Abstract
In a recent paper with Thomas Hoffmann-Ostenhof, we proved that the number of critical points \(\ell _k\) in the boundary set of a minimal k-partition tends to \(+\infty \) as \(k\rightarrow +\infty \). In this note, we show that \(\ell _k\) 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 \(\ell _k\) in the boundary set of a k-minimal partition tends to \(+\infty \) as \(k\rightarrow +\infty \). In this note, we show that \(\ell _k\) 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 \(\ell _k\) dans le bord d’une k-partition minimale tend vers \(+\infty \) lorsque \(k\rightarrow +\infty \). Dans cette note, nous montrons que \(\ell _k\) 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.
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Notes
Possibly after a modification of the open sets of the partition by capacity 0 subsets.
The half curves meet with equal angle at each critical point of N and also at the boundary together with the tangent to the boundary.
As always in the paper, our notion of partition is weak: the \(D_i\) are simply assumed to be mutually disjoint.
Pleijel’s inequality is correct if we add the condition \(t \ge 2\) which was not mentioned there.
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Acknowledgments
I would like to thank T. Hoffmann-Ostenhof for former discussions on the subject, V. Felli and L. Abatangelo for inviting me to give a course on the subject in Milano (February 2015) and the Isaac Newton Institute in Cambridge where the final version of this note was completed, the author being Simons Foundation visiting Fellow there. Finally I would like to thank the anonymous referee for his constructive critics.
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Helffer, B. Lower bound for the number of critical points of minimal spectral k-partitions for k large. Ann. Math. Québec 41, 111–118 (2017). https://doi.org/10.1007/s40316-016-0058-6
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DOI: https://doi.org/10.1007/s40316-016-0058-6