On a magnetic characterization of spectral minimal partitions

Helffer, 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 -