Optimal partitions for the sum and the maximum of eigenvalues

In this paper we compare the candidates to be spectral minimal partitions for two criteria: the maximum and the average of the first eigenvalue on each subdomains of the partition. We analyze in detail the square, the disk and the equilateral triangle. Using numerical simulations, we propose candidates for the max, prove that most of them can not be optimal for the sum and then exhibit better candidates for the sum.