On the Stokes equations with the Navier-type boundary conditions

Abstract

In a possibly multiply-connected three dimensional bounded domain, we prove in the L p theory the existence and uniqueness of vector potentials, associated with a divergence-free function and satisfying non homogeneous boundary conditions. Furthermore, we consider the stationary Stokes equations with nonstandard boundary conditions of the form u ·n = g and curlu ×n = h ×n on the boundary Γ. We prove the existence and uniqueness of weak, strong and very weak solutions. Our proofs are mainly based on Inf −Sup conditions.