JournalsjemsVol. 15, No. 6pp. 2081–2092

On a magnetic characterization of spectral minimal partitions

  • Bernard Helffer

    Université de Nantes, France
  • Thomas Hoffmann-Ostenhof

    Universität Wien, Austria
On a magnetic characterization of spectral minimal partitions cover
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Abstract

Given a bounded open set Ω\Omega in Rn\mathbb R^n (or in a Riemannian manifold) and a partition of Ω\Omega by kk open sets DjD_j, we consider the quantity maxjλ(Dj)\max_j \lambda(D_j) where λ(Dj)\lambda(D_j) is the ground state energy of the Dirichlet realization of the Laplacian in DjD_j. If we denote by Lk(Ω)\mathfrak L_k(\Omega) the infimum over all the kk-partitions of maxjλ(Dj)\max_j \lambda(D_j), a minimal kk-partition is then a partition which realizes the infimum. When k=2k=2, we find the two nodal domains of a second eigenfunction, but the analysis of higher kk'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=2n=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