Nodal and spectral minimal partitions -- The state of the art in 2015 --
It provides a comprehensive survey for mathematicians working on spectral geometry and partition problems, but is largely a review of existing results.
This paper reviews the state of the art on nodal and spectral minimal partitions, discussing Courant sharp cases, minimal k-partitions for specific domains, topology of regular partitions, and asymptotic behavior.
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.