Dice Question Streamline Icon: https://streamlinehq.com

Equality of Drábek and Krasnoselskii variational spectra for higher indices

Determine whether Drábek’s variational eigenvalues coincide with the Krasnoselskii (genus-based) variational eigenvalues for all k≥3 in the graph p-Laplacian setting.

Information Square Streamline Icon: https://streamlinehq.com

Background

Drábek introduced a family of variational eigenvalues using odd continuous surjections from spheres onto admissible sets, which is contained within the Krasnoselskii genus class. It is known that λ1=λ1D and λ2=λ2D, based on classical characterizations.

For higher indices (k≥3), it is not known whether the two constructions agree. Resolving this would either unify or separate these widely used variational frameworks.

References

However, the question of whether the higher variational eigenvalues satisfy $\lambda_k = \lambda_kD$ for $k \geq 3$ remains an open problem.

— Nonlinear spectral graph theory (2504.03566 - Deidda et al., 4 Apr 2025) in Subsubsection “Drábek Spectrum” (within Subsection 3.2, The variational spectrum)