Courant-sharp eigenvalues for the equilateral torus, and for the equilateral triangle

Abstract

We address the question of determining the eigenvalues $\lambda \_n$ (listed in nondecreasing order, with multiplicities) for which Courant's nodal domain theorem is sharp i.e., for which there exists an associated eigenfunction with $n$ nodal domains (Courant-sharp eigenvalues). Following ideas going back to Pleijel (1956), we prove that the only Courant-sharp eigenvalues of the flat equilateral torus are the first and second, and that the only Courant-sharp Dirichlet eigenvalues of the equilateral triangle are the first, second, and fourth eigenvalues. In the last section we sketch similar results for the right-angled isosceles triangle and for the hemiequilateral triangle.

Slight modifications and some misprints corrected

References in corpus (2)

• Nodal and spectral minimal partitions -- The state of the art in 2015 --
• Nodal domains in the square---the Neumann case

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• Courant-sharp eigenvalues of compact flat surfaces: Klein bottles and cylinders
• Optimal partitions for the sum and the maximum of eigenvalues
• Courant-sharp eigenvalues of a two-dimensional torus