On a magnetic characterization of spectral minimal partitions
Bernard Helffer, Thomas Hoffmann-Ostenhof · Journal of the European Mathematical Society · 2013
Given a bounded open set \Omega in \mathbb R^n (or in a Riemannian manifold) and a partition of \Omega by k open sets D_j , we consider the quantity \max_j \lambda(D_j) where \lambda(D_j) is the ground state energy of the Dirichlet realization of the Laplacian in D_j . If we denote by \mathfrak L_k(\Omega) the infimum over all the k -partitions of \max_j \lambda(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, we give the proof of one conjecture formulated in [5] and [16] about a magnetic characterization of the minimal partitions when n=2 .