  The authors study the Hodge theory of the exterior differential operator $d$
acting on $q$-forms on a smoothly bounded domain in $\RR^{N+1}$, and on the
half space $\rnp$. The novelty is that the topology used is not an $L^2$
topology but a Sobolev topology. This strikingly alters the problem as compared
to the classical setup. It gives rise to a boundary-value problem belonging to
a class of problems first introduced by Vi\v{s}ik and Eskin, and by Boutet de
Monvel.
