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 R3 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.