ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ASYMPTOTIC FORMULA FOR DETECTING INCLUSIONS VIA BOUNDARY MEASUREMENTS
KHALIFA KHELIFI, MOHAMED ABDELWAHED, NEJMEDDINE CHORFI, MAATOUG HASSINE
Communicated by Vicentiu D. Radulescu
Abstract. In this article, we are concerned with a geometric inverse problem related to the Laplace operator in a three-dimensional domain. The aim is to derive an asymptotic formula for detecting an inclusion via boundary measure- ment. The topological sensitivity method is applied to calculate a high-order topological asymptotic expansion of the semi-norm Kohn-Vogelius functional, when a Dirichlet perturbation is introduced in the initial domain.
1. Introduction
The detection of an object from boundary measurements is used in several ap- plications such as in fluid mechanics, electrical impedance tomography, electromag- netic casting, non-destructive testing [2, 11, 12, 15].
On the theoretical level, these applications correspond to geometric inverse prob- lems. Among the methods to solve this type of problems, there exist a method based on the Kohn-Vogelius formulation and the topological sensitivity method [5, 6, 10, 12, 17, 18, 20, 21, 30]. The majority of works interested to this method are based on the first-order asymptotic expansion of the Kohn-Vogelius functional [3, 4, 8, 9, 10, 12, 13, 14, 19, 22, 23]. This method is sufficient in the case of small unknown object far from the boundary.
In general application case the size of the object to detect is finite. For this reason, we consider high-order terms in the asymptotic expansion of the Kohn- Vogelius functional formula.
In this article we apply the topological sensitivity method and the Kohn-Vogelius formulation, to derive a high-order asymptotic formula connecting the known bound- ary data and the unknown inclusion properties; its location , size and shape. More precisely in this paper we derive a high-order topological asymptotic expansion of the semi-norm Kohn-Vogelius functional associated to the Laplace operator in three-dimensional domain, when a Dirichlet perturbation is introduced in the initial domain.
2010Mathematics Subject Classification. 35J15, 78M22.
Key words and phrases. Laplace operator; asymptotic analysis; topological gradient;
Kohn-Vogelius functional.
c
2018 Texas State University.
Submitted September 9, 2017. Published June 28, 2018.
1
The proposed approach permit to calculate the topological gradient for any or- der for the semi-norm Kohn-Vogelius functional. We present a general approach applicable to various problems such as elasticity, Stokes equations, Navier-Stokes equation, Maxwell’s equations, etc.
The remaining of this paper are organized as follows. We begin by presenting the inverse problem and the Kohn-Vogelius formulation in section 2. In section 3 we present the topological sensitivity method. In section 4, we establish a some preliminary results, where we derive an asymptotic formula describing the variation of the solutions of Neumann and Dirichlet problems when a Dirichlet perturbation is introduced in the initial domain. Section 5 presents the main result of the paper.
Finally, section 6 contains the proofs of the different results. The paper ends by some concluding remarks.
2. Inverse problem and the Kohn-Vogelius formulation
The geometric inverse Laplace problem related to the Laplace operator in three- dimensional domain is considered in this paper. Let Ω ⊂ R3 denote a bounded domain with smooth boundary∂Ω and satisfies ∂Ω = Γ1∪Γ2 with Γ1∩Γ2 =∅, Γ26=∅.
We suppose that there exist a sub-domain D∗ of Ω with a smooth boundary
∂D∗. The studied inverse problem can be formulated:
For regular given data F, V andψm, find the unknown domainD∗ such that ψ is solution of the following over determined problem
−∆ψ=F in Ω\D∗,
∇ψ·n=V on Γ1, ψ=ψm on Γ1,
ψ= 0 on Γ2, ψ= 0 on∂D∗.
To derive an asymptotic formula connecting the boundary measurements and the location of the unknown domain D∗, we propose in this work a new technique based on the Kohn-Vogelius formulation and the topological sensitivity technique.
The Kohn-Vogelius formulation is a self regularization method which transforms the geometric inverse problem to a shape optimization problem. It leads to define two problems for any given domain D ⊂ Ω. The first one, named the Neumann problem, is associated with the Neumann datumV:
−∆ψn=F in Ω\D
∇ψn·n=V on Γ1
ψn = 0 on Γ2 ψn= 0 on∂D.
(2.1)
The second one is associated to the measured ψm, which will be named as the Dirichlet problem:
−∆ψd=F in Ω\D ψd=ψm on Γ1
ψd= 0 on Γ2
ψd= 0 on∂D.
(2.2)
We remark that if the domains D and D∗ coincide then ψn = ψd. According to this observation, Kohn and Vogelius [25] proposed to change the inverse problem to the minimization of a function measuring the difference between the Dirichlet and Neumann solutions. We define the Kohn-Vogelius semi-norm function
J(Ω\D) = Z
Ω\D
|∇ψn− ∇ψd|2dx,
whereψn (resp. ψd) is solution to the Neumann (resp. Dirichlet) perturbed prob- lem.
3. Topological sensitivity method
To calculate a high-order topological asymptotic expansion of the semi-norm Kohn-Vogelius functional J, we apply the topological sensitivity method. It con- sists in calculating the variation ofJ regarding to a small perturbationBz,at the point z of the domain Ω. For z∈ Ω and > 0, we define Bz, =z+B, where B ⊂R3 is a bounded fixed regular domain which contains the origin. We define the perturbed domain Ωz,= Ω\Bz,Let us consider the following overdetermined boundary value problem
−∆ψ=F in Ω\Bz,,
∇ψ·n=V on Γ1, ψ=ψm on Γ1,
ψ= 0 on Γ2, ψ= 0 on∂Bz,.
(3.1)
We assume here that there exists Bz∗, = z∗+B ⊂ Ω such that there exists a solution to problem (3.1). Consequently, the following geometric inverse problem is considered:
FindBz,⊂Ω such that the solutionψsatisfies the overdetermined system (3.1).
The Kohn-Vogelius functional for the perturbed domain is defined by J(Ωz,) =
Z
Ωz,
|∇ψn,− ∇ψd,|2dx,
whereψn, is the solution to the perturbed Neumann problem
−∆ψn,=F in Ω\Bz,, ψn,= 0 on∂Bz,,
∇ψn,n=V on Γ1, ψn,= 0 on Γ2.
(3.2)
andψd,is the solution to the perturbed Dirichlet problem
−∆ψd,=F in Ω\Bz,, ψd,= 0 on∂Bz,,
ψd,=ψm on Γ1, ψd,= 0 on Γ2.
(3.3)
We remark that if= 0, Ωz,0= Ω andψ0 satisfies
−∆ψ0=F in Ω,
∇ψ0·n=V on Γ1, ψ0=ψm on Γ1,
ψ0= 0 on Γ2, ψn,0is solution to
−∆ψn,0=F in Ω
∇ψn,0n=V on Γ1 ψn,0= 0 on Γ2,
(3.4) andψd,0is solution to
−∆ψd,0=F in Ω ψd,0=ψm on Γ1
ψd,0= 0 on Γ2.
(3.5) As mentioned in the introduction the majority of works interested to this method are based on the first-order asymptotic expansions of the functional J presented by
J(Ωz,) =J(Ω) +f()δJ(z) +o(f()),
where δJ is the topological gradient and f is a positive scalar function with lim→0f() = 0. Then, for smallthe solution of the minimization problem
min
Bz,⊂ΩJ(Ω\ωz,),
is given byBz∗,, withz∗∈Ω whereδJ is the most negative. This is due to the fact that ifδJ(z∗)< δJ(z), we obtainJ(Ωz∗,)<J(Ωz,). The purpose of this work is to obtain an asymptotic expansion of higher order for the Kohn-Vogelius functional J to detect an object of finite size and valid when the topological gradient δJ vanishes at some critical points inside Ω, under the form:
J(Ωz,) =J(Ω) +
I
X
i=1
fi()δiJ(z) +o(fI()),
where fi, 1≤i≤I are scalar positives functions verifying fi+1() =o(fi()) and vanish with. δiJ is theithtopological derivative of the Kohn-Vogelius functional J.
To derive the expected expansion, we establish in the next section some prelim- inary results. The main results of this analysis will be presented in section 5.
4. Some preliminary results
The aim of this section is to present an asymptotic formula describing the vari- ation of the solutionsψn, andψd,caused by the perturbation of Ω byBz,.
In conductivity imperfections identification context, an asymptotic expansion describing the variation of the solutions for I = 1 was derived in [14, 19] for the Laplace equation. Another application was studied using Stokes system [1] for the detection of obstacles in a flow via the asymptotic expansion of the velocity filed.
In this work, to derive the desired formula, we need to find an asymptotic ex- pansion of the exterior problem solution for the Laplace equation defined inR3\B.
Let Φ∈H1/2(∂B), denoting byH the solution to
−∆H = 0 in R3\B, H→0 at∞,
H = Φ on∂B,
Resorting to the simple layer potential representation [16, 28],H can be written as H(y) =
Z
∂ω
E(y−t)q(t)ds(t), ∀y∈R3\B, (4.1) whereE is the Laplace equation fundamental solution inR3:
E(y) = 1 4πkyk, andqis the boundary integral equation unique solution
Z
∂B
E(y−t)q(t)ds(t) = Φ(y), ∀y∈∂B. (4.2) By the change of variablex=z+yand using the perturbationBz, is not close to the boundary∂Ω, we have
H((x−z)/) = Z
∂B
E(x−z− t)q(t)ds(t), ∀x∈R3\Bz,.
Denoting byϕx−z,tthe function
ϕx−z,t:7−→ϕx−z,t() =E((x−z)−t), ∀ >0.
Using the fact thatϕx−z,tis smooth regardingand satisfies the following behavior ϕx−z,t() =
I
X
p=1
p
p!ϕ(p)x−z,t(0) +O(I+1),
whereϕ(p)x−z,t(0) is thepth derivative of ϕx−z,t at= 0. Then the following lemma gives an asymptotic expansion of the function
x7→H(x−z ).
Lemma 4.1. For any I≥0, we have H((x−z)/) =
I
X
p=1
pH(p)(x−z) +O(I+1), ∀x∈R3\Bz,, whereH(p)is the smooth function defined by
H(p)(x−z) = 1 p!
Z
∂B
ϕ(p)x−z,t(0)q(t)ds(t).
Remark 4.2. The first-order asymptotic expansion of the exterior problem solution for the Laplace equation is proved by Guilaume and Sid Idris [21, page 1049].
4.1. Asymptotic formula of the Neumann problem solution. To present an asymptotic formula describing the variation of the Neumann Problem solution ψn,, we define the sequences functions (Ψn,i)0≤i≤I and (Wn,i)0≤i≤I, where for all 0 ≤i≤I, Ψn,i are smooth function defined in the initial domain Ω, obtained as the solution to a interior problem with Neumann boundary condition on Γ1 and Wn,i are smooth function defined inR3\B, obtained as the solution to a exterior problems. More precisely:
For i= 0: Ψn,0=ψn,0 andWn,0is the solution to
−∆Wn,0= 0 in R3\B, Wn,0→0 at ∞ Wn,0=−ψn,0(z) on∂B.
(4.3)
For i= 1: Ψn,1is the solution to
−∆Ψn,1= 0 in Ω,
∇Ψn,1·n=−∇Wn,0(1)(x−z)·n on Γ1, Ψn,1=−Wn,0(1)(x−z) on Γ2,
(4.4)
withWn,0(1) is defined by Lemma 4.1 in the particular casei= 0, φ=−ψn,0(z) and p= 1.
The functionWn,1 depends on Ψn,0and Ψn,1, it is solution of the exterior prob- lem
−∆Wn,1= 0 inR3\B, Wn,1→0 at ∞
Wn,1=−Ψn,1(z)−DΨn,0(z)(y) on∂B.
(4.5)
For1≤i≤I: The function Ψn,idepends onWn,j for 0≤j≤i−1 and is solution of the interior problem
−∆Ψn,i= 0 in Ω,
∇Ψn,i·n=−
i
X
p=1
∇Wn,i−p(p) (x−z)·n on Γ1,
Ψn,i=−
i
X
p=1
Wn,i−p(p) (x−z) on Γ2,
(4.6)
withWn,j(p)is defined by Lemma 4.1.
The functionWn,i depends on Ψn,j for 0≤j≤i and is solution of the exterior problem
−∆Wn,i= 0 inR3\B, Wn,i→0 at ∞ Wn,i=−Ψn,i(z)−
i
X
p=1
1
p!DpΨn,i−p(z)(yp) on∂B,
(4.7)
whereDpΨn,i−p(z) is thepthderivative of Ψn,i−p(the harmonic function) atz∈Ω andyp= (y, . . . ,y)∈(R3)p.
We are now ready to present an asymptotic formula describing the variation of the solutionψn, raised from the perturbation of Ω byBz,.Ω.
Theorem 4.3. In the perturbed domain Ωz,, the solution ψn of the Neumann Laplace equation has the asymptotic expansion
ψn,(x) =
I
X
i=0
i[Ψn,i(x) +Wn,i(x−z
)] +O(I+1) inΩz,.
4.2. Asymptotic formula of the Dirichlet problem solution. Similarly to the asymptotic of the Neumann solution, to present an asymptotic formula describing the variation of the Dirichlet Problem solutionψd,, we define the sequences func- tions (Ψd,i)0≤i≤I and (Wd,i)0≤i≤I, where for all 0≤i≤I, Ψd,iare smooth function defined in the initial domain Ω, obtained as the solution to a interior problem with Dirichlet boundary condition on Γ1andWd,iare smooth function defined inR3\B, obtained as the solution to a exterior problems. More precisely, the sequences func- tions (Ψd,i)0≤i≤I and (Wd,i)0≤i≤I, are defined as follow:
For i= 0: Ψd,0=ψd,0andWd,0 is the solution to
−∆Wd,0= 0 in R3\B, Wd,0→0 at∞ Wd,0=−ψd,0(z) on∂B.
(4.8)
For i= 1: Ψd,1is the solution to
−∆Ψd,1= 0 in Ω,
Ψd,1=−Wd,0(1)(x−z) on∂Ω, (4.9) withWd,0(1) is defined by Lemma 4.1 in the particular case i= 0,φ=−ψd,0(z) and p= 1.
The function Wd,1 depends on Ψd,0 and Ψd,1, and is solution of the exterior problem
−∆Wd,1= 0 in R3\B, Wd,1→0 at∞
Wd,1=−Ψd,1(z)−DΨd,0(z)(y) on∂B.
(4.10)
For1≤i≤I: The function Ψd,idepends onWd,j for 0≤j≤i−1 and is solution of the interior problem
−∆Ψd,i= 0 in Ω, Ψd,i=−
i
X
p=1
Wd,i−p(p) (x−z) on∂Ω, (4.11)
withWd,j(p)defined in Lemma 4.1.
The functionWd,i depends on Ψd,j for 0≤j ≤iand is solution of the exterior problem
−∆Wd,i= 0 in R3\B, Wd,i→0 at∞ Wd,i=−Ψd,i(z)−
i
X
p=1
1
p!DpΨd,i−p(z)(yp) on∂B.
(4.12)
We are now ready to present an asymptotic formula giving the variation ofψd,
raised from the perturbation of Ω byBz,.
Theorem 4.4. In the perturbed domain Ωz,, the solution ψd, of the Dirichlet Laplace equation has the asymptotic expansion
ψd,(x) =
I
X
i=0
i[Ψd,i(x) +Wd,i(x−z
)] +O(I+1) inΩz,. 5. Asymptotic formula
The main result is presented in this section. A high-order topological asymp- totic expansion is derived for the semi-norm Kohn-Vogelius functionalJ, when a Dirichlet perturbation is introduced in the initial domain. The functional J can be decomposed as
J(Ωz,) = Z
Ωz,
|∇ψd,|2dx + Z
Ωz,
|∇ψn,|2dx−2 Z
Ωz,
∇ψd,.∇ψn,dx
=Jd(Ωz,) +Jn(Ωz,) +Jdn(Ωz,), where
Jd(Ωz,) = Z
Ωz,
|∇ψd,|2dx, Jn(Ωz,) =
Z
Ωz,
|∇ψn,|2dx, Jd,n(Ωz,) =−2
Z
Ωz,
∇ψd,.∇ψn,dx.
The Dirichlet termJd has the following variation Jd(Ωz,)− Jd(Ω) =
Z
Ωz,
|∇ψd,|2dx− Z
Ω
|∇ψd,0|2dx
= Z
Ωz,
∇(ψd,+ψd,0)· ∇(ψd,−ψd,0)dx− Z
Bz,
|∇ψd,0|2dx.
By the Green formula, from the problems (3.3) and (3.3) with= 0, we deduce Z
Ωz,
∇(ψd,+ud,0)· ∇(ψd,−ψd,0)dx
=− Z
∂Bz,
∇(ψd,+ud,0)·nψd,0ds + 2 Z
Ωz,
F(ψd,−ψd,0)dx.
From problem (3.3) with= 0 we derive Z
Bz,
|∇ψd,0|2dx =− Z
∂Bz,
∇ψd,0·nψd,0ds + Z
Bz,
F ψd,0 dx, (5.1) then, we obtain
Jd(Ωz,)− Jd(Ω)
=− Z
∂Bz,
∇ψd,·nψd,0 ds + 2 Z
Ωz,
F(ψd,−ψd,0) dx − Z
Bz,
F ψd,0 dx
=− Z
∂Bz,
∇(ψd,−ψd,0)·nψd,0 ds− Z
∂Bz,
∇ψd,0·nψd,0 ds + 2
Z
Ωz,
F(ψd,−ψd,0) dx − Z
Bz,
F ψd,0 dx.
Then, from (5.1) it follows that Jd(Ωz,)− Jd(Ω) =−
Z
∂Bz,
(∇ψd,− ∇ψd,0).nψd,0 ds + Z
Bz,
|∇ψd,0|2dx + 2
Z
Ωz,
F(ψd,−ψd,0) dx −2 Z
Bz,
F ψd,0 dx. Similarly, the Neumann termJn has the variation
Jn(Ωz,)− Jn(Ω) = Z
Ωz,
|∇ψn,|2dx− Z
Ω
|∇ψn,0|2dx
= Z
∂Bz,
(∇ψn,− ∇ψn,0).nψn,0ds − Z
Bz,
|∇ψn,0|2 dx.
The Dirichlet/Neumann termJd,n has the variation Jd,n(Ωz,)− Jd,n(Ω) =
Z
Ωz,
∇ψd,.∇ψn,dx − Z
Ω
∇ψd,0.∇ψn,0dx
= Z
Ωz,
F(ψd,−ψd,0) dx − Z
Bz,
F ψd,0 dx. Then the functionalJ has the variation
J(Ωz,)− J(Ω) = Z
Bz,
|∇ψd,0|2dx− Z
Bz,
|∇ψn,0|2dx
− Z
∂Bz,
(∇ψd,− ∇ψd,0).nψd,0ds +
Z
∂Bz,
(∇ψn,− ∇ψn,0).nψn,0ds.
From Theorem 4.3, we have Z
∂Bz,
(∇ψn,− ∇ψn,0).nψn,0ds
=
I
X
i=1
i Z
∂Bz,
∇Ψn,i(x).n(x)ψn,0(x)ds
+
I
X
i=0
i Z
∂Bz,
∇xWn,i((x−z)/))·nψn,0ds+O(I+1), and using Theorem 4.4, we have
Z
∂Bz,
(∇ψd,− ∇ψd,0).nψd,0ds
=
I
X
i=1
i Z
∂Bz,
∇Ψd,i(x).n(x)ψd,0(x)ds
+
I
X
i=0
i Z
∂Bz,
∇xWd,i(x−z
)·nψd,0ds+O(I+1).
Consequently, the functionalJ has the following variation J(Ωz,)− J(Ω) =
I
X
i=0
i Z
∂Bz,
∇xWn,i(x−z
)·nψn,0ds
−
I
X
i=0
i Z
∂Bz,
∇xWd,i(x−z
)·nψd,0ds +
I
X
i=1
i Z
∂Bz,
∇Ψn,i(x)·n(x)ψn,0(x)ds
−
I
X
i=1
i Z
∂Bz,
∇Ψd,i(x)·n(x)ψd,0(x)ds +
Z
Bz,
|∇ψd,0|2dx− Z
Bz,
|∇ψn,0|2dx+O(I+1).
(5.2)
To present the desired asymptotic expansion of the Kohn-Vogelius functional J, for allz∈Ω we consider the following notation:
Tn,1i (z) =
i
X
p=0
1 p!
Z
∂B
∇yWn,i−p(y)·n(y)[∇(p)ψn,0(z)(yp)]ds(y),
Td,1i (z) =−
i
X
p=0
1 p!
Z
∂B
∇yWd,i−p(y)·n(y)[∇(p)ψd,0(z)(yp)]ds(y),
Tn,2i (z) =
i
X
p=0 p
X
q=0
1 q!(p−q)!
Z
∂B
[∇(q+1)Ψn,i−p+1(z)(yq)]·n(y)
×[∇(p−q)ψn,0(z)(yp−q)]ds(y), Td,2i (z) =−
i
X
p=0 p
X
q=0
1 q!(p−q)!
Z
∂B
[∇(q+1)Ψd,i−p+1(z)(yq)]·n(y)
×[∇(p−q)ψd,0(z)(yp−q)]ds(y), Td,3i (z) =
i
X
p=0
1 p!(i−p)!
Z
B
∇(p+1)ψd,0(z)(yp)· ∇(i−p+1)ψd,0(z)(yi−p)dy,
Tn,3i (z) =−
i
X
p=0
1 p!(i−p)!
Z
B
∇(p+1)ψn,0(z)(yp)· ∇(i−p+1)ψn,0(z)(yi−p)dy.
We can know derive the topological asymptotic expansion of the Kohn-Vogelius cost functional J by giving the variation J(Ωz,)− J(Ω) regarding to the geometric perturbation of the domain at any point. The main result is described by the following theorem.
Theorem 5.1. The topological asymptotic expansion of the Kohn-Vogelius func- tionalJ is given by
J(Ωz,)− J(Ω) =
I
X
i=1
iδiJ(z) +O(I+1),
where δiJ(z) =
(Tn,1i−1(z)−Td,1i−1(z) if i≤2, Tn,1i−1(z)−Td,1i−1(z) +Tn,2i−3(z)−Td,2i−3+Tn,3i−3−Td,3i−3(z) if 3≤i≤I.
6. Proofs
The aim of this section is to prove Theorems 4.3 and 4.4, and the main result described in Theorem 5.1.
Proofs of Theorems 4.3 and 4.4. To prove Theorem 4.3, in Ωz,we define the function
Rn,I = Ψn,0(x) +Wn,0(x−z
) +(Ψn,1(x) +Wn,1(x−z ) +· · ·+N(Ψn,I(x) +Wn,I(x−z
)−ψn(x).
We can easily show thatRn,I is harmonic in Ωz,. On∂Bz,we have
Rn,I(x) = Ψn,0(x) +Wn,0(x−z ) +
I
X
i=1
i[Ψn,i(x) +Wn,i(x−z )]
=
I
X
i=0
iΨn,i(x)−
I
X
i=0
ihXi
p=0
1
p!DpΨn,i−p(z)((x−z )p)i
.
(6.1)
From the multilinearity ofDpΨn,i−p(z), it follows that
I
X
i=1
ihXi
p=0
1
p!DpΨn,i−p(z)((x−z )p)i
=
I
X
i=0 i
X
p=0
i−p
p! DpΨn,i−p(z)((x−z)p)
=
I
X
i=0
i
N−i
X
p=0
1
p!DpΨn,i(z)((x−z)p).
Then, one can deduce Rn,I =
I
X
i=0
ih
Ψn,i(x)−
I−i
X
p=0
1
p!DpΨn,i(z)((x−z)p)i
. (6.2)
Using thatkx−zk=O() on∂Bz,and Taylor’s Theorem [29], we have Rn,I(x) =O(I+1), on∂Bz,.
On Γ2 we have Rn,I(x) =
I
X
i=0
iWn,i(x−z )−
I
X
i=1
i
i
X
p=1
Wn,i−p(p) (x−z)
=
I
X
i=0
iWn,i(x−z )−
I−1
X
i=0
i
I−i
X
p=1
pWn,i(p)(x−z) .
This equality can be written as Rn,I(x) =IWn,I(x−z
) +
I−1
X
i=0
i
Wn,i(x−z )−
I−i
X
p=1
pWn,i(p)(x−z) .
Then, by Lemma 4.1 we obtain
Rn,I =O(I+1) on Γ2. On Γn, using the same analysis we obtain
∇Rn,I·n=O(I+1) on Γ1.
Similarly, to prove Theorem 4.4, we define the functionRd,I in Ωz,by Rd,I = Ψd,0(x) +Wd,0(x−z
) +(Ψd,1(x) +Wd,1(x−z ) +· · ·+I(Ψd,I(x) +Wd,I(x−z
)−ψd(x)
and using the same analysis in the proof of Theorem 4.3 we deriveRd,I =O(I+1).
6.1. Proofs of the main results in Theorem 5.1. To prove Theorem 5.1, we have to estimate each term of the equality (5.2).
Estimate for the first and the second terms. By changing x =z+y, we have
Z
∂Bz,
∇xWn,i(x−z
)·n(x)ψn,0(x)ds= Z
∂B
∇yWn,i(y)·n(y)ψn,0(z+y)ds(y).
Sinceψn,0is smooth in a neighborhood ofz, one obtains ψn,0(z+y) =ψn,0(z) +
I−1
X
p=1
p
p!∇(p)ψn,0(z)(yp) +O(I)
=
I−1
X
p=0
p
p!∇(p)ψn,0(z)(yp) +O(I).
Then, we have Z
∂Bz,
∇xWn,i(x−z
)·n(x)ψn,0(x)ds
=
I−1
X
p=0
p+1 p!
Z
∂B
∇yWn,i(y)·n(y)[∇(p)ψn,0(z)(yp)]ds(y) +O(I+1).
Consequently,
I
X
i=0
i Z
∂Bz,
∇xWn,i((x−z)/))·nψn,0ds
=
I
X
i=0
i
I−1
X
p=0
p+1 p!
Z
∂B
∇yWn,i(y)·n(y)[∇(p)ψn,0(z)(yp)]ds(y) +O(I+1)
=
I
X
i=1
i
i−1
X
p=0
1 p!
Z
∂ω
∇yWn,i−p−1(y)·n(y)[∇(p)ψn,0(z)(yp)]ds(y) +O(I+1)
=
I
X
i=1
iTn,1i−1(z) +O(I+1),
Similarly, we obtain
I
X
i=0
i Z
∂Bz,
∇xWd,i((x−z)/))·nψd,0ds=−
I
X
i=1
iTd,1i−1(z) +O(I+1).
Estimate for the third and the fourth terms. By changingx =z+y, we have
Z
∂Bz,
∇Ψn,i(x)·n(x)ψn,0(x)ds=2 Z
∂B
∇Ψn,i(z+y)·n(z+y)un,0(z+y)ds(y).
Sinceψn,0is smooth in a neighborhood ofz, one obtains ψn,0(z+y) =ψn,0(z) +
I−1
X
p=1
p
p!∇(p)ψn,0(z)(yp) +O(I)
=
I−1
X
p=0
p
p!∇(p)ψn,0(z)(yp) +O(I).
Similarly, Ψi is smooth in a neighborhood ofz, then
∇Ψn,i(z+y) =
I−1
X
q=0
q
q!∇(q+1)Ψn,i(z)(yq) +O(I).
Then Z
∂Bz,
∇Un,i(x)·n(x)un,0(x)ds
=2 Z
∂B
[
I−1
X
q=0
q
q!∇(q+1)Un,i(z)(yq)]·n(y)[
I−1
X
p=0
p
p!∇(p)un,0(z)(yp)]ds(y) +O(I+1).
Using the Cauchy product formula, we derive Z
∂Bz,
∇Ψn,i(x)·n(x)ψn,0(x)ds
=
I−2
X
p=0
p+2
p
X
q=0
1 q!(p−q)!
Z
∂B
[∇(q+1)Ψn,i(z)(yq)]·n(y)
×[∇(p−q)ψn,0(z)(yp−q)]ds(y) +O(I+1).
Consequently,
I
X
i=1
i Z
∂Bz,
∇Ψn,i(x)·n(x)ψn,0(x)ds
=
I
X
i=1 I−2
X
p=0
i+p+2
p
X
q=0
1 q!(p−q)!
× Z
∂B
[∇(q+1)Ψn,i(z)(yq)]·n(y)[∇(p−q)ψn,0(z)(yp−q)]ds(y) +O(I+1)
=
I
X
i=3
i
i−3
X
p=0 p
X
q=0
1 q!(p−q)!
× Z
∂B
[∇(q+1)Ψn,i−p−2(z)(yq)]·n(y)[∇p−qψn,0(z)(y(p−q))]ds(y) +O(I+1)
=
I
X
i=3
iTn,2i−3(z) +O(I+1).
Similarly,
I
X
i=1
i Z
∂Bz,
∇Ψd,i(x)·n(x)ψd,0(x)ds
=
I
X
i=3
i
i−3
X
p=0 p
X
q=0
1 q!(p−q)!
× Z
∂B
[∇(q+1)Ψd,i−p−2(z)(yq)]·n(y)[∇p−qψd,0(z)(y(p−q))]ds(y) +O(I+1)
=−
I
X
i=3
iTd,2i−3(z) +O(I+1).
Estimate for the fifth and the sixth terms. Sinceψd,0andψn,0are sufficiently regular inBz,, we have
∇ψd,0(z+y) =∇ψd,0(z) +
I−1
X
i=1
i
i!∇(i+1)ψd,0(z)(yi) +O(I)
∇ψn,0(z+y) =∇ψn,0(z) +
I−1
X
i=1
i
i!∇(i+1)ψn,0(z)(yi) +O(I).
By the change of variablex=z+y, we obtain Z
Bz,
|∇ψd,0|2dx=3 Z
B
|∇ψd,0(z+y)|2dy
=3 Z
B
I−1X
i=0
i
i!|∇(i+1)ψd,0(z)(yi)
|2dy+O(I+1).
Using the Cauchy product formula, we obtain Z
Bz,
|∇ψd,0|2dx
=
I−3
X
i=0
i+3Xi
p=0
1 p!(i−p)!
Z
B
∇(p+1)ψd,0(z)(yp)· ∇(i−p+1)ψd,0(z)(yi−p)dy +O(I+1)
=
I
X
i=3
iTd,3i−3(z) +O(I+1).
Similarly, Z
Bz,
|∇ψn,0|2dx
=
I−3
X
i=0
i+3Xi
p=0
1 p!(i−p)!
Z
B
∇(p+1)ψn,0(z)(yp)· ∇(i−p+1)ψn,0(z)(yi−p)dy +O(I+1)
=−
I
X
i=3
iTn,3i−3(z) +O(I+1).
Finally, the desired result is obtained by using the above estimates.
Concluding remarks. This work is concerned with a geometric inverse problem related to the Laplace operator in three-dimensional domain. More precisely, the topological sensitivity method is applied to calculate a high-order topological as- ymptotic expansion of the semi-norm Kohn-Vogelius functional, when a Dirichlet perturbation is introduced in the initial domain.
The obtained expansion of the semi-norm Kohn-Vogelius functional is of higher interest and improves the detection of objects with any size of perturbation. The other advantage is when the topological derivative of order one is equal to zero for some critical points in the initial domain.
Acknowledgments. The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for funding this Research group No (RG-1435-026).
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Khalifa Khelifi
Department of Mathematics, College of Sciences, Monastir University, Monastir, Tunisia E-mail address:[email protected]
Mohamed Abdelwahed (corresponding author)
Department of Mathematics, College of Sciences, King Saud University, Riyadh, Saudi Arabia
E-mail address:[email protected]
Nejmeddine Chorfi
Department of Mathematics, College of Sciences, King Saud University, Riyadh, Saudi Arabia
E-mail address:[email protected]
Maatoug Hassine
Department of Mathematics, College of Sciences, Monastir University, Monastir, Tunisia E-mail address:[email protected]
