Nodal and spectral minimal partitions -- The state of the art in 2015 --

TL;DR

This paper reviews the state of the art in nodal and spectral minimal partitions as of 2015, discussing examples, topological properties, and asymptotic behavior, with connections to Courant sharp cases and Aharonov-Bohm approach.

Contribution

It provides a comprehensive overview of the latest developments in minimal partitions, including new examples, topological insights, and asymptotic analysis, linking spectral theory with geometric and topological methods.

Findings

Identification of Courant sharp cases in nodal partitions

Determination of minimal k-partitions for specific domains

Analysis of the asymptotic behavior of minimal partitions

Abstract

In this article, we propose a state of the art concerning the nodal and spectral minimal partitions. First we focus on the nodal partitions and give some examples of Courant sharp cases. Then we are interested in minimal spectral partitions. Using the link with the Courant sharp situation, we can determine the minimal k-partitions for some particular domains. We also recall some results about the topology of regular partitions and Aharonov-Bohm approach. The last section deals with the asymptotic behavior of minimal k-partition.