3 / 17
Let
H
1
be a Hilbert space of measurable functions
u
(
t
) : [0
, T
]
→
H
with a
scalar product
(
u, v
)
H
1
=
T
Z
0
(
u
(
t
)
, v
(
t
))
H
dt
and the norm
k
u
k
2
H
1
= (
u, u
)
H
1
.
We say that the function
u
(
t
)
has a derivative on
[0
, T
]
if the following
representation holds:
u
(
t
) =
u
(
t
0
) +
t
Z
t
0
v
(
τ
)
dt, t
0
∈
[0
, T
]
.
(5)
Here the function
v
(
t
)
∈
H
1
gives for almost all
t
∈
[0
, T
]
the value of
the derivative
du/dt
. Consider an operator
Au
= (
Su, u
(0)
, u
t
(0))
.
The
operator
A
defines on the domain
D
(
A
)
of all
u
∈
H
1
such that functions
du/dt
,
Lu
,
Ldu/dt
belong to
H
1
and continuous; the functions
d
2
u
dt
2
,
L
d
2
u
dt
2
belong to
H
1
and piecewise continuous. The image of operator
R
(
A
)
is a
linear manifold in a Hilbert space
W
=
H
1
×
D
(
√
L
)
×
H
where
D
(
√
L
)
is a Hilbert space of elements
u
=
L
−
1
2
ψ
,
ψ
∈
H
with the scalar product
(
u, v
)
D
(
√
L
)
=
√
Lu,
√
Lv
. It is possible to consider the solution of the
problem (3), (4) as a solution of the operator equation
Au
= (
h, f, g
)
,
where
(
h, f, g
)
∈
W
.
Theorem 1.
The operator
A
admits a closure
A
,
R
(
A
) =
R
(
A
) =
W
.
There exists a bounded inverse operator
A
−
1
on
W
and the problem
Au
= (
h, f, g
)
(6)
has the unique solution for all
h
∈
H
1
,
f
∈
D
(
√
L
)
,
g
∈
H
[8].
It is possible to prove that the solution of the operator equation (6) is
a weak solution of the problem (3), (4). It means that
u
∈
H
1
,
u
t
∈
H
1
,
√
Lu
∈
H
1
,
u
(0) =
f
and the following integral identity holds:
T
Z
0
(
u
t
, w
t
)
−
√
Lu,
√
Lw dt
+ (
g, w
(0)) = 0
(7)
for all
w
∈
H
1
,
w
t
∈
H
1
,
√
Lw
∈
H
1
,
w
(
T
) = 0
. The solution from such
class is unique.
Below we consider the case of hyperbolic problems (1), (2). The main
example of an elliptic second-order operator
L
is the Laplace operator
Lu
=
−
Δ
u
. Let
Ω
⊂
R
n
,
n
≥
2
, be an an arbitrary (may be unbounded)
domain with smooth boundary
Γ
. We consider the mixed problem for the
wave equation
u
tt
−
Δ
u
= 0
(8)
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2015. № 3
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