  This paper shows that the Stokes problem is well-posed when both velocity and
pressure vanish on the domain boundary. This result is achieved by extending
Ne\v{c}as' inequality to square-integrable functions that vanish in a small
band covering the boundary. It is found that the associated a priori pressure
estimate depends inversely on the volume of the band. Numerical experiments
confirm these findings. Based on these results, guidelines are provided for
applying vanishing pressure boundary conditions in model coupling and domain
decomposition methods.
