λ
N
+1
to the functional
f
N
, then along the direction of the middle point
ϕ
3
of segment joining
ϕ
α
1
and
ϕ
α
2
, also provides the supremum
λ
N
+1
to
f
N
. By the assumption
ϕ
1
=
ϕ
2
, there must be infinite amount of different
elements in
B
n
at each of them
f
N
attains its supremum. This contradicts
the compactness of gradient operators. That is, for
n
= 2
,
4
, and 6, the
multiplicity of any eigenvalue of (12) is no more than 1. This completes
the proof of the corollary.
The case
n
= 2
may cause particular theoretical interest [12,13]. Let
F
(
x, y
)
be defined on the unit rectangle
B
2
of the plane and be square-
integrable. By Theorem 2, it generates a gradient operator
Φ
, and (8) is
reduced to
Φ
ψ
=
1
0
F
(
x, y
)
ψ
(
y
)
dy
=
λϕ,
Φ
∗
ϕ
=
1
0
F
(
x, y
)
ϕ
(
x
)
dx
=
λψ.
(21)
Apparently,
ϕ
and
ψ
are eigenfunctions of self-adjoint operators
ΦΦ
∗
ϕ
=
λ
2
ϕ,
Φ
∗
Φ
ψ
=
λ
2
ψ.
Corollary 5 claims for this case that all eigenvalues of
Φ
have multiplicity
no more than 1. Indeed, suppose the contrary. Let
ϕ
1
(
x
)
ψ
1
(
y
)
and
ϕ
2
(
x
)
ψ
2
(
y
)
provide the same supremum
λ
1
on
B
2
,
λ
1
= sup
ϕψ
∈
B
2
Ω
F
(
x, y
)
ϕ
(
x
)
ψ
(
y
)
dxdy
=
=
Ω
F
(
x, y
)
ϕ
k
(
x
)
ψ
k
(
y
)
dxdy, k
= 1
,
2
.
(22)
By Proposition 4,
ϕ
1
⊥
ϕ
2
and
ψ
1
⊥
ψ
2
. Let
ϕ
3
ψ
3
= (
tϕ
1
+ (1
−
t
)
ϕ
2
)(
tψ
1
+ (1
−
t
)
ψ
2
)
,
0
t
1
.
It is easy to check,
Ω
F
(
x, y
)
ϕ
3
(
x
)
ψ
3
(
y
)
dxdy
= (1
−
2
t
+ 2
t
2
)
λ
1
,
ϕ
3
ψ
3
= 1
−
2
t
+ 2
t
2
,
thus
Ω
F
(
x, y
)
ϕ
3
(
x
)
ψ
3
(
y
)
ϕ
3
ψ
3
dxdy
=
λ
1
,
∀
t
∈
[0
,
1]
.
This means the functional reaches
λ
1
on
B
2
along all rays from the
origin and intersecting any point of segments joining
ϕ
1
,
ϕ
2
and
ψ
1
,
ψ
2
.
46
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2005. № 4
