for any orthonormal set
S
in a Hilbert space there is a complete orthonormal
basis that contains
S
as its subset [11]. The Proposition is thus justified.
Remark.
It appears the remarkable maximum property of gradient
operators expressed in (17), which is entirely analogous with compact self-
adjoint operators in Hilbert spaces. The Proposition also shows there are as
many different orthonormal bases as the cardinality of different elements
in
L
2
(Ω)
.
Corollary 5.
If the dimension of underlying space
R
n
with
n
= 2
,
4
and 6, all eigenvalues of gradient operators defined by (11) for arbitrarily
given
F
(
x
)
∈
L
2
(Ω)
,
F
(
x
) = 0
, have multiplicity no more than 1.
Proof (abridged).
Suppose the contrary, if there exist two different
eigenfunctions
ϕ
1
=
α
ϕ
α
N
+1
,
1
and
ϕ
2
=
α
ϕ
α
N
+1
,
2
corresponding to the
same eigenvalue
λ
N
+1
, which enable the following functional to achieve
its supremum on
B
n
,
λ
N
+1
= sup
ψ
α
∈
B
α
f
N
(
α
ψ
α
) =
= sup
ψ
α
∈
B
α
Ω
F
N
(
x
)
α
ψ
α
d
Ω =
f
N
(
α
ϕ
α
N
+1
,k
)
, k
= 1
,
2
.
(18)
Now construct a new element
ϕ
3
=
α
(
tϕ
α
1
+ (1
−
t
)
ϕ
α
2
)
,
t
∈
[0
,
1]
, and
put it into (18). After exposing the product, we have
f
N
(
ϕ
3
, t
) =
f
N
α
(
tϕ
α
1
+ (1
−
t
)
ϕ
α
2
) =
=
t
n
f
N
α
ϕ
α
1
+
t
n
−
1
(1
−
t
)
f
N
n
i
=1
ϕ
i
2
ˆ
i
ϕ
α
1
)+
+
t
n
−
2
(1
−
t
)
2
f
N
i
=
j
ϕ
i
2
ϕ
j
2
ˆ
i
ˆ
j
ϕ
α
1
+
. . .
. . .
+
t
2
(1
−
t
)
n
−
2
f
N
i
=
j
ϕ
i
1
ϕ
j
1
ϕ
α
2
+
t
(1
−
t
)
n
−
1
f
N
n
i
=1
ϕ
i
1
ˆ
i
ϕ
α
2
+ (1
−
t
)
n
f
N
α
ϕ
α
2
.
(19)
By Proposition 4,
ϕ
i
1
and
ϕ
i
2
are mutually orthogonal. Dropping index
N
for brevity, we have the necessary condition,
Df
=
f
i
h
i
1
ˆ
i
ϕ
α
1
=
i
ϕ
i
1
, h
i
1
λ
= 0
,
∀
h
i
1
∈
L
(
i
)
2
(0
,
1)
.
44
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2005. № 4
