# Moscow Mathematical Journal

Volume 15, Issue 3, July–September 2015 pp. 455–495.

Nodal Domains in the Square—the Neumann Case

**Authors**:
Bernard Helffer (1) and Mikael Persson Sundqvist (2)

**Author institution:**(1) Laboratoire de Mathématiques UMR CNRS 8628, Université Paris-Sud - Bật 425, F-91405 Orsay Cedex, France and Laboratoire Jean Leray, Université de Nantes, France

(2) Lund University, Department of Mathematical Sciences, Lund, Sweden

**Summary: **

Å. Pleijel has proved that in the case of the Laplacian on the square with Neumann condition, the equality in the Courant nodal theorem (Courant sharp situation) can only be true for a finite number of eigenvalues. We identify five Courant sharp eigenvalues for the Neumann Laplacian in the square, and prove that there are no other cases.

2010 Math. Subj. Class. 35B05; 35P20, 58J50.

**Keywords:**Nodal domains, Courant theorem, Square, Neumann.

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