Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 010, 22 pages      arXiv:2412.05612      https://doi.org/10.3842/SIGMA.2026.010

The Buckling and Clamped Plate Problems on Differential Forms

Fida El Chami a, Nicolas Ginoux b, Georges Habib ab, Ola Makhoul a and Simon Raulot c
a) Department of Mathematics, Faculty of Sciences II, Lebanese University, P.O. Box 90656 Fanar-Matn, Lebanon
b) Université de Lorraine, CNRS, IECL, 57000 Metz, France
c) Université de Rouen Normandie, CNRS, Normandie Univ, LMRS UMR 6085, 76000 Rouen, France

Received July 17, 2025, in final form January 22, 2026; Published online February 03, 2026

Abstract
We extend the buckling and clamped-plate problems to the context of differential forms on compact Riemannian manifolds with smooth boundary. We characterize their smallest eigenvalues and prove that, in the case of bounded Euclidean domains, their spectra without multiplicities on forms coincide with the spectra of the corresponding problems on functions. We obtain various estimates involving the first eigenvalues of the mentioned problems and the ones of the Hodge Laplacian with respect to Dirichlet and absolute boundary conditions on forms. These estimates generalize previous ones in the case of functions.

Key words: boundary value problem; eigenvalue estimate; buckling problem; clamped plate problem.

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