  The resolution of a very large class of linear and non-linear, stationary and
evolutive partial differential problems in the half-space (or similar) under
the slip boundary condition is reduced here to that of the corresponding
results for the same problem in the whole space. The approach is particularly
suitable for proving new results in strong norms. To determine whether this
extension is available, turns out to be a simple exercise. The verification
depends on a few general features of the functional space X related to the
space variables. Hence, we present an approach as much as possible independent
of the particular space X. We appeal to a reflection technique. Hence a crucial
assumption is to be in the presence of flat boundaries (see below). Instead of
stating "general theorems" we rather prefer to illustrate how to apply our
results by considering a couple of interesting problems. As a main example, we
show that the resolution of a class of problems for the evolution Navier-Stokes
equations under a slip boundary condition can be reduced to that of the
corresponding results for the Cauchy problem. In particular, we show that sharp
vanishing viscosity limit results that hold for the evolution Navier-Stokes
equations in the whole space can be extended to the boundary value problem in
the half-space. We also show some applications to non-Newtonian fluid problems.
