Isoperimetric relations between Dirichlet and Neumann eigenvalues

TL;DR

This paper investigates the relationship between Dirichlet and Neumann eigenvalues of the Laplacian, proposing a conjecture linking the count of certain Neumann eigenvalues to the domain's isoperimetric ratio, with implications for eigenfunction nodal sets.

Contribution

It introduces a new conjecture connecting eigenvalue counts to geometric properties of the domain, supported by analytical and numerical evidence.

Findings

Conjecture that the number of Neumann eigenvalues below the first Dirichlet eigenvalue is controlled by the isoperimetric ratio.

Numerical results support the proposed relationship.

Applications to nodal deficiency and connections to Yau's conjecture on nodal sets.

Abstract

Inequalities between the Dirichlet and Neumann eigenvalues of the Laplacian have received much attention in the literature, but open problems abound. Here, we study the number of Neumann eigenvalues no greater than the first Dirichlet eigenvalue. Based on a combination of analytical and numerical results, we conjecture that this number is controlled by the isoperimetric ratio of the domain. This has applications to the nodal deficiency of eigenfunctions and is closely related to a long-standing conjecture of Yau on the Hausdorff measure of nodal sets.