Elliptic partial differential operators and
symplectic algebra
1. Introduction: Organization of results
In this investigation we consider self-adjoint boundary value problems for linear
elliptic partial differential operators, review the classical Hilbert space theory for
such operators, and reinterpret this theory in the light of recent results in symplectic
linear algebra.
Consider a linear partial differential expression (or formal operator) of order
TV 2, in the Euclidean r-space E r with dimension r 2
(i.i)
Q\S\
A(x,Z) := ^2 as(*)DS = Yl aSuS2,..,Sr(xux2r-. xr)g sld s2 ...
Qx
sr
0\s\N 0 | s | i V l 2 V
with complex-valued coefficients as G C°°(0), in the compact closure O of a
bounded region 0 C E r , with C°°-smooth boundary dft. (See Appendix A for
explanation of symbols, such as multi-indices s = (si, S2, ? 5r)5 real coordinate
vectors x = (#i,£2,"' * ,xr) hi Euclidean space E r , and other notations; generally
in accord with the treatise [36, especially Chapters 1,2 and 3].)
We assume that the partial differential expression A(-,D) of (1.1) has even
order TV = 2m 2 and is (strongly, uniformly) elliptic in Ct. That is, the (highest
order) principal polynomial
(1-2)
AH
(*,£):= £
a'(x)tl1&---£r
\s\=2m
has real-valued coefficients {as(-) : \s\ = 2ra}, and also
(1.3)
AH(x,£)^0
for all x G Q and all non-zero real vectors £ = (£i, £2, , £r) £ ^ r -
In addition we assume that A(-,D) in (1.1) is Lagrange symmetric (or formally
self-adjoint) in fi, that is for all x G ft and all cp G
CQ°(0 )
(1.4) A(x,D)^ = A*(x,I%:= £
(~lpDs
fcfrjip) ,
0 | s | N
in terms of the Lagrange adjoint expression A*(-,D), or equally well (see Corollary
3.1 of Section 3)
(1.5) (Aip, ^) = (if, AiP) for all p, t/j G C£°(0)
in terms of the usual inner or scalar product (•, •) in the Hilbert space 1/2(0).
l
Previous Page Next Page