Inequalities between Neumann and Dirichlet Laplacian eigenvalues on planar domains

TL;DR

This paper extends a classical inequality relating Neumann and Dirichlet Laplacian eigenvalues from convex to all simply connected planar Lipschitz domains, using a new variational approach.

Contribution

It proves the inequality holds for all simply connected planar Lipschitz domains, broadening the class of domains where the classical eigenvalue relationship applies.

Findings

The inequality holds for all simply connected planar Lipschitz domains.

A novel variational principle is introduced for the proof.

The result generalizes Payne's classical inequality beyond convex domains.

Abstract

We generalize a classical inequality between the eigenvalues of the Laplacians with Neumann and Dirichlet boundary conditions on bounded, planar domains: in 1955, Payne proved that below the $k$-th eigenvalue of the Dirichlet Laplacian there exist at least $k + 2$ eigenvalues of the Neumann Laplacian, provided the domain is convex. It has, however, been conjectured that this should hold for any domain. Here we show that the statement indeed remains true for all simply connected planar Lipschitz domains. The proof relies on a novel variational principle.