A basic inequality for the Stokes operator related to the Navier boundary condition

We show that Au+Δu L2(Ωε) C(ε u L2(Ωε)+ u L2(Ωε)), where Ωε is a thin domain in of depth ε, the vector field u belongs to the domain of A, which is the Stokes operator for divergence-free vector fields on Ωε satisfying the Navier boundary condition