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...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:Journal of the European Mathematical Society : JEMS 2013-01, Vol.15 (6), p.2081-2092
Hauptverfasser: Helffer, Bernard, Hoffmann-Ostenhof, Thomas
Format: Artikel
Sprache:eng
Schlagworte:
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Beschreibung
Zusammenfassung: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 $n=2$.
ISSN:1435-9855
1435-9863
DOI:10.4171/JEMS/415