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

Quanyu Tang, Haiqi Zhang

Let $\Omega \subset {\mathbb{R}}^2$ be a bounded connected Lipschitz domain, and let $\{{\mu }_j(\Omega ){\}}_{j\ge 1}$ and $\{{\lambda }_j(\Omega ){\}}_{j\ge 1}$ denote the Neumann and Dirichlet Laplacian eigenvalues, r…