A Friedlander type estimate for Stokes operators

Abstract

Let $\Omega \subset {\mathbb{R}}^d$ be a bounded open connected set with Lipschitz boundary. Let $A^N$ and $A^D$ be the Neumann Stokes operator and Dirichlet Stokes operator on $\Omega$, respectively. Further let ${\lambda }_1^N\le {\lambda }_2^N\le \dots$ and ${\lambda }_1^D\le {\lambda }_2^D\le \dots$ be the eigenvalues of $A^N$ and $A^D$ repeated with multiplicity, respectively. Then \[ λ_{n+1}^N < λ_n^D \] for all $n\in \mathbb{N}$.