# On a magnetic characterization of spectral minimal partitions

Bernard Helffer; Thomas Hoffmann-Ostenhof

Journal of the European Mathematical Society (2013)

- Volume: 015, Issue: 6, page 2081-2092
- ISSN: 1435-9855

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topHelffer, Bernard, and Hoffmann-Ostenhof, Thomas. "On a magnetic characterization of spectral minimal partitions." Journal of the European Mathematical Society 015.6 (2013): 2081-2092. <http://eudml.org/doc/277728>.

@article{Helffer2013,

abstract = {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 $\texttt \{max\}_j\lambda (D_j)$ where $\lambda (D_j)$ is the ground state energy of the Dirichlet realization of the Laplacian in $D_j$. If we denote by $\mathcal \{L\}_k(\Omega )$ the infimum over all the $k$-partitions of $\texttt \{max\}_j\lambda (D_j)$, 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 [5] and [16] about a magnetic characterization of the minimal partitions when $n=2$.},

author = {Helffer, Bernard, Hoffmann-Ostenhof, Thomas},

journal = {Journal of the European Mathematical Society},

keywords = {minimal partitions; nodal sets; Aharonov-Bohm Hamiltonians; Courant's nodal theorem; minimal partitions; nodal sets; Aharonov-Bohm Hamiltonians; Courant's nodal theorem},

language = {eng},

number = {6},

pages = {2081-2092},

publisher = {European Mathematical Society Publishing House},

title = {On a magnetic characterization of spectral minimal partitions},

url = {http://eudml.org/doc/277728},

volume = {015},

year = {2013},

}

TY - JOUR

AU - Helffer, Bernard

AU - Hoffmann-Ostenhof, Thomas

TI - On a magnetic characterization of spectral minimal partitions

JO - Journal of the European Mathematical Society

PY - 2013

PB - European Mathematical Society Publishing House

VL - 015

IS - 6

SP - 2081

EP - 2092

AB - 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 $\texttt {max}_j\lambda (D_j)$ where $\lambda (D_j)$ is the ground state energy of the Dirichlet realization of the Laplacian in $D_j$. If we denote by $\mathcal {L}_k(\Omega )$ the infimum over all the $k$-partitions of $\texttt {max}_j\lambda (D_j)$, 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 [5] and [16] about a magnetic characterization of the minimal partitions when $n=2$.

LA - eng

KW - minimal partitions; nodal sets; Aharonov-Bohm Hamiltonians; Courant's nodal theorem; minimal partitions; nodal sets; Aharonov-Bohm Hamiltonians; Courant's nodal theorem

UR - http://eudml.org/doc/277728

ER -

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