HODGE THEORY IN THE SOBOLEV TOPOLOGY 3

M++1.

Section 4 applies a pseudodifferential formalism developed by

Boutet de Monvel to the study of our elliptic boundary value prob-

lem. This section is included essentially to show how the boundary

value problem under investigation can be view as an example of a very

general kind of problem first introduced in [BDM1]-[BDM3] and in

[ESKj.In Section 5 we explicitly construct the solution of the bound-

ary value problem on the half space in the case of functions. Section 6

studies the problem for q-forms.Sections 7 through 12 deals with the

problem on a bounded, smooth domain. In Section 7 we set up the

problem and calculate the domain of d* and the semi-explicit expres-

sion for d*. In Section 8 we introduce notation and some technical

facts needed in the sequel.Section 9 contains the proof of the coercive

estimate, and the proof of our result about existence of solutions for

the boundary value problem. Section 10 is devoted to the proof of

the a priori estimate in the case of functions. Section 11 gives the

proof of the regularity result in the case of g-forms. Finally, in Section

12, we conclude the proof of our main result in the case of a smoothly

bounded domain.

We are grateful to G. Grubb for helpful communications regarding

this work. We also thank Judy Kenney for reading the entire man-

uscript with care and pointing out a number of errors and ideas for

improvements; her contributions to Section 9 were decisive.

1. BASIC NOTATION AND DEFINITIONS

We use the symbol d to denote the usual operator of exterior dif-

ferentiation acting on g-forms. We let fl denote a smoothly bounded

domain in E^4"1. Usually, for simplicity only, a domain is assumed to be

connected. The symbol /\q(Q) denotes the q-forms on Q with smooth

coefficients. The symbol Ao(^) denotes the forms with coefficients that

are C°° and compactly supported in fi. We let A9(^) denote the q-forms

with coefficients that are smooth on Q, and Ao(^) denote the q-forms

with coefficients in C°°(Q) and having compact support in Q (that is,

the support may not be disjoint from the boundary of O).

Some of our explicit calculations will be performed on the special

domain R^ + 1 = {x = (x0,xu...,xN) e RN+l : x0 0}. The half

space is of course unbounded, so the function spaces we deal with must

take into account the integrability at infinity. On the other hand the

half space has the advantage of allowing explicit calculations.

If C is an operator on forms, then C denotes its formal adjoint, that

is, its adjoint calculated when acting on elements of Ao(^)-