The Buckling and Clamped Plate Problems on Differential Forms

TL;DR

This paper extends buckling and clamped-plate eigenvalue problems to differential forms on Riemannian manifolds, providing spectral characterizations and estimates that generalize classical results for functions.

Contribution

It introduces a framework for analyzing buckling and clamped-plate problems on differential forms, linking their spectra to those of functions and generalizing existing eigenvalue estimates.

Findings

Eigenvalues characterized for differential forms on manifolds.

Spectra on forms coincide with those on functions in Euclidean domains.

Derived estimates relate eigenvalues of these problems to the Hodge Laplacian.

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.