Remarks on the boundary set of spectral equipartitions

Given a bounded open set in R2 (or in a Riemannian manifold), and a partition of Ω by k open sets ωj, we consider the quantity maxj λ(ωj), where λ(ωj) is the ground state energy of the Dirichlet realization of the Laplacian in ωj. We denote by Lk(Ω) the infimum of maxj λ(ω) over all k-partitions. A minimal k-partition is a partition that realizes the infimum. Although the analysis of minimal k-partitions is rather standard when k=2 (we find the nodal domains of a second eigenfunction), the analysis for higher values of k becomes non-trivial and quite interesting. Minimal partitions are in particular spectral equipartitions, i.e. the ground state energies λ(ωj) are all equal. The purpose of this paper is to revisit various properties of nodal sets, and to explore if they are also true for minimal partitions, or more generally for spectral equipartitions. We prove a lower bound for the length of the boundary set of a partition in the two-dimensional situation. We consider estimates involving the cardinality of the partition.