Reproving Friedlander's inequality with the de Rham complex

TL;DR

The paper develops a de Rham complex–based framework with absolute boundary conditions to bound Neumann Laplacian eigenvalues by Dirichlet ones, providing a concise, higher-dimensional proof of Friedlander's inequality and a generalization of Rohleder’s curl–curl results. By expressing Laplacians on forms as direct sums of scalar Laplacians and exploiting Hilbert complex theory, the authors obtain sharp counting-function inequalities with binomial coefficients tied to the ambient dimension. The approach unifies known 2D vectorial results with higher-dimensional analogues, linking spectral bounds to topological invariants via the Euler characteristic and the cohomology of the domain. This framework suggests potential for stronger, dimension-driven improvements and broader applicability to related differential operators.

Abstract

Inequalities between Dirichlet and Neumann eigenvalues of the Laplacian and of other differential operators have been intensively studied in the past decades. The aim of this paper is to introduce differential forms and the de Rham complex in the study of such inequalities. We show how differential forms lie hidden at the heart of the work of Rohleder on inequalities between Dirichlet and Neumann eigenvalues for the Laplacian on planar domains. Moreover, we extend the ideas of Rohleder to a new proof of Friedlander's inequality for any bounded Lipschitz domain.

Theorems & Definitions (23)

• Lemma 2.1: Glazman's lemma
• Definition 2.2
• Lemma 2.3
• Theorem 3.1
• Lemma 3.2
• Theorem 3.3
• Remark 3.4
• Lemma 4.1
• proof
• Remark 4.2