# Domain Identification for Inverse Problem via Conformal Mapping and Fixed Point Methods in Two Dimensions

Authors: Fagueye Ndiaye
Journal: Abstract and Applied Analysis
 (2020)

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## Abstract

In this paper, we present a survey of the inverse eigenvalue problem for a Laplacian equation based on available Cauchy data on a known partΓ0 and a homogeneous Dirichlet condition on an unknown part Γ0 of the boundary of a bounded domain, Ω⊂ℝN. We consider variations in the eigenvalues and propose a conformal mapping tool to reconstruct a part of the boundary curve of the two-dimensional bounded domain based on the Cauchy data of a holomorphic function that maps the unit disk onto the unknown domain. The boundary values of this holomorphic function are obtained by solving a nonlocal differential Bessel equation. Then, the unknown boundary is obtained as the image of the boundary of the unit disk by solving an ill-posed Cauchy problem for holomorphic functions via a regularized power expansion. The Cauchy data were restricted to a nonvanishing function and to the normal derivatives without zeros. We prove the existence and uniqueness of the holomorphic function being considered and use the fixed-point method to numerically analyze the results of convergence. We’ll calculate the eigenvalues and compare the result with the shape obtained via minimization functional method, as developed in a previous study. Further, we’ll observe via simulations the shape of Γ and if it preserves its properties with varying the eigenvalues.


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## Body

## 1. Introduction

In the paper, we consider the mathematical modeling employed in electrostatic imaging during nondestructive testing. This analysis engenders an inverse boundary value problem for a Laplace equation. This problem entails the identification of an unknown domain,Γ, within a conducting host medium with constant conductivity based on known Cauchy data on the boundary of the medium, Γ0. For detailed examples, please consult [1].We suppose thatΩ is a simply connected domain in ℝN with C2 smooth boundaries ∂Ω and
(1)∂Ω=Γ0òΓ.The boundary Γ is assessed by imposing a voltage pattern at a number of electrodes attached to the boundary Γ0 and measuring the resultant current through the electrodes.The inverse problem we are concerned with is to determine the unknown part,Γ, of the boundary of the bounded domain Ω⊂ℝN, based on available Cauchy data g,h on Γ0:
(2)uΩ=gonΓ0,∂u∂ν=honΓ0,where u∈H1Ω denotes a solution of the Laplace equation:
(3)−ΔuΩ=λΩuΩinΩ,λΩeigenvalue,satisfying the condition:
(4)uΩ=0onΓ.where g denotes a continuous positive function on Γ0, h denotes a continuous negative function on Γ0, and ν denotes the outer normal vector unit.My thesis (please consult [2]) is aimed at identifying Γ via Cauchy data on Γ0 using methods based on conformal mapping and the minimization function. The results (please consult [3]) were published in AMO (Advanced Modeling and Optimization) in 2014.Here, we aim to compare the results obtained using conformal mapping and the fixed-point method and observe the variations in the shape ofΓ with varying eigenvalues.


## 2. Methods

### 2.1. Conformal Mapping Method

The conformal mapping method was presented in [3].This paper is primarily aimed at presenting iterative methods to solve the inverse problem1−2−3 by using ideas from conformal mapping theory that have been developed over the last decade. For detailed references, please consult [1, 4, 5].However, in our approach, we consider a spectral problem on the simply connected domainΩ in ℝN with C2 smooth boundary ∂Ω satisfying (1).The family of holomorphic functions corresponding to the eigenvalues of the spectrum was identified via the fixed-point method and Nyström method.The proposed reconstruction of the boundary curveΓ based on available Cauchy data on the accessible boundary, Γ0, is based on a conformal map Φ:D1→Ω. Thus, we consider ΓD1, the boundary of the complex disk D1 with radius one centered at the origin, and map ΓD1 onto ∂Ω via Φ. We introduce the following notations.
(5)Γ2=eiθ,0≤θ≤απ,0<α<2,Γ1=eiθ,απ<θ≤2π,0<α<2.Further,Γ and Γ0 are parameterized as follows:
(6)Γ0=Ψθ,θ∈0,απ,Γ=Ψθ,θ∈απ,2π,where Ψ:ℝ→ℂ is a continuous differentiable 2π−periodic from ℝ onto ℂ satisfying the property that Ψ0,2π is injective. This property implies that
(7)∣Ψ′θ∣≠0,∀θ∈0,2π.We can normalize the mappingΦ by prescribing Φ1=Ψ0 and define the bijective boundary correspondence function Ψ:0,2π⟶ℂ by setting
(8)Ψθ≔Φeiθ,θ∈0,2π.It is clear that the values ofΦ on the boundary produce a unique solution Ψ of the Cauchy problem, given by
(9)Φeiθ=Ψθ.Therefore, the operatorΨθ=Φeiθ,θ∈0,2π is well defined.The Cauchy-Riemann equations give rise to nonlinear ordinary differential equations via Bessel equations for the values on this part of the boundaryΨΓ2(10)Φ′eiθ=∂u^/∂νhoΦeiθonΓ2,which is to be determined via the fixed-point method.(10) is followed by the solution of several ill-posed Cauchy problems with respect to the holomorphic function Φ by using a regularized power series expansion to retrieve the unknown part of the boundary curve using ΦΓ1=Γ.


### 2.2. Position of Problem

The inverse problem1−2−3 is thereby reduced to the determination of a holomorphic function Φ that is a solution of
(11)iΦΓ2=Γ,iiu^=g^onΓ1,iiiu^=0onΓ2,iv−Δu^=λ^u^inD1,withλ^=λΦ′z2,v∂∂νu^eiθ=hΦeiθΦ′eiθonΓ2.with u^∈H1D1.Using the polar coordinates of the solutionu^ of the Dirichlet problem on the disk D1 with boundary conditions and the solution of the Bessel equation of order k, see [3], we have
(12)u^nr,θ=∑k=0+∞Ak,nJkrlk,neikθ,with
(13)Ak,n2≈8λk,nπ4λk,n−1+−1k,∀k≥0,n≥1where Jk denotes the Bessel function of order k and λk,n corresponds to u^k,n.Therefore, by(14)∂∂νu^eiθ=hΦeiθΦ′eiθonΓ2,we determine a family of holomorphic functions, Φn, such that
(15)Φn′eiθ≈∑k=0+∞pk,nλk,neikθhoΦneiθ,∀n≥1,with
(16)pk,nx=2x−1n2/πx−1/21+−1k,x≥π4.p is well defined. We present the details of this algorithm, including a convergence analysis result in the following sections.


## 2.1. Conformal Mapping Method

The conformal mapping method was presented in [3].This paper is primarily aimed at presenting iterative methods to solve the inverse problem1−2−3 by using ideas from conformal mapping theory that have been developed over the last decade. For detailed references, please consult [1, 4, 5].However, in our approach, we consider a spectral problem on the simply connected domainΩ in ℝN with C2 smooth boundary ∂Ω satisfying (1).The family of holomorphic functions corresponding to the eigenvalues of the spectrum was identified via the fixed-point method and Nyström method.The proposed reconstruction of the boundary curveΓ based on available Cauchy data on the accessible boundary, Γ0, is based on a conformal map Φ:D1→Ω. Thus, we consider ΓD1, the boundary of the complex disk D1 with radius one centered at the origin, and map ΓD1 onto ∂Ω via Φ. We introduce the following notations.
(5)Γ2=eiθ,0≤θ≤απ,0<α<2,Γ1=eiθ,απ<θ≤2π,0<α<2.Further,Γ and Γ0 are parameterized as follows:
(6)Γ0=Ψθ,θ∈0,απ,Γ=Ψθ,θ∈απ,2π,where Ψ:ℝ→ℂ is a continuous differentiable 2π−periodic from ℝ onto ℂ satisfying the property that Ψ0,2π is injective. This property implies that
(7)∣Ψ′θ∣≠0,∀θ∈0,2π.We can normalize the mappingΦ by prescribing Φ1=Ψ0 and define the bijective boundary correspondence function Ψ:0,2π⟶ℂ by setting
(8)Ψθ≔Φeiθ,θ∈0,2π.It is clear that the values ofΦ on the boundary produce a unique solution Ψ of the Cauchy problem, given by
(9)Φeiθ=Ψθ.Therefore, the operatorΨθ=Φeiθ,θ∈0,2π is well defined.The Cauchy-Riemann equations give rise to nonlinear ordinary differential equations via Bessel equations for the values on this part of the boundaryΨΓ2(10)Φ′eiθ=∂u^/∂νhoΦeiθonΓ2,which is to be determined via the fixed-point method.(10) is followed by the solution of several ill-posed Cauchy problems with respect to the holomorphic function Φ by using a regularized power series expansion to retrieve the unknown part of the boundary curve using ΦΓ1=Γ.


## 2.2. Position of Problem

The inverse problem1−2−3 is thereby reduced to the determination of a holomorphic function Φ that is a solution of
(11)iΦΓ2=Γ,iiu^=g^onΓ1,iiiu^=0onΓ2,iv−Δu^=λ^u^inD1,withλ^=λΦ′z2,v∂∂νu^eiθ=hΦeiθΦ′eiθonΓ2.with u^∈H1D1.Using the polar coordinates of the solutionu^ of the Dirichlet problem on the disk D1 with boundary conditions and the solution of the Bessel equation of order k, see [3], we have
(12)u^nr,θ=∑k=0+∞Ak,nJkrlk,neikθ,with
(13)Ak,n2≈8λk,nπ4λk,n−1+−1k,∀k≥0,n≥1where Jk denotes the Bessel function of order k and λk,n corresponds to u^k,n.Therefore, by(14)∂∂νu^eiθ=hΦeiθΦ′eiθonΓ2,we determine a family of holomorphic functions, Φn, such that
(15)Φn′eiθ≈∑k=0+∞pk,nλk,neikθhoΦneiθ,∀n≥1,with
(16)pk,nx=2x−1n2/πx−1/21+−1k,x≥π4.p is well defined. We present the details of this algorithm, including a convergence analysis result in the following sections.


## 3. Results and Discussion

### 3.1. Presentation of the Fixed Point Method

A fixed point is a solution of equationTx=x where T is an operator defined from a nonempty set M onto itself. The principal condition for a nonlinear operator to admit a fixed point is for it to be a strict contraction.The determination ofΦz is based on the Banach fixed-point theorem, where a contracting nonlinear application on a complete metric space admits a fixed point [1].General methods of construction of approximate conformal maps can be found in survey works by [1, 5, 6].


#### 3.1.1. Nonlocal Differential Equation

In this section, we attempt to determinateΦ, satisfying (15), using the fixed point method.LetA:H1/20,2π×H1/20,2π⟶H−1/20,2π denote the Dirichlet-Neumann operator on ΓD1, which maps u^1,u^2 onto the normal derivative of u^∈H1D1, defined as follows:
(17)Au^1,u^2≔∂u^∂νeiθ,θ∈0,2π,with the respective boundary values given by
(18)u^eiθ=u^1onΓ1,u^eiθ=u^2onΓ2.The boundary traces onΓ1 and Γ2 are, respectively, given by
(19)u^eiθ=g^onΓ1,u^eiθ=0onΓ2,∂u^∂νeiθ=Φn′eiθhΦneiθonΓ2.However, we have the nonlocal differential equation for the conformal mapΦn, which is a critical element in the inverse algorithm.
(20)Φn′eiθ=Au^1,u^2hΦneiθ.The boundary correspondence function,Ψn, is a solution of
(21)Ψn′θ=Au1Ψnθ,u2ΨnθhΨnθ.As h<0, equations (20) and (21) are well defined.Based on the available Cauchy data,g,h, we have (12)
(22)u^nr,θ≈∑k=0+∞2π2n+k−1coskπ/2−rπ/22n+k−1eikθπ22n+k−12−21+−1k,n≥1.the fundamental solution of (11).Let us discuss some definitions and properties related to fixed points before employing the fixed-point methods to determinateΦ.


#### 3.1.2. Fixed-Point Theorems (See [1])

Theorem 1.
(Fixed-Point theorem of Banach)
Letf:S⟶S be a nonlinear operator where S is a complete metric space. We assume that f is a strict contraction. Then, f admits a unique fixed point (see [7]).Proof of Theorem 2.
Consider any pointx0∈S and iteratively define
(23)xn+1=fxnforn=0,1,⋯.
Then,(24)fxn+1−fxn≤γxn+1−xn=γfxn−fxn−1,fxn+1−fxn≤γn+1fx0−x0pourn=0,2,⋯.
Ifn≥k,, then
(25)xn−xk=fxn−1−fxk−1,≤∑j=k−1n−2fxj+1−fxj,≤fx0−x0∑j=kn−1γj.
Hencexkk=1∞ is a Cauchy sequence in S, and, as S is complete, there exists x∈S such that xk⟶x in S. Thus, it is evident that
(26)fx=x.
Hence,x is fixed point for f and the fact that f is a strict contraction ensures its uniqueness.
The following is the version of the Banach fixed-point theorem and its corollary that is used in the subsequent proofs.Theorem 3.
LetS,d be a nonempty complete metric space. If f:S⟶S is such that, for at least one integer m≥1,fm is a strict contraction, then f admits an unique fixed point.Theorem 4.
LetX,d be a complete metric space and let T:X⟶X be a δ,L-almost contraction. Then,
(27)FixT=x∈X,Tx=x≠∅.(1)
For anyx0∈X, the Picard iteration xmm=0∞, xm=Tmx0 converges to some x∗∈FixT;(2)
The following estimate holds(28)dxm+i−1,x∗≤δi1−δdxm,xm−1,m=0,1,2,⋯,i=1,2.⋯


#### 3.1.3. Fixed-Point Equation and Iteration

LetS be an operator defined as follows:
(29)Sψθ=θ+ψθ,(30)withψ:H1/20,απ⟶H1/20,απ.We assume thatψ is not identically zero.Further, we introduce an operatorT:H010,απ⟶H010,απ,0<α<2 defined by
(31)Tψθ≔α2∫0θUψ−1απ∫0απUψdθdτ,θ∈0,απ,with
(32)Uψ≔Au1oSψ,u2oSψhoSψ.By integrating (32) and considering that ψ0=0 and ψαπ=0, we obtain
(33)1απ∫0απAu1oSψ,u2oSψhoSψ=1.Using the boundary conditions ofψ, we can express ψθ=beiθ2απ−θ2−1,b∈ℂ.It is easy to verify thatT is nonlinear and contracting.Now, we use the following theorem (see [1]).Theorem 5.
Letκ,τ be a pair of Cauchy data obtained from (8).Then, in terms of the holomorphic mapΦn:D1⟶Ω and its boundary correspondence function Ψn, the function ψn=S−1Ψn is a fixed point for T.We obtain a fixed point defined by successive iteration ofT beginning with an arbitrary approximation ψn0, for all n≥1. Therefore, Theorem 3.4 induces the following iteration:
(34)ψnm+1=Tψnm,m=0,1,2,⋯.The solution of the Cauchy problem is required to identify the boundary functionΨn from the function ψn. For this purpose, let us express Sψn using the following expansion:
(35)Sψnθ=∑k=0+∞an,keikθ,θ∈0,απ.UsingΦneiθ=Sψnθ, we obtain Φn using the following expansion:
(36)Φnz=∑k=0+∞an,keikθ,θ∈0,απ.The following convergence result justifies the procedure of iteration (31).Theorem 6 (see [1]).
LetD1 be the unit disk and g be a continuous positive function defined on it. Then, successive approximations
(37)ψnm+1=Tψnm,m=0,1,2,⋯,converge provided that the initial estimate Ψn0 of the boundary function is sufficiently close to the exact function ψn.Proof (see [4]).
Theorem6 is proved by first establishing that T′0≤1 in the case where Ω=D1. Thereafter, a continuity argument is applied to extend the estimate to the norm of T′ψn for ψn in a neighborhood of the fixed point defined in Theorem 3.4.Proof of Theorem 7.
We calculate the derivate of the nonlinear operatorT. We begin by estimating T′0. We have
(38)T′0=0.
Therefore, we have,T′0<1,∀0<α<2.
Using a continuity argument to estimate the norm ofT′ψnθ near the fixed point of Theorem 3.4, we have for all θ∈0,απ,(39)T′ψnθ=α2ψn′θ.
Thus,T′ψnθ<ψn′θ,∀θ∈0,απ and n≥1.
The approximation ofT′ψnθ near the fixed point of the theorem is ψn′θ,n≥1.
We now determinate a numerical approximation of the conformal mapΦn,k, for all n≥1.


#### 3.1.4. Numerical Approximation of the Conformal Map

In this section, we use an ordinary differential equation of the boundary correspondence function to obtain an approximation of the conformal map,Φn, for all n≥1.Let us introduce two boundary correspondence functions,Ψn2 and Ψn1, defined from 0,2π to ℂby(40)Ψn1θ≔Φneiθpourθ∈0,απ,Ψn2θ≔Φneiθpourθ∈απ,2π.OnΓ2, equation (21) becomes
(41)Ψn1′θ=∂u^/∂rhΨn1θ.Here, we use the length of arcΓ2 determined by dθ.By removing the identity,(42)1απ∫0απ∂u^/∂rhΨn1θdθ=1,that is obtained by integrating (41). Then, we have for ψn1θ≔Ψn1θ−θ, the equation
(43)ψn1′θ=Uψn1θ−1απ∫0απUψn1dθ.where we have for convenience
(44)Uψn1=∂u^/∂rhoSψn1,with S operator defined in (35).For(45)ψn1θ≔∫0θUψn1−1απ∫0απUψn1dθdτ,θ∈0,απ,the iterative solution of (43), we have the following convergence results:Theorem 8.
The successive approximations defined by (45) converge from the initial approximation ψn01=θαπ−θ provided that Ω is close enough to D1.
When we obtainψn1, that is, Φn, the boundary values of Φn on Γ2 to be used for the iteration of (45) are given by
(46)Φneiθ=∑k=0+∞an,keikθ,with
(47)2an,kπ=∫02πΦneiθe−ikθdθ,2an,kiπ=∫ΓD1z¯k+1Φnzdz,
So, onΓ2,
(48)an,kαπ=∫Γ2Φneiθe−ikθdθ.
We can express the expansion ofΦn in the form
(49)Φnz=∑k=0+∞an,kzk,which is numerically stable for ∣z∣<1. By construction, considering (40), we have Φn1=Ψn10 and Φneiαπ=Ψn1απ.
Now, we describe the numerical algorithm that we will implement in this paper. We use a method involving integral equations onΓ2. The integral equation obtained is resolved via the Nyström method.
We give ourselves a regular mesh,(50)θm=θ0+mh,1≤m≤N,h=απNon0,απ.
The approximationsΨnm1 of Ψn1θm at the nodes of mesh are calculated using the q previous approximate values ym−1,ym−2,⋯,ym−q.
The iterations allow approximations of the normal derivatives via trigonometric interpolation.
The integration interpolation polynomial on0,θm is used to associate Ψnm1 to Ψnm+11, via (43), with m=1,⋯,N.
Beginning with identity, we have(51)ψn1θ+t=ψn1θ+∫θθ+tUψn1τ−1απ∫0απUψn1dξdτ,which corresponds to
(52)ψnm+11=ψnm1+∫θmθm+1Uψn1τ−1dτ,or more precisely
(53)ψnm+11=ψnm1+∫θmθm+1∑k=0+∞pk,nλk,neikτhoSψn1τ−1dτ.
We substitute∑k=0+∞pk,nλk,neikτ/hoSψn1τ−1 in the integral via the Nyström method using the interpolation polynomial Pθ.


#### 3.1.5. Nyström Method (See [8])

This method, called the quadrature method, is the application of numerical methods of calculating integrals to obtain a linear system. We write(54)ψn1θm+1=ψn1θm−1+∫θm−1θm+1Pθdθ,where P denotes the interpolation polynomial of Lagrange of degree q−1, for the pair of values
(55)θm−j,fm−j,0≤j≤q−1,where we use
(56)Uψnj1=Uψn1θj,fm−j=fθm−j=∑k=0+∞pk,nλk,neikθm−jhoSψn1θm−j−1,Pθ=∑j=0q−1fθm−jlm−jθ,with the polynomials lm−jθ are Lagrange polynomials of ℝnX associated to θm,θm−1,θm−2,⋯,θm−q+1.That is,
(57)ψnm+11=ψnm−11+απN∑j=0q−1γq,jfm−j,N∈ℕ∗.Explicitly, we can express (57) in the form
(58)ψnm+11=ψnm−11+απN∑j=0q−1γq,j∑k=0+∞pk,nλk,neikθm−jhoSψn1θm−j−1,N∈ℕ∗.Therefore, we can setq=2,3,.⋯ for the approximations. Hence,
(59)Ifq=2thenψnm+11=ψnm−11+2απN∑k=0+∞pk,nλk,neikθmhoSψn1θm−1N∈ℕ∗.(60)Ifq=3then(61)ψnm+11=ψnm−11+απ3N7∑k=0+∞pk,nλk,neikθmhoSψn1θm−2∑k=0+∞pk,nλk,neikθm−1hoSψn1θm−1+∑k=0+∞pk,nλk,neikθm−2hoSψn1θm−2−6.Ifq=4, then
(62)ψnm+11=ψnm−11+απ3N8∑k=0+∞pk,nλk,neikθmhoSψn1θm−5∑k=0+∞pk,nλk,neikθm−1hoSψn1θm−1+4∑k=0+∞pk,nλk,neikθm−2hoSψn1θm−2−∑k=0+∞pk,nλk,neikθm−3hoSψn1θm−3−6.After fixingψn01, we calculate ψn11 using equation (45), for all q.The estimate ofψn1 allows us to calculate an,k,k=0,1,⋯,J using (29), (35), and (48) and, subsequently, the expression of ΦnJ1,J∈ℕ.We use(63)an,k=αn,k+iβn,kwith0≤k≤J.The traces of the curves corresponding to the equations defined below provide an approximate domain.(64)RΦnJ1=∑k=0Jαn,kcoskθ−βn,ksinkθ,IΦnJ1=∑k=0Jαn,ksinkθ+βn,kcoskθ.We attempt to determineψn1 for q=2. The approach for the other case is similar.We know that (59) is equivalent to
(65)ψnm+11=ψnm−11+2απNψnm1′,N∈ℕ∗,forq=2.After calculation, we have(66)ψnm+11=ψn01+2απN∑j=01/2m−1ψn2j+11′,or
(67)ψnm+11=ψn01+4πN∑j=01/2m−1Tψn2j+11′,N∈ℕ∗.


#### 3.1.6. Convergence Results ofψnm1m

We setUm+1=ψnm+11, as defined above.Forq=2, we have Umm≥0 a sequence defined in 0,απ,0<α<2, by
(68)Um+1=Um−1+4πNTUm′,N∈ℕ∗.The Picard theorem ensures the convergence ofUmm≥0.We can write(69)Um+1=U0+4πNTU1+TU3+TU5+⋯+TUm′,=U0+4πNTU11−α/24m−41−α/22′,N∈ℕ∗.Thus, we have(70)Umm≥0convergestoU0+4πα2U0′N4−α2forallθ∈0,απ,N∈ℕ∗.Finally, we can deduce the following result:Proposition 9.
LetT:H01a,b→H01a,b be a k-contraction and ψnn≥0 be a sequence of functions defined by
(71)ψn+1≔Tψnsothat,Tψn′≔kψn′,n≥0.
Ifψnn≥0 verifies this recurrence relation
(72)ψn+1=ψn−1+εTψn′,withε>0andn∈ℕ.
Then,(73)ψnn≥0convergestoψ0+Ck,εψ0′.C a constant depending on k and ε.Proof of Proposition 10.
The Picard theorem on the convergence ofψnn≥0.
We can write(74)ψn+1=ψ0+εTψ1+Tψ3+Tψ5+⋯+Tψn′n∈ℕ.
So,(75)ψn+1=ψ0+k2εψ01−k4n−41−k2′,n∈ℕ.
This yields the result.
Further, we setU0=ceiθ2απ−θ2−1,c∈ℂ. We have
(76)ψ1θ=ceiθ2απ−θ21+4πα2θαπ−θαπ−2θiN4−α2−c,withc∈ℂandN∈ℕ∗.
It is evident thatψ1 does not depend on n.
Therefore, according to (20) and (21), the map Φ1 is then defined by
(77)Φ1eiθ=θ+ceiθ2απ−θ21+4πα2θαπ−θαπ−2θiN4−α2−c,forallθ∈0;απ.with c∈ℂandN∈ℕ∗. The coefficient a is determined using the available Cauchy data and eigenvalues.
For allz0∈D1, Φ1′z0≠0 so Φ1 is holomorphic.
In the following argument, we poseΦ=Φn,k1,c=an,k because they are determined by eigenvalues.
Φn,k1 is analytic, continuous on D¯1; it realizes a conformal mapping from C1 onto ∂Ω, according to Theorem 1.4.5 (see [2] page 42). Further, it realizes a conformal mapping (univalent) from the domain D onto the domain Ω.
Hence, for all0<α<2, and all N∈ℕ∗, there exists an univalent holomorphic function Φn,k1, which realizes a conformal mapping from the domain D1 onto the domain Ω, which is a solution of problem (19), defined by
(78)Φn,k1Reiθ=θ+an,kReiθ2απ−θ21+4πα2θαπ−θαπ−2θiN4−α2−an,k.
For allθ∈0,απ with N∈ℕ∗,an,k,∈ℂ,0≤R≤1.an,k is determined using the available Cauchy data and eigenvalues.Proposition 11.
Consider the problem1−2−3 where
(79)∂Ω=Γ0òΓ,withΓtheunknownpart.∂Ω is a spiral defined on 0,2π by the following equations:
(80)X=θ−a1+a1−a2h1θcost−a1h1θ+a2sint,Y=−a2+a1h1θ+a2cost+a1−a2h1θsint,withan,k∈ℂ,a1=∣an,k∣cosargan,k,a2=∣an,k∣sinargan,k,h1θ=4πα2θαπ−θαπ−2θN4−α2andt=θ2απ−θ2.
Then,Γ is a part of the spiral, ∂Ω, image of 0,απ under
(81)Φn,k1Reiθ=θ+an,kReiθ2απ−θ21+4πα2θαπ−θαπ−2θiN4−α2−an,kdefined for all n≥1 and k≥0.Proof of the Proposition 12.
It suffices to determine the integral and decimal parts ofΦn,k1.


#### 3.1.7. Numerical Simulations

We determine the approximate shape of the unknown part of the boundary corresponding to the two first eigenvalues.LetΦn,k1 be the map defined in (78).We putgx,y=0 and hx,y=−1−x2−y2, where h is defined to be negative.First, we determinean,k of Φn,k1 corresponding to the first and second eigenvalues, after approximating the corresponding shapes of Γ2. Using the boundary values, we have previously established,
(82)Φn′eiθ≈∑k=0+∞pk,nλk,neikθhoΦneiθ,∀n≥1,uΦk,neiθ=0,foru^definedin16.The coefficientsa1 and a2 are determined based on the Cauchy data corresponding to each eigenvalue of the Laplacian operator. Hence, they are obtained by
(83)Φ1′eiθ≈∑k=0+∞2n+k−12π−1n2π/2n+k−12π2−21+−1keikθhoΦneiθ.Therefore, the calculation yields that corresponding toθ=0:
(84)a1=0,a2=N4−α24α4π31+∑k=0+∞2n+k−12π−1n2π2n+k−12π2−21+−1k.


#### 3.1.8. Conformal MapΦ0,1 Corresponding to First Eigenvalue λ0,1

Corresponding to the first eigenvalueλ0,1, we have
(85)a2=N4−α24α4π31−π2ππ2−4,(86)X=θ−N4−α24α4π31−π2ππ2−44πα2θαπ−θαπ−2θN4−α2cost+sint,Y=N4−α24α4π31−π2ππ2−4−1+cost−4πα2θαπ−θαπ−2θN4−α2sint.Therefore, we obtain the result for(1)
N=5 and α=0.5Figure1 is obtained by applying N=5 and α=0.5 in (86). Figure 1(a) shows the shapes of Γ and Γ0 in an orthogonal mark with graphic unit 20 cm, and Figure 1(b) shows these same shapes in an orthogonal mark with graphic unit 5 cm. It is this right part which gives us the enlargement of shapes of Γ and Γ0.
(2)
N=5 and α=1Figure 1
N5 Alpha0.5.
(a)(b)Figure2 is obtained by applying N=5 and α=1 in (86). Figure 2(a) shows the shapes of Γ and Γ0 in an orthogonal mark with graphic unit 20 cm, and Figure 2(b) shows these same shapes in an orthogonal mark with graphic unit 5 cm. It is this right part which gives us also the enlargement of shapes of Γ and Γ0.Figure 2
N5 Alpha1.
(a)(b)It is to be noted that, in each case (α=0.5 or α=1), Γ0 is sinusoïdal, Γ is a part of spiral, and ∂Ω converges to a finite limit. Therefore, ∂Ω is asymptotic to a circle and Ω is disk shaped.


#### 3.1.9. Conformal MapΦ1,1 Corresponding to the Second Eigenvalue λ1,1

Corresponding to the second eigenvalueλ1,1, we have
(87)a2=N4−α24α4π31−π2ππ2−4−22π,(88)X=θ−N4−α24α4π31−π2ππ2−4−22π4πα2θαπ−θαπ−2θN4−α2cost+sint,Y=N4−α24α4π31−π2ππ2−4−22π−1+cost−4πα2θαπ−θαπ−2θN4−α2sint.Thus, we conclude the result forN=5 and α=1.Figure3 is obtained by applying N=5 and α=1 in (87). Here too, Figure 3(a) shows the shapes of Γ and Γ0 in an orthogonal mark with graphic unit 20 cm, and Figure 3(b) shows these same shapes in an orthogonal mark with graphic unit 5 cm. Figure 3(a) gives us also the enlargement of shapes of Γ and Γ0.Figure 3
N5 Alpha1.
(a)(b)Here too,Γ0 is sinusoïdal, Γ is a part of spiral, and ∂Ω converges to a finite limit.


#### 3.1.10. Comparison Results forN=5 and α=1

The result for the first eigenvalue obtained via the minimization functional method forN=5 and α=1 in our previous study, and the result obtained using the fixed-point method for the same data has been depicted in the following diagram.In Figure4, we can observe that
(i)
Γ0 is sinusoïdal in both these studies(ii)
Γ is a part of spiral in the present study and is a part of spiral with a cusp, in the study in 2014(iii)
Both methods yielded a spiral shape for∂ΩFigure 4
Results forΓ of the previous study in 2014 black in (a). Results for Γ of the present study green to (b).
(a)(b)


#### 3.1.11. Variations ofΓ0 with respect to λn,k

The following diagram allows us to compare the shapes ofΓ0 corresponding the results in the first and the second study for the first eigenvalues.In Figure5, we obtain results of the variations of the shape of Γ0 with respect to λn,k. Figure 5(a) shows the variation of the shapes of Γ0 by applying N=5 and α=0.5 in (86) and in (87), i.e., for the first and second eigenvalues. Figure 5(b) shows the variation of the shapes of Γ0 by applying N=5 and α=1 in (86) and in (87), i.e., for the first and the second eigenvalue.Figure 5
Variations ofΓ0 with respect to λn,k.
(a)(b)


### 3.2. Results

All previous figures are obtained using “GeoGebra Geometry.”Figures1(b), 2(b), and 3(b) give us the enlargement of shapes of Γ and Γ0 found, and they allow us to see their exact shapes. Also, these exact shapes allow us to make a good interpretation of the results.So, the simulations yield the following results:(1)
The shape ofΓ0 is a spiral. When the value of α increases, its spiral shape is accentuated, and it is constituted of identical connected parts. Further, when the eigenvalue increases, the length of Γ0 is observed to increase accompanied by a change in its spiral shape (see Figure 5).(2)
However, the shape ofΓ is a spiral corresponding to all variations of α and the eigenvalues. Its shape does not vary and it converges to a finite limit, which is circular in shape.(3)
Figure4(a) shows the results of the previous study, which was conducted in 2014, for N=5 and α=1, and Figure 4(b) shows the results of the present study for N=5 and α=1. We can observe that the shape of ∂Ω is spiral in both the results; however, the present study provides a more precise shape of the spiral that is asymptotic to a circle. This shape can justify that Ω is a disk.Therefore, the result obtained using the fixed-point method is more precise and further identifiesΩ as a disk.


## 3.1. Presentation of the Fixed Point Method

A fixed point is a solution of equationTx=x where T is an operator defined from a nonempty set M onto itself. The principal condition for a nonlinear operator to admit a fixed point is for it to be a strict contraction.The determination ofΦz is based on the Banach fixed-point theorem, where a contracting nonlinear application on a complete metric space admits a fixed point [1].General methods of construction of approximate conformal maps can be found in survey works by [1, 5, 6].


### 3.1.1. Nonlocal Differential Equation

In this section, we attempt to determinateΦ, satisfying (15), using the fixed point method.LetA:H1/20,2π×H1/20,2π⟶H−1/20,2π denote the Dirichlet-Neumann operator on ΓD1, which maps u^1,u^2 onto the normal derivative of u^∈H1D1, defined as follows:
(17)Au^1,u^2≔∂u^∂νeiθ,θ∈0,2π,with the respective boundary values given by
(18)u^eiθ=u^1onΓ1,u^eiθ=u^2onΓ2.The boundary traces onΓ1 and Γ2 are, respectively, given by
(19)u^eiθ=g^onΓ1,u^eiθ=0onΓ2,∂u^∂νeiθ=Φn′eiθhΦneiθonΓ2.However, we have the nonlocal differential equation for the conformal mapΦn, which is a critical element in the inverse algorithm.
(20)Φn′eiθ=Au^1,u^2hΦneiθ.The boundary correspondence function,Ψn, is a solution of
(21)Ψn′θ=Au1Ψnθ,u2ΨnθhΨnθ.As h<0, equations (20) and (21) are well defined.Based on the available Cauchy data,g,h, we have (12)
(22)u^nr,θ≈∑k=0+∞2π2n+k−1coskπ/2−rπ/22n+k−1eikθπ22n+k−12−21+−1k,n≥1.the fundamental solution of (11).Let us discuss some definitions and properties related to fixed points before employing the fixed-point methods to determinateΦ.


### 3.1.2. Fixed-Point Theorems (See [1])

Theorem 1.
(Fixed-Point theorem of Banach)
Letf:S⟶S be a nonlinear operator where S is a complete metric space. We assume that f is a strict contraction. Then, f admits a unique fixed point (see [7]).Proof of Theorem 2.
Consider any pointx0∈S and iteratively define
(23)xn+1=fxnforn=0,1,⋯.
Then,(24)fxn+1−fxn≤γxn+1−xn=γfxn−fxn−1,fxn+1−fxn≤γn+1fx0−x0pourn=0,2,⋯.
Ifn≥k,, then
(25)xn−xk=fxn−1−fxk−1,≤∑j=k−1n−2fxj+1−fxj,≤fx0−x0∑j=kn−1γj.
Hencexkk=1∞ is a Cauchy sequence in S, and, as S is complete, there exists x∈S such that xk⟶x in S. Thus, it is evident that
(26)fx=x.
Hence,x is fixed point for f and the fact that f is a strict contraction ensures its uniqueness.
The following is the version of the Banach fixed-point theorem and its corollary that is used in the subsequent proofs.Theorem 3.
LetS,d be a nonempty complete metric space. If f:S⟶S is such that, for at least one integer m≥1,fm is a strict contraction, then f admits an unique fixed point.Theorem 4.
LetX,d be a complete metric space and let T:X⟶X be a δ,L-almost contraction. Then,
(27)FixT=x∈X,Tx=x≠∅.(1)
For anyx0∈X, the Picard iteration xmm=0∞, xm=Tmx0 converges to some x∗∈FixT;(2)
The following estimate holds(28)dxm+i−1,x∗≤δi1−δdxm,xm−1,m=0,1,2,⋯,i=1,2.⋯


### 3.1.3. Fixed-Point Equation and Iteration

LetS be an operator defined as follows:
(29)Sψθ=θ+ψθ,(30)withψ:H1/20,απ⟶H1/20,απ.We assume thatψ is not identically zero.Further, we introduce an operatorT:H010,απ⟶H010,απ,0<α<2 defined by
(31)Tψθ≔α2∫0θUψ−1απ∫0απUψdθdτ,θ∈0,απ,with
(32)Uψ≔Au1oSψ,u2oSψhoSψ.By integrating (32) and considering that ψ0=0 and ψαπ=0, we obtain
(33)1απ∫0απAu1oSψ,u2oSψhoSψ=1.Using the boundary conditions ofψ, we can express ψθ=beiθ2απ−θ2−1,b∈ℂ.It is easy to verify thatT is nonlinear and contracting.Now, we use the following theorem (see [1]).Theorem 5.
Letκ,τ be a pair of Cauchy data obtained from (8).Then, in terms of the holomorphic mapΦn:D1⟶Ω and its boundary correspondence function Ψn, the function ψn=S−1Ψn is a fixed point for T.We obtain a fixed point defined by successive iteration ofT beginning with an arbitrary approximation ψn0, for all n≥1. Therefore, Theorem 3.4 induces the following iteration:
(34)ψnm+1=Tψnm,m=0,1,2,⋯.The solution of the Cauchy problem is required to identify the boundary functionΨn from the function ψn. For this purpose, let us express Sψn using the following expansion:
(35)Sψnθ=∑k=0+∞an,keikθ,θ∈0,απ.UsingΦneiθ=Sψnθ, we obtain Φn using the following expansion:
(36)Φnz=∑k=0+∞an,keikθ,θ∈0,απ.The following convergence result justifies the procedure of iteration (31).Theorem 6 (see [1]).
LetD1 be the unit disk and g be a continuous positive function defined on it. Then, successive approximations
(37)ψnm+1=Tψnm,m=0,1,2,⋯,converge provided that the initial estimate Ψn0 of the boundary function is sufficiently close to the exact function ψn.Proof (see [4]).
Theorem6 is proved by first establishing that T′0≤1 in the case where Ω=D1. Thereafter, a continuity argument is applied to extend the estimate to the norm of T′ψn for ψn in a neighborhood of the fixed point defined in Theorem 3.4.Proof of Theorem 7.
We calculate the derivate of the nonlinear operatorT. We begin by estimating T′0. We have
(38)T′0=0.
Therefore, we have,T′0<1,∀0<α<2.
Using a continuity argument to estimate the norm ofT′ψnθ near the fixed point of Theorem 3.4, we have for all θ∈0,απ,(39)T′ψnθ=α2ψn′θ.
Thus,T′ψnθ<ψn′θ,∀θ∈0,απ and n≥1.
The approximation ofT′ψnθ near the fixed point of the theorem is ψn′θ,n≥1.
We now determinate a numerical approximation of the conformal mapΦn,k, for all n≥1.


### 3.1.4. Numerical Approximation of the Conformal Map

In this section, we use an ordinary differential equation of the boundary correspondence function to obtain an approximation of the conformal map,Φn, for all n≥1.Let us introduce two boundary correspondence functions,Ψn2 and Ψn1, defined from 0,2π to ℂby(40)Ψn1θ≔Φneiθpourθ∈0,απ,Ψn2θ≔Φneiθpourθ∈απ,2π.OnΓ2, equation (21) becomes
(41)Ψn1′θ=∂u^/∂rhΨn1θ.Here, we use the length of arcΓ2 determined by dθ.By removing the identity,(42)1απ∫0απ∂u^/∂rhΨn1θdθ=1,that is obtained by integrating (41). Then, we have for ψn1θ≔Ψn1θ−θ, the equation
(43)ψn1′θ=Uψn1θ−1απ∫0απUψn1dθ.where we have for convenience
(44)Uψn1=∂u^/∂rhoSψn1,with S operator defined in (35).For(45)ψn1θ≔∫0θUψn1−1απ∫0απUψn1dθdτ,θ∈0,απ,the iterative solution of (43), we have the following convergence results:Theorem 8.
The successive approximations defined by (45) converge from the initial approximation ψn01=θαπ−θ provided that Ω is close enough to D1.
When we obtainψn1, that is, Φn, the boundary values of Φn on Γ2 to be used for the iteration of (45) are given by
(46)Φneiθ=∑k=0+∞an,keikθ,with
(47)2an,kπ=∫02πΦneiθe−ikθdθ,2an,kiπ=∫ΓD1z¯k+1Φnzdz,
So, onΓ2,
(48)an,kαπ=∫Γ2Φneiθe−ikθdθ.
We can express the expansion ofΦn in the form
(49)Φnz=∑k=0+∞an,kzk,which is numerically stable for ∣z∣<1. By construction, considering (40), we have Φn1=Ψn10 and Φneiαπ=Ψn1απ.
Now, we describe the numerical algorithm that we will implement in this paper. We use a method involving integral equations onΓ2. The integral equation obtained is resolved via the Nyström method.
We give ourselves a regular mesh,(50)θm=θ0+mh,1≤m≤N,h=απNon0,απ.
The approximationsΨnm1 of Ψn1θm at the nodes of mesh are calculated using the q previous approximate values ym−1,ym−2,⋯,ym−q.
The iterations allow approximations of the normal derivatives via trigonometric interpolation.
The integration interpolation polynomial on0,θm is used to associate Ψnm1 to Ψnm+11, via (43), with m=1,⋯,N.
Beginning with identity, we have(51)ψn1θ+t=ψn1θ+∫θθ+tUψn1τ−1απ∫0απUψn1dξdτ,which corresponds to
(52)ψnm+11=ψnm1+∫θmθm+1Uψn1τ−1dτ,or more precisely
(53)ψnm+11=ψnm1+∫θmθm+1∑k=0+∞pk,nλk,neikτhoSψn1τ−1dτ.
We substitute∑k=0+∞pk,nλk,neikτ/hoSψn1τ−1 in the integral via the Nyström method using the interpolation polynomial Pθ.


### 3.1.5. Nyström Method (See [8])

This method, called the quadrature method, is the application of numerical methods of calculating integrals to obtain a linear system. We write(54)ψn1θm+1=ψn1θm−1+∫θm−1θm+1Pθdθ,where P denotes the interpolation polynomial of Lagrange of degree q−1, for the pair of values
(55)θm−j,fm−j,0≤j≤q−1,where we use
(56)Uψnj1=Uψn1θj,fm−j=fθm−j=∑k=0+∞pk,nλk,neikθm−jhoSψn1θm−j−1,Pθ=∑j=0q−1fθm−jlm−jθ,with the polynomials lm−jθ are Lagrange polynomials of ℝnX associated to θm,θm−1,θm−2,⋯,θm−q+1.That is,
(57)ψnm+11=ψnm−11+απN∑j=0q−1γq,jfm−j,N∈ℕ∗.Explicitly, we can express (57) in the form
(58)ψnm+11=ψnm−11+απN∑j=0q−1γq,j∑k=0+∞pk,nλk,neikθm−jhoSψn1θm−j−1,N∈ℕ∗.Therefore, we can setq=2,3,.⋯ for the approximations. Hence,
(59)Ifq=2thenψnm+11=ψnm−11+2απN∑k=0+∞pk,nλk,neikθmhoSψn1θm−1N∈ℕ∗.(60)Ifq=3then(61)ψnm+11=ψnm−11+απ3N7∑k=0+∞pk,nλk,neikθmhoSψn1θm−2∑k=0+∞pk,nλk,neikθm−1hoSψn1θm−1+∑k=0+∞pk,nλk,neikθm−2hoSψn1θm−2−6.Ifq=4, then
(62)ψnm+11=ψnm−11+απ3N8∑k=0+∞pk,nλk,neikθmhoSψn1θm−5∑k=0+∞pk,nλk,neikθm−1hoSψn1θm−1+4∑k=0+∞pk,nλk,neikθm−2hoSψn1θm−2−∑k=0+∞pk,nλk,neikθm−3hoSψn1θm−3−6.After fixingψn01, we calculate ψn11 using equation (45), for all q.The estimate ofψn1 allows us to calculate an,k,k=0,1,⋯,J using (29), (35), and (48) and, subsequently, the expression of ΦnJ1,J∈ℕ.We use(63)an,k=αn,k+iβn,kwith0≤k≤J.The traces of the curves corresponding to the equations defined below provide an approximate domain.(64)RΦnJ1=∑k=0Jαn,kcoskθ−βn,ksinkθ,IΦnJ1=∑k=0Jαn,ksinkθ+βn,kcoskθ.We attempt to determineψn1 for q=2. The approach for the other case is similar.We know that (59) is equivalent to
(65)ψnm+11=ψnm−11+2απNψnm1′,N∈ℕ∗,forq=2.After calculation, we have(66)ψnm+11=ψn01+2απN∑j=01/2m−1ψn2j+11′,or
(67)ψnm+11=ψn01+4πN∑j=01/2m−1Tψn2j+11′,N∈ℕ∗.


### 3.1.6. Convergence Results ofψnm1m

We setUm+1=ψnm+11, as defined above.Forq=2, we have Umm≥0 a sequence defined in 0,απ,0<α<2, by
(68)Um+1=Um−1+4πNTUm′,N∈ℕ∗.The Picard theorem ensures the convergence ofUmm≥0.We can write(69)Um+1=U0+4πNTU1+TU3+TU5+⋯+TUm′,=U0+4πNTU11−α/24m−41−α/22′,N∈ℕ∗.Thus, we have(70)Umm≥0convergestoU0+4πα2U0′N4−α2forallθ∈0,απ,N∈ℕ∗.Finally, we can deduce the following result:Proposition 9.
LetT:H01a,b→H01a,b be a k-contraction and ψnn≥0 be a sequence of functions defined by
(71)ψn+1≔Tψnsothat,Tψn′≔kψn′,n≥0.
Ifψnn≥0 verifies this recurrence relation
(72)ψn+1=ψn−1+εTψn′,withε>0andn∈ℕ.
Then,(73)ψnn≥0convergestoψ0+Ck,εψ0′.C a constant depending on k and ε.Proof of Proposition 10.
The Picard theorem on the convergence ofψnn≥0.
We can write(74)ψn+1=ψ0+εTψ1+Tψ3+Tψ5+⋯+Tψn′n∈ℕ.
So,(75)ψn+1=ψ0+k2εψ01−k4n−41−k2′,n∈ℕ.
This yields the result.
Further, we setU0=ceiθ2απ−θ2−1,c∈ℂ. We have
(76)ψ1θ=ceiθ2απ−θ21+4πα2θαπ−θαπ−2θiN4−α2−c,withc∈ℂandN∈ℕ∗.
It is evident thatψ1 does not depend on n.
Therefore, according to (20) and (21), the map Φ1 is then defined by
(77)Φ1eiθ=θ+ceiθ2απ−θ21+4πα2θαπ−θαπ−2θiN4−α2−c,forallθ∈0;απ.with c∈ℂandN∈ℕ∗. The coefficient a is determined using the available Cauchy data and eigenvalues.
For allz0∈D1, Φ1′z0≠0 so Φ1 is holomorphic.
In the following argument, we poseΦ=Φn,k1,c=an,k because they are determined by eigenvalues.
Φn,k1 is analytic, continuous on D¯1; it realizes a conformal mapping from C1 onto ∂Ω, according to Theorem 1.4.5 (see [2] page 42). Further, it realizes a conformal mapping (univalent) from the domain D onto the domain Ω.
Hence, for all0<α<2, and all N∈ℕ∗, there exists an univalent holomorphic function Φn,k1, which realizes a conformal mapping from the domain D1 onto the domain Ω, which is a solution of problem (19), defined by
(78)Φn,k1Reiθ=θ+an,kReiθ2απ−θ21+4πα2θαπ−θαπ−2θiN4−α2−an,k.
For allθ∈0,απ with N∈ℕ∗,an,k,∈ℂ,0≤R≤1.an,k is determined using the available Cauchy data and eigenvalues.Proposition 11.
Consider the problem1−2−3 where
(79)∂Ω=Γ0òΓ,withΓtheunknownpart.∂Ω is a spiral defined on 0,2π by the following equations:
(80)X=θ−a1+a1−a2h1θcost−a1h1θ+a2sint,Y=−a2+a1h1θ+a2cost+a1−a2h1θsint,withan,k∈ℂ,a1=∣an,k∣cosargan,k,a2=∣an,k∣sinargan,k,h1θ=4πα2θαπ−θαπ−2θN4−α2andt=θ2απ−θ2.
Then,Γ is a part of the spiral, ∂Ω, image of 0,απ under
(81)Φn,k1Reiθ=θ+an,kReiθ2απ−θ21+4πα2θαπ−θαπ−2θiN4−α2−an,kdefined for all n≥1 and k≥0.Proof of the Proposition 12.
It suffices to determine the integral and decimal parts ofΦn,k1.


### 3.1.7. Numerical Simulations

We determine the approximate shape of the unknown part of the boundary corresponding to the two first eigenvalues.LetΦn,k1 be the map defined in (78).We putgx,y=0 and hx,y=−1−x2−y2, where h is defined to be negative.First, we determinean,k of Φn,k1 corresponding to the first and second eigenvalues, after approximating the corresponding shapes of Γ2. Using the boundary values, we have previously established,
(82)Φn′eiθ≈∑k=0+∞pk,nλk,neikθhoΦneiθ,∀n≥1,uΦk,neiθ=0,foru^definedin16.The coefficientsa1 and a2 are determined based on the Cauchy data corresponding to each eigenvalue of the Laplacian operator. Hence, they are obtained by
(83)Φ1′eiθ≈∑k=0+∞2n+k−12π−1n2π/2n+k−12π2−21+−1keikθhoΦneiθ.Therefore, the calculation yields that corresponding toθ=0:
(84)a1=0,a2=N4−α24α4π31+∑k=0+∞2n+k−12π−1n2π2n+k−12π2−21+−1k.


### 3.1.8. Conformal MapΦ0,1 Corresponding to First Eigenvalue λ0,1

Corresponding to the first eigenvalueλ0,1, we have
(85)a2=N4−α24α4π31−π2ππ2−4,(86)X=θ−N4−α24α4π31−π2ππ2−44πα2θαπ−θαπ−2θN4−α2cost+sint,Y=N4−α24α4π31−π2ππ2−4−1+cost−4πα2θαπ−θαπ−2θN4−α2sint.Therefore, we obtain the result for(1)
N=5 and α=0.5Figure1 is obtained by applying N=5 and α=0.5 in (86). Figure 1(a) shows the shapes of Γ and Γ0 in an orthogonal mark with graphic unit 20 cm, and Figure 1(b) shows these same shapes in an orthogonal mark with graphic unit 5 cm. It is this right part which gives us the enlargement of shapes of Γ and Γ0.
(2)
N=5 and α=1Figure 1
N5 Alpha0.5.
(a)(b)Figure2 is obtained by applying N=5 and α=1 in (86). Figure 2(a) shows the shapes of Γ and Γ0 in an orthogonal mark with graphic unit 20 cm, and Figure 2(b) shows these same shapes in an orthogonal mark with graphic unit 5 cm. It is this right part which gives us also the enlargement of shapes of Γ and Γ0.Figure 2
N5 Alpha1.
(a)(b)It is to be noted that, in each case (α=0.5 or α=1), Γ0 is sinusoïdal, Γ is a part of spiral, and ∂Ω converges to a finite limit. Therefore, ∂Ω is asymptotic to a circle and Ω is disk shaped.


### 3.1.9. Conformal MapΦ1,1 Corresponding to the Second Eigenvalue λ1,1

Corresponding to the second eigenvalueλ1,1, we have
(87)a2=N4−α24α4π31−π2ππ2−4−22π,(88)X=θ−N4−α24α4π31−π2ππ2−4−22π4πα2θαπ−θαπ−2θN4−α2cost+sint,Y=N4−α24α4π31−π2ππ2−4−22π−1+cost−4πα2θαπ−θαπ−2θN4−α2sint.Thus, we conclude the result forN=5 and α=1.Figure3 is obtained by applying N=5 and α=1 in (87). Here too, Figure 3(a) shows the shapes of Γ and Γ0 in an orthogonal mark with graphic unit 20 cm, and Figure 3(b) shows these same shapes in an orthogonal mark with graphic unit 5 cm. Figure 3(a) gives us also the enlargement of shapes of Γ and Γ0.Figure 3
N5 Alpha1.
(a)(b)Here too,Γ0 is sinusoïdal, Γ is a part of spiral, and ∂Ω converges to a finite limit.


### 3.1.10. Comparison Results forN=5 and α=1

The result for the first eigenvalue obtained via the minimization functional method forN=5 and α=1 in our previous study, and the result obtained using the fixed-point method for the same data has been depicted in the following diagram.In Figure4, we can observe that
(i)
Γ0 is sinusoïdal in both these studies(ii)
Γ is a part of spiral in the present study and is a part of spiral with a cusp, in the study in 2014(iii)
Both methods yielded a spiral shape for∂ΩFigure 4
Results forΓ of the previous study in 2014 black in (a). Results for Γ of the present study green to (b).
(a)(b)


### 3.1.11. Variations ofΓ0 with respect to λn,k

The following diagram allows us to compare the shapes ofΓ0 corresponding the results in the first and the second study for the first eigenvalues.In Figure5, we obtain results of the variations of the shape of Γ0 with respect to λn,k. Figure 5(a) shows the variation of the shapes of Γ0 by applying N=5 and α=0.5 in (86) and in (87), i.e., for the first and second eigenvalues. Figure 5(b) shows the variation of the shapes of Γ0 by applying N=5 and α=1 in (86) and in (87), i.e., for the first and the second eigenvalue.Figure 5
Variations ofΓ0 with respect to λn,k.
(a)(b)


## 3.1.1. Nonlocal Differential Equation

In this section, we attempt to determinateΦ, satisfying (15), using the fixed point method.LetA:H1/20,2π×H1/20,2π⟶H−1/20,2π denote the Dirichlet-Neumann operator on ΓD1, which maps u^1,u^2 onto the normal derivative of u^∈H1D1, defined as follows:
(17)Au^1,u^2≔∂u^∂νeiθ,θ∈0,2π,with the respective boundary values given by
(18)u^eiθ=u^1onΓ1,u^eiθ=u^2onΓ2.The boundary traces onΓ1 and Γ2 are, respectively, given by
(19)u^eiθ=g^onΓ1,u^eiθ=0onΓ2,∂u^∂νeiθ=Φn′eiθhΦneiθonΓ2.However, we have the nonlocal differential equation for the conformal mapΦn, which is a critical element in the inverse algorithm.
(20)Φn′eiθ=Au^1,u^2hΦneiθ.The boundary correspondence function,Ψn, is a solution of
(21)Ψn′θ=Au1Ψnθ,u2ΨnθhΨnθ.As h<0, equations (20) and (21) are well defined.Based on the available Cauchy data,g,h, we have (12)
(22)u^nr,θ≈∑k=0+∞2π2n+k−1coskπ/2−rπ/22n+k−1eikθπ22n+k−12−21+−1k,n≥1.the fundamental solution of (11).Let us discuss some definitions and properties related to fixed points before employing the fixed-point methods to determinateΦ.


## 3.1.2. Fixed-Point Theorems (See [1])

Theorem 1.
(Fixed-Point theorem of Banach)
Letf:S⟶S be a nonlinear operator where S is a complete metric space. We assume that f is a strict contraction. Then, f admits a unique fixed point (see [7]).Proof of Theorem 2.
Consider any pointx0∈S and iteratively define
(23)xn+1=fxnforn=0,1,⋯.
Then,(24)fxn+1−fxn≤γxn+1−xn=γfxn−fxn−1,fxn+1−fxn≤γn+1fx0−x0pourn=0,2,⋯.
Ifn≥k,, then
(25)xn−xk=fxn−1−fxk−1,≤∑j=k−1n−2fxj+1−fxj,≤fx0−x0∑j=kn−1γj.
Hencexkk=1∞ is a Cauchy sequence in S, and, as S is complete, there exists x∈S such that xk⟶x in S. Thus, it is evident that
(26)fx=x.
Hence,x is fixed point for f and the fact that f is a strict contraction ensures its uniqueness.
The following is the version of the Banach fixed-point theorem and its corollary that is used in the subsequent proofs.Theorem 3.
LetS,d be a nonempty complete metric space. If f:S⟶S is such that, for at least one integer m≥1,fm is a strict contraction, then f admits an unique fixed point.Theorem 4.
LetX,d be a complete metric space and let T:X⟶X be a δ,L-almost contraction. Then,
(27)FixT=x∈X,Tx=x≠∅.(1)
For anyx0∈X, the Picard iteration xmm=0∞, xm=Tmx0 converges to some x∗∈FixT;(2)
The following estimate holds(28)dxm+i−1,x∗≤δi1−δdxm,xm−1,m=0,1,2,⋯,i=1,2.⋯


## 3.1.3. Fixed-Point Equation and Iteration

LetS be an operator defined as follows:
(29)Sψθ=θ+ψθ,(30)withψ:H1/20,απ⟶H1/20,απ.We assume thatψ is not identically zero.Further, we introduce an operatorT:H010,απ⟶H010,απ,0<α<2 defined by
(31)Tψθ≔α2∫0θUψ−1απ∫0απUψdθdτ,θ∈0,απ,with
(32)Uψ≔Au1oSψ,u2oSψhoSψ.By integrating (32) and considering that ψ0=0 and ψαπ=0, we obtain
(33)1απ∫0απAu1oSψ,u2oSψhoSψ=1.Using the boundary conditions ofψ, we can express ψθ=beiθ2απ−θ2−1,b∈ℂ.It is easy to verify thatT is nonlinear and contracting.Now, we use the following theorem (see [1]).Theorem 5.
Letκ,τ be a pair of Cauchy data obtained from (8).Then, in terms of the holomorphic mapΦn:D1⟶Ω and its boundary correspondence function Ψn, the function ψn=S−1Ψn is a fixed point for T.We obtain a fixed point defined by successive iteration ofT beginning with an arbitrary approximation ψn0, for all n≥1. Therefore, Theorem 3.4 induces the following iteration:
(34)ψnm+1=Tψnm,m=0,1,2,⋯.The solution of the Cauchy problem is required to identify the boundary functionΨn from the function ψn. For this purpose, let us express Sψn using the following expansion:
(35)Sψnθ=∑k=0+∞an,keikθ,θ∈0,απ.UsingΦneiθ=Sψnθ, we obtain Φn using the following expansion:
(36)Φnz=∑k=0+∞an,keikθ,θ∈0,απ.The following convergence result justifies the procedure of iteration (31).Theorem 6 (see [1]).
LetD1 be the unit disk and g be a continuous positive function defined on it. Then, successive approximations
(37)ψnm+1=Tψnm,m=0,1,2,⋯,converge provided that the initial estimate Ψn0 of the boundary function is sufficiently close to the exact function ψn.Proof (see [4]).
Theorem6 is proved by first establishing that T′0≤1 in the case where Ω=D1. Thereafter, a continuity argument is applied to extend the estimate to the norm of T′ψn for ψn in a neighborhood of the fixed point defined in Theorem 3.4.Proof of Theorem 7.
We calculate the derivate of the nonlinear operatorT. We begin by estimating T′0. We have
(38)T′0=0.
Therefore, we have,T′0<1,∀0<α<2.
Using a continuity argument to estimate the norm ofT′ψnθ near the fixed point of Theorem 3.4, we have for all θ∈0,απ,(39)T′ψnθ=α2ψn′θ.
Thus,T′ψnθ<ψn′θ,∀θ∈0,απ and n≥1.
The approximation ofT′ψnθ near the fixed point of the theorem is ψn′θ,n≥1.
We now determinate a numerical approximation of the conformal mapΦn,k, for all n≥1.


## 3.1.4. Numerical Approximation of the Conformal Map

In this section, we use an ordinary differential equation of the boundary correspondence function to obtain an approximation of the conformal map,Φn, for all n≥1.Let us introduce two boundary correspondence functions,Ψn2 and Ψn1, defined from 0,2π to ℂby(40)Ψn1θ≔Φneiθpourθ∈0,απ,Ψn2θ≔Φneiθpourθ∈απ,2π.OnΓ2, equation (21) becomes
(41)Ψn1′θ=∂u^/∂rhΨn1θ.Here, we use the length of arcΓ2 determined by dθ.By removing the identity,(42)1απ∫0απ∂u^/∂rhΨn1θdθ=1,that is obtained by integrating (41). Then, we have for ψn1θ≔Ψn1θ−θ, the equation
(43)ψn1′θ=Uψn1θ−1απ∫0απUψn1dθ.where we have for convenience
(44)Uψn1=∂u^/∂rhoSψn1,with S operator defined in (35).For(45)ψn1θ≔∫0θUψn1−1απ∫0απUψn1dθdτ,θ∈0,απ,the iterative solution of (43), we have the following convergence results:Theorem 8.
The successive approximations defined by (45) converge from the initial approximation ψn01=θαπ−θ provided that Ω is close enough to D1.
When we obtainψn1, that is, Φn, the boundary values of Φn on Γ2 to be used for the iteration of (45) are given by
(46)Φneiθ=∑k=0+∞an,keikθ,with
(47)2an,kπ=∫02πΦneiθe−ikθdθ,2an,kiπ=∫ΓD1z¯k+1Φnzdz,
So, onΓ2,
(48)an,kαπ=∫Γ2Φneiθe−ikθdθ.
We can express the expansion ofΦn in the form
(49)Φnz=∑k=0+∞an,kzk,which is numerically stable for ∣z∣<1. By construction, considering (40), we have Φn1=Ψn10 and Φneiαπ=Ψn1απ.
Now, we describe the numerical algorithm that we will implement in this paper. We use a method involving integral equations onΓ2. The integral equation obtained is resolved via the Nyström method.
We give ourselves a regular mesh,(50)θm=θ0+mh,1≤m≤N,h=απNon0,απ.
The approximationsΨnm1 of Ψn1θm at the nodes of mesh are calculated using the q previous approximate values ym−1,ym−2,⋯,ym−q.
The iterations allow approximations of the normal derivatives via trigonometric interpolation.
The integration interpolation polynomial on0,θm is used to associate Ψnm1 to Ψnm+11, via (43), with m=1,⋯,N.
Beginning with identity, we have(51)ψn1θ+t=ψn1θ+∫θθ+tUψn1τ−1απ∫0απUψn1dξdτ,which corresponds to
(52)ψnm+11=ψnm1+∫θmθm+1Uψn1τ−1dτ,or more precisely
(53)ψnm+11=ψnm1+∫θmθm+1∑k=0+∞pk,nλk,neikτhoSψn1τ−1dτ.
We substitute∑k=0+∞pk,nλk,neikτ/hoSψn1τ−1 in the integral via the Nyström method using the interpolation polynomial Pθ.


## 3.1.5. Nyström Method (See [8])

This method, called the quadrature method, is the application of numerical methods of calculating integrals to obtain a linear system. We write(54)ψn1θm+1=ψn1θm−1+∫θm−1θm+1Pθdθ,where P denotes the interpolation polynomial of Lagrange of degree q−1, for the pair of values
(55)θm−j,fm−j,0≤j≤q−1,where we use
(56)Uψnj1=Uψn1θj,fm−j=fθm−j=∑k=0+∞pk,nλk,neikθm−jhoSψn1θm−j−1,Pθ=∑j=0q−1fθm−jlm−jθ,with the polynomials lm−jθ are Lagrange polynomials of ℝnX associated to θm,θm−1,θm−2,⋯,θm−q+1.That is,
(57)ψnm+11=ψnm−11+απN∑j=0q−1γq,jfm−j,N∈ℕ∗.Explicitly, we can express (57) in the form
(58)ψnm+11=ψnm−11+απN∑j=0q−1γq,j∑k=0+∞pk,nλk,neikθm−jhoSψn1θm−j−1,N∈ℕ∗.Therefore, we can setq=2,3,.⋯ for the approximations. Hence,
(59)Ifq=2thenψnm+11=ψnm−11+2απN∑k=0+∞pk,nλk,neikθmhoSψn1θm−1N∈ℕ∗.(60)Ifq=3then(61)ψnm+11=ψnm−11+απ3N7∑k=0+∞pk,nλk,neikθmhoSψn1θm−2∑k=0+∞pk,nλk,neikθm−1hoSψn1θm−1+∑k=0+∞pk,nλk,neikθm−2hoSψn1θm−2−6.Ifq=4, then
(62)ψnm+11=ψnm−11+απ3N8∑k=0+∞pk,nλk,neikθmhoSψn1θm−5∑k=0+∞pk,nλk,neikθm−1hoSψn1θm−1+4∑k=0+∞pk,nλk,neikθm−2hoSψn1θm−2−∑k=0+∞pk,nλk,neikθm−3hoSψn1θm−3−6.After fixingψn01, we calculate ψn11 using equation (45), for all q.The estimate ofψn1 allows us to calculate an,k,k=0,1,⋯,J using (29), (35), and (48) and, subsequently, the expression of ΦnJ1,J∈ℕ.We use(63)an,k=αn,k+iβn,kwith0≤k≤J.The traces of the curves corresponding to the equations defined below provide an approximate domain.(64)RΦnJ1=∑k=0Jαn,kcoskθ−βn,ksinkθ,IΦnJ1=∑k=0Jαn,ksinkθ+βn,kcoskθ.We attempt to determineψn1 for q=2. The approach for the other case is similar.We know that (59) is equivalent to
(65)ψnm+11=ψnm−11+2απNψnm1′,N∈ℕ∗,forq=2.After calculation, we have(66)ψnm+11=ψn01+2απN∑j=01/2m−1ψn2j+11′,or
(67)ψnm+11=ψn01+4πN∑j=01/2m−1Tψn2j+11′,N∈ℕ∗.


## 3.1.6. Convergence Results ofψnm1m

We setUm+1=ψnm+11, as defined above.Forq=2, we have Umm≥0 a sequence defined in 0,απ,0<α<2, by
(68)Um+1=Um−1+4πNTUm′,N∈ℕ∗.The Picard theorem ensures the convergence ofUmm≥0.We can write(69)Um+1=U0+4πNTU1+TU3+TU5+⋯+TUm′,=U0+4πNTU11−α/24m−41−α/22′,N∈ℕ∗.Thus, we have(70)Umm≥0convergestoU0+4πα2U0′N4−α2forallθ∈0,απ,N∈ℕ∗.Finally, we can deduce the following result:Proposition 9.
LetT:H01a,b→H01a,b be a k-contraction and ψnn≥0 be a sequence of functions defined by
(71)ψn+1≔Tψnsothat,Tψn′≔kψn′,n≥0.
Ifψnn≥0 verifies this recurrence relation
(72)ψn+1=ψn−1+εTψn′,withε>0andn∈ℕ.
Then,(73)ψnn≥0convergestoψ0+Ck,εψ0′.C a constant depending on k and ε.Proof of Proposition 10.
The Picard theorem on the convergence ofψnn≥0.
We can write(74)ψn+1=ψ0+εTψ1+Tψ3+Tψ5+⋯+Tψn′n∈ℕ.
So,(75)ψn+1=ψ0+k2εψ01−k4n−41−k2′,n∈ℕ.
This yields the result.
Further, we setU0=ceiθ2απ−θ2−1,c∈ℂ. We have
(76)ψ1θ=ceiθ2απ−θ21+4πα2θαπ−θαπ−2θiN4−α2−c,withc∈ℂandN∈ℕ∗.
It is evident thatψ1 does not depend on n.
Therefore, according to (20) and (21), the map Φ1 is then defined by
(77)Φ1eiθ=θ+ceiθ2απ−θ21+4πα2θαπ−θαπ−2θiN4−α2−c,forallθ∈0;απ.with c∈ℂandN∈ℕ∗. The coefficient a is determined using the available Cauchy data and eigenvalues.
For allz0∈D1, Φ1′z0≠0 so Φ1 is holomorphic.
In the following argument, we poseΦ=Φn,k1,c=an,k because they are determined by eigenvalues.
Φn,k1 is analytic, continuous on D¯1; it realizes a conformal mapping from C1 onto ∂Ω, according to Theorem 1.4.5 (see [2] page 42). Further, it realizes a conformal mapping (univalent) from the domain D onto the domain Ω.
Hence, for all0<α<2, and all N∈ℕ∗, there exists an univalent holomorphic function Φn,k1, which realizes a conformal mapping from the domain D1 onto the domain Ω, which is a solution of problem (19), defined by
(78)Φn,k1Reiθ=θ+an,kReiθ2απ−θ21+4πα2θαπ−θαπ−2θiN4−α2−an,k.
For allθ∈0,απ with N∈ℕ∗,an,k,∈ℂ,0≤R≤1.an,k is determined using the available Cauchy data and eigenvalues.Proposition 11.
Consider the problem1−2−3 where
(79)∂Ω=Γ0òΓ,withΓtheunknownpart.∂Ω is a spiral defined on 0,2π by the following equations:
(80)X=θ−a1+a1−a2h1θcost−a1h1θ+a2sint,Y=−a2+a1h1θ+a2cost+a1−a2h1θsint,withan,k∈ℂ,a1=∣an,k∣cosargan,k,a2=∣an,k∣sinargan,k,h1θ=4πα2θαπ−θαπ−2θN4−α2andt=θ2απ−θ2.
Then,Γ is a part of the spiral, ∂Ω, image of 0,απ under
(81)Φn,k1Reiθ=θ+an,kReiθ2απ−θ21+4πα2θαπ−θαπ−2θiN4−α2−an,kdefined for all n≥1 and k≥0.Proof of the Proposition 12.
It suffices to determine the integral and decimal parts ofΦn,k1.


## 3.1.7. Numerical Simulations

We determine the approximate shape of the unknown part of the boundary corresponding to the two first eigenvalues.LetΦn,k1 be the map defined in (78).We putgx,y=0 and hx,y=−1−x2−y2, where h is defined to be negative.First, we determinean,k of Φn,k1 corresponding to the first and second eigenvalues, after approximating the corresponding shapes of Γ2. Using the boundary values, we have previously established,
(82)Φn′eiθ≈∑k=0+∞pk,nλk,neikθhoΦneiθ,∀n≥1,uΦk,neiθ=0,foru^definedin16.The coefficientsa1 and a2 are determined based on the Cauchy data corresponding to each eigenvalue of the Laplacian operator. Hence, they are obtained by
(83)Φ1′eiθ≈∑k=0+∞2n+k−12π−1n2π/2n+k−12π2−21+−1keikθhoΦneiθ.Therefore, the calculation yields that corresponding toθ=0:
(84)a1=0,a2=N4−α24α4π31+∑k=0+∞2n+k−12π−1n2π2n+k−12π2−21+−1k.


## 3.1.8. Conformal MapΦ0,1 Corresponding to First Eigenvalue λ0,1

Corresponding to the first eigenvalueλ0,1, we have
(85)a2=N4−α24α4π31−π2ππ2−4,(86)X=θ−N4−α24α4π31−π2ππ2−44πα2θαπ−θαπ−2θN4−α2cost+sint,Y=N4−α24α4π31−π2ππ2−4−1+cost−4πα2θαπ−θαπ−2θN4−α2sint.Therefore, we obtain the result for(1)
N=5 and α=0.5Figure1 is obtained by applying N=5 and α=0.5 in (86). Figure 1(a) shows the shapes of Γ and Γ0 in an orthogonal mark with graphic unit 20 cm, and Figure 1(b) shows these same shapes in an orthogonal mark with graphic unit 5 cm. It is this right part which gives us the enlargement of shapes of Γ and Γ0.
(2)
N=5 and α=1Figure 1
N5 Alpha0.5.
(a)(b)Figure2 is obtained by applying N=5 and α=1 in (86). Figure 2(a) shows the shapes of Γ and Γ0 in an orthogonal mark with graphic unit 20 cm, and Figure 2(b) shows these same shapes in an orthogonal mark with graphic unit 5 cm. It is this right part which gives us also the enlargement of shapes of Γ and Γ0.Figure 2
N5 Alpha1.
(a)(b)It is to be noted that, in each case (α=0.5 or α=1), Γ0 is sinusoïdal, Γ is a part of spiral, and ∂Ω converges to a finite limit. Therefore, ∂Ω is asymptotic to a circle and Ω is disk shaped.


## 3.1.9. Conformal MapΦ1,1 Corresponding to the Second Eigenvalue λ1,1

Corresponding to the second eigenvalueλ1,1, we have
(87)a2=N4−α24α4π31−π2ππ2−4−22π,(88)X=θ−N4−α24α4π31−π2ππ2−4−22π4πα2θαπ−θαπ−2θN4−α2cost+sint,Y=N4−α24α4π31−π2ππ2−4−22π−1+cost−4πα2θαπ−θαπ−2θN4−α2sint.Thus, we conclude the result forN=5 and α=1.Figure3 is obtained by applying N=5 and α=1 in (87). Here too, Figure 3(a) shows the shapes of Γ and Γ0 in an orthogonal mark with graphic unit 20 cm, and Figure 3(b) shows these same shapes in an orthogonal mark with graphic unit 5 cm. Figure 3(a) gives us also the enlargement of shapes of Γ and Γ0.Figure 3
N5 Alpha1.
(a)(b)Here too,Γ0 is sinusoïdal, Γ is a part of spiral, and ∂Ω converges to a finite limit.


## 3.1.10. Comparison Results forN=5 and α=1

The result for the first eigenvalue obtained via the minimization functional method forN=5 and α=1 in our previous study, and the result obtained using the fixed-point method for the same data has been depicted in the following diagram.In Figure4, we can observe that
(i)
Γ0 is sinusoïdal in both these studies(ii)
Γ is a part of spiral in the present study and is a part of spiral with a cusp, in the study in 2014(iii)
Both methods yielded a spiral shape for∂ΩFigure 4
Results forΓ of the previous study in 2014 black in (a). Results for Γ of the present study green to (b).
(a)(b)


## 3.1.11. Variations ofΓ0 with respect to λn,k

The following diagram allows us to compare the shapes ofΓ0 corresponding the results in the first and the second study for the first eigenvalues.In Figure5, we obtain results of the variations of the shape of Γ0 with respect to λn,k. Figure 5(a) shows the variation of the shapes of Γ0 by applying N=5 and α=0.5 in (86) and in (87), i.e., for the first and second eigenvalues. Figure 5(b) shows the variation of the shapes of Γ0 by applying N=5 and α=1 in (86) and in (87), i.e., for the first and the second eigenvalue.Figure 5
Variations ofΓ0 with respect to λn,k.
(a)(b)


## 3.2. Results

All previous figures are obtained using “GeoGebra Geometry.”Figures1(b), 2(b), and 3(b) give us the enlargement of shapes of Γ and Γ0 found, and they allow us to see their exact shapes. Also, these exact shapes allow us to make a good interpretation of the results.So, the simulations yield the following results:(1)
The shape ofΓ0 is a spiral. When the value of α increases, its spiral shape is accentuated, and it is constituted of identical connected parts. Further, when the eigenvalue increases, the length of Γ0 is observed to increase accompanied by a change in its spiral shape (see Figure 5).(2)
However, the shape ofΓ is a spiral corresponding to all variations of α and the eigenvalues. Its shape does not vary and it converges to a finite limit, which is circular in shape.(3)
Figure4(a) shows the results of the previous study, which was conducted in 2014, for N=5 and α=1, and Figure 4(b) shows the results of the present study for N=5 and α=1. We can observe that the shape of ∂Ω is spiral in both the results; however, the present study provides a more precise shape of the spiral that is asymptotic to a circle. This shape can justify that Ω is a disk.Therefore, the result obtained using the fixed-point method is more precise and further identifiesΩ as a disk.


## 4. Conclusion

In summary,Φn,k preserves the spiral shape of Γ and its properties but it varies those of Γ0 based on variations in eigenvalues.Therefore,∂Ω is a spiral, which is asymptotic to a circle and Ω is identified to be a disk.This result can be applied to the one-dimensional case, but it is more interesting to study this phenomenon in dimensions three and four.In an inverse problem with similar conditions, it is more relevant to use conformal mapping and the fixed-point method to yield interesting numerical results. The general case thatΩ is a disk remains an open problem.



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