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arXiv · 1504.01015

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

Bernard Helffer
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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.

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Bernard Helffer. 2015-04-04. Lower bound for the number of critical points of minimal spectral k-partitions for k large. https://arxiv.org/abs/1504.01015

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