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The Buckling and Clamped Plate Problems on Differential Forms

Fida El Chami, Nicolas Ginoux, Georges Habib, Ola Makhoul, Simon Raulot

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.

The Buckling and Clamped Plate Problems on Differential Forms

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.

Paper Structure

This paper contains 9 sections, 14 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. The buckling and clamped plate problems
  3. The buckling problem
  4. The clamped plate problem
  5. Eigenvalues of the buckling and clamped plate operators
  6. Eigenvalues for bounded Euclidean domains
  7. General estimates
  8. Manifolds with Weitzenböck curvature operator bounded from below
  9. Domains in the unit sphere

Key Result

Lemma 2.1

Let $(M^n,g)$ be a compact Riemannian manifold with smooth boundary $\partial M$ and let $\nu$ be the inward unit normal vector field to the boundary. The following boundary value problem: for given $f\in\Omega^p(M)$, $\omega_1\in\Omega^p(M)_{|_{\partial M}}$, $\omega_2\in\Omega^{p-1}(\partial M)$, $\omega_3\in\Omega^p(\partial M)$, is elliptic in the sense of Lopatinskiı̆--Shapiro.

Theorems & Definitions (20)

  • Lemma 2.1
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Proposition 3.1
  • Theorem 3.3
  • Theorem 3.5
  • Corollary 3.7
  • Remark 3.8
  • ...and 10 more