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).