Luigi Fontana and Steven G. Krantz and Marco M. Peloso
\def\RR{{\mathbb R}} \def\rnp{\RR^{N+1}_+} 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.