Lower bound for the number of critical points of minimal spectral k -partitions for k large
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Lower bound for the number of critical points of minimal spectral k-partitions for k large Bernard Helffer1,2
Received: 12 September 2015 / Accepted: 14 January 2016 © Fondation Carl-Herz and Springer International Publishing Switzerland 2016
Abstract In a recent paper with Thomas Hoffmann-Ostenhof, we proved that the number of critical points k in the boundary set of a minimal k-partition tends to +∞ as k → +∞. In this note, we show that k 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 k in the boundary set of a k-minimal partition tends to +∞ as k → +∞. In this note, we show that k 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. Résumé Dans un article récent avec Thomas Hoffmann-Ostenhof, nous avons démontré que le nombre de points critiques k dans le bord d’une k-partition minimale tend vers +∞ lorsque k → +∞. Dans cette note, nous montrons que k croît linéairement avec k comme le suggère la conjecture hexagonale sur le comportement asymptotique de l’énergie de ces partitions minimales. Comme dans la preuve originelle par Pleijel de son théorème, la démonstration s’appuie sur l’inégalité de Faber-Krahn et la formule de Weyl, mais ici, en utilisant la caractérisation magnétique des partitions minimales, nous avons à établir une formule de Weyl pour un opérateur d’Aharonov-Bohm contrôlé par rapport au nombre k de pôles.
B
Bernard Helffer [email protected]
1
Laboratoire de mathématique, Univ Paris-Sud, CNRS, Université Paris-Saclay, Bâtiment 425, 91405 Orsay Cedex, France
2
Laboratoire de Mathématiques Jean Leray, Université de Nantes, 44322 Nantes, France
123
B. Helffer
Keywords
Minimal partitions · Spectral theory · Laplacian · Aharonov-Bohm
Mathematics Subject Classification
35P05
1 Introduction We consider the Dirichlet Laplacian in a bounded regular domain ⊂ R2 . In [10] we have analyzed the relations between the nodal domains of the real-valued eigenfunctions of this Laplacian and the partitions of by k disjoint open sets Di which are minimal in the sense that the maximum over the Di ’s of the ground state energy (or smallest eigenvalue) λ(Di ) of the Dirichlet realization of the Laplacian in Di
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