Lower bound for the number of critical points of minimal spectral k-partitions for k large

TL;DR

This paper demonstrates that the number of critical points in minimal spectral k-partitions grows linearly with k, supporting a hexagonal conjecture, by extending Weyl's formula to Aharonov-Bohm operators with multiple poles.

Contribution

It establishes a linear lower bound on the critical points of minimal spectral partitions for large k, using a magnetic characterization and Weyl's formula for Aharonov-Bohm operators.

Findings

Critical points grow linearly with k

Supports hexagonal conjecture on asymptotic behavior

Extends Weyl's formula to magnetic operators

Abstract

In a recent paper with Thomas Hoffmann-Ostenhof, we proved that the number of critical points in the boundary set of a k-minimal partition tends to infinity as k tends to infinity. In this note, we show that this number increases linearly with k as suggested by a hexagonal conjecture about the asymptotic behavior of the energy of these minimal partitions. As the original proof by Pleijel, this involves Faber-Krahn's inequality and Weyl's formula, but this time, due to the magnetic characterization of the minimal partitions, we have to establish a Weyl's formula for Aharonov-Bohm operator controlled with respect to a k-dependent number of poles.