2 ELLIPTIC PARTIAL DIFFERENTIAL OPERATORS AND SYMPLECTIC ALGEBRA

In such a case

(1.6) A\ip-+A{-,D)p

is a symmetric (unbounded and non-closed) linear operator on the classical domain

CQ°(£1) C £2(0) . Of course, the operator A can also be extended to be a continuous

map of Sobolev Hilbert spaces (see (A.22) of Appendix A)

(1.7) A: W2m(Q)-L2(n),

(indeed, a continuous map A : W^™(Q) — L2,ioc(^))-

Our goal is to describe and classify all self-adjoint linear operators T on domains

D(T) C £2(^2), which are generated by any given Lagrange symmetric elliptic

differential expression A(-,D) of the form (1.1); that is, which are extensions of

the corresponding symmetric operator A on C Q ° ( 0 ) , as in (1.6); hence, extensions

of the unique symmetric closure of this operator A. We observe that the order

2m 2 of the elliptic expression A(-,D), and the dimension r 2 of the prescribed

bounded region Q C E r are entirely arbitrary; except we note the strong smoothness

hypotheses, for instance that the coefficients {as} £ C°°(Q) and that the boundary

dQ is C^-smooth, which are assumed throughout. (In any case the standard

theories require at least {as} £ C2(Q) and dVt in class C2rn+1, see [36, Section 13]).

We have made this choice of exposition because our investigations emphasize the

new concepts of symplectic algebra, and their application to boundary value theory

of linear elliptic operators; and our methods are not intended as a contribution to

regularity theory or to analytic considerations like non-uniform ellipticity of A(-, D)

in fi, or weak cone conditions of dd. In accord with this philosophy, we occasionally

review introductory expository materials on symplectic algebra (say, in Section 2)

and on elliptic boundary value theory (say, in the first part of Section 3) in order

to make this paper self-contained and accessible to a wider audience.

Of course this announced goal is only partially achieved, within certain specified

frameworks. However, as an example of some new results of our investigations, we

obtain self-adjoint extensions that are determined by non-local conditions and, in

particular, are not defined by differential operators on the boundary dQ of ft.

The content of the paper: this Introduction Section 1 is followed by a re-

view Section 2, devoted to symmetric operators in Hilbert space and properties

of the associated complex symplectic spaces, and the relation of these topics to

elliptic boundary value problems as illuminated by some examples of classical par-

tial differential operators; Section 3 is devoted to the symplectic theory of general

elliptic partial differential operators in Hilbert spaces; in Sections 4 and 5 we illus-

trate the general theory of such operators with several important special boundary

value problems, both locally- and globally-determined, for the classical second-order

Laplace operator, the fourth-order bi-harmonic operator and higher order polyhar-

monic operators; the paper concludes with the Appendix A concerning information

on symbols and notations.

2. R e v i e w of Hilbert and symplecti c space theor y

For the purposes outlined in Section 1 we augment here the classical theory for

self-adjoint extensions of symmetric operators as familiarly based on Hilbert space

constructions, see [3], [5], [27], [31] and [35], so as to incorporate new methods

and concepts of symplectic algebra and geometry, as defined and described below,

see [9], [10], [11] and [12].