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- W3085716922 abstract "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, Mobius strips,ldots . 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 $(0,pi) times mathbb{S}^1_r$ where $r in {0.5,1}$ is the radius of the circle $mathbb{S}^1_r$, and we show that the only Courant-sharp Dirichlet eigenvalues of these cylinders are the first and second eigenvalues." @default.
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- W3085716922 date "2022-01-01" @default.
- W3085716922 modified "2023-10-16" @default.
- W3085716922 title "Courant-sharp eigenvalues of compact flat surfaces: Klein bottles and cylinders" @default.
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