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The question of determining for which eigenvalues there exists an eigenfunction which has the same number of nodal domains as the label of the associated eigenvalue (Courant-sharp property) was motivated by the analysis of minimal spectral partitions. In previous works, many examples have been analyzed corresponding to squares, rectangles, disks, triangles, tori, Möbius strips, and so forth. A natural toy model for further investigations is the flat Klein bottle, a non-orientable surface with Euler characteristic 0, and particularly the Klein bottle associated with the square torus, whose eigenvalues have higher multiplicities. In this note, we prove that the only Courant-sharp eigenvalues of the flat Klein bottle associated with the square torus (resp. with square fundamental domain) are the first and second eigenvalues. We also consider the flat cylinders (0,π) × S1r where r ∈ {0.5,1} is the radius of the circle S1r, and we show that the only Courant-sharp Dirichlet eigenvalues of these cylinders are the first and second eigenvalues. [ABSTRACT FROM AUTHOR]

The question of determining for which eigenvalues there exists an eigenfunction which has the same number of nodal domains as the label of the associated eigenvalue (Courant-sharp property) was motivated by the analysis of minimal spectral partitions. In previous works, many examples have been analyzed corresponding to squares, rectangles, disks, triangles, tori, . . . . A natural toy model for further investigations is the Mo¨ bius strip, a non-orientable surface with Euler characteristic 0, and particularly the "square" Mo¨ bius strip whose eigenvalues have higher multiplicities. In this case, we prove that the only Courant-sharp Dirichlet eigenvalues are the first and the second, and we exhibit peculiar nodal patterns. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we determine, in the case of the Laplacian on the flat three-dimensional torus (ℝ/ℤ)³, all the eigenvalues having an eigenfunction which satisfies the Courant nodal domain theorem with equality (Courant- sharp situation). Following the strategy of Å. Pleijel (1956), the proof is a combination of an explicit lower bound of the counting function and a Faber-Krahn-type inequality for domains in the torus, deduced, as in the work of P. Bérard and D. Meyer (1982), from an isoperimetric inequality. [ABSTRACT FROM AUTHOR]