The de Rham complex on unbounded domains with Sobolev space topology

This paper studies the operator dd∗+d∗d acting on q-forms on an unbounded domain with smooth boundary, where d is the exterior derivative and d∗ is the adjoint of d calculated using the Sobolev space topology. The domain of d∗ is determined and an expression for d∗ is obtained. The operator dd∗+d∗d gives rise to a boundary value problem. Global regularity is obtained using weighted norms and global existence is obtained by using the theory of compact operators.