10
1. PROBLEMS ON THE HALF-AXIS
where B(Dt) denotes the vector of the operators Bi(Dt), ... , 5
m +
j(Dt) , / is a
given function on R
+
, and g is a given vector from C m + J . We seek a function u
on M+ and a vector u = (u\, ... ,uj) such that u is a solution of the differential
equation (1.1.2), and the pair (u,u) satisfies the boundary conditions (1.1.3) which
can be written in the coordinate form as
J
Bk(Dt)u{t)\t^oJrY^Ck^u3 = 9k, k = 1, ... ,ra + J.
3 = 1
REMARK
1.1.1. Here and in the following we will not make a distinction be-
tween column and row vectors. In (1.1.3) u and g are considered as column vectors.
1.1.2. The Green formula and the formally adjoint problem. In order
to define the formally adjoint problem to (1.1.2), (1.1.3), we use a modification of
the classical Green formula.
First we consider the case jik 2m. Let
2ra
j=0
be the formally adjoint operator to L. Furthermore, let V denote the vector
(1.1.4) p = ( i , A , . . . , A
2 m
"
1
) -
Then the operator B(Dt) can be written in the form
(1.1.5) B(Dt) = Q-V
(here V is considered as a column vector), where the elements of the (ra + J) x 2m
matrix
Q = \Qk,j)
V ,J /ifc
lkm+J, lj2ra
are defined by the coefficients of the operators B^ as follows:
I 0
for j = 1, ... ,/x*. -hi ,
qk*
^
n
for j / i
f c
+ l.
THEOREM
1.1.1. The following Green formula is satisfied for all infinitely dif-
ferentiable functions u, v on M+ with compact support and all vectors u £ C
J
,
v E C
m + J
:
oo
(1.1.6) f Lu -vdt + (B(Dt)u\t=o + Cu, v)Cm+J
0
oo
= fu.L+v~dt+ ((Vu)(0), P(Dt)v\t=o + Q* v)Q2m + fe, C*t;)c, .
o
ifere P(Dt) denotes the vector with the components
2m-j
(1.1.7) Pj{Dt) = -i J2
*J+sDst
j = l,...,2m ,
and Q*, C* are the adjoint matrices to Q and C, respectively.
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