HODGE THEORY IN THE SOBOLEV TOPOLOGY FOR THE DE RHAM COMPLEX

LUIGI FONTANA STEVEN G. KRANTZ MARCO M. PELOSO

Abstract. The authors study the Hodge theory of the exterior differential operator d acting on q-forms on a smoothly bounded domain in ℝ^{N+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 Višik and Eskin, and by Boutet de Monvel.