  We study the domain structure in ferroelectric thin films with a `passive'
layer (material with damaged ferroelectric properties) at the interface between
the film and electrodes within a continuous medium approximation. An abrupt
transition from a monodomain to a polydomain state has been found with the
increase of the `passive' layer thickness $d$. The domain width changes very
quickly at the transition (exponentially with $d^{-2}$). We have estimated the
dielectric response $dP/dE$ (the slope of the hysteresis loop) in the
`fatigued' multidomain state and found that it is in agreement with experiment,
assuming realistic parameters of the layer. We derive a simple universal
relation for the dielectric response, which scales as $1/d$, involving only the
properties of the passive layer. This relation qualitatively reproduces the
evolution of the hysteresis loop in fatigued samples and it could be tested
with controlled experiments. It is expected that the coercive field should
increase with decreasing lateral size of the film. We believe that specific
properties of the domain structure under bias voltage in ferroelectrics with a
passive layer can resolve the long-standing `paradox of the coercive field'.
