Nodal minimal partitions in dimension $3$

Bernard Helffer; Thomas Hoffmann-Ostenhof; Susanna Terracini

doi:10.3934/dcds.2010.28.617

Article Contents

2010, Volume 28, Issue 2: 617-635. Doi: 10.3934/dcds.2010.28.617 open access

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Nodal minimal partitions in dimension $3$

1.
Département de Mathématiques, Bat. 425, Université Paris-Sud, 91 405 Orsay Cedex
2.
Institut für Theoretische Chemie, Universit¨at Wien, Währinger Strasse 17, A-1090 Wien
3.
Dipartimento di Matematica, Università di Milano Bicocca, Via Cozzi, 53 20125 Milano

Received: December 31, 2009

Revised: March 31, 2010

Published: June 2010

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Abstract

In continuation of [20], we analyze theproperties of spectral minimal $k$-partitions of an open set$\Omega$ in$\mathbb R^3$ which are nodal, i.e. produced by the nodal domains of an eigenfunction of the Dirichlet Laplacian in $\Omega$. We show that such a partition is necessarily a nodal partition associated with a $k$-th eigenfunction. Hence we have in this case equality in Courant's nodal theorem.

Keywords:

Optimal partitions; Eigenvalues; Nodal domains; Courant nodal Theorem; Spectral minimal partitions.

Mathematics Subject Classification: 35Q40, 35P20, 47F05, 81Q05.

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