Hodge theory in the Sobolev topology for the de Rham complex

The authors study the Hedge theory of the exterior differential operator d acting on q-forms on a smoothly bounded domain in RN+1, and on the half space R-+(N+1). 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 Visik and Eskin, and by Boutet de Monvel.