Introduction to some conjectures for spectral minimal partitions

Abstract

Abstract Given a bounded open set Ω in R n (or in a Riemannian manifold) and a partition of Ω by k open sets D j , we consider the quantity max j λ ( D j ) where λ ( D j ) is the ground state energy of the Dirichlet realization of the Laplacian in D j . If we denote by L k ( Ω ) the infimum over all the k -partitions of max j λ ( D j ), 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.