  The well-known Wall theorem states a simple and precise relation among
temperature, pressure and density of a fluid at contact with a confining hard
wall in thermodynamic equilibrium. In this Letter we develop an extension of
the Wall theorem to out-of-equilibrium conditions, providing an exact relation
between pressure, density and temperaure at the wall, valid for strong
non-equilibrium situations. We derive analytically this Non-equilibrium Wall
theorem for stationary states and validate it with non-equilibrium event-driven
molecular-dynamics simulations. We compare the analytical expression with
simulations by direct evaluation of temperature, density and pressure on the
wall in linear regime, medium and very strong out-of-equilibrium conditions of
a nanoconfined liquid under flow in stationary state, presenting viscous
heating and heat transport. The agreement between theory and simulation is
excellent, allowing for a conclusive validation. In addition, we explore the
degree of accuracy of using the equilibrium Wall theorem and different
expressions for the local temperature, employed in non-equilibrium
molecular-dynamics simulations.
