Biharmonic Steklov problems with Neumann boundary conditions and spectral inequalities on differential forms

Technical Summary: Biharmonic Steklov Problems with Neumann Boundary Conditions and Spectral Inequalities on Differential Forms

Problem Statement
This paper addresses the spectral theory of the biharmonic operator on compact Riemannian manifolds with boundary, specifically focusing on differential forms of arbitrary degree $p$. While the scalar biharmonic Steklov problem with Dirichlet boundary conditions (BSD) has been generalized to differential forms in prior work, the corresponding problem with Neumann-type boundary conditions (BSN) lacked a definitive formulation for forms. The authors introduce a specific biharmonic Steklov problem with Neumann boundary conditions for differential $p$-forms, denoted as (BSN), and investigate its well-posedness, spectral properties, and relationship to other classical spectral problems (Dirichlet, Neumann, Steklov, and BSD).

The primary mathematical challenge involves defining boundary conditions that render the problem well-posed and possess a discrete spectrum, while simultaneously allowing for the derivation of spectral inequalities that generalize the classical Kuttler-Sigillito inequalities known for scalar functions on planar domains.

Methodology
The authors employ a combination of geometric analysis, elliptic operator theory, and variational methods:

1. Formulation and Well-Posedness: The BSN problem is defined by the system $\Delta^2 \omega = 0$ in the interior $M$, subject to boundary conditions involving the inward unit normal $\nu$, the exterior derivative $d$, the codifferential $\delta$, and the tangential component $\iota^*$. Specifically, the conditions are $\nu \lrcorner \omega = 0$, $\nu \lrcorner d\omega = 0$, $\nu \lrcorner \Delta \omega = 0$, and $\nu \lrcorner \Delta d\omega + \ell \iota^* \omega = 0$ on $\partial M$.
2. Ellipticity Analysis: The authors verify that the problem satisfies the Shapiro-Lopatinskij condition for ellipticity. This involves analyzing the principal symbols of the boundary operators and demonstrating that the associated map on the space of bounded solutions to the ordinary differential equation generated by the principal symbol is an isomorphism. This ensures the existence of smooth solutions and the regularity of eigenfunctions.
3. Variational Characterization: Using the min-max principle, the authors establish variational characterizations for the eigenvalues. They define appropriate Sobolev spaces (e.g., $\check{H}^2_N(M)$) and bilinear forms to characterize the eigenvalues $\ell_{k,p}$ as extrema of Rayleigh quotients involving the $L^2$ norms of $\Delta \omega$ and the tangential trace $\iota^* \omega$.
4. Spectral Comparison: The core of the analysis involves comparing the eigenvalues of the BSN problem ($\ell_{k,p}$) with those of the Dirichlet Laplacian ($\lambda_{k,p}$), Neumann Laplacian ($\mu_{k,p}$), Steklov operator ($\sigma_{k,p}$), and the BSD problem ($q_{k,p}$). This is achieved by constructing specific subspaces and utilizing the variational definitions to bound the products and sums of these eigenvalues.

Key Contributions and Results

• Definition of the BSN Problem for Forms: The paper defines a specific boundary-value problem (BSN) for differential forms that generalizes the scalar Neumann biharmonic Steklov problem. It is shown that this problem admits a discrete spectrum consisting of an unbounded, non-decreasing sequence of positive real eigenvalues with finite multiplicities. The kernel of the operator is identified as the absolute de Rham cohomology space $H^p_A(M)$.
• Variational Characterizations: The authors provide rigorous variational characterizations for the eigenvalues of the BSN problem. Specifically, the first strictly positive eigenvalue $\ell_{1,p}$ is characterized as the infimum of the ratio $\|\Delta \omega\|^2_{L^2(M)} / \|\iota^* \omega\|^2_{L^2(\partial M)}$ over a suitable space of forms.
• Generalization of Kuttler-Sigillito Inequalities: The paper establishes a series of spectral inequalities linking the BSN eigenvalues to those of other problems. For $k \ge 1$, the following inequalities are proven:
  • $\mu_{k,p} \sigma_{1,p} \le \ell_{k,p}$ and $\mu_{1,p} \sigma_{k,p} \le \ell_{k,p}$.
  • For the first eigenvalue, the inequalities are strict: $\mu_{1,p} \sigma_{1,p} < \ell_{1,p}$.
  • A relation involving the BSD eigenvalue $q_{1,p}$: $q_{1,p} \sigma^2_{1,p} < \ell_{1,p}$.
  • Inequalities relating the inverse eigenvalues: $\mu^{-1}_{1,p} < \lambda^{-1}_{1,p} + (q_{1,p} \ell_{1,p})^{-1/2}$ and $\mu^{-1}_{1,p} < \lambda^{-1}_{1,p} + (q_{1,p} \sigma_{1,p})^{-1}$.
• Comparison of Alternative Formulations: The authors briefly discuss two alternative boundary condition sets (BSN2 and BSN3) that also yield discrete spectra. They demonstrate that the eigenvalues of the proposed BSN problem satisfy $\ell^{BSN3}_{k,p} \le \ell_{k,p} \le \ell^{BSN2}_{k,p}$, indicating that the chosen formulation (BSN) provides the tightest bounds necessary for the derived inequalities.

Significance and Claims
The paper claims to extend the theory of biharmonic Steklov problems from scalar functions to differential forms of arbitrary degree on compact Riemannian manifolds. By establishing the well-posedness of the Neumann-type problem and proving the generalized Kuttler-Sigillito inequalities, the work provides a quantitative bridge between different physical models (represented by Dirichlet, Neumann, Steklov, and biharmonic spectra) in the context of differential forms.

The authors note that while some of these inequalities were previously known for scalar functions on planar domains or compact Riemannian manifolds, this work generalizes them to the more complex setting of differential forms. The results rely on the specific geometric and analytic properties of the Hodge-de Rham Laplacian and the interplay between the interior and boundary behavior of differential forms. The paper does not propose new physical applications or experimental setups but rather contributes to the mathematical foundation of spectral geometry on manifolds with boundary.