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

In a recent paper with Thomas Hoffmann-Ostenhof, we proved that the number of critical points ℓk in the boundary set of a minimal k-partition tends to + ∞ as k→ + ∞. In this note, we show that ℓk increases linearly with k as suggested by a hexagonal conjecture about the asymptotic behavior of the energy of these minimal partitions. As in the original proof by Pleijel of his celebrated theorem, 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. In a recent paper with Thomas Hoffmann-Ostenhof, we proved that the number of critical points ℓk in the boundary set of a k-minimal partition tends to + ∞ as k→ + ∞. In this note, we show that ℓk 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. © 2016, Fondation Carl-Herz and Springer International Publishing Switzerland.