1. Introduction The initial boundary value problems for Schr¨odinger equations in the cylinders with base containing conical points were established in [2,3]. Such problems for parabolic systems have been studied in Sobolev spaces with weights [4,5]. We are concerned with initial boundary value problems for Schr¨odinger systems in cylinders with base containing conical point. The paper is organized is the following way. In Section 2 we define the problem. In Section 3 we establish the unique existence of the generalized solution of the problem. Finally, in Section 4 we apply the obtained results to a problem of mathematical physics.

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JOURNAL OF SCIENCE OF HNUE
Natural Sci., 2010, Vol. 55, No. 6, pp. 82-89
ON THE SOLVABILITY OF THE INITIAL BOUNDARY
VALUE PROBLEM FOR SCHRO¨DINGER SYSTEMS
IN CONICAL DOMAINS
Nguyen Thi Lien
Hanoi National University of Education
E-mail: Lienhnue@gmail.com
Abstract. In this paper, we consider the initial boundary value problem
for Schro¨dinger systems in the cylinders with base containing the conical
point. The existence and the uniqueness of the generanized solution of this
problem are given.
Keywords: Initial boundary value problem, generalized solution, cylinders
with conical base.
1. Introduction
The initial boundary value problems for Schro¨dinger equations in the cylinders
with base containing conical points were established in [2,3]. Such problems for
parabolic systems have been studied in Sobolev spaces with weights [4,5].
We are concerned with initial boundary value problems for Schro¨dinger sys-
tems in cylinders with base containing conical point.
The paper is organized is the following way. In Section 2 we define the prob-
lem. In Section 3 we establish the unique existence of the generalized solution of
the problem. Finally, in Section 4 we apply the obtained results to a problem of
mathematical physics.
2. Notations and formulation of the problem
Let Ω be a bounded domain in Rn (n ≥ 2) with the boundary ∂Ω. We
suppose that S = ∂Ω \ {0} is a smooth manifold and Ω is in a neighbourhood U of
the origin 0 coincides with the cone K = {x : x/ | x |∈ G}, where G is a smooth
domain on the unit sphere Sn−1 in Rn. Let T be a positive real number or T =∞.
Set Ωt = Ω× (0, t), St = S × (0, t).
For each multi-index α = (α1, . . . , αn) ∈ Nn, |α| = α1 + · · ·+ αn, the symbol
Dα = ∂|α|/∂xα11 ...∂x
αn
n denotes the generalized derivative of order α with respect
to x = (x1, ..., xn); ∂
k/∂tk is the generalized derivative of order k with respect to
t. Let u = (u1, ..., us) be a complex-valued vector function defined on ΩT . We use
notation: Dαu = (Dαu1, ..., D
αus); utj = ∂
ku/∂tk = (∂ju1/∂t
j , .., ∂jus/∂t
j).
Let us introduce some functional spaces used in this paper (see [1]):
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On the solvability of the initial boundary value problem for Schro¨dinger systems...
We use H l(Ω) be the space of functions defined in Ω with the norm
‖u‖Hl(Ω) =
( l∑
|α|=0
∫
Ω
|Dαu|2dx) 12
Let X, Y be Banach spaces. Denote by L2((0, T );X) the space consisting of all
measureable functions u : (0, T ) −→ X with the norm
‖u‖L2((0,T );X) =
( ∫ T
0
‖u(t)‖2Xdt
) 1
2
and by H((0, T );X, Y ) the space consisting of all functions u ∈ L2((0, T );X) such
that the generalized derivative ut exists and belongs to L2((0, T );Y ). The norm in
H((0, T );X, Y ) is defined by
‖u‖H((0,T );X,Y ) =
(‖u‖2L2((0,T );X) + ‖ut‖2L2((0,T );Y )) 12
Now we introduce a differential operator of order 2m
L(x, t,D) =
m∑
|p|,|q|=0
(−1)|p|Dp(apq(x, t)Dq),
where apq are s×s matrices smooth elements of which are in ΩT , apq = (−1)|p|+|q|a?pq
(a∗qp is the transportated conjugate matrix to apq).We assume there exists a constant
c0 > 0 independing on t such that: ∀ξ ∈ Rn \ {0}, ∀η ∈ Cs \ {0} :∑
|p|=|q|=m
apq(x, t)ξ
pξqηη ≥ c0|ξ|2m|η|2, (2.1)
where ξp = ξp11 ...ξ
pn
n , ξ
q = ξq11 ...ξ
qn
n .
We introduce also a system of boundary operators
Bj = Bj(x, t,D) =
∑
|p|≤µj
bj,p(x, t)D
p, j = 1, ..., m,
on S. Suppose that bj,p(x, t) are s× s matrices smooth elements of which are in ΩT
and
ordBj = µj ≤ m− 1 for j = 1, ..., χ,
m ≤ ordBj = µj ≤ 2m− 1 for j = χ+ 1, ..., m.
Assume that coefficients ofBj are independent of t if ordBj < m and {Bj(x, t,D)}mj=1
is a normal system on S for all t ∈ [0, T ], i.e., the two following conditions are
satisfied:
83
Nguyen Thi Lien
(i) µj 6= µk for j 6= k,
(ii) Boj (x, t, ν(x)) 6= 0 for all (x, t) ∈ ST , j = 1, ..., m.
Here ν(x) is the unit outer normal to S at point x and Boj (x, t,D) is the principal
part of Bj(x, t,D),
Boj = B
o
j (x, t,D) =
∑
|p|=µj
bj,p(x, t)D
p, j = 1, ..., m.
Furthermore, Boj (0, t, ν(x)) 6= 0 for all x ∈ S closed enough to the origin 0.
We set
HmB (Ω) =
{
u ∈ Hm(Ω) : Bju = 0 on S for j = 1, .., χ
}
with the same norm in Hm(Ω) and
B(t, u, v) =
m∑
|p|,|q|=0
∫
Ω
apqD
quDpvdx, t ∈ [0, T ].
Doing the same in the Garding’s inequality, we have:
Lemma 2.1. Suppose that coefficients of the operator L(x,t,D) satisfy condition
(2.1). Then there exists two constant µ0 and λ0 such that
(−1)mB(t, u, u) ≥ µ0‖u(x, t)‖2Hm(Ω) − λ0‖u(x, t)‖2L2(Ω)
for all functions u(x, t) ∈ H((0, T );HmB (Ω), H−mB (Ω)).
Thus, set u = eiλ0tv if necessary, we can suppose that
(−1)mB(t, u, u) ≥ µ0‖u(x, t)‖2Hm(Ω) (2.2)
for all u ∈ HmB (Ω) and t ∈ [0, T ]. Applying Green’s formula, we can assume that it
can be choose boundary operators Φj on ST , j = 1, ..., m such that
B(t, u, v) =
∫
Ω
Luv +
χ∑
j=1
∫
S
ΦjBjvds+
m∑
j=χ+1
∫
S
BjΦjvds. (∗)
Denote H−mB (Ω) the dual space to H
m
B (Ω). We write
〈
., .
〉
to stand for the pair-
ing between HmB (Ω) and H
−m
B (Ω), and (., .) to define the inner product in L2(Ω).
We then have the continuous imbeddings HmB (Ω) ↪→ L2(Ω) ↪→ H−mB (Ω) with the
equation 〈
f, v
〉
= (f, v) for f ∈ L2(Ω) ⊂ H−mB (Ω), v ∈ HmB (Ω).
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On the solvability of the initial boundary value problem for Schro¨dinger systems...
We study the following problem in the cylinder ΩT :
(−1)m−1iL(x, t,D)u− ut = f(x, t) in ΩT , (2.3)
Bju = 0, on ST , j = 1, ..., m, (2.4)
u |t=0= φ, on Ω, (2.5)
where f ∈ L2((0, T );HmB (Ω)) and φ ∈ L2(Ω) are given functions.
The solution u(x, t) is searched in the generalized sense. That means
u ∈ H((0, T );HmB (Ω), H−mB (Ω)) is called a generalized solution of the problem (2.3)-
(2.5) if u(., 0) = φ and the equality
(−1)m−1iB(t, u, v)− 〈ut, v〉 = 〈f(t), v〉 (2.6)
holds for a.e. t ∈ (0, T ) and all v ∈ HmB (Ω).
3. The unique solvability of the problem
Theorem 3.1. Suppose that coefficients of the operator L(x,t,D) satisfy condition
(2.2). Then problem (2.3)-(2.5) has at most one generalized solution in the space
generalized solution u ∈ H((0, T );HmB (Ω), H−mB (Ω)).
Proof. Suppose u1(x, t), u2(x, t) are two generalized solutions of problem (2.3)-(2.5)
inH((0, T );HmB (Ω), H
−m
B (Ω)). Denote u(x, t) = u
1(x, t)−u2(x, t). Arccording to the
denifition of generalized solutions, substituting v := u into (2.6), then integrating
both sides of the obtained equality with respect to t from 0 to b (b > 0), we arrive
at
(−1)m−1i
∫ b
0
B(t, u(., t), u(., t))dt−
∫ b
0
〈
ut, u(., t)
〉
dt = 0.
Thus
(−1)m
∫ b
0
B(t, u(., t), u(., t))dt− i
∫ b
0
〈
ut, u(., t)
〉
dt = 0. (3.1)
Since ∫ b
0
〈
ut, u(., t)
〉
dt = ‖u(b)‖2L2(Ω) −
∫ b
0
〈
u, ut(., t)
〉
dt,
we get ∫ b
0
〈
ut, u(., t)
〉
dt =
1
2
‖u(b)‖2L2(Ω).
Adding (3.1) with its complex conjugate, we discover∫ b
0
B(t, u(., t), u(., t))dt = 0
Using the inequality (2.2), we have∫ b
0
‖u‖2Hm
B
(Ω)dt ≤
∫ b
0
B(t, u(., t), u(., t))dt = 0,
85
Nguyen Thi Lien
so
‖u‖2L2((0,b);HmB (Ω)) =
∫ b
0
‖u‖2L2((0,T );HmB (Ω))dt = 0.
This implies u ≡ 0 on [0, b]. Therefore, u ≡ 0 on ΩT . The proof of the uniquence of
generalized solution is completed.
Theorem 3.2. Suppose that f ∈ L2((0, T );H−mB (Ω)), φ ∈ L2(Ω) and the conditions
of Theorem 3.1 is fulfilled. Then there exists a generalized solution in generalized
solution u ∈ H((0, T );HmB (Ω), H−mB (Ω)) of the problem (2.3)-(2.5) which satisfies
‖u‖2
H((0,T );Hm
B
(Ω),H−m
B
(Ω))
≤ C(‖φ‖2L2(Ω) + ‖f‖2L2((0,T );H−mB (Ω))),
where C is a constant independent of φ, f and u.
Proof. Suppose {ψk(x)}∞k=1 be a system functions in HmB (Ω), which is orthonormal
in L2(Ω) and its linear closure is just H
m
B (Ω). We look for u
N(x, t) in the form:
uN(x, t) =
N∑
k=1
CNk (t)ψk(x), where {CNk (t)}Nk=1 is the solution of the ordinary differ-
ential system:
(−1)m−1i
m∑
|p|,|q|=0
∫
Ω
apqD
quNDpψldx−
∫
Ω
uNt ψldx =
〈
f, ψl
〉
, l = 1, ..., N (3.2)
CNk (0) = Ck = (φ, ψk), k = 1, ..., N. (3.3)
After multiplying both sides of (3.2) by CNl (t), taking sum with respect to l from 1
to N and integrating with respect to t from 0 to τ (τ > 0), we get
(−1)m−1i
τ∫
0
B(t, uN , uN)dt−
τ∫
0
(uNt , u
N)dt =
τ∫
0
〈
f, uN
〉
dt.
From this equality we obtain
(−1)m
τ∫
0
B(t, uN , uN)dt− 1
2
i
(‖u(τ)‖2L2(Ω) − ‖u(0)‖2L2(Ω)) = i
τ∫
0
〈
f, uN
〉
dt. (3.4)
Adding (3.4) with its complex conjugate, we have
(−1)m−1
τ∫
0
B(t, uN , uN)dt = Im
τ∫
0
〈
f, uN
〉
dt
1
2
(‖u(τ)‖2L2(Ω) − ‖u(0)‖2L2(Ω)) = −Re
τ∫
0
〈
f, uN
〉
dt.
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On the solvability of the initial boundary value problem for Schro¨dinger systems...
Noting that
|(−1)m−1
τ∫
0
B(t, uN , uN)dt| ≥ µ‖uN‖2L2((0,τ);HmB (Ω))
‖uN(0)‖2L2(Ω) = ‖
N∑
k=1
(φ, ψk)ψk‖2L2(Ω) ≤ ‖φ‖2L2(Ω)
and
|Im
τ∫
0
〈
f, uN
〉
dt| − |Re
τ∫
0
〈
f, uN
〉
dt| ≤ 2
τ∫
0
‖f‖H−m
B
(Ω)‖uN‖HmB (Ω)
≤ ‖uN‖2L2((0,τ);HmB (Ω)) +
1
‖f‖2
L2((0,τ);H
−m
B
(Ω))
,
So we have
‖uN‖2L2((0,τ);HmB (Ω)) ≤ C
(‖f‖2
L2((0,τ);H
−m
B
(Ω))
+ ‖φ‖2L2(Ω)
)
.
Letting τ tend to T , we get
‖uN‖2L2((0,T );HmB (Ω)) ≤ C
(‖f‖2
L2((0,T );H
−m
B
(Ω))
+ ‖φ‖2L2(Ω)
)
. (3.5)
Now, fix any v ∈ HmB (Ω) with ‖v‖HmB (Ω) ≤ 1 and write v = v1 + v2, where
v1 ∈ span{ψl}Nl=1, (v2, ψl)L2(Ω) = 0, l = 1, ..., N. Since ‖v‖HmB (Ω) ≤ 1, ‖v1‖HmB (Ω) ≤ 1.
We obtain from (3.2) that
−(uNt , v1) + (−1)m−1iB(t, uN , v1) =
〈
f, v1
〉
.
Thus, 〈
uNt , v
〉
= (uNt , v) = (u
N
t , v1) = (−1)m−1iB(t, uN , v1)−
〈
f, v1
〉
.
Since ‖v1‖Hm
B
(Ω) ≤ 1,
‖uNt ‖H−m
B
(Ω) ≤ |
〈
uNt , v
〉| ≤ |B(t, uN , v1)|+ |〈f, v1〉|
≤ C(‖f‖H−m
B
(Ω) + ‖uN‖HmB (Ω)
)
.
Therefore, by (3.5),
‖uNt ‖2L2((0,T );H−mB (Ω)) ≤ C
(‖f‖2
L2((0,T );H
−m
B
(Ω))
+ ‖uN‖2L2((0,T );HmB (Ω))
)
≤ C(‖f‖2
L2((0,T );H
−m
B
(Ω))
+ ‖φ‖2L2(Ω)
)
.
From this inequality and (3.5) we get
‖uNt ‖2H((0,T );Hm
B
(Ω),H−m
B
(Ω))
≤ C(‖f‖2
L2((0,T );H
−m
B
(Ω))
+ ‖φ‖2L2(Ω)
)
, (3.6)
87
Nguyen Thi Lien
where C is the constant independent of φ, f and N .
Because {uN} is bounded in Hilbert space H((0, T );HmB (Ω), H−mB (Ω)), we can
choose a subsequence weakly convergent to u(x, t) ∈ H((0, T );HmB (Ω), H−mB (Ω)).
We will prove that u(x, t) is a generalized solution of problem (2.3)-(2.5). Fix a
positive real number τ, τ ≤ T and a positive integer h. Take a function η ∈
L2((0, τ);H
m
B (Ω)) in the form
η(x, t) =
h∑
l=1
dl(t)ψl(x), (3.7)
where dl(t) are smooth functions defined on [0, τ ]. Multiplying both sides of (3.2)
with N ≥ h by dl(t), taking sum with respect to l from 1 to h and integrating with
respect to t from 0 to τ , we have
(−1)m−1i
τ∫
0
B(t, uN , η)dt−
τ∫
0
〈
uNt , η)
〉
dt =
τ∫
0
〈
f, η
〉
dt.
Letting N tend to ∞, we have
(−1)m−1i
τ∫
0
B(t, u, η)dt−
τ∫
0
(ut, η)dt =
τ∫
0
〈
f, η
〉
dt. (3.8)
Since the set of functions of the form (3.7) is dense in L2((0, τ);H
m
B (Ω)), the equality
(3.8) holds for all η ∈ L2((0, τ);HmB (Ω)). This implies
(−1)m−1iB(t, u, v)− 〈ut, v〉 = 〈f(t), v〉
holds for a.e. t ∈ (0,+∞) and all v ∈ HmB (Ω).The inequality in the theorem is
followed from (3.6).
Now, we will prove that u(., 0) = φ. An intergration by parts from (3.8) yields
(−1)m−1i
τ∫
0
B(t, u, η)dt−
τ∫
0
(u, ηt)dt+ (u(., 0), η(., 0)) =
τ∫
0
〈
f, η
〉
dt (3.9)
holds for all η ∈ C1([0, τ ], HmB (Ω)) satisfying η(., τ) = 0. We have
(−1)m−1i
τ∫
0
B(t, uN , η)dt−
τ∫
0
(uN , ηt)dt+ (u
N(., 0), η(., 0)) =
τ∫
0
〈
f, η
〉
dt.
Passing to the limit as N →∞ with noting that uN(., 0)→ φ in L2(Ω), we get
(−1)m−1i
τ∫
0
B(t, u, η)dt−
τ∫
0
(u, ηt)dt+ (φ, η(., 0)) =
τ∫
0
〈
f, η
〉
dt. (3.10)
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On the solvability of the initial boundary value problem for Schro¨dinger systems...
Comparing (3.9) and (3.10),we obtain
(u(., 0), η(., 0)) = (φ, η(., 0)).
Since η(., 0) ∈ HmB (Ω) is arbitrary u(., 0) = φ.Theorem 3.1 is proved.
4. An example
In this section we apply the previous results to the Cauchy-Dirichlet problem
for the wave equation. We consider the following problem:
4u− utt = f(x, t), (x, t) ∈ ΩT , (4.1)
u|t=0 = ut|t=0 = 0, x ∈ Ω, (4.2)
u|ST = 0, (4.3)
where 4 is the Laplace operator. By
o
H1(Ω) we denote the completion of
o
C∞(Ω) in
the norm of the spaceH1(Ω). ThenH((0, T );H1B(Ω), H
−1
B (Ω) = H((0, T );
o
H1(Ω),
o
H−1(Ω)).
From this fact and Theorem 3.1 and 3.2 we obtain following results.
Theorem 4.1. Suppose that f ∈ L2((0, T );
o
H−1(Ω)), φ ∈ L2(Ω). Then problem
(4.1)-(4.3) has a unique generalized solution u in the spaceH((0, T );
o
H1(Ω),
o
H−1(Ω))
and
‖u‖2
H((0,T );
o
H1(Ω),
o
H−1(Ω))
≤ C
(
‖φ‖2L2(Ω) + ‖f‖2
L2((0,T );
o
H−1(Ω))
)
,
where C is a constant independent of φ, f and u.
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