On Spectral Minimal Partitions: A Survey

Abstract

Given a bounded open set Ω in \({\mathbb{R}^n}\) (or a Riemannian manifold) and a partition of Ω by k open sets Dj, we can consider the quantity maxjλ(Dj) where λ(Dj) is the groundstate energy of the Dirichlet realization of the Laplacian in Dj. If we denote by \({\mathfrak{L}_k(\Omega)}\) the infimum over all the k-partitions of maxjλ(Dj), a minimal (spectral) k-partition is then a partition which realizes the infimum. Although the analysis is rather standard when k = 2 (we find the nodal domains of a second eigenfunction), the analysis of higher k’s becomes non trivial and quite interesting.