A Representation Formula for the Voltage Perturbations Caused by Diametrically Small Conductivity Inhomogeneities. Proof of Uniform Validity

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A representation formula for the voltage perturbations caused by diametrically small conductivity inhomogeneities.

Proof of uniform validity

Hoai-Minh Nguyen, Michael S. Vogelius

Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA Received 5 October 2008; received in revised form 5 March 2009; accepted 8 March 2009

Available online 7 April 2009

Abstract

We revisit the asymptotic formulas originally derived in [D.J. Cedio-Fengya, S. Moskow, M.S. Vogelius, Identification of con- ductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction, Inverse Problems 14 (1998) 553–595; A. Friedman, M. Vogelius, Identification of small inhomogeneities of extreme conductivity by boundary measurements: A theorem on continuous dependence, Arch. Ration. Mech. Anal. 105 (1989) 299–326]. These formu- las concern the perturbation in the voltage potential caused by the presence of diametrically small conductivity inhomogeneities.

We significantly extend the validity of the previously derived formulas, by showing that they are asymptotically correct, uniformly with respect to the conductivity of the inhomogeneities. We also extend the earlier formulas by allowing the conductivities of the inhomogeneities to be completely arbitraryL^∞, positive definite, symmetric matrix-valued functions. We briefly discuss the relevance of the uniform asymptotic validity, and the admission of arbitrary anisotropically conducting inhomogeneities, as far as applications of the perturbation formulas to “approximate cloaking” are concerned.

©2009 Elsevier Masson SAS. All rights reserved.

Keywords:Voltage representation formulas; Small inhomogeneities; Cloaking

Contents

1. Introduction . . . 2284

2. A preliminary uniform estimate . . . 2286

3. The leading order asymptotics whenD_ρ=ρD. . . 2291

3.1. A uniform estimate for the first remainder term . . . 2294

3.2. A uniform estimate for the second remainder term . . . 2298

3.3. Proof of the main theorem . . . 2302

4. Other inhomogeneitiesDρ . . . 2305

5. Some remarks on cloaking . . . 2307

* Corresponding author.

E-mail address:[email protected] (M.S. Vogelius).

0294-1449/$ – see front matter ©2009 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2009.03.005

Acknowledgements . . . 2308 Appendix A. Auxiliary results for exterior domains . . . 2308 References . . . 2314

1. Introduction

Asymptotic formulas that quantify the effect of small conductivity inhomogeneities on the voltage potential of an electrical conductor have recently received quite a bit of attention, see for instance [1,4,5] and references therein. One important application of such formulas has been the approximate solution of the electrical impedance tomography problem, namely: “to determine the location and (some) geometric properties of the inhomogeneities from boundary measurements of voltages and current fluxes” [3]. Another more recent application is the precise estimation of the degree of near-invisibility associated with approximate cloaks obtained by so-called “mapping techniques”, see [9].

LetΩ be a connected, bounded, smooth domain inR^d,d=2,3, letγ₀be a smooth background conductivity, and letf ∈H^−1/2(∂Ω)be a prescribed normal boundary flux (with

∂Ωf dσ =0).u₀denotes the background voltage potential,i.e., the solution to

∇ ·(γ₀∇u₀)=0 inΩ, (γ₀∇u₀)·n=f on∂Ω, (1.1)

say, with

∂Ωu₀dσ =0. We could allow any finite number of (well separated) conductivity inhomogeneities in- sideΩ — but for simplicity let us assume there is only one (the principal effect of a finite number would simply be the sum of the individual effects). This conductivity inhomogeneity has small diameter (say, of magnitude 0< ρ1).

We shall denote the open set occupied by the inhomogeneityD_ρ. The conductivity insideD_ρis given by the symmet- ric, positive definite matrix-valued functionγ_1,ρ. We defineγ_ρ to be the conductivity

γ_ρ=

γ₀ inΩ\D_ρ,

γ_1,ρ inD_ρ. (1.2)

u_ρdenotes the voltage potential corresponding to the conductivity distributionγ_ρ,i.e., the solution to

∇ ·(γ_ρ∇u_ρ)=0 inΩ, (γ_ρ∇u_ρ)·n=f on∂Ω, (1.3)

with

∂Ωu_ρdσ=0. Initially we shall just assume that D_ρ is contained in a small ball,i.e., in a set of the form x₀+ρB₁, wherex₀is a point inΩ,B₁is the unit ball, centered at the origin, andρis taken sufficiently small that x₀+ρB₁Ω. For simplicity let us assumeΩ contains the origin, and thatx₀=0. ByB_δwe shall denote the ball of radiusδ, centered at the origin. The first result in this paper (Theorem 1 in Section 2) asserts that, given any positive sandδ

uρ−u0_H^s_(Ω\Bδ)Cρ^dfH^−1/2(∂Ω), (1.4)

with a constantCthat isindependentofγ1,ρ,ρandf, but depends onΩ,δ,s, and the background conductivityγ0. The novelty here is that the constantCis independent ofγ1,ρ, the (ρ dependent) conductivity of the inhomogeneity.

If we only consider a conductivityγ1, that is isotropic and independent ofρ, and we do not insist that the constantC be independent ofγ1, then the estimate (1.4) follows immediately from the representation formula(s) proven in [4,5].

However, it is exactly the dependence ofγ1,ρonρ, and the independence of the constantCofγ1.ρthat is important for applications to cloaking. More precisely: to obtain meaningful estimates of the degree of near-invisibility associated with certain approximate cloaks constructed by “mapping techniques” it is most convenient to have an estimate for uρ−u0that is uniform in γ1,ρ. For instance, in [9] one could have used the fact thatC is independent of γ1,ρ to give a much more direct proof of the near-invisibility estimate — as it were (without recognizing this uniformity) the analysis in [9] relies on a somewhat indirect argument based on monotonicity and the validity of the estimate (1.4) in the two degenerate isotropic cases,γ1=0 andγ1= ∞.

Having proven the uniform estimate (1.4) we then return to consider the question of uniform validity of asymp- totic formulas such as that derived in [5]. For that purpose we considerD_ρ of the formD_ρ=ρDΩ, whereDis

a bounded, simply connected, smooth domain. For simplicity we takeγ₀to be a constant positive definite symmet- ric matrix, whereas we permit γ_1,ρ to be an arbitrary positive definite symmetric matrix-valued L^∞ function. The representation formula from [5] asserts that for constant, isotropicγ0andγ1(ρindependent)

uρ(x)=u0(x)+ρ^d|D|∇yG(x,0)·(γ0−γ1)M∇u0(0)+o ρ^d

, (1.5)

whereo(ρ^d)/ρ^d→0 asρ→0. The functionG(x, y)is a particular Green’s function (the so-called Neumann func- tion) for the Laplacian on the domainΩ, namely the solution to

_xG(x, y)= −δ_{x=y}, with ∂

∂n_xG(x, y)= − 1

|∂Ω|, (1.6)

normalized by

∂ΩG(x, y) dσ_x=0. In order to calculate the matrixMletφ_k denote the solution to

∇ ·(γ∇φ_k)=0 inR^d, φ_k(z)−z_k→0 as|z| → ∞, (1.7) withγ denoting the rescaled conductivity function

γ=

γ₀ inR^d\D, γ₁ inD.

The matrixMis given by M_{j k}= 1

|D|

D

∂φ_k

∂z_j dz.

In [5] it is shown that the matrixM(corresponding to constant isotropicγ₀andγ₁) is symmetric and positive definite.

Furthermore, in this caseM is a function of the single scalar variablec=γ₁/γ₀, and it is shown that(1−^γ_γ¹₀)M= (1−c)M(c)has finite limits asc→0 andc→ ∞. These limits are exactly the symmetric, positive definite matrices that appear in the asymptotic formulas for the two degenerate cases,γ1=0 andγ1= ∞(see [6]). The second term in the right-hand side of (1.5) is therefore of orderρ^duniformly inγ1— it will also, uniformly inγ1, represent the leading term ofu_ρ−u₀provided the remainder termo(ρ^d)can be shown to have the property thato(ρ^d)/ρ^d→0, uniformly inγ₁as ρ→0. This uniform smallness assertion (for variable,ρ dependent, anisotropicγ_1,ρ) is exactly the content of Theorem 2 in Section 3.

It is well known that a formula similar to (1.5) (and a bound similar to (1.4)) holds for (subsequences of) ar- bitrarily shaped, volumetrically small inhomogeneities, withρ^d|D|replaced by |Dρ|, providedγ1,ρ stays bounded and bounded away from zero (cf. [4]). However, for these results to be valid uniformly in γ_1,ρ a condition of the typeD_ρ=ρD (orD_ρ⊂ρD) is absolutely essential. To see this considerD_ρ in the form of a thin, “square” sheet (−1,1)^d−1×(−ρ, ρ)(or a smoothed-out version of this). For a fixedρthe solutions corresponding to a sequence of conductivity problems with (isotropic) conductivitiesγ1approaching+∞will converge to a solution to the conduc- tivity problem inΩ\Dρthat is constant on∂Dρ. It is therefore not very difficult to see that we may pick a sequence ρ_n→0 and a sequence of isotropic (constant) conductivitiesγ_1,n→ ∞such that the corresponding sequence of solu- tions to the conductivity problems approaches a function that solves∇ ·(γ₀∇w₀)=0 inΩ\([−1,1]^d−1× {0}), and is constant on[−1,1]^d−1× {0}. Since this limit generically is notγ₀-harmonic in all ofΩ, and thus not equal tou₀, it follows that the estimate (1.4), or a formula like (1.5) (withρ^d|D|replaced by|D_ρ|) cannot hold uniformly inγ₁ forDρ of the formDρ=(−1,1)^d−1×(−ρ, ρ).

In Section 4 we show that the conditionD_ρ=ρDmay be slightly relaxed without affecting the uniform validity of the principal two terms of the asymptotic expansion of u_ρ. To be precise, Theorem 3 asserts that the result in Theorem 2 still remains valid for domainsD_ρ that satisfy(1−r_ρ)ρD⊂D_ρ⊂(1+r_ρ)ρDwithr_ρ→0 asρ→0.

We conclude the main part of this paper with a brief discussion of potential applications of our results to ap- proximate cloaking. Appendix A of this paper contains a number of results concerning solvability, uniqueness and representation formulas for exterior problems, that were crucial for the analysis in Section 3.

2. A preliminary uniform estimate

In this section,γ₀denotes a smooth (say,C^∞) symmetric, positive definite matrix-valued function, defined onΩ, andγ_1,ρ denotes a symmetric (uniformly) positive definite matrix-valuedL^∞function defined onD_ρ. The conduc- tivityγ_ρ is given by (1.2). To simplify notation concerning we introduce

C₊^∞(Ω)=

C^∞(Ω)d×d

∩

γ (x)symmetric, positive definite, minγ >0 and

L^∞₊(D_ρ)=

L^∞(D_ρ)d×d

∩

γ (x)symmetric, positive definite, ess infγ >0 .

Here minγsignifies the largest real numberm, such thatξ^tγ (x)ξm|ξ|²for allξ∈R^d, and allx∈Ω, and ess infγ denotes the supremum of the set of real numbersmfor whichξ^tγ (x)ξm|ξ|²for allξ∈R^d, and almost allx∈D_ρ. LetF be an element ofL²(Ω), with support insideΩ\B_δ, for someδ >0, and letf be an element ofH^−1/2(∂Ω), with

ΩF dx−

∂Ωf dσ=0. Consider the standard weak solution,v_ρ∈H¹(Ω), to the boundary value problem

∇ ·(γρ∇vρ)=F inΩ, (γρ∇vρ)·n=f on∂Ω, (2.1)

normalized by

∂Ωv_ρdσ=0. Letv₀∈H¹(Ω)denote the solution to the corresponding problem withγ_ρ replaced by the background conductivityγ₀,

∇ ·(γ₀∇v₀)=F inΩ, (γ₀∇v₀)·n=f on∂Ω, (2.2)

normalized by

∂Ωv0dσ=0. These two solutions are also the minimizers of the corresponding energies E_ρ(v)=1

2

Ω

γ_ρ∇v,∇vdx+

Ω

F v dx−

∂Ω

f v dσ

and

E0(v)=1 2

Ω

γ0∇v,∇vdx+

Ω

F v dx−

∂Ω

f v dσ,

inH¹(Ω)∩ {

∂Ωv dσ=0}. The smoothness of γ0 is needed to insure thatv0 be smooth, and that all the “error”

norms are equivalent by elliptic regularity estimates. As a first result in this section we shall prove

Lemma 1.SupposeDρ⊂BKρ for some positive constantK, independent ofρ. Letγρ be given by(1.2), withγ0∈ C₊^∞(Ω)andγ1,ρ∈L^∞₊(Dρ). For a fixedδ >0, letF be an element ofL²(Ω), with support insideΩ\Bδ, and let f be an element ofH^−1/2(∂Ω), with

ΩF dx−

∂Ωf dσ =0. Letv_ρ andv₀denote the solutions to(2.1)and(2.2) respectively, normalized by

∂Ωv_ρdσ=

∂Ωv₀dσ=0. There exist a constantρ₀, independent ofγ_1,ρ,F andf, and a constantC, independent ofγ_1,ρ,ρ,F andf, such that

E_ρ(v_ρ)−E₀(v₀) Cρ^d

F²_L2(Ω\B_δ)+ f²_H−1/2(∂Ω)

, forρ < ρ₀. (2.3)

The constantsρ₀andCdepend onγ₀,Ω,δandK.

Proof. It obviously suffices to considerδsufficiently small (say, thatB_δ⊂Ω or evenB_2δ⊂Ω). Pickρ₀< δ/2Kso thatB_2Kρis contained inB_δforρ < ρ₀. We divide the proof of the estimate (2.3) into two separate cases.

The caseEρ(vρ)E0(v0). In this case

E_ρ(v_ρ)−E₀(v₀) =E_ρ(v_ρ)−E₀(v₀)E_ρ v^∗

−E₀(v₀) ∀v^∗∈H¹(Ω), (2.4)

and we proceed to construct an appropriatev^∗. Let 0χρ1 denote a smooth cut-off function with χ_ρ≡1 inB_Kρ, χ_ρ≡0 inΩ\B_2Kρ, and |∇χ_ρ|C

ρ everywhere.

Using thisχ_ρwe define v^∗=χ_ρ(x)v₀(0)+

1−χ_ρ(x) v₀(x).

Then

∇v^∗≡0 inB_Kρ(which containsD_ρ), ∇v^∗= ∇v₀ ∀x∈Ω\B_2Kρ, (2.5) and furthermore

∇v^∗(x) ²= ∇χ_ρ(x)

v₀(0)−v₀(x) +

1−χ_ρ(x)

∇v₀(x) ² 2 ∇χ_ρ(x) ² v₀(0)−v₀(x) ²+ 1−χ_ρ(x) ² ∇v₀(x) ²

2

C∇v₀²_C0(B_2Kρ)+ ∇v₀²_C0(B_2Kρ)

, (2.6)

for allx ∈B_2Kρ. The constantC is independent ofρ andγ_1,ρ. We note thatv^∗(x)=v₀(x)for x ∈∂Ω, and for x∈Ω\B_δ, and as a consequence of this and (2.5), (2.6),

2 E_ρ

v^∗

−E₀(v₀)

=

Ω

γ_ρ∇v^∗,∇v^∗ dx−

Ω

γ₀∇v₀,∇v₀dx

=

B_2Kρ\B_Kρ

γ₀∇v^∗,∇v^∗ dx−

B_2Kρ

γ₀∇v₀,∇v₀dx

B2Kρ\BKρ

γ₀∇v^∗,∇v^∗ dx

Cρ^d∇v₀²_C0(B_2Kρ)Cρ^d

F²_L2(Ω\B_δ)+ f²_H−1/2(∂Ω)

,

withC independent ofρandγ_1,ρ. This verifies (2.3) in caseE_ρ(u_ρ) > E₀(v₀).

The caseEρ(vρ) < E0(v0). In this case

E_ρ(v_ρ)−E₀(v₀) = −E_ρ(v_ρ)+E₀(v₀), (2.7)

and to get an estimate of the typeCρ^d(F²_L2(Ω\Bδ)+ f²_H−1/2(∂Ω))that is independent ofγ1,ρ, we shall introduce the dual variational principle. LetV denote the set

V = σ∈

L²(Ω)d

: ∇ ·σ=F inΩ, σ·n=f on∂Ω . Then it is well known that

Eρ(vρ)= −1 2

Ω

γρ∇vρ,∇vρdx

=max

σ∈V−1 2

Ω

γ_ρ⁻¹σ, σ dx

−1

2

Ω

γ_ρ⁻¹σ^∗, σ^∗

dx ∀σ^∗∈V . (2.8)

Since

E0(v0)= −1 2

Ω

γ0∇v0,∇v0dx it follows from (2.7) and (2.8) that

E_ρ(v_ρ)−E₀(v₀) 1 2

Ω

γ_ρ⁻¹σ^∗, σ^∗ dx−1

2

Ω

γ₀∇v₀,∇v₀dx, (2.9)

for any σ^∗∈V. We proceed to construct σ^∗∈V for which

Ωγ_ρ⁻¹σ^∗, σ^∗dx is near

Ωγ0∇v0,∇v0dx. Let Wρ denote the solution to

W_ρ=0 inB_2K\B_K,

∂W_ρ

∂n =0 on∂B_K,

∂Wρ

∂n =

γ₀∇v₀(ρx)

·n on∂(B_2K).

This problem has a solution, since

∂(B2Kρ)(γ₀∇v₀)·n dσ=0. The solution is unique up to a constant, and it satisfies the estimate

∇Wρ²_L2(B2K\BK)C∇v0²_C0(∂(B_2Kρ))C

F²_L2(Ω\Bδ)+ f²_H−1/2(∂Ω)

.

It follows immediately by rescaling thatw_ρ=ρW_ρ(^x_ρ)satisfies wρ=0 inB2Kρ\BKρ,

∂w_ρ

∂n =0 on∂(BKρ),

∂w_ρ

∂n =(γ₀∇v₀)·n on∂(B_2Kρ) and

∇w_ρ²_L2(B_2Kρ\B_Kρ)Cρ^d

F²_L2(Ω\B_δ)+ f²_H−1/2(∂Ω)

, (2.10)

with a constantCthat is independent ofρandγ_1,ρ. We now define the fieldσ^∗by the formula σ^∗=

⎧⎨

⎩

0 inBKρ,

∇w_ρ inB_2Kρ\B_Kρ, γ₀∇v₀ inΩ\B_2Kρ.

This field is clearly in[L²(Ω)]^d, and it satisfies∇ ·σ^∗=F inΩ as well asσ^∗·n=(γ₀∇v₀)·n=f on∂Ω,i.e., σ^∗is an element ofV. By usingσ^∗as a test field in (2.9) we get

E_ρ(v_ρ)−E₀(v₀) 1 2

Ω

γ_ρ⁻¹σ^∗, σ^∗ dx−1

2

Ω

γ₀∇v₀,∇v₀dx

=1 2

B2Kρ\BKρ

γ₀⁻¹∇w_ρ,∇w_ρ dx−1

2

B2Kρ

γ₀∇v₀,∇v₀dx

B2Kρ\BKρ

γ₀⁻¹∇w_ρ,∇w_ρ dx

Cρ^d

F²_L2(Ω\Bδ)+ f²_H−1/2(∂Ω)

,

withCindependent ofρand the conductivityγ_1,ρof the inhomogeneityD_ρ. For the last inequality we have used the estimate (2.10). This verifies (2.3) in caseE_ρ(u_ρ) < E₀(v₀), and thus completes the proof of Lemma 1. 2

LetF,f andγ_ρbe as in the preceding lemma. We easily calculate that

−

Ω\B_δ

F v_ρdx+

∂Ω

f v_ρdσ=

Ω

γ_ρ∇v_ρ,∇v_ρdx

= −

Ω

γρ∇vρ,∇vρdx−2

Ω

F vρdx+2

∂Ω

f vρdσ

= −2Eρ(v_ρ), and similarly

−

Ω\Bδ

F v0dx+

∂Ω

f v0dσ=

Ω

γ0∇v0,∇v0dx

= −

Ω

γ₀∇v₀,∇v₀dx−2

Ω

F v₀dx+2

∂Ω

f v₀dσ

= −2E0(v0).

As a consequence

Ω\Bδ

F (v_ρ−v₀) dx−

∂Ω

f (v_ρ−v₀) dσ =2

E_ρ(v_ρ)−E₀(v₀) ,

and so, due to Lemma 1

Ω\Bδ

F (vρ−v0) dx−

∂Ω

f (vρ−v0) dσ Cρ^d

F²_L2(Ω\Bδ)+ f²_H−1/2(∂Ω)

, (2.11)

withC independent ofρ,F,f andγ1,ρ. If we define the bounded linear operatorA_ρ:(F, f )→((v_ρ−v0)|_Ω\Bδ,

−(v_ρ−v0)|∂Ω)fromL²(Ω\B_δ)×H^−1/2(∂Ω)intoL²(Ω\B_δ)×H^1/2(∂Ω), then (2.11) simply asserts that A_ρ(F, f ), (F, f ) Cρ^d

F²_L2(Ω\B_δ)+ f²_H−1/2(∂Ω)

,

where·,·denotes the natural duality betweenL²(Ω\B_δ)×H^1/2(∂Ω)andL²(Ω\B_δ)×H^−1/2(∂Ω). We note that Aρis self-adjoint, and by “polarization” it now follows that

|||(F,f )sup|||1

|||(G,g)sup|||1

A_ρ(F, f ), (G, g) Cρ^d,

with|||(F, f )||| =(F²_L2(Ω\Bδ)+ f²_H−1/2(∂Ω))^1/2. In other words v_ρ−v0²_L2(Ω\Bδ)+ v_ρ−v0²_H1/2(∂Ω)

1/2

= sup

|||(G,g)|||1

A_ρ(F, f ), (G, g) Cρ^d, for all(F, f )withF²_L2(Ω\Bδ)+ f²_H−1/2(∂Ω)1, or

vρ−v0²_L2(Ω\Bδ)+ vρ−v0²_H1/2(∂Ω)

1/2

Cρ^d

F²_L2(Ω\Bδ)+ f²_H−1/2(∂Ω)

1/2

, withC independent ofρ,F,f andγ_1,ρ. Sincev_ρ−v₀solves the equation

∇ ·

γ0∇(vρ−v0)

=0 inΩ\Bδ, with

γ0∇(vρ−v0)

·n=0 on∂Ω,

elliptic regularity theory implies that the above estimate also holds for the Sobolev norm (v_ρ −v₀²_Hs(Ω\B_2δ)+ vρ−v0²_H^s_(∂Ω))^1/2,i.e.,

vρ−v0²_Hs(Ω\B2δ)+ vρ−v0²H^s(∂Ω)

1/2

Cρ^d

F²_L2(Ω\Bδ)+ f²_H−1/2(∂Ω)

1/2

Cρ^d

F²_L2(Ω\B_2δ)+ f²_H−1/2(∂Ω)

1/2

,

for anys∈R₊and anyF having support insideΩ\B_2δ. Replacingδwithδ/2 we have therefore established

Theorem 1. SupposeD_ρ ⊂B_Kρ for some positive constant K, independent of ρ. Let γ_ρ be given by(1.2), with γ₀∈C₊^∞(Ω)andγ_1,ρ∈L^∞₊(D_ρ). For a fixedδ >0, letF be an element ofL²(Ω), with support insideΩ\B_δ, and let f be an element ofH^−1/2(∂Ω), with

ΩF dx−

∂Ωf dσ =0. Letvρ andv0denote the solutions to(2.1)and(2.2) respectively, normalized by

∂Ωv_ρdσ =

∂Ωv₀dσ =0. Given any s∈R+ there exist a constantρ₀, independent ofγ_1,ρ,F andf, and a constantC, independent ofγ_1,ρ,ρ,F andf, such that

vρ−v0²_Hs(Ω\Bδ)+ vρ−v0²H^s(∂Ω)

1/2

Cρ^d

F²_L2(Ω\Bδ)+ f²_H−1/2(∂Ω)

1/2

, for allρ < ρ0. The constantsρ0andCdepend onγ0,Ω,δandK.

Theorem 1 is of independent importance. However for the purpose of this paper it is two corollaries, both pertaining to the special caseF =0, that are particularly relevant. The first corollary is of direct relevance to the estimation of the effectivity of approximate cloaks, as briefly discussed in Section 5.

Corollary 1.Suppose D_ρ⊂B_Kρ for some positive constant K, independent of ρ. Let γ_ρ be given by(1.2), with γ₀∈C₊^∞(Ω)andγ_1,ρ ∈L^∞₊(D_ρ). Letf be an element ofH^−1/2(∂Ω), with

∂Ωf dσ =0. Letu_ρ andu₀denote the solutions to(1.3)and(1.1)respectively, normalized by

∂Ωu_ρdσ=

∂Ωu₀dσ=0. Given anys∈R₊, and any δ∈R₊, there exist a constantρ0, independent ofγ1,ρ andf, and a constantC, independent ofγ1,ρ,f, andρ, such that

u_ρ−u_0H^s_(Ω\Bδ)+ u_ρ−u₀H^s(∂Ω)Cρ^dfH^−1/2(∂Ω) ∀ρ < ρ₀.

The second corollary, which shall prove essential for our analysis in Section 3.2, estimates the combined perturba- tion caused by the small inhomogeneity and a change in the normal flux. In order to formulate this corollary we need some additional notation. Letw_ρbe the solution to

∇ ·(γ_ρ∇w_ρ)=0 inΩ,

(γ_ρ∇w_ρ)·n=g on∂Ω, (2.12)

normalized by

∂Ωw_ρdσ=0. Here we assume thatgis an element ofH^−1/2(∂Ω), with

∂Ωg dσ=0. It follows immediately that

Ω\D_ρ

γ0∇(wρ−uρ),∇(wρ−uρ) dx

∂Ω

(g−f )(wρ−uρ) dσ

g−fH^−1/2(∂Ω)w_ρ−u_ρH^1/2(∂Ω)

Cg−fH^−1/2(∂Ω)w_ρ−u_ρH¹_(Ω\Bδ). (2.13) Due to the fact that

∂Ωw_ρdσ=

∂Ωu_ρdσ=0 we have wρ−uρ_H¹_(Ω\Bδ)C∇(wρ−uρ)

L²(Ω\Bδ).

A combination of this with (2.13) immediately gives that there exist constantsρ₀andCsuch that

w_ρ−u_ρH1(Ω\B_δ)Cg−f_H−1/2(∂Ω), forρ < ρ₀. (2.14) The constantsρ0andCare independent ofγ1,ρ,f andg. If we decompose

wρ−u0=(wρ−uρ)+(uρ−u0), and combine Corollary 1 with (2.14) we obtain

Corollary 2.Suppose D_ρ⊂B_Kρ for some positive constant K, independent of ρ. Let γ_ρ be given by(1.2), with γ0∈C₊^∞(Ω)andγ1,ρ∈L^∞₊(D_ρ). Letf, g∈H⁻¹²(∂Ω)with

∂Ωf dσ =

∂Ωg=0, and letu0and w_ρ inH¹(Ω) denote the solutions to(1.1)and(2.12), normalized by

∂Ωu0dσ=

∂Ωwρdσ=0. Given anyδ >0 there exist a constantρ₀, independent ofγ_1,ρ,f andg, and a constantC, independent ofγ_1,ρ,f,gandρ, such that

wρ−u0_H¹_(Ω\Bδ)C

g−fH^−1/2(∂Ω)+ρ^dfH^−1/2(∂Ω)

, for allρ < ρ0.

3. The leading order asymptotics whenDρ=ρD

We shall now consider the case when the inhomogeneityDρ is of the formDρ=ρD, for some bounded, simply connected, smooth domainD⊂BK. We shall examine issues related to the principal term of the expression

1

ρ^d(u_ρ−u₀)

asρ→0. We briefly describe the structure of the expression 1

ρ^d(u_ρ−u₀), ρ→0,

a structure that (forγ0andγ1,ρisotropic) is already well known,cf.[5]. What is not at all known, and what we shall prove here is that for inhomogeneities that are dilatations of a fixed setD, we can describe the expression(u_ρ−u₀)/ρ^d by an explicit (bounded) formula that is asymptotically correctuniformlyinγ_1,ρ. As pointed out in the introduction such a formula is not available for generalD_ρ. For simplicity we shall restrict attention to the case

γ_ρ(x)=

γ₀ ifx∈Ω\ρD, γ_1,ρ(x) ifx∈ρD,

whereγ0is a constant, symmetric, positive definite matrix, andγ1,ρ is an arbitrary symmetric (uniformly) positive definite matrix-valued function defined onρD, in other words

γ1,ρ∈L^∞₊(ρD)=

L^∞(ρD)d×d

∩

γ (x)symmetric, positive definite, ess infγ >0 .

By a simple linear change of variables x=Lx,γρ changes (up to a constant scalar multiple) intoLγρL^T ◦L⁻¹, Dchanges intoL(D), andΩintoL(Ω). Since we can chooseLsuch thatLγ0L^T =I we may thus, without loss of generality, assume thatγ_ρis of the form

γ_ρ(x)=

I ifx∈Ω\ρD,

γ1,ρ(x) ifx∈ρD, (3.1)

whereγ1,ρis an arbitrary matrix-valued function inL^∞₊(ρD). The novelty of the present results is the fact thatγ1,ρis an arbitrary matrix valued function inL^∞₊(ρD), and the fact that all the estimates and convergence (approximation) statements are uniform with respect toγ_1,ρ∈L^∞₊(ρD).

Letγ_ρ^∗denote the rescaled coefficient γ_ρ^∗(z)=

I ifz∈R^d\D, γ1,ρ(ρz) ifz∈D, and letφ_kbe the solution to

∇ · γ_ρ^∗∇φk

=0 inR^d, withφk(z)−zk→0 as|z| → ∞. (3.2) We note thatφk(z)=ψk(z)+zk, whereψksatisfies

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

ψk=0 inR^d\D,

∇ ·

γ_1,ρ(ρ·)∇ψ_k

= −∇ ·

γ_1,ρ(ρ·)∇z_k

inD,

∂ψk

∂n

ext

−

γ_1,ρ(ρ·)∇ψ_k

·n

int=

γ_1,ρ(ρ·)n

k−n_k on∂D,

[ψ_k] =0 on∂D,

(3.3)

andψ_k(z)→0 as|z| → ∞. Here[ψ] =ψ|ext−ψ|intdenotes the jump of the functionψacross∂D. It is easy to see that the solutionψ_k(up to a constant, ford=2) coincides with the unique solution to (3.3) inW¹(R^d), the existence of which is guaranteed by Proposition 4 in Appendix A. We define the functionL_ρ:

L_ρ(x)= ∇yΦ(x,0)·

D

I−γ_1,ρ(ρz)

∇φ_k(z) dz ∂

∂xk

u₀(0), (3.4)

and the functionW_ρ:

⎧⎨

⎩

W_ρ=0 inΩ,

∂Wρ

∂n = −∂Lρ

∂n on∂Ω, (3.5)

normalized by

∂ΩW_ρdσ= −

∂ΩL_ρdσ. HereΦ(x, y)is the free space fundamental solution for the Laplacian Φ(x, y)=

−_2π¹ ln|x−y| ifd=2,

1

4π|x−y| ifd=3.

We note thatψk,φk,Lρ andWρ generically all depend onρ; they would be independent ofρif we only considered γ1,ρ∈L^∞₊(ρD)of the formγ1,ρ(x)=γ1(x/ρ)for a given fixedγ1∈L^∞₊(D)(for instance if we considered inho- mogeneitiesρDof fixed constant conductivityγ1). It is easy to see that the matrix

D(I−γ1,ρ(ρz))ij∂jφk(z) dzis symmetric.

In the caseγ1,ρ is constant (but possibly dependent onρ) we define the matrix M_{j k}= 1

|D|

D

∂φk

∂zj

dz. (3.6)

We note thatMmay depend onρ, and that it is not necessarily symmetric (though(I−γ_1,ρ)Mis). Then Lρ(x)= |D|(I−γ1,ρ)ijMj k

∂

∂x_ku0(0) ∂

∂y_iΦ(x,0), and it is easy to see that

L_ρ(x)+W_ρ(x)= |D|(I−γ1,ρ)_ijM_{j k} ∂

∂x_ku0(0) ∂

∂y_iG(x,0), (3.7)

whereGis the special Green’s function introduced in (1.6). Here and in the future we use the Einstein summation convention,i.e., repeated indices (representing integers) in a single term implies summation from 1 tod.

Theorem 2.Supposef is inH^−1/2(∂Ω)with

∂Ωf dσ=0, andγρ is given by(3.1)withγ1,ρ∈L^∞₊(ρD). Letu0

andu_ρ denote the solutions to(1.1)and(1.3), normalized by

∂Ωu₀dσ=

∂Ωu_ρdσ=0. LetL_ρ(x)andW_ρ(x)be given by(3.4)and(3.5). Then for any fixedδ >0,

ρlim→0

Ω\Bδ

1

ρ^d∇(u_ρ−u₀)− ∇L_ρ− ∇W_ρ

²dx=0.

The limiting process is uniform in γ_1,ρ∈L^∞₊(ρD)and inf ∈ {f

H⁻¹²(∂Ω)1,

∂Ωf dσ =0}. That is, for any >0andδ >0, there exists a positive constantρ₀(, δ), such that

Ω\Bδ

1

ρ^d∇[u_ρ−u₀] − ∇L_ρ− ∇W_ρ

²dx < , for allρ < ρ0, allγ1,ρ∈L^∞₊(ρD)and allf∈ {f

H⁻¹2(∂Ω)1,

∂Ωf dσ=0}. The term∇Lρ+ ∇Wρ is bounded inL²(Ω\Bδ), uniformly with respect toρ,γ1,ρ∈L^∞₊(ρD), andf ∈ {f

H⁻¹²(∂Ω)1,

∂Ωf dσ=0}.

We may without loss of generality suppose thatBδ⊂Ω and thatKρ0< δ/2. Then the function ¹

ρ^d(uρ−u0)− (L_ρ+W_ρ)is harmonic inΩ\B_δ/2, with _∂n^∂ ((u_ρ−u₀)−(L_ρ+W_ρ))=0 on∂Ω. A combination of standard elliptic theory and the energy estimate of Theorem 2 therefore yields similar estimates in anyH^s(Ω\B_δ)(orH^s(∂Ω)) norm, as stated in the following corollary.