]).…”

A theory of spectral partitions of metric graphs

“…3.13] shows that things are not that simple. Even the case of p ∈ (0, ∞) requires a relatively fine control of the behaviour of the optimal partitions, and will be deferred to a later work [HKMP20].…”