Germán Miranda: Eigenvalues inequalities for the Laplacian and the de Rham complex

Abstract:

In this talk, we begin with a historical overview of eigenvalue inequalities for the Laplacian and other differential operators on bounded domains. Next, we combine into a common framework the ideas of Rohleder with differential forms and the de Rham complex. 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 can use the de Rham complex to give a generalization of the work of Rohleder on inequalities between curl-curl and Dirichlet Laplacian eigenvalues.

Let \(\lambda_j(\Delta_N)\) and \(\lambda_j(\Delta_D)\) be the jth eigenvalue of the Neumann and Dirichlet Laplacian on a bounded Lipschitz domain in \(\mathbb{R}^d\) with \(d\geq 2\). Then, using this common framework, we can give a new proof of Friedlander's inequality

\[\lambda_{j+1}(\Delta_N) \leq \lambda_j (\Delta_D) ,\]

for all \(j=1, 2, \ldots\).

This is joint work with Magnus Fries and Magnus Goffeng.