Introduction to some conjectures for spectral minimal partitions

Given a bounded open set X in Rn (or in a Riemannian manifold) and a partition of X by
k open sets Dj, we consider the quantity maxjk(Dj) where k(Dj) is the ground state energy of the
Dirichlet realization of the Laplacian in Dj. If we denote by LkðXÞ the infimum over all the k-partitions of maxjk(Dj), a minimal k-partition is then a partition which realizes the infimum. When k =2, we find the two nodal domains of a second eigenfunction, but the analysis of higher k’s is non trivial and quite interesting. In this paper, which is complementary of the survey [20], we consider the two-dimensional case and present the properties of minimal spectral partitions, illustrate
the difficulties by considering simple cases like the disk, the rectangle or the sphere (k = 3). We will present also the main conjectures in this rather new subject.