# American Institute of Mathematical Sciences

June  2010, 28(2): 617-635. doi: 10.3934/dcds.2010.28.617

## Nodal minimal partitions in dimension $3$

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

Received  January 2010 Revised  April 2010 Published  April 2010

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.
Citation: Bernard Helffer, Thomas Hoffmann-Ostenhof, Susanna Terracini. Nodal minimal partitions in dimension $3$. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 617-635. doi: 10.3934/dcds.2010.28.617
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