Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 199, pp. 1–7.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
SOLVABILITY OF SECOND-ORDER BOUNDARY-VALUE
PROBLEMS ON NON-SMOOTH CYLINDRICAL DOMAINS
BELKACEM CHAOUCHI
Abstract. In this note, we present an abstract approach for the study of
a second-order boundary-value problem on cusp domain. This study is performed in the framework of anisotropic little Hölder spaces. Our strategy is
to use of the commutative version of the well known sums of operators theory.
This technique allows us to obtain the space regularity of the unique strict
solution for our problem.
1. Introduction
3
Let Ω ⊂ R a cusp domain defined by
x1
x2
Ω = (x1 , x2 , x3 ) ∈ R3 : 0 < x3 < d0 , (
,
) ∈ Ω0 ,
α
α
(x3 ) (x3 )
where Ω0 ⊂ R2 is a smooth domain of class C ∞ , α > 1 and d0 > 0. This article concerns the solvability of the boundary-value problem of the second-order differential
equation
∂t2 u(t, x) + ∆u(t, x) − λu(t, x) = h(t, x),
(t, x) ∈ Π =]0, 1[×Ω,
(1.1)
subject to the following boundary value conditions
a(x)u(0, x) − b(x)∂t u(0, x) = 0
x ∈ Ω,
u(1, x) = 0 x ∈ Ω,
u(t, x) = 0
(1.2)
(t, x) ∈]0, 1[×∂Ω.
Here, x = (x1 , x2 , x3 ) represents a generic point of R3 and λ is a fixed positive
spectral parameter.
The main assumptions on the functions a and b are
a, b > 0,
a, b ∈ C 1 (Ω).
We are especially interested with the case when the right hand side of (1.1) is taken
in the anisotropic little Hölder space
h2ν,2σ (Π) = h2ν ([0, 1]; h2σ (Ω)),
ν, σ ∈]0, 1/2[,
2000 Mathematics Subject Classification. 35L05, 46E35, 47A62.
Key words and phrases. Little Hölder space; sum of linear operators; cusp domain.
c
2013
Texas State University - San Marcos.
Submitted May 2, 2013. Published September 11, 2013.
1
2
B. CHAOUCHI
EJDE-2013/199
more details about these spaces are given in Section 2. We assume also that the
right hand side h satisfies the condition
h=0
on ∂Π.
(1.3)
Note that in our situation, the classical arguments such as the variational method
does not apply. Consequently, we opt for the use of the technique of the sum of
linear operators. Fore more details about this technique, we refer the reader to
[6, 7, 10, 11, 12]. In the literature, we find several regularity results concerning
elliptic and parabolic problems which have been obtained via this technique, see
[1, 2, 8, 9]. In this paper, we will use the commutative version developed in [6].
Our main result on the existence, uniqueness and regularity of the strict solution
of (1.1)-(1.2) is stated in the following theorem.
Theorem 1.1. Let h ∈ h2ν,2σ (Π) with ν, σ ∈]0, 1/2[, satisfying Assumption (1.3).
Then, under conditions (1.2), Problem (1.1) has a unique strict solution u such
that
(x3 )4σα+2α ∂t2 u and (x3 )4σα (∆ − λ)u ∈ h2ν,2σ (Π).
This article is organized as follows: In section 2, we introduce the necessary
notation and some definitions related to the functional framework of anisotropic
little Hölder spaces. In section 3, we recall the main results of the sum’s operators
theory. In section 4, using a suitable change of variables our concrete problem is
transformed into a new one posed in a cylindrical domain. Next, thanks to the
sums technique, we will give a complete study of our transformed problem which
allows us to justify our main result.
2. Little Hölder spaces
We briefly recall the definition of the anisotropic little Hölder spaces. We will
denote by Cb2σ (Ω) the space of the bounded and 2σ-Hölder continuous functions
defined on Ω. The little Hölder space h2σ (Ω) is defined by
o
n
|f (x0 ) − f (x)|
=
0
,
h2σ (Ω) = f ∈ Cb2σ (Ω) : lim+ sup
ε→0 x0 6=x kx0 − xk2σ
endowed with the norm
|f (x0 ) − f (x)|
.
0
2σ
x0 6=x kx − xk
kukh2σ (Ω) = max |f (x)| + sup
x∈Ω
The anisotropic little Hölder space h2ν,2σ (Π) is defined by
h2ν,2σ (Π) = f ∈ C 2σ ([0, 1]; h2σ (Ω)) : lim
ε→0+
sup
kf (t) − f (t0 )kh2σ (Ω)
|t −
0<|t−t0 |≤ε
t0 |2ν
=0 .
We endow h2ν,2σ (Π) with the norm
kukh2ν,2σ (Π) = max ku(t)kh2σ (Ω) + sup
t∈[0,1]
t0 6=t
ku(t0 ) − u(t)kh2σ (Ω)
|t0 − t|2ν
,
more details about little Hölder spaces are given in [13, 14].
Remark 2.1. It is necessary to note that any function of h2ν,2σ (Π), can be extended
to a function of h2ν,2σ (Π). This is why we shall write in the sequel h2ν,2σ (Π) or
h2ν,2σ (Π).
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SOLVABILITY OF SECOND-ORDER BVPS
3
3. On the sum of linear operators
Let E a complex Banach space and A, B two closed linear operators with domains
D(A), D(B). Let L be the operator defined by
Lu = Bu + Au,
u ∈ D(L) = D(A) ∩ D(B).
(3.1)
where A and B satisfy the assumptions
(H1) (i) ρ(A) ⊃ ΣA = {µ : |µ| ≥ r, | Arg(µ)| < π − A },
k(A − µI)−1 kL(E) ≤ CA /|µ|,
∀µ ∈ ΣA ;
(ii) ρ(B) ⊃ ΣB = {µ : |µ| ≥ r, | Arg(µ)| < π − B },
k(B − µI)−1 kL(E) ≤ CB /|µ|,
∀µ ∈ ΣB ;
(iii) A + B < π;
(iv) D(A) + D(B) = E.
(H2) for all µ1 ∈ ρ(A) and all µ2 ∈ ρ(B),
(A−µ1 I)−1 (B −µ2 I)−1 −(B −µ2 I)−1 (A−µ1 I)−1 = [(A−µ1 I)−1 ; (B −µ2 I)−1 ] = 0,
where ρ(A) and ρ(B) are the resolvent sets of A and B.
The main result proved in [6] reads as follows:
Theorem 3.1. Let % ∈]0, 1[. Assume (H1), (H2) hold and f ∈ DA (%). Then, the
problem
Au + Bu = f,
has a unique strict solution u ∈ D(A) ∩ D(B), given by
Z
1
(B + µ)−1 (A − µ)−1 f dµ,
u=−
2iπ Γ
where Γ is a sectorial curve lying in (ΣA ) ∩ (Σ−B ) oriented from ∞e+iθ0 to ∞e−iθ0
with B < θ0 < π − A . Moreover, Au, Bu ∈ DA (%).
Remark 3.2. The interpolation spaces DA (ρ), with % ∈]0, 1[, are defined as follows
DA (ρ) = {ξ ∈ E : lim krρ A(A − rI)−1 ξkE = 0};
r→0+
for more details, see [13, 14].
4. Applications of the sums theory
4.1. Change of variables. As in [1], consider the change of variables T : Π → Q,
(t, x1 , x2 , x3 ) 7→ (t, ξ1 , ξ2 , ξ3 ) = (t,
x1
x2
1
(x3 )1−α ),
,
,
α
α
(x3 ) (x3 ) α − 1
where
Q =]0, 1[×D,
D = Ω0 ×]d1 , +∞[,
d1 =
1
(d0 )1−α > 0,
α−1
Let us introduce the following change of functions
v(t, ξ) = u(t, x),
e
a(ξ) = a(x),
g(t, ξ) = h(t, x),
eb(ξ) = b(x).
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B. CHAOUCHI
EJDE-2013/199
Consequently, our problem (1.1) becomes
φ(ξ3 )∂t2 v(t, ξ) + [P − λφ(ξ3 )]v(t, ξ) = f (t, ξ),
e
a(ξ)v(0, ξ) − eb(ξ)∂t v(0, ξ) = 0,
v(1, ξ) = 0,
ξ ∈ D,
v(., ξ) = 0,
ξ ∈ ∂D,
(t, ξ) ∈ Q,
ξ ∈ D,
with
(4.1)
2α
ξ = (ξ1 , ξ2 , ξ3 ), f (t, ξ) = φ(ξ3 )g(t, ξ), φ(ξ3 ) = (ξ3 ) 1−α .
Here P is the second order differential operator with C ∞ -bounded coefficients on
Q given by
2α
P v(t, ξ) = (α − 1) α−1 (∂ξ21 v + ∂ξ22 v + ∂ξ23 v)
2α α 2 ξ1 2 2
ξ2
) ( ) ∂ξ1 v + ( )2 ∂ξ22 v
+ (α − 1) α−1 (
α−1
ξ3
ξ3
2α α 2 ξ1 ξ2 2
)
∂ξ1 ξ2 v
+ (α − 1) α−1 2(
2
α − 1 (ξ3 )
2α 2α ξ1
ξ1
ξ2
+ (α − 1) α−1
∂ξ21 ξ3 v + ∂ξ21 ξ3 v + ∂ξ22 ξ3 v
α − 1 ξ3
ξ3
ξ3
2α α(α + 1) ξ1
ξ
2
∂ξ1 v +
∂ξ2 v
+ (α − 1) α−1
2
2
2
(α − 1) (ξ3 )
(ξ3 )
2α α
1
+ (α − 1) α−1
∂ξ v .
α − 1 ξ3 3
(4.2)
Remark 4.1. Observe that the functions e
a and eb are necessarily bounded on Q.
In fact, one has
α
α
1
|e
a(ξ1 , ξ2 , ξ3 )| = |a(((α − 1)ξ3 ) 1−α ξ1 , ((α − 1)ξ3 ) 1−α ξ2 , ((α − 1)ξ3 ) 1−α )|
≤C
max
|a(x1 , x2 , x3 )| .
(x1 ,x2 ,x3 )∈Ω
The following lemma specifies the impact of the change of variables on the functional framework
Lemma 4.2. Let ν, σ ∈]0, 1/2[. Then
(1) h ∈ h2ν,2σ (Π) ⇒ g ∈ h2ν,2σ (Q);
(2) h ∈ h2ν,2σ (Π) ⇒ f ∈ h2ν,2σ (Q);
(3) f ∈ h2ν,2σ (Q) ⇒ (x3 )4σα h ∈ h2ν,2σ (Π);
a ∈ C 1 (D);
(4) a ∈ C 1 (Ω) ⇒ e
1
e
(5) b ∈ C (Ω) ⇒ b ∈ C 1 (D);
For the prof of the above lemma, it suffices to use the same arguments as in [5,
Proposition 3.1].
4.2. Abstract formulation of the transformed problem. Let E = h2σ (Ω), we
choose X = C([0, 1]; E) equipped with its natural norm
kf kX = sup kf kE .
0≤t≤1
Let us define the vector-valued functions
v : [0, 1] → E; t → v(t);
v(t)(ξ) = v(t, ξ),
EJDE-2013/199
SOLVABILITY OF SECOND-ORDER BVPS
f : [0, 1] → E; t → f (t);
5
f (t)(ξ) = f (t, ξ).
Let (P, D(P )) be the linear operator given by (4.2) where
D(P ) = {ψ ∈ C(D) ∩ W 2,q (D), q ≥ 3 : P ψ ∈ C(D), ψ = 0 on ∂D}.
Now, for ξ ∈ D and 0 ≤ t ≤ 1, define the two operators A and B by
D(A) = {v ∈ X : φ(ξ3 )∂t2 v ∈ X, e
a(ξ)v(0) − eb(ξ)∂t v(0) = 0, v(1) = 0},
(Av)(t) = φ(ξ3 )∂t2 v(t),
ξ3 ≥ d1 ,
and
D(B) = {v ∈ X : v(t) ∈ D(P )}
(4.3)
(Bv)(t) = [P − λφ(ξ3 )](v(t)).
Consequently, the abstract version of Problem (1.1) is
Av + Bv = f.
(4.4)
Proposition 4.3. The operator B satisfy Assumption (H1).
Proof. The operator B has the same properties as the operator P − λφ(ξ3 )I. We
are then concerned with the study of the spectral problem
(P − λφ(ξ3 ))v − µv = ϕ ∈ hσ (D)
v=0
on ∂D
ϕ=0
on ∂D.
with the necessary condition
(4.5)
Due to [4], there exist K > 0 and C > 0 such that for Re µ > 0 one has
kvkhσ (D) ≤
K
K
kϕkhσ (D) ≤
kϕkhσ (D)
|Cλ + µ| + 1
|µ| + 1
which implies that the operator (4.3) is the generator of an analytic semigroup
(T (s))s≥0 strongly continuous, therefore there exists B ∈]0, π2 [ such that B satisfies
(H1).
Proposition 4.4. The operator A satisfies Assumption (H1).
Proof. For simplicity, we use the same argument as in [3]. The study of operator
A given by (4.3) is based essentially on the study of the spectral problem
v 00 (t) − zv(t) = φ(t)
(4.6)
e
a(ξ)v(0) − eb(ξ)∂t v(0) = 0,
v(1) = 0
For z ∈ C \ R + − the unique solution v is
Z 1
v(t) = (A − z)−1 φ =
K√z (t, ξ, s)ϕ(s)ds,
(4.7)
0
where
K√z (t, ξ, s) =
√
√
√
[e
a(ξ) sinh zs+e
b(ξ) z cosh zs]
√ e √
√
z[e
a(ξ) sinh z+b(ξ) z cosh z]
√
√
√
√
sinh
z(1−s)
[e
a(ξ) sinh zt+e
b(ξ) z cosh zt]


√
√
√
√
z[e
a(ξ) sinh z+e
b(ξ) z cosh z]

√

 sinh z(1−t)
√
if 0 ≤ s ≤ t
if t ≤ s ≤ 1,
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B. CHAOUCHI
EJDE-2013/199
√
with < z > 0. One has
√
1 e
√
√
√
√ √
a(ξ) sinh z(exp( z) − exp(− z)) + z (exp( z) + exp(− z))
e
2 b(ξ)
2
√
√
e
a(ξ)
≥|
+ Re z| sinh Re z.
eb(ξ)
Then
Z
1
K√z (t, ξ, s)ϕ(s)ds
0
Rt
√
√
√
√
cosh Re z(1 − t) 0 [e
a(ξ) cosh Re zs + eb(ξ)| z| cosh Re zs]ds
≤
eb(ξ)|√z||( ea(ξ) + Re √z)| sinh Re √z
e
b(ξ)
√
√ R1
√
[e
a(ξ) cosh Re zt + eb(ξ)| z| cosh Re zt] t cosh Re z(1 − s)ds
+
eb(ξ)|√z||( ea(ξ) + Re √z)| sinh Re √z
e
√
b(ξ)
and
Z
|
0
1
eb(ξ)( ea(ξ) + Re √z)
e
b(ξ)
K√z (t, ξ, s)ϕ(s)ds| ≤
eb(ξ)|√z||( ea(ξ) + Re √z)| Re √z|√z|
e
b(ξ)
1
≤
,
cos(θ/2)|z|
which means that Hypothesis (H1) is satisfied with A ∈]0, π/2[.
Remark 4.5. It is important to note that: 1. Thanks to [13], we have
DA (ν) = ϕ ∈ h2ν ([0, 1]; E) : ϕ(0) = ϕ(1) = 0 .
2. Hypothesis (H2) is checked in a similar way as in [6] and [12].
Applying the sums technique, we obtain the following maximal regularity results
Proposition 4.6. Let f ∈ h2ν ([0, 1]; h2σ (D)), ν, σ ∈]0; 1/2[. Then, for λ > 0
Problem (4.4) has a unique strict solution v satisfying
Av ∈ DA (ν)
Bv ∈ DA (ν)).
As in [8], to prove our main result, that is theorem 1.1, it suffices to use the
inverse change of variables T −1 : Q → Π,
α
α
1
(t, ξ1 , ξ2 , ξ3 ) 7→ (t, x1 , x2 , x3 ) = (t, ((α−1)ξ3 ) 1−α ξ1 , ((α−1)ξ3 ) 1−α ξ2 , ((α−1)ξ3 ) 1−α ).
Acknowledgments. The author is thankful to the anonymous referees for their
careful reading of a previous version of the manuscript, which led to a substantial
improvements.
EJDE-2013/199
SOLVABILITY OF SECOND-ORDER BVPS
7
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Belkacem Chaouchi
Lab. de l’Energie et des Systèmes, Intelligents Khemis Miliana University, Algeria
E-mail address: chaouchicukm@gmail.com