# On a magnetic characterization of spectral minimal partitions

### Bernard Helffer

Université de Nantes, France### Thomas Hoffmann-Ostenhof

Universität Wien, Austria

## 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, we give the proof of one conjecture formulated in [5] and [16] about a magnetic characterization of the minimal partitions when $n=2$.

## Cite this article

Bernard Helffer, Thomas Hoffmann-Ostenhof, On a magnetic characterization of spectral minimal partitions. J. Eur. Math. Soc. 15 (2013), no. 6, pp. 2081–2092

DOI 10.4171/JEMS/415