Inequalities between Dirichlet and Neumann eigenvalues on Carnot groups

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.

10 pages; dedicated to Nikolai Nadirashvili, on the occasion of his 70th birthday