Hodge theory in the Sobolev topology for the de Rham complex
Luigi Fontana and Steven G. Krantz and Marco M. Peloso
The authors study the Hodge theory of the exterior differential operator $d$ acting on $q$-forms on a smoothly bounded domain in ${\mathbb R}^{N+1}$, and on the half space ${\mathbb R}^{N+1}_+$. 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 Vi\v{s}ik and Eskin, and by Boutet de Monvel.