arXiv research

Researchers found the first and second eigenvalues are Courant-sharp on a Möbius strip.

We address the question of determining the eigenvalues $λ\_n$ (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), …

Study finds only first and second eigenvalues are Courant-sharp for flat Klein bottle and cylinders.

Study how nodal domains change on surfaces under perturbations.

Let $Ω\subset \mathbb R^d\,, d\geq 2$, be a bounded open set, and denote by $λ\_j(Ω), j\geq 1$, the eigenvalues of the Dirichlet Laplacian arranged in nondecreasing order, with multiplicities. The weak form of Pleijel's theorem states that the number of eigenvalues $λ\_j(Ω)$, for which there exists an associated eigenf…

Proves an Euler-type formula for Möbius strip partitions.

Detailed proofs and extensions of eigenvalue multiplicity bounds for spheres and plane domains.