Hodge Theory in the Sobolev Topology for the De Rham Complex on a Smoothly Bounded Domain in Euclidean Space

The Hodge theory of the de Rham complex in the setting of the Sobolev topology is studied. As a result, a new elliptic boundaryvalue problem is obtained. Next, the Hodge theory of the @-Neumann problem in the Sobolev topology is studied. A new @-Neumann boundary condition is obtained, and the corresponding subelliptic estimate derived. The classical Hodge theory on a domain in R N+1 (or, more generally, on a real manifold) is based on the complex ^ 0 d ! ^ 1 d ! ^ 2 d ! In the topology of L 2 ((), one can calculate both the domain of existence and the actual form of d , the Hilbert space adjoint of d. It turns out that d equals d 0 , the formal adjoint of d on the domain of d. See FOK] for details. Of course these facts are well known. They lead to the study of the second order, self-adjoint operator 2 d d + d d: Said operator 2 makes sense precisely on those forms that lie both in the domain of d and in the domain of d. Calculated on a domain or region in space, the operator 2 turns out to be the (negative of the) ordinary Laplacian 4. The Hodge theory of 2, which by today's standards is rather straightforward, shows that 2 has closed range in L 2 ((). The orthogonal complement of the range is of course the kernel of the adjoint of 2 (which is nothing other than 2 itself). The Neumann operator N for the d-complex is a right inverse for 2. It turns out that the operator N is, essentially, a pseudodiierential operator of degree ?2. The regularity theory for the operator 2, and also for the operator d, may be read oo from the mapping properties of N. However, because the operator d has a large kernel, one must choose a solution to the equation du = f carefully.