We study the one-dimensional stationary solutions of an integro-differential
equation derived by Giacomin and Lebowitz from Kawasaki dynamics in Ising
systems with Kac potentials, \cite{GiacominLebowitz}. We construct stationary
solutions with non zero current and prove the validity of the Fourier law in
the thermodynamic limit showing that below the critical temperature the limit
equilibrium profile has a discontinuity (which defines the position of the
interface) and satisfies a stationary free boundary Stefan problem.
Under-cooling and over-heating effects are also studied. We show that if
metastable values are imposed at the boundaries then the mesoscopic stationary
profile is no longer monotone and therefore the Fourier law is not satisfied.
It regains however its validity in the thermodynamic limit where the limit
profile is again monotone away from the interface
