Friedlander's inequality and the de Rham complex

TL;DR

This paper explores the use of differential forms and the de Rham complex to analyze inequalities between Dirichlet and Neumann eigenvalues of the Laplacian, revealing their underlying role in spectral inequalities.

Contribution

It introduces the application of differential forms and the de Rham complex to the study of eigenvalue inequalities, connecting geometric analysis with spectral theory.

Findings

Differential forms are fundamental in understanding eigenvalue inequalities.

The de Rham complex provides a new perspective on classical spectral inequalities.

Connections are established between geometric structures and spectral bounds.

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.