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Nodal domains and spectral minimal partitions. (English) Zbl 1171.35083
Two-dimensional Schrödinger operators in bounded domains are considered. Results concerning the local properties of the nodal set of an eigenfunction are extended up to the boundary. For the two-dimensional case, it is shown that every open optimal partition is regular and strong. The relations between the nodal domains of eigenfunctions, spectral minimal partitions and spectral properties of the corresponding operator are analized. The existence and regularity results of the minimal partitions and the characterization of the minimal partitions, associated with nodal sets as the nodal domains of Courant-sharp eigenfunctions, represent the main result of the paper.

MSC:
35P05 General topics in linear spectral theory for PDEs
35J10 Schrödinger operator, Schrödinger equation
35B65 Smoothness and regularity of solutions to PDEs
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