The Stokes Dirichlet-to-Neumann operator

Let \(\Omega \subset \mathbb {R}^d\) be a bounded open connected set with Lipschitz boundary. Let \(A^N\) and \(A^D\) be the Stokes Neumann operator and Stokes Dirichlet operator on \(\Omega \), 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.