Neumann--Dirichlet eigenvalue comparison at the first Dirichlet threshold in the plane

Abstract

Let $\Omega\subset\mathbb R^2$ be a bounded connected Lipschitz domain, and let $\{\mu_j(\Omega)\}_{j\geq1}$ and $\{\lambda_j(\Omega)\}_{j\geq1}$ denote the Neumann and Dirichlet Laplacian eigenvalues, respectively, counted with multiplicity. We prove that $$ \mu_3(\Omega)<\lambda_1(\Omega), $$ thereby removing the simple-connectivity assumption from the previously known planar result at the first Dirichlet threshold. We also establish a three-spectrum counting inequality relating the Dirichlet, Neumann, and conductivity spectra.