The Buckling and Clamped Plate Problems on Differential Forms

Technical Summary: The Buckling and Clamped Plate Problems on Differential Forms

Problem Statement
This paper extends the classical buckling and clamped plate problems, traditionally formulated for scalar functions on Riemannian manifolds, to the context of differential forms. Specifically, the authors consider an $n$-dimensional compact Riemannian manifold $(M, g)$ with a smooth boundary $\partial M$. They investigate two eigenvalue problems for $p$-forms $\omega$:

1. The Clamped Plate Problem:
   $$\Delta^2 \omega = \Gamma \omega \quad \text{on } M, \quad \omega = 0, \quad \nabla_\nu \omega = 0 \quad \text{on } \partial M$$
2. The Buckling Problem:
   $$\Delta^2 \omega = \Lambda \Delta \omega \quad \text{on } M, \quad \omega = 0, \quad \nabla_\nu \omega = 0 \quad \text{on } \partial M$$

Here, $\Delta = d\delta + \delta d$ is the Hodge Laplacian, $\nu$ is the inward unit normal vector field, and the boundary conditions $\omega|_{\partial M} = 0$ and $\nabla_\nu \omega|_{\partial M} = 0$ are shown to be equivalent to the vanishing of tangential and normal components of the form and its codifferential ($\iota^*\omega = 0, \nu \lrcorner \omega = 0, \iota^*\delta\omega = 0, \nu \lrcorner d\omega = 0$). These conditions are established as elliptic in the sense of Lopatinski\u{ı}–Shapiro.

Methodology
The authors employ a variational approach combined with spectral theory and geometric analysis.

• Spectral Existence: They define appropriate Hilbert spaces of $L^2$-integrable forms satisfying the boundary conditions. Using integration by parts formulas (specifically leveraging the vanishing of boundary terms for forms in the defined space $Z$), they establish the self-adjointness and compactness of the associated operators. This allows them to invoke the spectral theorem for compact self-adjoint operators to prove the existence of discrete spectra consisting of eigenvalues with finite multiplicities.
• Variational Characterization: The first eigenvalues $\Gamma_{1,p}$ and $\Lambda_{1,p}$ are characterized via Rayleigh quotients involving the $L^2$-norms of $\Delta \omega$, $d\omega$, and $\delta \omega$.
• Comparison with Scalar Problems: For bounded Euclidean domains, the authors utilize the existence of parallel $p$-forms to construct a mapping between eigenfunctions of the scalar problems and eigenforms of the differential form problems.
• Geometric Inequalities: To derive estimates, the authors utilize the Weitzenb"ock formula ($\Delta = \nabla^*\nabla + W^{[p]}$) and the Reilly formula. They also employ test-forms constructed from the contraction of eigenforms with parallel vector fields (in the Euclidean case) or tangential projections of ambient vector fields (in the spherical case).

Key Contributions and Results

1. Spectral Equivalence in Euclidean Domains:
   The paper proves that for a bounded domain $M \subset \mathbb{R}^n$, the spectra (without multiplicities) of the buckling and clamped plate problems on $p$-forms coincide exactly with the spectra of the corresponding problems on functions (0-forms). Specifically, $\Lambda_{i,p} = \Lambda_{i,0}$ and $\Gamma_{i,p} = \Gamma_{i,0}$. The multiplicity of the eigenvalues for $p$-forms is $\binom{n}{p}$ times that of the scalar case. This result allows for the explicit calculation of eigenvalues for Euclidean balls using known Bessel function zeros.

2. General Eigenvalue Estimates:
   The authors establish a hierarchy of inequalities relating the first eigenvalues of the buckling ($\Lambda_{1,p}$), clamped plate ($\Gamma_{1,p}$), Dirichlet Hodge Laplacian ($\lambda_{1,p}$), and absolute boundary condition Laplacian ($\mu_{1,p}$) problems:

   • $\Gamma_{1,p} < \Lambda_{1,p}^2$
   • $\Lambda_{1,p} \lambda_{1,p} < \Gamma_{1,p}$
   • $\inf(\lambda_{1,p+1}, \lambda_{1,p-1}) \leq \Lambda_{1,p}$
   • $\lambda_{1,1} \leq \Lambda_{1,0}$
   • $\max(\mu_{1,p}, \mu_{1,n-p}) \leq \Lambda_{1,p}$

   These inequalities generalize known results for scalar functions (e.g., those by Payne, Ashbaugh, Laugesen, and Chen et al.) to the setting of differential forms. The paper notes that equalities in these inequalities generally do not hold for bounded Euclidean domains.

3. Curvature-Dependent Lower Bounds:
   Under the assumption that the Weitzenb"ock curvature operator $W^{[p]}$ is bounded from below by a positive constant $\gamma p(n-p)$, the authors derive new lower bounds for the first eigenvalues:

   • $\Lambda_{1,p} > \gamma p(n-p+1)$
   • $\Gamma_{1,p} > \gamma^2 p^2(n-p+1)^2$
     These results generalize previous estimates by Chen, Cheng, Wang, and Xia for scalar functions.

4. Results on the Unit Sphere:
   For domains within the unit round sphere $S^n$, the paper derives inequalities relating the first eigenvalues of different degrees ($p-1, p, p+1$). Specifically, for $1 \leq p \leq [n/2]$:
   $$p\Gamma_{1,p-1} + (n-p)\Gamma_{1,p+1} < C_{n,p} \Gamma_{1,p}$$
   $$\min(p\Lambda_{1,p-1}, (n-p)\Lambda_{1,p+1}) < \frac{C_{n,p}}{2} \Lambda_{1,p}$$
   where $C_{n,p}$ is an explicit constant depending on $n$ and $p$.

Significance and Claims
The authors state that their work provides the first systematic extension of the buckling and clamped plate problems to differential forms. They claim that the boundary conditions chosen are the "most natural" in this context. The primary significance of the work lies in:

• Generalization: Successfully extending classical spectral inequalities from functions to forms, thereby unifying the treatment of these problems across different degrees of differential forms.
• Characterization: Providing a precise characterization of the first eigenvalues and proving the discreteness of the spectra.
• Unification: Demonstrating that in the Euclidean setting, the spectral properties of forms are directly determined by the scalar case, scaled by combinatorial factors.
• New Estimates: Offering novel lower bounds dependent on curvature conditions (Weitzenb"ock operator) and establishing relationships between different boundary value problems (Dirichlet, Absolute, Buckling, Clamped Plate) on general manifolds.

The paper concludes by noting that while universal inequalities have been established for the scalar case, extending those to the form context remains an open question for future work.