Stability of Displacement to the Second Fundamental Problem in Plane Elasticity

Acta Mathematica Scientia 2014,34B(1):125–140

http://actams.wipm.ac.cn

STABILITY OF DISPLACEMENT TO THE

SECOND FUNDAMENTAL PROBLEM IN

PLANE ELASTICITY∗

Juan LIN (��)1,2 Jinyuan DU (��)1†

1. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

2. Department of Foundation, Fujian Commercial College, Fuzhou 350012, China

E-mail: [email protected]; [email protected]

Abstract In this article, by using the stability of Cauchy type integral when the smooth

perturbation for integral curve and the Sobolev type perturbation for kernel density hap-

pen, we discuss the stability of the second fundamental problem in plane elasticity when

the smooth perturbation for the boundary of the elastic domain (unit disk) and the Sobolev

type perturbation for the displacement happen. And the error estimate of the displacement

between the second fundamental problem and its perturbed problem is obtained.

Key words elastic domain; Cauchy type integral; displacement; perturbation

2010 MR Subject Classification 30E20; 30E25; 45E99

1 Introduction

It is well known that complex variable method is one of the effective methods for solving

various problems in plan elasticity mechanics (see, e.g. [1–4]). By introducing two analytic

functions, the fundamental problem can be transferred to a boundary value problem for analytic

function and then further reduced to a integral equation called Sherman-Lauricella equation.

When the elastic domain is a unit disk, we can get the closed form of solution for Sherman-

Lauricella equation. In general case, it is difficult to get the closed form of solution for Sherman-

Lauricella equation. So, the research on the stability of the second fundamental problem on the

elastic domain aroused the interest of all. In such discussion, the theory for boundary value of

the Cauchy type integral and when the smooth perturbation for integral curve and the Sobolev

type perturbation for the kernel density happen the stability of the Cauchy type integral play

an important role (see, e.g. [5–9]).

In [10], the authors discussed the stability of the second fundamental problem when the

smooth perturbation for the boundary circle of the elastic domain disk and the Sobolev type

∗Received September 24, 2012; revised January 25, 2013. This work was supported by NNSF of China

(11171260), RFDP of Higher Education of China (20100141110054), NSF of Fujian Province, China (2008J0187)

and STF of Education Department of Fujian Province, China (JA11341).†Corresponding author: Jinyuan DU.

126 ACTA MATHEMATICA SCIENTIA Vol.34 Ser.B

perturbation for the displacement given in the boundary circle happen by the results in [7]. But

the discussion in [10] was based on a closed set in the elastic domain and displacement with the

trivial perturbation. In the present article, we remove the limitation of a closed set and obtain

a better conclusion under the displacement with the Sobolev perturbation.

We assume that the elastic domain S is a bounded simply connected region with boundary

Γ, a smooth closed contour, oriented counter-clockwisely. And let O = (0, 0) ∈ S. Assume

all functions (and their derivatives) given in S or on Γ are Holder continuous. We introduce

holomorphic functions ϕ(z) and ψ(z), also called complex stress functions. The displacement

u+ iv at the point z in S could be expressed by ϕ(z) and ψ(z) [1, 2]

2μ(u+ iv) = κϕ(z)− zϕ′(z)− ψ(z), z ∈ S, (1.1)

where κ, μ are the elastic constants and 1 < κ < 3. So the second fundamental problem about

S can be reduced to the boundary problem for analytic functions [1, 2]: for given displacement

function g(t) = u(t) + iv(t) (t ∈ Γ), to find two holomorphic functions ϕ(z) and ψ(z) in S,

satisfying the boundary value condition

κ ϕ(t)− tϕ′(t)− ψ(t) = 2μg(t), t ∈ Γ (1.2)

and

ϕ(0) = 0 (or ψ(0) = 0). (1.3)

We introduce a new unknown function ω(t) defined on Γ. Let

ϕ(z) =1

2πi

∫Γ

ω(t)

t− zdt, z ∈ S, (1.4)

ψ(z) = − κ

2πi

∫Γ

ω(t)

t− zdt− 1

2πi

∫Γ

t ω′(t)

t− zdt, z ∈ S. (1.5)

By (1.4), we can obtain

ϕ′(z) =1

2πi

∫Γ

ω(t)

(t− z)2dt =

1

2πi

∫Γ

ω′(t)

t− zdt, z ∈ S. (1.6)

According to the Plemelj formula and (ϕ+)′(t) = (ϕ′)+(t), we can get ϕ(t), ϕ′(t), ψ(t) from

(1.4), (1.6) and (1.5), substituting them into (1.2), we gain the following integral equation [1, 2]

which be called the Sherman-Lauricella equation

κ ω(t) +κ

2πi

∫Γ

ω(τ) d

[ln

τ − t

τ − t

]+

1

2πi

∫Γ

ω(τ) d

[τ − t

τ − t

]= 2μg(t), t ∈ Γ. (1.7)

Remark 1.1 It is true that the ϕ and ψ given in (1.4) and (1.5) satisfy (1.2) while ω is

the solution of the Sherman-Lauricella equation (1.7) and ω′ ∈ H(Γ).

Assume Γ is the unit circle L : t = eiθ, 0 ≤ θ ≤ 2π, then S is the unit disk U = {z : |z| <1}. Under this case, we have

ττ = 1,τ − t

τ − t= −τt, d

[τ − t

τ − t

]= −t dτ, d

[ln

τ − t

τ − t

]=1

τdτ, while τ, t ∈ L, (1.8)

No.1 J. Lin & J.Y. Du: STABILITY OF DISPLACEMENT TO FUNDAMENTAL PROBLEM 127

so, by (1.8), equation (1.7) (Γ = L) can be changed into

ω(t) = a+ bt+2μ

κg(t), (1.9)

where

a = − μ

2κπi

∫L

g(τ)

τdτ, b =

2μ

κ2 − 1

[1

2πi

∫L

g(τ)dτ − 1

2κπi

∫L

g(τ)dτ

]. (1.10)

Remark 1.2 (1.9) show that g ∈ C(n)(L) is equivalent to ω ∈ C(n)(L) and g(n) ∈ H(L)

is equivalent to ω(n) ∈ H(L).

Remark 1.3 (1.9) and (1.10) yield∣∣a∣∣ ≤ μ

κ

∥∥g∥∥L,

∣∣b∣∣ ≤ 2μ

κ(κ− 1)

∥∥g∥∥L, (1.11)

and ∥∥ω∥∥L≤ (3κ− 1)μ

κ(κ− 1)

∥∥g∥∥L,

∥∥ω′∥∥L≤ 2μ

κ(κ− 1)

∥∥g∥∥L+2μ

κ

∥∥g′∥∥L, (1.12)

where ‖ · ‖L = maxL| · |.

2 Perturbation of the Circle

If the function δ(τ) (τ ∈ L) has the continuous derivative on L up to order n, we denote

it by δ ∈ Cn(L) and call it a n-smooth perturbation of the circle L. If δ ∈ Cn(L), its Sobolev

norm reads ∥∥δ∥∥n= max

{∥∥δ∥∥L,∥∥δ′∥∥

L, · · · ,

∥∥δ(n)∥∥L

}, (2.1)

where∥∥δ(k)∥∥

L= max

{∣∣δ(k)(τ)∣∣, τ ∈ L}(k = 0, 1, · · · , n) denotes the Chebyshev norm of the

function δ(k) defined on the curve L.

We set

σ(t) = t+ δ(t), t ∈ L, (2.2)

then

Lδ : λδ(θ) = σ(eiθ)= eiθ + δ

(eiθ), 0 ≤ θ ≤ 2π (2.3)

is also a simple and closed smooth contour under a small perturbation δ while ‖δ‖1 < 2π. And

we also call Lδ the perturbation of L with δ. Let Uδ be the region bounded by Lδ. With the

length of minor arc between t1 and t2 denoted by [t1, t2]Lminarc, then

2

π

[t1, t2

]Lminarc

≤∣∣t1 − t2

∣∣ ≤ [t1, t2]Lminarc for any t1, t2 ∈ L (2.4)

by the Newton-Leibniz formula, we get

∣∣δ(t1)− δ(t2)∣∣ = ∣∣∣∣ ∫[ �

t1t2

]L

minarc

δ′(τ)dτ

∣∣∣∣ ≤ ∥∥δ′∥∥L

[t1, t2

]Lminarc

≤ π

2

∥∥δ∥∥1

∣∣t1 − t2∣∣, (2.5)

so we have ∣∣σ(t1)− σ(t2)∣∣ ≥ [1− π

2

∥∥δ∥∥1

] ∣∣t1 − t2∣∣ while

∥∥δ∥∥1

<2

π. (2.6)


This inequality shows that σ(t) is 1-1 continuous mapping from L to Lδ, namely, there exist

ξ = σ(t), t ∈ L (2.7)

and

t = σ−1(ξ), ξ ∈ Lδ, (2.8)

which are inverse to each other.

We denote the interior region with the boundary curve Lδ as Uδ. The second fundamental

problem about Uδ can be reduced to the boundary problem for analytic functions: to find two

holomorphic functions ϕ∗(z) and ψ∗(z) in Uδ, satisfying the boundary value condition

κϕ∗(ξ) − ξϕ′∗(ξ)− ψ∗(ξ) = 2μg∗(ξ), ξ ∈ Lδ. (2.9)

As before, we introduce a new function ω∗(ζ), such that

ϕ∗(z) =1

2πi

∫Lδ

ω∗(ζ)

ζ − zdζ, z ∈ Uδ, (2.10)

ψ∗(z) = − κ

2πi

∫Lδ

ω∗(ζ)

ζ − zdζ − 1

2πi

∫Lδ

ζω′∗(ζ)

ζ − zdζ, z ∈ Uδ. (2.11)

By (2.10) and (2.11), (2.9) can be changed into the integral equation on Lδ

κ ω∗(ξ) +κ

2πi

∫Lδ

ω∗(ζ) d

[ln

ζ − ξ

ζ − ξ

]+

1

2πi

∫Lδ

ω∗(ζ) d

[ζ − ξ

ζ − ξ

]= 2μg∗(ξ), ξ ∈ Lδ, (2.12)

which is the Sherman-Lauricella equation of (2.9). Denoting

ω∗(ξ) = ω∗(σ(t)) = ωδ(t), g∗(ξ) = g∗(σ(t)) = gδ(t), (2.13)

(2.12) can be expressed as

κωδ(t) +κ

2πi

∫L

ωδ(τ)d

[ln

σ(τ) − σ(t)

σ(τ) − σ(t)

]+

1

2πi

∫L

ωδ(τ)d

[σ(τ) − σ(t)

σ(τ) − σ(t)

]= 2μgδ(t), t ∈ L,

(2.14)

then

κ ωδ(t) +κ

2πi

∫L

ωδ(τ) d

[ln

τ − t

τ − t

]+

1

2πi

∫L

ωδ(τ) d

[τ − t

τ − t

]= 2μgδ(t) +G(t), t ∈ L, (2.15)

where

G(t) =κ

2πi

∫L

ωδ(τ) d

⎡⎣ln (τ − t) (

σ(τ) − σ(t))

(τ − t

)(σ(τ) − σ(t)

)⎤⎦+ 1

2πi

∫L

ωδ(τ) d

[τ − t

τ − t− σ(τ) − σ(t)

σ(τ) − σ(t)

].

(2.16)

Remark 2.1 G is called the term of perturbation for equation (2.15). In the sequel, we

will see that its stability play an important role.

From (2.15) and (1.8), we get

ωδ(t) = aδ + bδt+2μ

κgδ(t) +

1

κG(t), (2.17)


where

aδ = − μ

2κπi

∫L

gδ(τ)

τdτ − 1

4κπi

∫L

G(τ)

τdτ, (2.18)

bδ =2μ

κ2 − 1

[1

2πi

∫L

gδ(τ)dτ −1

2κπi

∫L

gδ(τ)dτ

]+

1

κ2 − 1

[1

2πi

∫L

G(τ)dτ − 1

2κπi

∫L

G(τ)dτ

]. (2.19)

Remark 2.2 By (2.17), we get

ω′∗(ξ) =bδ

1 + δ′(t)+2μ

κg′∗(ξ) +

G′(t)

κ(1 + δ′(t)), t = σ−1(ξ), ξ ∈ Lδ, (2.20)

and we will prove that ω′∗ ∈ H(Lδ) while g′∗ ∈ H(Lδ) and δ(3) ∈ C(L) in Section 4.

3 Stability of the Term of Perturbation

Suppose that Γ is a smooth curve and f ∈ Cn(Γ). We set its difference quotient function

of n-th order by

(Dn[f ]

)(τ, t) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩1

(τ − t)n

[f(τ)−

n−1∑j=0

f (j)(t)

j!(τ − t)j

], τ �= t,

f (n)(t)

n!, τ = t,

(τ, t) ∈ Γ× Γ. (3.1)

Lemma 3.1 [11] If f ∈ Cn+m(Γ), Γ is smooth, its natural equation is t = φ(s), t ∈Γ, s ∈ [0, �Γ], �Γ is the length of Γ, then Dn[f ] ∈ Cm(Γ × Γ). More precisely, all its partial

derivatives up to order m inclusive are continuous and independent of the order of τ and t,

∥∥(Dn[f ])(m)∥∥

Γ×Γ≤ 2m

n!

[1

CΓ

]n+m ∥∥f (n+m)∥∥Γ, (3.2)

ω((

Dn[f ])(m)

;h)≤[1

CΓ

]n+m+1 [2m

n!ω

(f (n+m);

h

CΓ

)+

2m+2

(n− 1)!

∥∥f (n+m)∥∥Γω

(φ′;

h

CΓ

)],

(3.3)

where CΓ is a chord-arc ratio of Γ.

Remark 3.1 This lemma generalizes and improves the results in some earlier literatures

[12–15]. In particular, if f (n) ∈ H(Γ), φ′ ∈ H , then Dn[f ] ∈ H(Γ× Γ).

Lemma 3.2 [11] If f ∈ Cn+1(Γ) and Γ is smooth, then

∂Dn[f ]

∂τ= Dn[f

′]− nDn+1[f ],∂Dn[f ]

∂t= nDn+1[f ]. (3.4)

Lemma 3.3 [11] Suppose that h is defined in the unit circle L. If h′(t) exists, then[h(t)

]′= −t2h′(t),

[Re[h(t)

]]′= i t Im

[h′(t)t

],[Im[h(t)

]]′= −i tRe

[h′(t)t

]. (3.5)

If h′′(t) exists, then[Re[h(t)

]]′′= t3 h′(t) + t2Re

[h′′(t)t2

],[Im[h(t)

]]′′= it3 h′(t) + t2Im

[h′′(t)t2

]. (3.6)


By Lemma 3.2 and Lemma 3.3, we have

d

[ln(τ − t)(σ(τ) − σ(t))

(τ − t)(σ(τ) − σ(t))

]= d

[ln(D1[σ]

)(τ, t)− ln

(D1[σ]

)(τ, t)

]= τ G1(τ, t) dτ, (3.7)

where

G1(τ, t) = 2Re

[(D2[δ]

)(τ, t)−

(D1[δ

′])(τ, t)

1 +(D1[δ]

)(τ, t)

τ

], (3.8)

and

d

[τ − t

τ − t− σ(τ) − σ(t)

σ(τ) − σ(t)

]= t

[(D1[σ]

)(τ, t)(

D1[σ])(τ, t)

− 1

]dτ + τtd

[(D1[σ])(τ, t)(D1[σ]

)(τ, t)

](3.9)

with (D1[σ]

)(τ, t)

(D1[σ])(τ, t)− 1 =

(D1[δ])(τ, t) − (D1[δ])(τ, t)

1 +(D1[δ]

)(τ, t)

=: G2(τ, t), (3.10)

d

[(D1[σ])(τ, t)

(D1[σ])(τ, t)

]=(D1[σ])(τ, t)

(D1[σ])(τ, t)d

[ln(D1[σ])(τ, t)

(D1[σ])(τ, t)

]= −τ G1(τ, t)

(G2(τ, t) + 1

)dτ. (3.11)

Lemma 3.4 Suppose that G is given in (2.16). If δ ∈ C2(L) and∥∥δ′∥∥

L≤ ρ < 2

πthen∥∥G∥∥

L≤ M0

∥∥ωδ

∥∥L

∥∥δ∥∥2

with M0 =2(2κ+ 3)π

2− πρ. (3.12)

If δ ∈ C3(L) and∥∥δ∥∥

2≤ ρ < 2

πthen∥∥G′∥∥

L≤ M1

∥∥ωδ

∥∥L

∥∥δ∥∥3

(3.13)

with

M1 =1

2− πρ

[π3

6(κ+ 1) +

π2

2(κ+ 2) + 6π

]+(κ+ 3)π3ρ

[2− πρ]2 . (3.14)

If δ ∈ C4(L) and∥∥δ∥∥

2≤ ρ < 2

πthen∥∥G′′∥∥

L≤ M2

∥∥ωδ

∥∥L

∥∥δ∥∥4

(3.15)

with

M2 =π2

2− πρ

[(1 + κ)π2

16+(5 + 2κ)π

6+

κ+ 7

2

]+(6 + κ)π5ρ2

2 [2− πρ]3

+π3ρ

[2− πρ]2

[(3 + κ)π2

12+(51 + 14κ)π

24+ 11 + κ

]. (3.16)

Proof By (3.7) and (3.9), we can rewrite (2.16) as the following

G(t) = κ I1(t) + t I2(t)− t I3(t), (3.17)

where

I1(t) =1

2πi

∫L

ωδ(τ)τ G1(τ, t)dτ, (3.18)

I2(t) =1

2πi

∫L

ωδ(τ)G2(τ, t)dτ. (3.19)

I3(t) =1

2πi

∫L

ωδ(τ)(G2(τ, t) + 1)G1(τ, t)dτ. (3.20)


Let

Υ(τ, t) =(D2[δ])(τ, t)− (D1[δ

′])(τ, t)

(D1[σ])(τ, t), (3.21)

by (3.2) and ‖δ′‖L ≤ ρ, we have

∥∥Υ∥∥L×L

≤ 2

2− π‖δ′‖L

[π

2+(π

2

)2 12!

]∥∥δ′′∥∥L≤ 2π

2− πρ

∥∥δ′′∥∥L, (3.22)

and by (3.8), we get ∥∥G1

∥∥L×L

≤ 4π

2− πρ

∥∥δ′′∥∥L, (3.23)

which results in ∥∥I1∥∥L≤ 4π

2− πρ

∥∥ωδ

∥∥L

∥∥δ′′∥∥L,∥∥I3∥∥L

≤ 4π

2− πρ

∥∥ωδ

∥∥L

∥∥δ′′∥∥L. (3.24)

(3.10) and (3.2) imply ∥∥G2

∥∥L×L

≤ 2π

2− πρ

∥∥δ′∥∥L, (3.25)

which results in ∥∥I2∥∥L≤ 2π

2− πρ

∥∥ωδ

∥∥L

∥∥δ′∥∥L. (3.26)

(3.17), (3.24) and (3.26) yield (3.12).

By (3.21) and (3.4), we obtain

∂Υ

∂t(τ, t) =

2(D3[δ]

)(τ, t)−

(D2[δ

′])(τ, t)−

(D2[δ]

)(τ, t)Υ(τ, t)

1 +(D1[δ]

)(τ, t)

, (3.27)

while ‖δ‖2 ≤ ρ, (3.22) and (3.2) result in∥∥∥∥∂Υ

∂t

∥∥∥∥L×L

≤ 1

2

[π2

2− π‖δ′‖L

(π

6+1

2

)+

π3

(2− π‖δ′‖L)2

∥∥δ′′∥∥L

]∥∥δ∥∥3≤ 1

2C1

∥∥δ∥∥3, (3.28)

where

C1 =π2

2− πρ

[π

6+1

2

]+

π3ρ

[2− πρ]2. (3.29)

By (3.5), we have ∥∥∥∥∂G1

∂t

∥∥∥∥L×L

≤ C1

∥∥δ∥∥3, (3.30)

so, we get ∥∥I ′1∥∥L≤ C1

∥∥ωδ

∥∥L

∥∥δ∥∥3. (3.31)

From (3.10)

∂G2

∂t(τ, t) =

∂

∂t

{(D1[σ])(τ, t)

(D1[σ])(τ, t)

}=

t(D1[σ])(τ, t)

(D1[σ])(τ, t)2Re

[(D2[δ])(τ, t)

1 + (D1[δ])(τ, t)t

], (3.32)

by (3.2) and ‖δ′‖L ≤ ρ, we get∥∥∥∥∂G2

∂t

∥∥∥∥L×L

≤ π2

2(2− π‖δ′‖L

)∥∥∥∥δ′′∥∥∥∥L

≤ C2

∥∥∥∥δ′′∥∥∥∥L

, (3.33)


where

C2 =π2

2(2− πρ

) . (3.34)

Thus, ∥∥I ′2∥∥L≤ C2

∥∥ωδ

∥∥L

∥∥δ′′∥∥L. (3.35)

(3.20), (3.23), (3.30) and (3.33) result in∥∥I ′3∥∥L≤ C3

∥∥ωδ

∥∥L

∥∥δ∥∥3, (3.36)

where

C3 =π2

2− πρ

[π

6+1

2

]+

3π3ρ

[2− πρ]2. (3.37)

By (3.17), (3.24), (3.26), (3.31), (3.35) and (3.36), we get

∥∥G′∥∥L≤[κ C1 + C2 + C3 +

6π

2− πρ

] ∥∥ωδ

∥∥L

∥∥δ∥∥3= M1

∥∥ωδ

∥∥L

∥∥δ∥∥3, (3.38)

where M1 is just that given in (3.14).

By (3.27), we have

∂2Υ

∂t2=6D4[δ]− 2D3[δ

′]− 2D3[δ]Υ

1 +D1[δ]− 2D2[δ]

1 +D1[δ]

∂Υ

∂t. (3.39)

(3.22), (3.28), (3.2) and (3.39) result in∥∥∥∥∂2Υ

∂t2

∥∥∥∥L×L

≤ 1

2C∗4∥∥δ∥∥

4while ‖δ‖2 ≤ ρ, (3.40)

where

C∗4 =π3

2− πρ

[π

16+1

6

]+

π4ρ

[2− πρ]2

[π

12+

7

12

]+

π5ρ2

2 [2− πρ]3 . (3.41)

Thus, by (3.6), (3.28) and (3.40),∥∥∥∥∂2G1

∂t2

∥∥∥∥L×L

≤ 2

[∥∥∥∥∂Υ

∂t

∥∥∥∥L×L

+

∥∥∥∥∂2Υ

∂t2

∥∥∥∥L×L

]≤ C4‖δ‖4, (3.42)

where

C4 =π2

2− πρ

[π2

16+

π

3+1

2

]+

π3ρ

[2− πρ]2

[π2

12+7π

12+ 1

]+

π5ρ2

2 [2− πρ]3. (3.43)

By (3.18) and (3.42), we get ∥∥I ′′1 ∥∥L≤ C4

∥∥ωδ

∥∥L

∥∥δ∥∥4. (3.44)

Let

Λ(τ, t) =(D2[δ])(τ, t)

(D1[σ])(τ, t), (3.45)

then, by Lemma 3.2,

∂Λ

∂t(τ, t) = − [(D2[δ])(τ, t)]

2

[(D1[σ])(τ, t)]2 +

2(D3[δ])(τ, t)

(D1[σ])(τ, t)= −

[Λ(τ, t)

]2+2(D3[δ])(τ, t)

(D1[σ])(τ, t), (3.46)


and by (3.2), we get∥∥∥∥∂Λ

∂t

∥∥∥∥L×L

≤[

π3

12 (2− π‖δ′‖L)+

π4

16 (2− π‖δ′‖L)2

∥∥δ′′∥∥L

]∥∥δ∥∥3≤ C∗5

∥∥δ∥∥3

while ‖δ‖2 ≤ ρ,

(3.47)

where C∗5 =π3

12[2−πρ] +π4ρ

16[2−πρ]2. By (3.32), (3.5), (3.33), (3.47) and (3.2), we obtain∥∥∥∥∂2G2

∂t2

∥∥∥∥L×L

≤ C5

∥∥δ∥∥3, (3.48)

where

C5 =π2

2− πρ

[π

6+1

2

]+

3π4ρ

8 [2− πρ]2, (3.49)

which results in ∥∥I ′′2 ∥∥L≤ C5

∥∥ωδ

∥∥L

∥∥δ∥∥3. (3.50)

By (3.20), (3.23), (3.30), (3.33), (3.42) and (3.48), we have∥∥I ′′3 ∥∥L≤ C6

∥∥ωδ

∥∥L

∥∥δ∥∥4

while∥∥δ∥∥

2≤ ρ, (3.51)

where

C6 =π2

2− πρ

[π2

16+

π

3+1

2

]+

π3ρ

[2− πρ]2

[π2

4+7π

4+ 5

]+

3π5ρ2

[2− πρ]3 . (3.52)

By (3.17), (3.35), (3.36), (3.44), (3.50) and (3.51), we get∥∥G′′∥∥L≤[κ C4 + C5 + C6 + 2C2 + 2C3

]∥∥ωδ

∥∥L

∥∥δ∥∥4= M2

∥∥ωδ

∥∥L

∥∥δ∥∥4, (3.53)

where M2 is just that given in (3.16). �

4 Stability of the Displacement

Denote

AΦ,υ = supt1,t2∈L

{ |Φ(t1)− Φ(t2)||t1 − t2|υ

, t1 �= t2

}, AΦ∗,υ = sup

ξ1,ξ2∈Lδ

{ |Φ∗(ξ1)− Φ∗(ξ2)||ξ1 − ξ2|υ

, ξ1 �= ξ2

},

(4.1)

which are respectively the Holder semi-norm of Φ and the Holder semi-norm of Φ∗.

Definition 4.1 Let δ ∈ Sn(ρ) = {δ| ‖δ‖n ≤ ρ, δ ∈ Cn(Γ)}, Γδ is the perturbation for the

curve Γ by δ, Φ ∈ Hυ(Γ) and Φ∗ ∈ Hυ(Γδ

)(0 < υ ≤ 1). If there are constants A+(Φ, n, ρ, υ)

and R(Φ, n, ρ, υ) independent of the perturbation δ, sometime simply denoted as A+Φ,υ and

RυΦ,n, such that

max{AΦ∗,υ, AΦ,υ} ≤ A+(Φ, n, ρ, υ) = A+Φ,υ , (4.2)

and ∥∥Φ∗ ◦ σ − Φ∥∥

L≤ R(Φ, n, ρ, υ)

∥∥δ∥∥υ

n= Rυ

Φ,n

∥∥δ∥∥υ

n, (4.3)

then, the function Φ∗ is called a Sobolev type perturbation of n-th order for Φ by δ with the

index υ. In particular, we simply say that Φ∗ is the Sobolev type perturbation of Φ by δ while


n = 0. The constant A+(Φ, n, ρ, υ) is called a Holder super norm of Φ∗ and Φ. The constant

R(Φ, n, ρ, υ) is called a Sobolev perturbation coefficient of Φ∗ for Φ.

Remark 4.1 Let Ω is a region and Γ∪Γδ ⊂ Ω. If f ∈ Hυ(Ω) then f |Γδis just a Sobolev

type perturbation of f |Γ by δ, since we may take A+(f |Γ, 0, ρ, υ) = R(f |Γ, 0, ρ, υ) = Af,υ where

Af,υ is the Holder semi-norm of f on Ω. Under this case, we call f |Γδthe trivial perturbation

for f |Γ.Lemma 4.1 [11] Suppose that δ ∈ Sn(ρ), Γδ is the perturbation for the curve Γ by δ,

Φ ∈ Hυ(Γ)(0 < υ ≤ 1) and Φ∗ is a Sobolev type perturbation of n-th order for Φ by δ.

D+(D−)and D+

δ

(D−δ)denote respectively the interior (exterior) region bounded by Γ and the

interior (exterior) region bounded by Γδ. Let the Cauchy integral operators

(S[Φ]

)(z) =

⎧⎪⎪⎨⎪⎪⎩1

2πi

∫Γ

Φ(τ)

τ − zdτ, z �∈ Γ,

1

πi

∫Γ

Φ(τ)

τ − tdτ, z = t ∈ Γ,

(4.4)

(Sδ[Φ∗]

)(z) =

⎧⎪⎪⎨⎪⎪⎩1

2πi

∫Γδ

Φ∗(ζ)

ζ − zdζ, z �∈ Γδ,

1

πi

∫Γδ

Φ∗(τ)

τ − tdτ, z = t ∈ Γδ,

(4.5)

and their projection operators

(S±[Φ]

)(z) =

⎧⎪⎪⎨⎪⎪⎩1

2πi

∫Γ

Φ(τ)

τ − zdτ, z ∈ D±,

1

2

[(S[Φ]

)(t)± Φ(t)

], z = t ∈ Γ,

(4.6)

(S±δ [Φ∗]

)(z) =

⎧⎪⎪⎨⎪⎪⎩1

2πi

∫Γδ

Φ∗(ζ)

ζ − zdζ, z ∈ D±δ ,

1

2

[(Sδ[Φ∗]

)(t)± Φ∗(t)

], z = t ∈ Γδ,

(4.7)

then, for 0 < ν < υ,∥∥∥S±[Φ]− S±δ [Φ∗]∥∥∥

D±∩D±

δ

≤ C(υ, ν, ρ, �Γ, CΓ

)∥∥δ∥∥υ−ν

max{n,1}, (4.8)

where �Γ denotes the length of the curve Γ, CΓ is the Chord-Arc ratio for Γ,

C(υ, ν, ρ, �Γ, CΓ) = C1(υ, ν, ρ, �Γ, CΓ) + C2(υ, ν, ρ, �Γ, CΓ) + C3(υ, ν, ρ, �Γ, CΓ) (4.9)

with

C1(υ, ν, ρ, �Γ, CΓ) =�νΓ

2νπνC1−νΓ

{[(1 +

ρ

CΓ

)υ

+ 1

]A+Φ,υ

} νυ [2Rυ

Φ,n

]1− νυ

+�υΓ

2υπυ

[1 +

1

CΓ

]ρ1−υ+ν

[CΓ − ρ]1−υA+Φ,υ +Rυ

Φ,nρν , (4.10)

C2(υ, ν, ρ, �Γ, CΓ) = 3ρν

[22υ(1 + ρ)

πυ(CΓ − ρ)+ 2υ

]A+Φ,υ , (4.11)


C3(υ, ν, ρ, �Γ, CΓ) =

⎧⎪⎪⎨⎪⎪⎩22+υ32−υ(1 + ρ)ρν

π(1− υ)(CΓ − ρ)A+Φ,υ, 0 < υ < 1,

24

π

{ρν ln [1 + (1 + ρ)�ΓCΓ] +

1

ν e

}1 + ρ

CΓ − ρA+Φ,υ , υ = 1.

(4.12)

Theorem 4.1 Assume that δ ∈ C3(L) and ‖δ‖3 ≤ ρ ≤ 1.92κ(κ−1)(3κ−1)(2κ+3)π+0.96κ(κ−1)π . If g∗

is the Sobolev type perturbation of g by δ, then ω∗ is the the Sobolev type perturbation of

2-nd order for ω by δ, where ω and ω∗ are respectively the normal solutions for the Sherman-

Lauricella equations (1.7) (Γ = L) and (2.12).

Proof First, we have, from (1.10), (2.18) and (2.19)∣∣aδ − a∣∣ ≤ μ

κ

∥∥gδ − g∥∥

L+

1

2κ

∥∥G∥∥L,∣∣bδ − b

∣∣ ≤ 2μ

κ(κ− 1)

∥∥gδ − g∥∥

L+

1

κ(κ− 1)

∥∥G∥∥L, (4.13)

so, by (1.9), (2.17) and (4.13), we get∥∥ωδ − ω∥∥

L≤∣∣aδ − a

∣∣+ ∣∣bδ − b∣∣+ 2μ

κ

∥∥gδ − g∥∥

L+1

κ

∥∥G∥∥L

≤ μ(3κ− 1)

κ(κ− 1)

∥∥gδ − g∥∥

L+

3κ− 1

2κ(κ− 1)

∥∥G∥∥L. (4.14)

(3.12) and (4.14) result in

(1− ρ N)∥∥ωδ − ω

∥∥L≤ μ(3κ− 1)

κ(κ− 1)

∥∥gδ − g∥∥

L+N

∥∥δ∥∥2

∥∥ω∥∥L, (4.15)

where N = 3κ−12κ(κ−1)M0, M0 is given in (3.12). Noting

ρ ≤ 1.92κ(κ− 1)

(3κ− 1)(2κ+ 3)π + 0.96κ(κ− 1)π, (4.16)

we get

ρ N =3κ− 1

2κ(κ− 1)

2(2κ+ 3)π

2− πρρ < 0.96. (4.17)

Obviously, there is a Sobolev perturbation coefficient Rυg,0 of g∗ for g such that∥∥gδ − g

∥∥L≤ Rυ

g,0

∥∥δ∥∥υ

L. (4.18)

(4.15), (4.18), (1.12) and (2.13) result in∥∥ω∗ ◦ σ − ω∥∥

L≤ (3κ− 1)μ

κ(κ− 1) (1− ρ N)

(Rυ

g,0 + ρ1−υN∥∥g∥∥

L

) ∥∥δ∥∥υ

2= Rυ

ω,2

∥∥δ∥∥υ

2. (4.19)

This inequality and (1.12) imply∥∥ωδ

∥∥L≤ (3κ− 1)μ

κ(κ− 1) (1− ρ N)

(ρυ Rυ

g,0 +∥∥g∥∥

L

), (4.20)

so, we have ∣∣bδ − b∣∣ ≤ 2μ

(Rυ

g,0 + ρ1−υN∥∥g∥∥

L

)κ (κ− 1) (1− ρN)

∥∥δ∥∥υ

2(4.21)

and ∣∣bδ

∣∣ ≤ 2μ(ρυRυ

g,0 +∥∥g∥∥

L

)κ (κ− 1) (1− ρN)

. (4.22)


Second, for ξj = tj + δ(tj)(tj ∈ L

), by (2.17), we have

∣∣ω∗(ξ2)− ω∗(ξ1)∣∣ ≤ ∣∣bδ

∣∣∣∣t2 − t1∣∣+ 2μ

κ

∣∣g∗(ξ2)− g∗(ξ1)∣∣+ 1

κ

∣∣G(t2)−G(t1)∣∣

= l1 + l2 + l3. (4.23)

(2.6) implies ∣∣t2 − t1∣∣ ≤ 2

2− πρ

∣∣σ(t2)− σ(t1)∣∣ = 2

2− πρ

∣∣ξ2 − ξ1∣∣. (4.24)

(4.24), (4.22) and∣∣t2 − t1

∣∣ = ∣∣t2 − t1∣∣1−υ∣∣t2 − t1

∣∣υ result in

l1 ≤4μ

κ(κ− 1) (1− ρ N) (2− πρ)υ[ρυ Rυ

g,0 +∥∥g∥∥

L

] ∣∣ξ2 − ξ1∣∣υ. (4.25)

By (3.2), (3.13) and (4.20), we obtain

∣∣G(t2)−G(t1)∣∣ ≤ ∥∥ (D1[G]])

∥∥L

∣∣t2−t1∣∣ ≤ πρμ(3κ− 1)M1

2κ(κ− 1) (1− ρ N)

[ρυ Rυ

g,0 +∥∥g∥∥

L

] ∣∣t2−t1∣∣, (4.26)

so,

l3 ≤πρμ(3κ− 1)M1

κ2(κ− 1) (1− ρ N) (2− πρ)υ[ρυ Rυ

g,0 +∥∥g∥∥

L

] ∣∣ξ2 − ξ1∣∣υ. (4.27)

Obviously,

l2 ≤2μ

κAg∗,υ

∣∣ξ2 − ξ1∣∣υ. (4.28)

Hence, by (4.23), (4.25), (4.27) and (4.28), we get

Aω∗,υ ≤4κμ+ πρμ(3κ− 1)M1

κ2(κ− 1) (1− ρ N) (2− πρ)υ[ρυ Rυ

g,0 +∥∥g∥∥

L

]+2μ

κAg∗,υ. (4.29)

From (1.9), we have ∣∣ω(t2)− ω(t1)∣∣ ≤ ∣∣b∣∣ |t2 − t1|+

2μ

κ|g(t2)− g(t1)| . (4.30)

(4.30) and (1.11) result in

Aω,υ ≤22−υμ

κ(κ− 1)

∥∥g∥∥L+2μ

κAg,υ . (4.31)

From (4.29) and (4.31), we know that

max{Aω∗,υ, Aω,υ} ≤4κμ+ πρμ(3κ− 1)M1

κ2(κ− 1) (1− ρ N) (2 − πρ)υ[ρυ Rυ

g,0 +∥∥g∥∥

L

]+2μ

κA+

g,υ = A+ω,υ. (4.32)

�

Theorem 4.2 Assume that δ ∈ C4(L) and ‖δ‖4 ≤ ρ ≤ 1.92κ(κ−1)(3κ−1)(2κ+3)π+0.96κ(κ−1)π . If

g∗ and g′∗ are respectively the Sobolev type perturbation of g by δ and the Sobolev type

perturbation of g′ by δ, then ω′∗ is the Sobolev type perturbation of 3-rd order for ω′ by δ,

where ω and ω∗ are respectively the normal solutions for the Sherman-Lauricella equations

(1.7) (Γ = L) and (2.12).

Proof On the one hand, by (1.9) and (2.20), we have∣∣∣ω′∗(ξ)− ω′(t)∣∣∣ ≤ ∣∣∣∣ bδ

1 + δ′(t)− b

∣∣∣∣+ 1

κ

∣∣∣∣ G′(t)

1 + δ′(t)

∣∣∣∣+ 2μ

κ

∣∣∣g′∗(ξ)− g′(t)∣∣∣ = �1 + �2 + �3. (4.33)


By (4.21) and (1.11),

�1 ≤∣∣∣∣ bδ − b

1 + δ′(t)

∣∣∣∣+ ∣∣∣∣ b δ′(t)

1 + δ′(t)

∣∣∣∣≤

2μ(Rυ

g,0 + ρ1−υN∥∥g∥∥

L

)κ (κ− 1) (1− ρN) (1− ρ)

∥∥δ∥∥υ

2+

2μ∥∥g∥∥

L

κ(κ− 1)(1− ρ)

∥∥δ′∥∥L

≤ 2μ

κ (κ− 1) (1− ρN) (1− ρ)

[Rυ

g,0 + ρ1−υ(1 +N − ρ N

)∥∥g∥∥L

]∥∥δ∥∥υ

2. (4.34)

By (3.13) and (4.20),

�2 ≤(3κ− 1)μM1

κ2(κ− 1) (1− ρ N) (1− ρ)

(ρυ Rυ

g,0 +∥∥g∥∥

L

) ∥∥δ∥∥3. (4.35)

Obviously, there is a Sobolev perturbation coefficient Rυg′,0 of g′∗ for g′ such that

�3 ≤2μ

κRυ

g′,0

∥∥δ∥∥υ

L. (4.36)

By (4.33), (4.34), (4.35) and (4.36), we get∥∥ω′∗ ◦ σ − ω′∥∥

L≤ Rυ

ω′,3

∥∥δ∥∥υ

3, (4.37)

where

Rυω′,3 =

μ [2κ+ (3κ− 1)ρM1]Rυg,0 + μρ1−υ

[2κ(1 +N − ρN

)+ (3κ− 1)M1

] ∥∥g∥∥L

κ2 (κ− 1) (1− ρN) (1− ρ)+2μ

κRυ

g′,0.

(4.38)

On the other hand, by (2.17), we have∣∣∣ω′∗(ξ)− ω′∗(ζ)∣∣∣ ≤ ∣∣bδ

∣∣ ∣∣∣∣ 1

1 + δ′(t)− 1

1 + δ′(τ)

∣∣∣∣+ 1

κ

∣∣∣∣ G′(t)

1 + δ′(t)− G′(τ)

1 + δ′(τ)

∣∣∣∣+ 2μ

κ

∣∣∣g′∗(ξ)− g′∗(ζ)∣∣∣

= k1 + k2 + k3. (4.39)

By Lemma 3.1, we obtain∣∣∣∣ 1

1 + δ′(t)− 1

1 + δ′(τ)

∣∣∣∣ ≤ 1

(1 − ρ)2π

2

∣∣∣δ′′∣∣∣L

∣∣∣t− τ∣∣∣ ≤ πρ

2(1− ρ)2

∣∣∣t− τ∣∣∣, (4.40)

and ∣∣∣G′(t)−G′(τ)∣∣∣ ≤ ∥∥ (D1[G

′]])∥∥

L

∣∣∣t− τ∣∣∣ ≤ π

2

∥∥G′′∥∥L

∣∣∣t− τ∣∣∣. (4.41)

By (4.22), (4.40), (3.13), (4.20), (4.41), (4.24) and (3.15),

k1 ≤2πρμ

κ (κ− 1) (1− ρN) (1− ρ)2 (2− πρ)υ

[ρυRυ

g,0 +∥∥g∥∥

L

]∣∣ξ − ζ∣∣υ, (4.42)

and

k2 ≤1

κ

[∣∣G′(t)∣∣ ∣∣∣∣ 1

1 + δ′(t)− 1

1 + δ′(τ)

∣∣∣∣+ ∣∣∣∣ 1

1 + δ′(τ)

∣∣∣∣ |G′(t)−G′(τ)|]

≤ πρ

2κ(1− ρ)2[ρM1 + (1− ρ)M2

]∥∥ωδ

∥∥L

∣∣t− τ∣∣

≤ πρμ(3κ− 1) [ρM1 + (1− ρ)M2]

κ2(κ− 1) (1− ρN) (1 − ρ)2 (2− πρ)υ

[ρυRυ

g,0 +∥∥g∥∥

L

]∣∣ξ − ζ∣∣υ. (4.43)


Obviously,

k3 ≤2μ

κAg′

∗,υ

∣∣ξ − ζ∣∣υ. (4.44)

So, (4.39), (4.42), (4.43) and (4.44) result in

Aω′

∗,υ ≤

πρμ[2κ+ (3κ− 1) (ρM1 + (1− ρ)M2)

]κ2 (κ− 1) (1− ρN) (1− ρ)2 (2− πρ)

υ

[ρυRυ

g,0 +∥∥g∥∥

L

]+2μ

κAg′

∗,υ. (4.45)

From (1.9), we have

|ω′(t2)− ω′(t1)| =2μ

κ|g′(t2)− g′(t1)| , (4.46)

obviously,

Aω′,υ =2μ

κAg′,υ. (4.47)

By (4.45) and (4.47), we get

max{Aω′

∗,υ, Aω′,υ} ≤ A+

ω′,υ, (4.48)

where

A+ω′,υ ≤

πρμ {2κ+ (3κ− 1) [ρM1 + (1− ρ)M2]}κ2 (κ− 1) (1− ρN) (1− ρ)2 (2− πρ)

υ

(ρυRυ

g,0 +∥∥g∥∥

L

)+2μ

κA+

g′,υ. (4.49)

�

Corollary 4.1 Assumptions are the same to that in Theorem 4.2, then �∗(ξ) = ξω′∗(ξ)

is the Sobolev type perturbation of 3-rd order for �(t) = tω′(t) by δ.

Proof We know that

|�∗(ξ)−�(t)| ≤ |ξ| |ω′∗(ξ)− ω′(t)|+ |ξ − t| |ω′(t)| , (4.50)

namely,

‖�∗ ◦ σ −�‖L ≤ (1 + ρ) ‖ω′∗ ◦ σ − ω′‖L + ‖δ‖L ‖ω′‖L . (4.51)

By (4.37) and (1.12), we get

‖�∗ ◦ σ −�‖L ≤ Rυ�,3

∥∥δ∥∥υ

3, (4.52)

where

Rυ�,3 = (1 + ρ)Rυ

ω′,3 +2μρ1−υ

κ(κ− 1)

∥∥g∥∥L+2μρ1−υ

κ

∥∥g′∥∥L

(4.53)

with Rυω′,3 is given in (4.38).

We also have∣∣�∗(ξ2)−�∗(ξ1)∣∣ ≤ ∣∣ξ2∣∣ |ω′∗(ξ2)− ω′∗(ξ1)|+

∣∣ξ2 − ξ1∣∣ |ω′∗(ξ1)| , (4.54)

namely, |�∗(ξ2)−�∗(ξ1)| ≤ (1 + ρ)Aω′

∗,υ |ξ2 − ξ1|υ + |ξ2 − ξ1|

[‖ω′δ ◦ σ − ω′‖

L+ ‖ω′‖L

]. By

(4.48), (4.37) and (1.12), we get

A�∗,υ ≤ (1 + ρ)A+ω′,υ + (2 + πρ)1−υ

[Rυ

ω′,3ρυ +

2μ

κ(κ− 1)

∥∥g∥∥L+2μ

κ

∥∥g′∥∥L

](4.55)

with A+ω′,υ and Rυ

ω′,3 are given in (4.49) and (4.38), respectively.

From (1.9), we have

|�(t2)−�(t1)| ≤ |ω′(t2)− ω′(t1)|+ |t2 − t1| |ω′(t1)| . (4.56)


(4.56), (4.48) and (1.12) result in

A�,υ ≤ A+ω′,υ +

22−υμ

κ(κ− 1)

∥∥g∥∥L+22−υμ

κ

∥∥g′∥∥L. (4.57)

By (4.55), (4.57) and (4.49), we get

max{A�∗,υ, A�,υ} ≤ (1 + ρ)A+ω′,υ + (2 + πρ)1−υ

[Rυ

ω′,3ρυ +

2μ

κ(κ− 1)

∥∥g∥∥L+2μ

κ

∥∥g′∥∥L

]= A+

�,υ. (4.58)

�

Theorem 4.3 Assume that δ ∈ C4(L) and ‖δ‖4 ≤ ρ ≤ 1.92κ(κ−1)(3κ−1)(2κ+3)π+0.96κ(κ−1)π . If

g∗ and g′∗ are respectively the Sobolev type perturbation of g by δ and the Sobolev type

perturbation of g′ by δ, then, for 0 < ν < υ ≤ 1,∥∥[u∗ + iv∗]− [u+ iv]∥∥

U∩Uδ≤ D(υ, ν, ρ)

∥∥δ∥∥υ−ν

3, (4.59)

where D(υ, ν, ρ) is the constant independent of δ. More precisely,

D(υ, ν, ρ) =κ

μLω(υ, ν, ρ) +

1

2μLω′(υ, ν, ρ) +

1

2μL�(υ, ν, ρ), (4.60)

where

Lω(υ, ν, ρ) = α(υ, ν, ρ)A+ω,υ + β(υ, ν, ρ)

(A+

ω,υ

) νυ(Rυ

ω,2

)1− νυ + ρν Rυ

ω,2, (4.61)

Lω′(υ, ν, ρ) = α(υ, ν, ρ)A+ω′,υ + β(υ, ν, ρ)

(A+

ω′,υ

) νυ (

Rυω′,3


ω′,3, (4.62)

L�(υ, ν, ρ) = α(υ, ν, ρ)A+�,υ + β(υ, ν, ρ)

(A+

�,υ

) νυ(Rυ

�,3


�,3 (4.63)

with

α(υ, ν, ρ) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩3ρν

[22υ(1 + ρ)

υ(2− πρ)+ 2υ +

22+υ31−υ(1 + ρ)

(1 − υ)(2− πρ)+

ρ1−υ (π + 2)

6υ (2− πρ)1−υ

], 0 < υ < 1,

12ρν

[1 + ρ

2− πρ+

π + 14

24+2(1 + ρ) ln(5 + 4ρ)

2− πρ

]+

24(1 + ρ)

νe(2− πρ), υ = 1,

β(υ, ν, ρ) =1

ν2νυ−1

[ (1 +

π

2ρ)υ

+ 1

] νυ

,

Rυω,2, A

+ω,υ, Rυ

ω′,3, A+ω′,υ, Rυ

�,3 and A+�,υ are given in (4.19), (4.32), (4.38), (4.49), (4.53) and

(4.58), respectively.

Proof From (1.1), (1.4), (1.5), (1.6) and (4.6), we know

u+ iv =κ

2μϕ(z)− z

2μϕ′(z)− 1

2μψ(z),

and

ϕ(z) =(S+[ω]

)(z), ϕ′(z) =

(S+[ω′]

)(z), ψ(z) = −κ

(S+[ω]

)(z)−

(S+[�]

)(z), z ∈ U,


analogously,

u∗ + iv∗ =κ

2μϕ∗(z)−

z

2μϕ′∗(z)−

1

2μψ∗(z),

and

ϕ∗(z) =(S+δ [ω∗]

)(z), ϕ′∗(z) =

(S+δ [ω

′∗])(z), ψ∗(z) = −κ

(S+δ [ω∗]

)(z)−

(S+δ [�∗]

)(z), z ∈ Uδ.

So, we have∥∥[u∗ + iv∗]− [u+ iv]∥∥

U∩Uδ≤ κ

μ

∥∥S+δ [ω∗]− S+[ω]∥∥

U∩Uδ+

1

2μ

∥∥S+δ [ω′∗]− S+[ω′]∥∥

U∩Uδ

+1

2μ

∥∥S+δ [�∗]− S+[�]∥∥

U∩Uδ.

By Theorem 4.1, Theorem 4.2, Corollary 4.1 and Lemma 4.1, we get⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

∥∥S+δ [ω∗]− S+[ω]∥∥

U∩Uδ≤ Lω(υ, ν, ρ)

∥∥δ∥∥υ−ν

2,∥∥S+δ [ω′∗]− S+[ω′]

∥∥U∩Uδ

≤ Lω′(υ, ν, ρ)∥∥δ∥∥υ−ν

3,∥∥S+δ [�∗]− S+[�]

∥∥U∩Uδ

≤ L�(υ, ν, ρ)∥∥δ∥∥υ−ν

3,

where Lω(υ, ν, ρ), Lω′(υ, ν, ρ) and L�(υ, ν, ρ) are given in (4.61), (4.62) and (4.63).

Finally, (4.59) is obtained. �

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Stability of Displacement to the Second Fundamental Problem in Plane Elasticity