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].
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