On a magnetic characterization of spectral minimal partitions

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 previously by V. Bonnaillie-Noel and B. Helffer about a magnetic characterization of the minimal partitions when $n=2$.