The Stokes Dirichlet-to-Neumann operator

Let \(\Omega \subset \mathbb {R}^d\) $\mathrm{\Omega }\subset {\mathbb{R}}^d$ be a bounded open connected set with Lipschitz boundary. Let \(A^N\) $A^N$ and \(A^D\) $A^D$ be the Stokes Neumann operator and Stokes Dirichlet operator on \(\Omega \) $\mathrm{\Omega }$ , respectively. We study the associated Stokes version of the Dirichlet-to-Neumann operator and show a Krein formula which relates these three Stokes version operators. We also prove a Stokes version of the Friedlander inequalities, which relates the Dirichlet eigenvalues and the Neumann eigenvalues.