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...
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Veröffentlicht in: | Journal of the European Mathematical Society : JEMS 2013-01, Vol.15 (6), p.2081-2092 |
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Sprache: | eng |
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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 $n=2$. |
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ISSN: | 1435-9855 1435-9863 |
DOI: | 10.4171/JEMS/415 |