Courant-sharp property for Dirichlet eigenfunctions on the Möbius strip

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, \ldots . A natural toy model for further investigations is the Möbius strip, a non-orientable surface with Euler characteristic $0$, and particularly the "square" Möbius 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.

Revised version prior to publication. Accepted for publication in Portugaliae Mathematica

References in corpus (2)

• An Euler-type formula for partitions of the Möbius strip
• Courant-sharp eigenvalues of compact flat surfaces: Klein bottles and cylinders

Cited by in corpus (2)

• Stability of spectral partitions and the Dirichlet-to-Neumann map
• Courant-sharp eigenvalues of compact flat surfaces: Klein bottles and cylinders