The Stokes Dirichlet-to-Neumann operator

Abstract

Let Ω⊂Rd be a bounded open connected set with Lipschitz boundary. Let AN and AD be the Stokes Neumann operator and Stokes Dirichlet operator on Ω, 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.