1.1. TH E BOUNDARY VALUE PROBLEM AND ITS FORMALLY ADJOIN T 11
Proof: Let Lj be the following differential operators:
i - i
(1.1.8) Lj = 2_] asDl for j = 1,... , 2m, LQ = 0.
s=0
We prove by induction that
oo oo
(1.1.9) IuT^vdt = [{Ljii
-v-iDjtu-
i ) dt -
Y^(Dst-lu)(d)
(Pai/)(0)
0 0 s = 1
for smooth functions it, v with compact support. Obviously, (1.1.9) is satisfied for
j = 0. Suppose that (1.1.9) is valid for a given nonnegative integer j = j
0
2m.
Using the equations
= CLj D3t U + £ j + l U ,
= —i aj v + Dt Pj+i v
and integrating by parts, we get
LjU
P3v
(1.1.10)
oc
/
{LjU v i D\u -
PJV)
dt
= [(Lj+w v-i DJt+1u T^v) dt - (DJtu)(0) Pj+iv){0).
o
Consequently, (1.1.9) is satisfied for j = jo + 1 and therefore for each nonnegative
integer j 2m. In particular, for j = 2m we have
°? y 2m
(1.1.11) u-L+vdt= Lu vdt - 5^(A*~M(0) (Psv)(0).
0 0 s = 1
Furthermore, we have
(1.1.12) ( B ( A ) 4 = o , t / )
C m + J
= {Q'(Tu)(Q),v)Cm+J - ((Pti)(0), Q*v)c2m
and
(1.1-13) (Cu,v)Cm+J = (u,C*v)CJ.
The equalities (1.1.11)-(1.1.13) yield (1.1.6).
Let P(Dt) be the operator given in the Green formula (1.1.6). By (1.1.7), there
is the representation
P = TV
with the triangular matrix
(1.1.14)
/ Q1
a2
02m-l a2m \
«2m
0
V
Q2r)
0 0 /
It is natural to define the formally adjoint problem to (1.1.2)—(1.1.3) by the
operators on the right-hand side of (1.1.6).
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