Journal of the European Mathematical Society

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Volume 15, Issue 6, 2013, pp. 2081–2092
DOI: 10.4171/JEMS/415

Published online: 2013-10-16

On a magnetic characterization of spectral minimal partitions

Bernard Helffer[1] and Thomas Hoffmann-Ostenhof[2]

(1) Université de Nantes, France
(2) Universität Wien, Austria

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 \cite{BH} and \cite{HeEg} about a magnetic characterization of the minimal partitions when $n=2$.

Keywords: minimal partitions, nodal sets, Aharonov-Bohm Hamiltonians, Courant's nodal theorem

Helffer Bernard, Hoffmann-Ostenhof Thomas: On a magnetic characterization of spectral minimal partitions. J. Eur. Math. Soc. 15 (2013), 2081-2092. doi: 10.4171/JEMS/415