By the weak compactness of
B
α
, for any fixed point
(
x
1
,
· · ·
, x
ˆ
i
,
· · ·
, x
n
)
in
Ω
i
,
F
k
(
x
1
,
· · ·
, x
ˆ
i
,
· · ·
, x
n
)
→
0
as
(
ψ
i
k
(
x
i
)
−
ϕ
i
(
x
i
))
tends weakly to
0. By the Dominated Convergence Theorem [7],
F
k L
2
(Ω
i
)
→
0
,
k
= 1, 2,
. . . ,
n
, hence
Ω
F
(
x
1
,
· · ·
, x
n
)
α
ψ
α
k
−
α
ϕ
α
d
Ω
≤
n
i
=1
F
k L
2
(Ω
i
)
→
0
as
k
→ ∞
. This is to be shown for the first part of the Lemma.
To verify the second statement we take an orthonormal basis
e
β
α
(
x
α
)
,
β
= 1
,
2
,
· · ·
in each
L
(
α
)
2
(0
,
1)
and construct a set
E
,
E
=
n
α
=1
e
γ
α
α
(
x
α
)
,
γ
α
= 1
,
2
,
· · ·
, each
γ
α
runs over
N
independently. Since
L
n
2
(0
,
1)
is dense
in
L
2
(Ω)
,
E
becomes an orthonormal basis of the latter [4, 8]. Any element
F
∈
L
2
(Ω)
can be expressed uniquely in the form of Fourier series,
F
(
x
1
,
· · ·
, x
n
) =
∞
β
=1
P
β
n
α
=1
e
β
α
α
, P
β
=
F,
α
e
β
α
α
L
2
(Ω)
.
By assumption
F
=
0, there must be some
P
k
=0
. Let
ψ
k
=(sign
P
k
)
α
e
k
α
.
Substitute the above series into (6), and take inner product with just defined
ψ
k
, we obtain immediately
λ >
|
P
k
|
>
0
, what is claimed in the Lemma.
Now we proceed to establish the necessary conditions which a solution
of (4),
α
ϕ
α
, should satisfy. Suppose the expression (4) achieves its
supremum at some element
α
ϕ
α
∈
B
n
. According to Lagrange Principle,
α
ϕ
α
must satisfy the following conditions with a multiplier
λ
[5, 9]:
D
Ω
F
(
x
1
,
· · ·
, x
n
)
α
ϕ
α
dx
−
λ
α
ϕ
α
,
α
ϕ
α
L
2
(Ω)
−
1 = 0
,
(7)
where
D
denotes the Gateaux directional derivative with respect to all
ϕα
,
λ
is a real to be determined. According to the rules of differentiation,
Df
=
n
i
=1
D
i
f
,
D
i
f
is the partial derivative with respect to
ϕ
i
. Let
h
i
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2005. № 4
37
