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COEFFICIENTS OF SINGULARITIES OF THE BIHARMONIC PROBLEM OF NEUMANN TYPE: CASE OF THE CRACK
WIDED CHIKOUCHE and AISSA AIBÈCHE Received 13 April 2001
This paper concerns the biharmonic problem of Neumann type in a sector V. We give a representation of the solutionuof the problem in a form of a series u=
α∈Ecαrαφα, and the functionsφα are solutions of an auxiliary problem obtained by the separation of variables.
2000 Mathematics Subject Classification: 35B65, 35C10, 35J25.
1. Introduction. LetVbe a sector of angleω≤2π defined by
V=
(rcosθ,rsinθ)∈R2; 0< r < ρ, 0< θ < ω
(1.1)
andΣthe circular boundary part defined by
Σ=
(ρcosθ,ρsinθ)∈R2; 0< θ < ω
. (1.2)
We are interested in the study of functionsu, belonging to the Sobolev spaces H2(V ), solutions of
∆2u=0 inV ,
Mu=T u=0 forθ=0,ω, (1.3)
where
Mu=νu+(1−ν)
∂12un21+2∂12un1n2+∂22un22
,
T u= −∂u
∂n +(1−ν) d ds
∂12un1n2−∂12u n21−n22
−∂22un1n2
, (1.4)
νis a real number called Poisson coefficient(0< ν <1/2). We show that these functionsuare written under the form
u(r ,θ)=
α∈E
cαrαφα(θ), (1.5)
Eis the set of solutions of the equation inα
sin2(α−1)ω=1−ν 3+ν
2
(α−1)2sin2ω, Reα >1. (1.6)
For the study of the solutions of (1.6), see, for example, Blum and Rannacher [1] and Grisvard [2].
We are going to calculate the coefficientscαof development (1.5). These cal- culations have already been done by Tcha-Kondor [3] for the Dirichlet’s bound- ary conditions. He has established, thanks to the Green’s formula, a relation of biorthogonality between the functionsφαandφβallowing him to calculate the coefficientscβ. We follow the same approach. This needs the writing in the domainVof an appropriate Green formula. Using this formula, we establish a relation of biorthogonality between the functionsφαandφβ, which is reduced under some conditions to the simple relation obtained by Tcha-Kondor, which enables us to calculate the coefficientscβin the particular case of the crack (ω=2π).
2. Separation of variables. Replacingubyrαφα(θ)in problem (1.3) leads us to the boundary value problem
φ(4)α (θ)+ α2+(α−2)2
φα(θ)+α2(α−2)2φα(θ)=0, (2.1) να2+(1−ν)α
φα+φα=0, θ=0, θ=ω, (2.2) (2−ν)α2−3(1−ν)α+2(1−ν)
φα+φ(3)α =0, θ=0, θ=ω. (2.3)
The relation similar to orthogonality for the biharmonic operator is given by the following theorem.
Theorem2.1. Letφαandφβbe solutions of (2.1) withαandβsolutions of (1.6). So, forα≠β, we have the following relation:
φα,φβ
= ω
0
α2−2α φα−ν
α+β
+(3−ν)−2α α−β φα
φβ
+ β2−2β
φβ+ν α+β
+(3−ν)−2β α−β φβ
φα
dθ
=0.
(2.4)
Proof. We use the following Green formula:
V
v2u−u2v dx=
Γ
uT v+∂u
∂nMv
−
vT u+∂v
∂nMu
dσ , (2.5)
...
(Γ is the boundary ofV). For two functionsu,v solutions of (1.3), we get the Green’s formula in the following form:
Σ
uT v+∂u
∂nMv
−
vT u+∂v
∂nMu
dσ=0. (2.6)
OnΣ, we have, for the functionuα=rαφα,
∂uα
∂n =∂uα
∂r =αrα−1φα, Muα=rα−2 α2−(1−ν)α
φα+νφα
, T uα=rα−3
−α2(α−2)φα+ (ν−2)α+(3−ν) φα
.
(2.7)
The theorem results from the application of formula (2.6) to the biharmonic functionsuα=rαφαanduβ=rβφβ, and by using relations (2.7).
Remark2.2. This relation between the functionsφαandφβis similar to the relation of biorthogonality obtained when the functionsφαandφβfulfill (2.1) with the Dirichlet’s boundary conditionsφα=φα=φβ=φβ=0 forθ=0 andθ=ω. In this case, the relation is simplified because we have
ω
0 φαφβdθ= ω
0 φαφβdθ. (2.8)
Remark2.3. By a double integration by parts, we get ω
0 φαφβdθ= ω
0 φαφβdθ+ φα,φβω
0− φα,φβ
ω
0. (2.9)
Corollary2.4. Letφαandφβbe solutions of (2.1) withαandβsolutions of (1.6); in addition,
φα,φβ
ω
0− φα,φβ
ω
0 =0. (2.10)
So, forα≠β, we have the following relation:
φα,φβ
= ω
0 α2−2α φα+φα
φβ+
β2−2β φβ+φβ
φα
dθ=0. (2.11) Remark2.5. Foruα=rαφα, we have
uα−2 r
∂uα
∂r =rα−2 α2−2α
φα+φα
. (2.12)
LetPbe the operatorP=−(2/r )(∂/∂r ). FromCorollary 2.4andRemark 2.5, we deduce the following corollary.
Corollary2.6. Setuα=rαφα(θ)anduβ=rβφβ, whereφαandφβare solutions of (2.1) withαandβsolutions of (1.6); in addition,
φα,φβω
0− φα,φβω
0 =0. (2.13)
Ifα≠β, we have the following relation:
Σ
P uαuβ+uαP uβ
dσ=0. (2.14)
Now, usingCorollary 2.6, we calculate the coefficientscαof the development of the solutionuof (1.3). The calculations will be done in the case of the crack (ω=2π), which is a very important case of singularity of domains. The explicit knowledge of the roots manifestly simplifies the calculations.
3. Case of a crack. The crack corresponds to ω=2π; if we replace this value in (1.6), we find that solutionsαof (1.6) are the real valuesk/2. In this case, all the roots are of multiplicity 2.
We are going to representuas follows:
u=
α∈E
cαuα+
α∈E
dαvα, E= k
2, k >2
, uα=rαϕα, vα=rαψα,
(3.1)
ϕαare the even solutions inθ
ϕα(θ)=rα
cos(α−2)θ+4−(1−ν)α (1−ν)α cosαθ
, (3.2)
andψαthe odd solutions inθ
ψα(θ)=rα
sin(α−2)θ−4+(1−ν)(α−2) (1−ν)α sinαθ
. (3.3)
In this case(ω=2π), we haveα=k/2, then
ϕα(ω)=ϕα(0)=0, ψα(ω)=ψα(0)=0; (3.4)
...
hence, [ϕα,ϕβ]ω0 = [ϕα,ϕβ]ω0 = 0 and [ψα,ψβ]ω0 = [ψα,ψβ]ω0 = 0. But [ϕα,ψβ]ω0 =0 and[ϕα,ψβ]ω0 ≠0. From here comes the idea of decompos- ing the solutionuof (1.3) to its even and odd parts with respect tok
u=u1+u2, ui=
α∈Ei
cαuα+dαvα
, i=1,2,
E1= {n, n >1}, E2=
2n+1
2 ,2n >1
.
(3.5)
3.1. Calculation ofcβanddβ. We consider the integrals
Σ
P uiuβ+uiP uβ
dσ ,
Σ
P uivβ+uiP vβ
dσ , (3.6)
ifα∈E1, thenϕα(ω)=ϕα(0), ψα(ω)=ψα(0),
ifα∈E2, thenϕα(ω)= −ϕα(0), ψα(ω)= −ψα(0). (3.7) Equations (3.4) and (3.7) allow us to applyCorollary 2.6to functionsuα and uβ(resp.,uα,vβandvα,vβ); then, we obtain
Σ
P uiuβ+uiP uβ
dσ=2cβ
ΣuβP uβdσ+dβ
Σ
P vβuβ+vβP uβ
dσ ,
Σ
P uivβ+uiP vβ
dσ=cβ
Σ
P uβvβ+uβP vβ
dσ+2dβ
ΣvβP vβdσ . (3.8) Direct calculation gives us
Σ
P uβvβ+uβP vβ
dσ=0,
ΣuβP uβdσ= 2ρ2β−1ω
(1−ν)2β β(1−ν)(3+ν)−8 ,
ΣvβP vβdσ= −2ρ2β−1ω
(1−ν)2β (1−ν)(3+ν)(β−2)+8 .
(3.9)
So, we have just established the following proposition.
Proposition3.1. Letube the solution of (1.3) written in the form
u=u1+u2, (3.10)
where
ui=
α∈Ei
cαuα+dαvα
. (3.11)
Suppose that the series that givesuiis uniformly convergent; so, ifβ∈Ei,i= 1,2, then
cβ= (1−ν)2βρ1−2β 4ω (1−ν)(3+ν)β−8
Σ
P uiuβ+uiP uβ
dσ ,
dβ= −(1−ν)2βρ1−2β 4ω (1−ν)(3+ν)(β−2)+8
Σ
P uivβ+uiP vβ
dσ .
(3.12)
Remark3.2. We haveζ∈H3/2(Σ), the trace ofuonΣandχ∈H1/2(Σ), the trace ofP uonΣ. Ifζbelongs to the spaceH4(]0,2π[)andχtoH2(]0,2π[), then we have a uniform convergence of the series inVρ0for allρ0≤ρ, [3].
3.2. Independence of the coefficients. We are going to prove that the co- efficientscβ(resp.,dβ) of the development of the solutionuof (1.3) are inde- pendent fromρ.
We have the following result.
Theorem3.3. The coefficientscβanddβare independent fromρ.
Proof. In order to prove thatcβ is independent fromρ, we are going to show that its derivative with respect toρis null, and by observing the expres- sion ofcβ(Proposition 3.1), we just have to prove that
γβ=ρ1−2β
Σ
P uiuβ+uiP uβ
dσ (3.13)
has the null derivative with respect toρ. By derivation in regard tor, we have
γβ= ω
0
∂ui
∂r r2−βϕβ+
(2−β)ui−2∂2ui
∂r2 +
β2−21 r
∂ui
∂r
r1−βϕβ
+∂ui
∂r r−βϕβ−βuir−β−1 β2−2β
ϕβ+ϕβ dθ.
(3.14)
OnΣ, we have
∂ui
∂r = −T ui+(1−ν) 1
r3
∂2ui
∂θ2 − 1 r2
∂3ui
∂r ∂θ2
,
(2−β)ui−2∂2ui
∂r2 = −βMui+ 2−(1−ν)β1 r
∂ui
∂r + 1 r2
∂2ui
∂θ2
.
(3.15)
...
Reinjecting these formulas in the expression ofγβ, we obtain
γβ= − ω
0
Muiβr1−βϕβ+T uir2−βϕβ
dθ
+ ω
0 β2−(1−ν)β
ϕβ+ϕβ
∂ui
∂r −(1−ν) ∂3ui
∂r ∂θ2ϕβ
r−βdθ
+ ω
0 2−(1−ν)(β−1)∂2ui
∂θ2 ϕβ−βui β2−2β
ϕβ+ϕβ
r−1−βdθ.
(3.16)
By a double integration by parts, we verify that ω
0
∂2ui
∂θ2ϕβdθ= ω
0 uiϕβdθ, (3.17)
ω
0
∂3ui
∂r ∂θ2ϕβdθ= ω
0
∂ui
∂r ϕβdθ. (3.18)
Reinjecting in the expression ofγβ and puttingρ1−2β·ρin factor, we obtain
γβ= −ρ1−2β ω
0 Mui
βrβ−1ϕβ
+T ui
rβϕβ
ρ dθ
+ρ1−2β ω
0
β2−(1−ν)β
ϕβ+νϕβ
rβ−2∂ui
∂r ρ dθ +ρ1−2β
ω
0
−β2(β−2)ϕβ+ −(2−ν)β+(3−ν) ϕβ
rβ−3uiρ dθ.
(3.19)
By taking account of (2.7), whose expressions appear explicitly inγβ, we obtain
γβ=ρ1−2β
−
Σ
Mui∂uβ
∂n +T uiuβ
dσ+
Σ
Muβ∂ui
∂n +T uβui
dσ
=0 (3.20)
since we come back to the Green’s formula (2.6) applied touianduβ. We follow the same analysis to prove the independence ofdβwith respect toρ.
3.3. Convergence of the series. We writecαanddαin the form
cα=Iiρ−α, dα=jiρ−a, (3.21)
where
Ii=ρ (1−ν)2α
4ω (1−ν)(3+ν)α−8
Σ
P uiϕα+ui α2−2α
ϕα+ϕα
ρ−2 dσ , Ji=−ρ (1−ν)2α
4ω(1−ν)(3+ν)(α−2)+8
Σ
P uiψα+ui α2−2α ψα+ψα
ρ−2 dσ . (3.22)
The solutionuof (1.3) is then written as follows:
u=u1+u2, (3.23)
ui=
α∈Ei
r ρ
α
Iiϕα+ r
ρ α
Jiψα
. (3.24)
We have the following result.
Theorem3.4. The series (3.24) converges as soon asr < ρ. Proof. Set
Ni,α= ω
0
P uiϕα+ui α2−2α
ϕα+ϕα
ρ−2 dθ
= ω
0 P uiϕαdθ+
α2−2α ρ−2
ω
0 uiϕαdθ+ρ−2 ω
0 uiϕαdθ.
(3.25)
We show thatNi,αis a product of 1/αby limited term forαlarge.
According to (3.17), we have ω
0 uiϕαdθ= ω
0 uiϕαdθ. (3.26)
Replacingϕαby its expression and integrating by parts, we get ω
0 uiϕαdθ=1 α
−α α−2
ω
0 ui sin(α−2)θ dθ−4−(1−ν)α (1−ν)α
ω
0 ui sinαθ dθ
. (3.27)
On the other hand, by a triple integration by parts, we have α2−2αω
0 uiϕαdθ= 1 α
α2 (α−2)2
ω
0 ui sin(α−2)θ dθ +α−2
α
4−(1−ν)α (1−ν)α
ω
0 ui sinαθ dθ
.
(3.28)
...
Also, by an integration by parts, we get ω
0
∆ui−2 r
∂ui
∂r
ϕαdθ= − 1 α−2
ω
0
∂∆ui
∂θ −2 r
∂2ui
∂r ∂θ
sin(α−2)θ dθ
−1 α
4−(1−ν)α (1−ν)α
ω
0
∂∆ui
∂θ −2 r
∂2ui
∂r ∂θ
sinαθ dθ.
(3.29) Then, we deduce the existence of a constantC0so as
Ni,α≤C0
α. (3.30)
Using this last inequality and remarking thatϕαis limited, as well as the term (1−ν)2α
4ω (1−ν)(ν+3)α−8 (3.31)
for largeα, we deduce the existence of a constantCso as
α∈Ei
cαrαϕα
≤
α∈Ei
C α
r ρ
α
, (3.32)
which converges as soon asr < ρ.
In the same way, we prove the convergence of the series
α∈Eidαrαψα.
References
[1] H. Blum and R. Rannacher,On the boundary value problem of the biharmonic op- erator on domains with angular corners, Math. Methods Appl. Sci.2(1980), no. 4, 556–581.
[2] P. Grisvard,Singularities in Boundary Value Problems, Recherches en Mathéma- tiques Appliquées, vol. 22, Masson, Paris, 1992 (French).
[3] O. Tcha-Kondor,Nouvelles séries trigonométriques adaptées à l’étude de problèmes aux limites pour l’équation biharmonique. Étude du cas de la fissure[New trigonometric series adapted to the study of boundary value problems for the biharmonic equation. The case of the crack], C. R. Acad. Sci. Paris Sér. I Math.315(1992), no. 5, 541–544 (French).
Wided Chikouche: Département de Mathématiques, Centre Universitaire de Jijel, Ouled Aissa, BP 98, 18000 Jijel, Algeria
E-mail address:[email protected]
Aissa Aibèche: Département de Mathématiques, Centre Universitaire de Jijel, Ouled Aissa, BP 98, 18000 Jijel, Algeria
E-mail address:[email protected]