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Hodge-Laplacian Eigenvalues on Surfaces with Boundary

Muravyev Mikhail

TL;DR

The paper develops a differential-forms framework for comparing Neumann and Dirichlet eigenvalues of the Hodge-Laplacian on compact manifolds with boundary. It shows a decomposition $spec_A(\\Omega_N^p(M)) = spec_A(cE_N^p(M)) ⊔ spec_A(C_N^p(M))$ and, when $M$ is simply connected, $spec_A(cE_N^p(M)) = spec_A(C_N^{p+1}(M))$, enabling a unified variational approach. For Euclidean domains with smooth boundary, it derives the inequality $\\theta_{(n-1)m+1} ≤ λ_m$, relating the absolute spectrum on $(n-1)$-forms to the Dirichlet spectrum and recovering Rohleder's results as corollaries. The work extends Rohleder's ideas to arbitrary dimension and form degree, highlighting a cohesive structure for boundary-value problems in Hodge theory.

Abstract

Recently Rohleder proposed a new variational approach to an inequality between the Neumann and Dirichlet eigenvalues in the simply connected planar case using the language of classical vector analysis. Writing his approach in terms of differential forms permits to generalize these results to a much broader context. The spectrum of the absolute boundary problem for the Hodge-Laplacian on a Riemannian manifold with boundary is presented as a union of the spectra of the absolute boundary problem on the spaces of closed and co-exact forms. An inequality for the eigenvalues of the absolute boundary problem for the Hodge-Laplacian and the Dirichlet boundary problem for the Laplace-Beltrami operator in the Euclidean case is obtained using this presentation. The Rohleder's results are obtained as corollaries of a more general theorem.

Hodge-Laplacian Eigenvalues on Surfaces with Boundary

TL;DR

The paper develops a differential-forms framework for comparing Neumann and Dirichlet eigenvalues of the Hodge-Laplacian on compact manifolds with boundary. It shows a decomposition and, when is simply connected, , enabling a unified variational approach. For Euclidean domains with smooth boundary, it derives the inequality , relating the absolute spectrum on -forms to the Dirichlet spectrum and recovering Rohleder's results as corollaries. The work extends Rohleder's ideas to arbitrary dimension and form degree, highlighting a cohesive structure for boundary-value problems in Hodge theory.

Abstract

Recently Rohleder proposed a new variational approach to an inequality between the Neumann and Dirichlet eigenvalues in the simply connected planar case using the language of classical vector analysis. Writing his approach in terms of differential forms permits to generalize these results to a much broader context. The spectrum of the absolute boundary problem for the Hodge-Laplacian on a Riemannian manifold with boundary is presented as a union of the spectra of the absolute boundary problem on the spaces of closed and co-exact forms. An inequality for the eigenvalues of the absolute boundary problem for the Hodge-Laplacian and the Dirichlet boundary problem for the Laplace-Beltrami operator in the Euclidean case is obtained using this presentation. The Rohleder's results are obtained as corollaries of a more general theorem.
Paper Structure (6 sections, 26 theorems, 154 equations)

This paper contains 6 sections, 26 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Basic facts and notations
  3. Boundary problems
  4. Absolute boundary condition
  5. Second boundary condition
  6. Variational principles comparison

Key Result

Theorem 1.2

Let $M\subset\mathbb{R}^2$ be a bounded simply connected planar domain. Then where $\omega(v) =\partial_1 v_2 - \partial_2 v_1$ is the vorticity of a vector field $v$ and $\mathcal{H}_{a}$ is the space of all vector fields $v\in L^2(M)^2$ such that ${\mathrm{div}}\! v,$$\omega(v) \in L^2(M)$ and $v|_{\partial M}$ is tangent to the boundary.

Theorems & Definitions (36)

  • Definition 1.1
  • Theorem 1.2: Rohleder, Rohleder
  • Theorem 1.3: Rohleder, Rohleder_2
  • Theorem 1.4: Theorem \ref{['main_result']}
  • Theorem 1.5: Theorem \ref{['comparisson']}
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 2.1
  • Proposition 2.2: Olle
  • ...and 26 more