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Nodal domains and spectral minimal partitions. (Domaines nodaux et partitions spectrales minimales.) (French) Zbl 1216.35020

This expository paper is devoted to the study of the Dirichlet problem for a Laplacian in a bounded domain \(\Omega\). The author discusses the connections between the partitions of \(\Omega\) by nodal domains of a characteristic function of this Laplacian and that of \(k\) open sets \(D_i\) which are minimal, for a fixed \(k\), the maximum of the smallest characteristic value \(\lambda(D_i)\) of the Dirichlet realization of the Laplacian on \(D_i\).
There are presented results of Conti-Terracini-Verzini, Helffler-Hoffmann-Ostenhof-Terracini and Pleijel. The author emphasizes on minimal partitions and the strictly Courant case. He discusses the examples of the disk and of the rectangle and presents an approach for the regular case.

MSC:

35J25 Boundary value problems for second-order elliptic equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35P05 General topics in linear spectral theory for PDEs
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