  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.
