A review on large k minimal spectral kpartitions and Pleijel's Theorem
Abstract
In this survey, we review the properties of minimal spectral $k$partitions in the twodimensional case and revisit their connections with Pleijel's Theorem. We focus on the large $k$ problem (and the hexagonal conjecture) in connection with two recent preprints by J. Bourgain and S. Steinerberger on the Pleijel Theorem. This leads us also to discuss some conjecture by I. Polterovich, in relation with square tilings. We also establish a Pleijel Theorem for AharonovBohm Hamiltonians and deduce from it, via the magnetic characterization of the minimal partitions, some lower bound for the number of critical points of a minimal partition.
 Publication:

arXiv eprints
 Pub Date:
 September 2015
 DOI:
 10.48550/arXiv.1509.04501
 arXiv:
 arXiv:1509.04501
 Bibcode:
 2015arXiv150904501H
 Keywords:

 Mathematics  Spectral Theory;
 35B05
 EPrint:
 Conference in honor of James Ralston's 70th birthday