Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 181, pp. 1–6.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
TRIGONOMETRIC SERIES ADAPTED FOR THE STUDY OF
DIRICHLET BOUNDARY-VALUE PROBLEMS OF LAMÉ
SYSTEMS
BOUBAKEUR MEROUANI, RAZIKA BOUFENOUCHE
Abstract. Several authors have used trigonometric series for describing the
solutions to elliptic equations in a plane sector; for example, the study of
the biharmonic operator with different boundary conditions, can be found in
[2, 9, 10]. The main goal of this article is to adapt those techniques for the
study of Lamé systems in a sector.
1. Introduction
Let S be the truncated plane sector of angle ω ≤ 2π, and positive radius ρ:
S = {(r cos θ, r sin θ) ∈ R2 , 0 < r < ρ, 0 < θ < ω} .
(1.1)
Let Σ be the circular boundary part
Σ = {(ρ cos θ, ρ sin θ) ∈ R2 , 0 < θ < ω}.
(1.2)
We are interested in the study of functions u belonging to the Sobolev space
[H 1 (S)]2 , and that are solutions to the Lamé type system
Lu = ∆u + ν0 ∇(div u) = 0
u=0
in S
for θ = 0, ω ,
(1.3)
where
λ+µ
,
µ
λ, µ are the Lamé constants, with λ ≥ 0, µ > 0 and ν is a real number (0 < ν < 1/2)
called Poisson coefficient.
We shall analyze the solutions u of this problem which can be written in series
of the form:
X
u(r, θ) =
cα rα vα (θ).
(1.4)
ν0 = (1 − 2ν)−1 =
α∈E
Here E stands for the set of solutions of the equation in a complex variable α(ν0 ),
ν
2
0
sin2 αω =
α sin2 ω, Re α 0
(1.5)
ν0 + 2
For further studies of the set E, see, for example Lozi [5].
2010 Mathematics Subject Classification. 35B40, 35B65, 35C20.
Key words and phrases. Sector; crack; singularity; Lamé system; series.
c
2015
Texas State University - San Marcos.
Submitted February 6, 2015. Published July 1, 2015.
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B. MEROUANI, R. BOUFENOUCHE
EJDE-2015/181
We will adapt the technique used in [2, 9, 10], for the bilaplacian. The novelty
is that, here, we treat a system of PDE’s instead of the equations studied before.
For this purpose we introduce a Betti formula instead of a Green formula. To
compute the coefficients of the the singularities which can occur in the solutions,
this technique is easier and more direct than the classical one used in [4].
We will focus on the important case of the crack, i.e. ω = 2π. The calculations
in that case are more explicit and give the known results for the Laplacian as just
a particular case.
2. Separation of variables
α
Replacing u by r vα (θ) = rα (v1,α (θ), v2,α (θ)) in the problem (1.3) and using the
change of variables
w1,α (θ) = cos θv1,α (θ) + sin θv2,α (θ),
(2.1)
w2,α (θ) = − sin θv1,α (θ) + cos θv2,α (θ)
leads us the system
00
0
w1,α
(θ) + (ν0 + 1)(α2 − 1)w1,α (θ) + (ν0 (α − 1) − 2)w2,α
(θ) = 0;
00
0
(ν0 + 1)w2,α
(θ) + (α2 − 1)w2,α (θ) + (ν0 (α + 1) + 2)w1,α
(θ) = 0.
w1,α (0) = w2,α (0) = 0.
(2.2)
cos ωw1,α (ω) − sin ωw2,α (ω) = sin ωw1,α (ω) + cos ωw2,α (ω) = 0.
By Merouani [7], the solutions of (2.2) are linear combination of the functions
2αv0 [cos(α − 2)θ − cos αθ]
ϕα (θ) =
(2.3)
−2αv0 sin(α − 2)θ + 2(v0 (α − 2) − 4) sin αθ
and
2αv0 sin(α − 2)θ − 2(v0 (α + 2) + 4) sin αθ
ψα (θ) =
2αv0 [cos(α − 2)θ − cos αθ]
0 < θ < ω,
(2.4)
A relationship, similar to classical orthogonality, for this system is given by the
following theorem.
Theorem 2.1. Let wα = (w1,α , w2,α ) and wβ = (w1,β , w2,β ) be solutions of (2.2)
with α and β solutions of (1.5). Then, for β =
6 α, we have
[wα , wβ ]
Z ωh
=
0
w1,β i
1
0
0
ν0 (w2,α , w1,α ) + ((v0 + 1)w1,α , w2,α )
dθ = 0
w2,β
(β − α)
Proof. We shall use Betti formula
Z
Z
(vLu − uLv)dx = [vσ(u) · η − uσ(v) · η]dσ
S
(2.5)
(2.6)
Γ
where
σ11 (u) σ12 (u)
σ(u) =
σ12 (u) σ22 (u)
cos θ
is the tensor of stress, η =
is the outward unit vector normal to Σ, and
sin θ
Γ is the boundary of S. For two functions u, v which are solutions of (1.3), using
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TRIGONOMETRIC SERIES
3
Betti’s formula we obtain
Z
[vσ(u).η − uσ(v).η]dσ = 0
(2.7)
Σ
α
on Σ, for the function u = r ϕα , taking account of the change of variables (2.1),
we have
rα−1
σ(u) · η =
Mα,v0 (wα )
(2.8)
µ
with Mα,v0 (wα ) being the matrix
0
0
((v0 − 1)w2,α
+ (α(v0 + 1) + (v0 − 1))w1,α ) cos θ − (w1,α
+ (α − 1)w2,α ) sin θ
0
0
(w1,α
+ (α − 1)w2,α ) cos θ + ((v0 − 1)w2,α
+ (α(v0 + 1) + (v0 − 1))w1,α ) sin θ
The results follow from the application of formula (2.7) to the functions u = rα ϕα
and u = rβ ψβ , and by using relation (2.8).
Corollary 2.2. Let wα and wβ be solution of (2.2) with α and β solution of (1.5).
Suppose in addition that
Z ω
w1,β
0
0
(w2,α
, w1,α
)
=0
(2.9)
w2,β
0
and α 6= β, then
Z
[wα , wβ ] =
0
ω
w1,β
[((v0 + 1)w1,α , w2,α )]
dθ = 0
w2,β
(2.10)
Proof. Substituting (2.9) in (2.5), we obtain (2.10).
α
Remark 2.3. For wα = r (w1,α , w2,α ), we define the operator
(v0 + 1)w1,α
T wα = rα−1
w2,α
Corollary 2.4. From corollary 2.2, if α 6= β, we have
Z
(T wα wβ + wα T wβ )dσ = 0.
(2.11)
Σ
Proof. From the definition of the operator T and Corollary 2.2 we have
Z
Z ω
(T wα .wβ + wα T wβ )dσ = 2rα+β−1
((v0 + 1)w1,α w1,β + w2,α w2,β )dθ = 0.
Σ
0
Corollary 2.5. Suppose that u = α∈E cα rα ϕα is uniformly convergent in S. If
[ϕβ , ϕβ ] 6= 0 then
R
(T uuβ + uT uβ )dσ
1
cβ = ρ−2β+1 Σ
.
2
[ϕβ , ϕβ ]
P
Proof. For u = α∈E cα rα ϕα and taking account the definition of the operator T
we have
Z
(T u.uβ + u.T uβ )dσ
Σ
Z ω X
α−1 (v0 + 1)ϕ1,α
=
cα r
rβ ϕβ
ϕ2,α
0
P
α∈E
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B. MEROUANI, R. BOUFENOUCHE
EJDE-2015/181
(v0 + 1)ϕ1,β cα rα ϕα rβ−1
dθ
ϕ2,β
α∈E
Z ω X
(v0 + 1)ϕ1,β (v0 + 1)ϕ1,α
β+α−1
ϕβ + ϕα
=
cα r
dθ.
ϕ2,α
ϕ2,β
0
+
X
α∈E
From Corollary 2.2, if α 6= β, then
Z
(T uuβ + uT uβ )dσ = 2C β [ϕβ , ϕβ ]ρ2β−1 .
Σ
Expression cβ of Corollary 2.5 results from this last equality.
Remark 2.6. The technique we develop for the study of the trigonometric series is
based on Theorem 2.1 and Corollary 2.5. To illustrate this, we study the following
trigonometric series in the particular case of the crack (ω = 2π), which is an
important case of singular domains. The explicit knowledge of the roots of (1.5)
simplifies the computations.
3. Complete case study of the crack
To simplify the calculations, we decompose every solution u of (1.3) into two
parts with respect to θ
u = U1 + U2 .
3.1. Study of first part. The first part is the expression ϕα and is given by (2.3)
where
k
E = { , k ∈ N∗ } because ω = 2π.
2
After some calculation, we obtain that
[ϕβ , ϕβ ] = [β 2 v02 (2v0 + 3) + 4(v0 (β − 2) − 4)2 ]πρ2β−1 6= 0 .
Define the sub-sector
Sρ0 = S ∩ {(r cos θ, r sin θ) ∈ R2 , r < ρ0 }, ρ0 < ρ.
We define the traces on Σ,
U1 = ξ1 ∈ H̃ 1/2 (Σ)
2
2
and T U1 = φ1 ∈ H̃ 1/2 (Σ) .
Let
Z
2π
cα = Aα,v0
0
with
Aα,v0 =
(v0 + 1)ϕ1,α
ϕ1,α
ξ1
+ ρ0
φ1 (ρ0 , θ)dθ
ϕ2,α
ϕ2,α
(3.1)
ρ−α
0
2[α2 v02 (2v0 + 3) + 4(v0 (α − 2) − 4)2 ]π
Corollary 3.1. If U1 is solution of (1.3), then
X
U1 =
cα rα ϕα
(3.2)
α∈E
where cα is given by (3.1). The series converges uniformly in S ρ0 for all ρ0 < ρ.
Moreover (3.2) converges globally in (H 1 (Sρ ))2 , if α3/2 cα ρα ∈ l2 .
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TRIGONOMETRIC SERIES
5
Proof. (i) if (3.2) occurs, then cα is expressed by (3.1) under Corollary 2.5.
(ii) if U1 is solution of (1.3) and cα given by (3.1) then cα = ◦(αρ−α
0 ). This
implies the uniform convergence of the series in S ρ0 towards some W1 satisfying
(1.3).
From Grisvard-Geymonat [3], there exists a positive ε, sufficiently small such
that the solution of (1.3) is written as
X
U1 =
Kα rα ϕα
α∈E
which converges for r < ε. Theorem 2.1 implies that Kα = cα therefore W1 and U1
coincide in Sε . They coincide in Sρ0 since they are real analytic.
Remark 3.2. If ξ1 belongs to the space (H 2 (]0, 2π[))2 and φ1 to (H 1 (]0, 2π[))2 ,
then cα = ◦(αρ−α
0 ) and we have uniform convergence of the series in S ρ0 for all
ρ0 ≤ ρ.
3.2. Study of the second part. The second part is the expression ψα given by
(2.4) where
k
E = { , k ∈ N∗ }
2
because ω = 2π. After some calculations, we obtain
[ψα , ψα ] = [(v0 + 3)α2 v02 + 4(v0 + 1)(v0 (α + 2) + 4)2 ]πρ2α−1 6= 0 .
We define the following trace on Σ,
U2 = ξ2 ∈ (H̃ 1/2 (Σ))2
and T U2 = φ2 ∈ (H̃ 1/2 (Σ))2 .
Let
Z
dα = Bα, v0
0
with
Bα,v0 =
2π
ξ1
(v0 + 1)ψ1,α
ψ2,α
+ ρ0
ψ1,α
φ1 )(ρ0 , θ)dθ.
ψ2,α
(3.3)
ρ−α
0
2[(v0 + 3)α2 v02 + 4(v0 + 1)(v0 (α + 2) + 4)2 ]π
Corollary 3.3. If U2 is solution of (1.3) then
X
U2 =
dα rα ψα
(3.4)
α∈E
where dα is given by (3.3). The series converges uniformly in S ρ0 for all ρ0 < ρ.
Moreover (3.4) converges globally in (H 1 (Sρ ))2 , if α3/2 dα ρα ∈ l2 .
Remark 3.4. For v0 = 0 we obtain the trigonometric series for the Laplace equation in a sector. This is compatible with (1.3) with v0 = 0.
References
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EJDE-2015/181
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Boubakeur Merouani
Department of Mathemaics, Univ. Setif 1, Algeria
E-mail address: mermathsb@hotmail.fr
Razika Boufenouche
Department of Mathemaics, Univ. Jijel, Algeria
E-mail address: r boufenouche@yahoo.fr