Introduction In many inverse problems the reconstruction of the Dirichlet-to-Neumann oper- ator has paramount importance

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ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

DIRICHLET-TO-NEUMANN OPERATOR ON THE PERTURBED UNIT DISK

HASSAN EMAMIRAD, MOHAMED-REZA MOKHTARZADEH

Abstract. This article concerns the Laplacian on a perturbed unit disk Ω= {z=rexp(iθ) :r <1 +f(θ)}, with dynamical boundary condition whose solution can be represented by a Dirichlet-to-Neumann semigroup. By neglecting the terms of order², we obtain a simple expression which allows us to use Chernoff’s theorem for its approximation. As a motivation for this research, we present an example which shows the feasibility of applying Chernoff’s The- orem.

1. Introduction

In many inverse problems the reconstruction of the Dirichlet-to-Neumann operator has paramount importance. For example, in [1, 2, 10] a nondestructive testing technique is used to determine the distribution of electrical conductivity within some inaccessible region from electrostatic measurements at its boundary. It has been able to detect the presence and the approximate location of some object of unknown shape buried in a medium of constant background conductivity. In the present work we elaborate the same shape of the buried object, that is a star-shaped perturbation of a circle (see Figure 1). On this domain our problem is generated by an elliptic equation with a time dependent boundary condition as follows.

Let Ω ={z =rexp(iθ) : r < 1 +f(θ)} be the perturbed unit disk. In the cylinder Γ= (0,∞)×Ωwith boundary∂Γ= (0,∞)×∂Ω, we consider the linear elliptic differential equation with dynamic boundary conditions

∆u= 0, in Γ,

∂u

∂t +∂u

∂ν = 0, on∂Γ, u(0) =g, on∂Ω,

(1.1)

We will see in the next section that by making the change of variables to polar coordinates and neglecting the terms of order², the system (1.1) can be transformed

2000Mathematics Subject Classification. 35J25, 47F05, 47D06.

Key words and phrases. Dirichlet-to-Neumann operator,γ-harmonic lifting.

c

2012 Texas State University - San Marcos.

Submitted June 19, 2012. Published September 18, 2012.

1

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into the system

∆_γv= 0, in Γ,

∂v

∂t + ∂v

∂νγ

= 0, on∂Γ, v(0) =h, on∂D,

(1.2)

whereDis the unit disk, Γ the cylinder (0,∞)×Dwith boundary∂Γ = (0,∞)×∂D,

∆γ= div(γ∇), γ=

1 −f⁰

−f⁰ 1

and h(ρ, θ) =g(ρ(1 +f(θ)), θ). The Dirichlet-to-Neumann operator Λγ for this new system is defined by

[Λγh](ω) =− ∂v

∂νγ

=−ω·γ∇v

_∂D, (1.3)

whereω is the outer normal vector atω∈∂D, which coincides withω.

It is well known that the operator Λ_γ generates a (C₀) semigroup in L²(∂Ω) (see [9]) or in C(∂Ω) (see [7]). In a simple case, whereγ =I and Ω is the unit ball B in Rⁿ this semigroup has a trivial explicit representation given by Lax [8], which is

e^−tΛ^Ih(ω) =w(e^−tω), for anyω∈∂D.

wherewis the harmonic lifting ofh; i.e., the solution of

∆Iw= 0, inD, w

_∂D=h, on∂D, (1.4)

In [6], it is shown that this representation cannot be generalized if Ωhas other geometry than a ball inRⁿ. This consideration motivates the authors of [5] to use a new technique for having an approximation of the solution via Chernoff’s Theorem.

In fact, Chernoff [3] proves the following result.

Theorem 1.1. If X is a Banach space and {V(t)}_t≥0 is a family of contractions onX withV(0) =I. Suppose that the derivative V⁰(0)f exists for allf in a setD and the closureΛofV⁰(0)|_D generates aC0-semigroupS(t)of contractions. Then, for eachf ∈X,

n→∞lim kV(t

n)ⁿf−S(t)fk= 0, (1.5) uniformly fort in compact subsets ofR+.

In the next section we define all the ingredients of the above Theorem in the framework of our problem and we prove how we can approximate the solution of the system (1.2) by mean of this Theorem. Section 3 is devoted to give a bench mark example for justifying the feasibility of our method. Finally we generalize in section 4 this problem to 3D-consideration and take Ωas a perturbed unit sphere.

Because of the extreme complexity of the system we restrict ourself to the discrete values ofφ,{φjj= 1, . . . , N}, which achieves toN times 2D system, each of which can be computed as above.

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2. Modified system By making the change of variables

ρ= r

1 +f(θ)

andθremains invariant, the Laplacian in the new coordinates will be (see [4]),

∆ρ,θ = 1

(1 +f)² + ²(f⁰)² (1 +f)⁴

∂²

∂ρ² +1 ρ

1

(1 +f)² + 2²(f⁰)²

(1 +f)⁴ − f⁰⁰ (1 +f)³

∂

∂ρ

−1 ρ

2f⁰ (1 +f)³

∂²

∂ρ∂θ + 1 ρ²

1 (1 +f)²

∂²

∂θ².

(2.1) By factoring the term _(1+f)¹ ₂ and neglecting the terms of order ², we obtain

∆⁽⁾_ρ,θ = ∂²

∂ρ² +1

ρ(1−f⁰⁰) ∂

∂ρ−1

ρ(2f⁰) ∂²

∂ρ∂θ + 1 ρ²

∂²

∂θ². (2.2) Here we have replaced _1+f¹ by 1−f+. . .. Now let

γ=

1 −f⁰

−f⁰ 1

and ∆γ=div(γ∇). Since in polar coordinates,

∇=^t(∂

∂ρ,1 ρ

∂

∂θ), div(F1, F2) = 1 ρ

∂

∂ρρF1+ ∂

∂θF2

,

we obtain ∆⁽⁾_ρ,θ = ∆γ. Consequently the Dirichlet-to-Neumann operator will be [Λγh](1,0) =−(1,0)·

1 −f⁰

−f⁰ 1 vρ v_θ ρ

(ρ,θ)=(1,0)

. (2.3)

Remark 2.1. Since γ is not continuous at the origin, even for f ∈ C¹([0,2π]), in the sequel we replace Ω by Ω \ {0}, because for any periodic function f ∈ C^∞([0,2π]), we haveγ∈C^∞(Ω\ {0}).

Now, let us define∂X :=C(∂D) and the operatorV(t) by V(t)h(ω) :=v(e^−tγω)

_∂D, for any h∈∂X, (2.4) wherev is the solution of (1.3).

Theorem 2.2. The family of the operators {V(t)}_t≥0 defined above satisfies the assumptions of the Chernoff ’s Theorem 1.1.

Proof. Forsmall enough, the eigenvalues ofγ,λ= 1±f⁰(θ) are positive implies that for anyω∈∂D, we havee^−tγω∈D. Hence the fact thatV(t) is a contraction follows from the maximum principle. It is obvious that V(0) =I on the Banach space∂X. Since

D(Λγ) ={h∈∂X: Λγh∈∂X}, for anyh∈D(Λγ),_dt^dV(t)h

_t=0=−ω·γ∇v(ω) = Λγh. This shows that we can apply Theorem 1.1 for the family of operators{V(t)}_t≥0.

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3. An example Let us denote ∆0u= div(∇u) and ∆1u= div(f⁰

0 1 1 0

∇u) in such a way that

∆γ= ∆0−∆1. Now let us defineu0andu1 such that by neglecting the terms of order², v=u0+u1 satisfies the system (1.2). In fact by takingu0the solution of the Dirichlet problem

∆0u0= 0, inD, u₀

∂D=g₀, on∂D, (3.1)

andu1the solution of the Poisson problem

∆0u1= ∆1u0, in D, u1

_∂D=g1, on∂D. (3.2)

Since

∆_γ(u₀+u₁) = (∆₀−∆₁)(u₀+u₁)

= ∆₀u₀+(∆₀u₁−∆₁u₀)−²∆₁u₁

by takingh=g0+g1,w=u0+u1approximates the solution of (1.2). Now by choosingg0(θ) = cos(2θ), the solution of (1.3) will beu0(ρ, θ) =ρ²cos(2θ), hence

Φ(ρ, θ) = ∆1u0= div(f⁰ 0 1

1 0

∇u0)

= (−8f⁰(θ) sin(2θ) + 2f⁰⁰(θ) cos(2θ))

Taking f(θ) = cos(4θ) (see Figure 1), one gets Φ(ρ, θ) = −32 cos(6θ). Putting g₁(θ) = cos(6θ) in (2.3) its solutionu₁(ρ, θ) =ρ²cos(6θ).

-1.0 -0.5 0.5 1.0 x

-1.0 -0.5 0.5 1.0 y

Figure 1. Ωforf(θ) = cos(4θ) and= 0.1

Hence forf(θ) = cos(4θ) andh(θ) = cos(2θ) +cos(6θ), the function

w(ρ, θ) =ρ²[cos(2θ) +cos(6θ)] (3.3) approximates the solution of (1.2).

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Since forf(θ) = cos(4θ), e^tγ=e^t

coshψ(t) sinhψ(t)) sinhψ(t) coshψ(t)

,

whereψ(t) = 4tsin(4θ). In this example the approximating family given by (2.4) can be constructed as

[V(t)h](1, θ) =w

e^−t

coshψ(t) −sinhψ(t)

−sinhψ(t) coshψ(t) 1 0

and by replacing in it in (3.3), we obtain [V(t)h](1, θ)

=w e^−t(coshψ(t)), e^−t(−sinhψ(t))

= e^−t(coshψ(t))² cosh

2e^−t(−sinhψ(t))i

+cosh

6e^−t(−sinhψ(t))i .

(3.4)

Consequently,

d

dt[V(0)h](1, θ) =−2(1 +)

which is exactly the directional derivative obtained by the following standard for- mula

[Λγh](1,0) =−(1,0)·

1 4sin 4θ

4sin 4θ 1

2ρ(cos 2θ+cos 6θ) ρ(−2 sin 2θ−6sin 6θ)

(ρ,θ)=(1,0)

. This proves thatV(t) given by (3.4), can be used in Chernoff’s Theorem in order to compute an approximating expression for the system (1.2).

4. Generalization to the perturbed unit sphere Let

Ω:={(r, θ, φ)∈[0,1 +f(θ, φ)]×[0,2π)×[−π/2, π/2)}

be a perturbed unit sphere. Then the spherical coordinate the system (1.1) can be written as

1 r²

∂

∂r

r²∂u

∂r

+ 1

r²sin²φ

∂²u

∂θ² + 1 r²sinφ

∂

∂φ

sinφ∂u

∂φ

= 0, in Γ,

∂u

∂t +∂u

∂r = 0, on∂Γ, u(0) =g, on∂Ω,

(4.1)

where Γ= [0,∞)×Ω. As in Section 2, we change variables:

ρ= r

1 +f(θ, φ)

where θ and φ remain invariant. Then the Laplacian in the new coordinates will be ∆_ρ,θ,φ, which has a rather complicated expression, but by neglecting the terms of order ², we transform ∆ρ,θ,φ to ∆⁽⁾_ρ,θ,φ. Our goal at this stage is to solve the problem,

∆⁽⁾_ρ,θ,φv= 0, in Γ,

∂v

∂t +∂v

∂ρ = 0, on∂Γ, v(0) =g, on∂Ω,

(4.2)

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where Γ = [0,∞)×Ω, with Ω being the unit ball in R³. This system is still quite complicate to be solved; so by taking N discrete values of φ, {φj := jh : j = 1, . . . , N} with h= π/N, replacing _∂φ^∂v by (v(ρ, θ, φ_j+1)−v(ρ, θ, φ_j))/h and replacing ^∂_∂φ²^v₂ by (v(ρ, θ, φ_j+1)−2v(ρ, θ, φj) +v(ρ, θ, φ_j−1))/h²we retrieve a system of N equations of (ρ, θ) parameters, each of which has an expression as we have already discussed in the preceding sections.

We postpone the numerical resolution of this system for a future article, with numerical illustrations.

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Hassan Emamirad

School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O.

Box 19395-5746, Tehran, Iran E-mail address:emamirad@ipm.ir

Mohamed-Reza Mokhtarzadeh

School of Mathematics, Institute for Research in FundamentalSciences (IPM), P.O.

Box 19395-5746, Tehran, Iran

E-mail address:mrmokhtarzadeh@ipm.ir