Optimal partitions for the sum and the maximum of eigenvalues

TL;DR

This paper investigates optimal spectral partitions for minimizing the maximum and sum of the first eigenvalues across subdomains, analyzing specific shapes and proposing improved partition candidates through numerical simulations.

Contribution

It introduces new candidate partitions for the sum criterion and proves the non-optimality of many for the max criterion, advancing understanding of spectral partitioning.

Findings

Proposed new candidates for minimal sum partitions.

Proved most candidates are not optimal for the maximum eigenvalue.

Numerical simulations support the analysis and candidate proposals.

Abstract

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.