Inequalities between Dirichlet and Neumann eigenvalues on Carnot groups

TL;DR

This paper establishes a comparison between Dirichlet and Neumann eigenvalues for the sub-Laplacian on Carnot groups, extending known Euclidean and Heisenberg results with a simple proof.

Contribution

It proves that the j-th Dirichlet eigenvalue exceeds the (j+1)-st Neumann eigenvalue on Carnot groups, generalizing previous Euclidean and Heisenberg findings.

Findings

Dirichlet eigenvalues are greater than Neumann eigenvalues in this setting

The result generalizes classical inequalities to Carnot groups

The proof is notably simple and elegant

Abstract

We show that the $j$-th Dirichlet eigenvalue of the sub-Laplacian on an open set of a Carnot group is greater than the $(j+1)$-st Neumann eigenvalue. This extends earlier results in the Euclidean and Heisenberg case and has a remarkably simple proof.