High-Order Terms in the Asymptotic Expansions of the Steady-State Voltage Potentials in the Presence of Conductivity Inhomogeneities of Small Diameter ^{∗}

Habib Ammari ^{†}

Centre de Mathématiques Appliquées

Ecole Polytechnique 91128 Palaiseau Cedex, France ammari@cmapx.polytechnique.fr

Hyeonbae Kang^{‡} School of Mathematical Sciences Seoul National University Seoul 151-747, Korea hkang@math.snu.ac.kr

December 12, 2001

Abstract

We derive high-order terms in the asymptotic expansions of the steadystate voltage potentials in the presence of a finite number of diametrically small inhomogeneities with conductivities different from the background conductivity. Our derivation is rigorous, and based on layer potential techniques. The asymptotic expansions in this paper are valid for inhomogeneities with Lipschitz boundaries and those with extreme conductivities.

Mathematics subject classification (MSC2000): 35B30 Keywords: small conductivity inhomogeneities, asymptotic expansions, generalized polarization tensors Short title: Small conductivity inhomogeneities

1 Introduction

Let Ω be a bounded domain in ℝ^{d}, d ≥ 2, with a connected Lipschitz boundary ∂Ω. Let ν denote the unit outward normal to ∂Ω. Suppose that Ω contains a finite

∗This work was completed while both authors were at the Mathematical Sciences Research Institute (MSRI) during the special program on inverse problems. They would like to thank Gunther Uhlmann for invitation and the Institute for partial support and providing stimulating environment.

†partly supported by ACI Jeunes Chercheurs (0693) from the Ministry of Education and Scientific Research, France.

‡partly supported by KOSEF 98-0701-03-5 and BK21 at the School of Mathematical Sciences of SNU.

number m of small inhomogeneities (D_{l})_{l=1}^{m}, each of the form D_{l} = z_{l} + ϵB_{l}, where B_{l}, l = 1, …, m, is a bounded Lipschitz domain in ℝ^{d} containing the origin. We assume that the domains (D_{l})_{l=1}^{m} are separated apart from each other and apart from the boundary. More precisely, we assume that there exists a constant c_{0} > 0 such that

(1.1)  |z_{l} − z_{l′}| ≥ 2c_{0} > 0, ∀ l ≠ l′ and dist(z_{l}, ∂Ω) ≥ 2c_{0} > 0, ∀ l,

and ϵ, the common order of magnitude of the diameters of the inhomogeneities, is sufficiently small, that these inhomogeneities are disjoint, and that their distance to ℝ^{d} \\overline{Ω} is larger than c_{0}. We also assume that the ”background” is homogeneous with conductivity 1 and the inhomogeneities D_{l} have conductivities k_{l}, k_{l} ≠ 1, 1 ≤ l ≤ m.

Let u_{ϵ} denote the steady-state voltage potential in the presence of the conductivity inhomogeneities, i.e., the solution to

(1.2)  \{ \begin{matrix}∇·(χ(Ω \ ⋃_{l=1}^{m} \overline{D_{l}}) + ∑_{l=1}^{m} k_{l}χ(D_{l}))∇u_{ϵ} = 0 in Ω, \\ \frac{∂u_{ϵ}}{∂ν} |_{∂Ω} = g. \end{matrix}

Let U denote the ”background” potential, that is, the solution to

(1.3)  \begin{cases}ΔU = 0 in Ω, \\ \frac{∂U}{∂ν} |_{∂Ω} = g. \end{cases}

The function g represents the applied boundary current; it belongs to L_{0}^{2}(∂Ω) = {g ∈ L^{2}(∂Ω), ∫_{∂Ω} g = 0}. The potentials, u_{ϵ} and U, are normalized by ∫_{∂Ω} u_{ϵ} = ∫_{∂Ω} U = 0.

The main achievement of this paper is a rigorous derivation, based on layer potential techniques, of high-order terms in the asymptotic expansion of u_{ϵ}|_{∂Ω} as ϵ → 0. The leading order term in this asymptotic formula has been derived by Cedio-Fengya et al. [7]; see also the prior work of Friedman and Vogelius [14] for the case of perfectly conducting or insulating inhomogeneities. The main result of this paper is the following full asymptotic expansion of the solution for the case m = 1:

Theorem 1.1 Suppose that the inhomogeneity consist of single component and let u_{ε} be the solution of (1.2). The following pointwise asymptotic expansion on ∂Ω holds for d = 2, 3:

(1.4)  \begin{aligned}u_{ε}(x) & = U(x) − ϵ^{d−2} ∑_{|i|=1}^{n} ∑_{|j|=1}^{n−|i|+1} \frac{1}{j!}ϵ^{|i|+|j|}× \\ & [((I + ∑_{p=1}^{n+2−|i|−|j|−d} ϵ^{d+p−1}Q_{p})(∂^{l}U(z)))_{i}M_{ij}∂_{z}^{j}N(x, z)] \\ & +O(ϵ^{d+n}), \end{aligned}

where the remainder O(ϵ^{d+n}) is dominated by Cϵ^{d+n}∥g∥_{L^{2}(∂Ω)} for some C independent of x ∈ ∂Ω. Here N(x, z) is the Neumann function, M_{ij}, i, j ∈ ℕ^{d}, are the generalized polarization tensors defined in (3.2), and the operator Q_{p} is defined in (4.12).

We have a similar expansion for the solutions of the Dirichlet problem (Theorem 4.2.

The derivation of the asymptotic expansions for any fixed number m of well separated inhomogeneities (these are a fixed distance apart) follows by iteration of the arguments that we will present for the case m = 1. In other words, we may develop asymptotic formulas involving the difference between the fields u_{ϵ} and U on ∂Ω with l inhomogeneities and those with l − 1 inhomogeneities, l = m, …, 1, and then at the end essentially form the sum of these m formulas (the reference fields change, but that may easily remedied). The derivation of each of the m formulas is virtually identical.

We also note that the asymptotic expansion (1.4) is valid for inhomogeneities with zero or infinity conductivity (cavity or perfect conductor). Precise definitions of generalized polarization tensors (GPT) associated with the domains B_{l} and the conductivities k_{l} will be given at the end of section 3. These GPT seem to be natural generalizations of the tensors that have been introduced by Schiffer and Szegö [23] and thoroughly studied by many other authors [22], [18], [14], [7]. (See section 3.)

The higher-order terms are essential when ∇U(z_{l}) = 0, for then the leading order term in the asymptotic expansion of u_{ϵ}|_{∂Ω}, given in [7], vanishes. We remind the reader that, for general current inputs g, ∇U vanishes at some ”critical points” inside Ω.

The proof of our asymptotic expansion is radically different from the ones in [14], [7], and [26]. It is based on layer potential techniques and a decomposition formula of the steady-state voltage potential into a harmonic part and a refraction part. This formula is due to Kang and Seo [15]. What makes our proof particularly original and elegant is that the rigorous derivation of high-order terms follows almost immediately. The extension of the techniques used in [14], [7], and [26] to construct higher-order terms in the expansion of u_{ϵ}|_{∂Ω} as ϵ → 0 seems to be laborious. Furthermore, the general approach developed in this paper could be carried out to obtain more precise asymptotic formulas for the full Maxwell’s equations and for the equations of linear elasticity than those derived in [3] and [1]. The method of this paper also enables us to extend the asymptotic expansions to the cases of inhomogeneities with Lipschitz boundaries. Previously, the leading order term was derived under the assumption that inhomogeneities are C^{1,α} smooth [14] [7]. We note that our method works as well even when the inhomogeneities have extreme conductivities (k = 0 or k = ∞).

Let us now explain what makes this asymptotic formula interesting in the electrical impedance tomography (EIT). It is well-known that the ultimate objective of EIT is to recover, most efficiently and accurately, the conductivity distribution inside a body from measurements of current flows and voltages on the body’s surface. The vast and growing literature reflects the many possible applications of EIT, e.g. for medical diagnosis or nondestructive evaluation of materials [6]. In its most general form EIT is severely ill-posed and nonlinear. Taking advantage of the smallness of the inhomogeneities, Cedio-Fengya et al. [7] have used the leading order term in the asymptotic expansion of u_{ϵ}|_{∂Ω} to find the locations z_{l}, l = 1, …, m of the inhomogeneities and certain properties of the domains B_{l}, l = 1, …, m (relative size, orientation). The algorithm proposed by Cedio-Fengya et al. [7] is based on a least-square algorithm. Ammari et al. [2] have also utilized this leading order term to design a variationally based direct reconstruction method. The new idea in [2] is to form the integral of the ”measured boundary data” against harmonic test functions and choose the input current g so as to obtain expression involving the inverse Fourier transform of distributions supported at the locations z_{l}, l = 1, …, m. Applying a direct Fourier transform to this data then pins down the locations. This approach is similar to the ideas used by Calderón [5] in his proof of uniqueness of the linearized conductivity problem and later, by Sylvester and Uhlmann in their important work [24] on uniqueness of the three-dimensional inverse conductivity problem. An other algorithm that makes use of an asymptotic expansion of the voltage potentials has been derived by Brühl et al. [4]. This algorithm is in the spirit of the linear sampling method of Colton and Kirsch [9].

In all of these algorithms, the locations z_{l}, l = 1, …, m of the inhomogeneities are found with an error O(ϵ) and little about the domains B_{l} can be reconstructed. Making use of higher-order terms in the asymptotic expansion of u_{ϵ}|_{∂Ω} we certainly would be able to reconstruct the small inhomogeneities with higher resolution from boundary information about specific solutions to (1.2). Perhaps, more importantly, this would allow us to identify quite general conductivity inhomogeneities without restrictions on their sizes.

The use of higher-order terms in the asymptotic expansion of u_{ϵ}|_{∂Ω} may also be decisive in dramatically improving the algorithm of Kwon et al. [19], that is based on the observation of the pattern of a simple weighted combination of an input current g of the form g = a · ν for some constant vector a and the corresponding output voltage. We also believe that the use of such higher-order terms would improve the algorithm of Mast et al. [20], that uses eigenfunctions of the scattering operator.

This paper is organized as follows. In section 2, we collect some notations and preliminary regarding layer potentials. In section 3, we introduce the generalized polarization tensors associated with the domains D_{l} and the conductivities k_{l}. In section 4, we provide a rigorous derivation of high-order terms in the asymptotic expansion of the output voltage potentials. For reasons of brevity we restrict a significant part of this derivation to the case of a single inhomogeneity (m = 1). The proof in the case of multiple well-separated inhomogeneities may be derived by a fairly straightforward iteration of the arguments we present, however we leave the details to the reader.

2 Layer Potentials for the Laplacian

Let us first review some well-known properties of the layer potentials for the Laplacian and prove some useful identities.

The theory of layer potentials has been developed in relation to the boundary value problems. Let D be a bounded domain in ℝ^{d}, d ≥ 2. We assume that ∂D is Lipschitz. Let Γ(x) be the fundamental solution of the Laplacian Δ:

(2.1)  Γ(x) = \begin{cases}\frac{1}{2π} ln |x|, & d = 2, \\ \frac{1}{(2 − d)ω_{d}} |x|^{2−d}, & d ≥ 3, \end{cases}

where ω_{d} is the area of (d − 1) dimensional unit sphere. The single and double layer potentials of the density function ϕ on D is defined by

(2.2)  S_{D}ϕ(x) := ∫_{∂D} Γ(x − y)ϕ(y)dσ(y), x ∈ ℝ^{d},

(2.3)  D_{D}ϕ(x) := ∫_{∂D} \frac{∂}{∂ν_{y}} Γ(x − y)ϕ(y)dσ(y), x ∈ ℝ^{d} \ ∂D.

For a function u defined on ℝ^{d} \ ∂D, we denote

\frac{∂}{∂ν^{±}} u(x) := lim_{t→0^{+}}⟨∇u(x ± tν_{x}), ν_{x}⟩, x ∈ ∂D

if the limit exists. Here ν_{x} is the outward unit normal to ∂Ω at x.

The proof of the following trace formula can be found in [11, 13, 21] (for Lipschitz domains, see [25]):

(2.4)  \frac{∂}{∂ν^{±}} S_{D}ϕ(x) = (±\frac{1}{2}I + K_{D}^{∗})ϕ(x),
(2.5)  (Dϕ)|_{±} = (∓\frac{1}{2}I + K_{D})ϕ(x), x ∈ ∂D,

where

K_{D}ϕ(x) = \frac{1}{ω_{d}} p.v. ∫_{∂D} \frac{⟨x − y, ν_{y}⟩}{|x − y|^{d}}ϕ(y)dσ(y)

and K_{D}^{∗} is the L^{2}-adjoint of K_{D}. When ∂D is Lipschitz, K_{D} is a singular integral operator and known to be bounded on L^{2}(∂Ω) [8]. Let L_{0}^{2}(∂D) := {f ∈ L^{2}(∂D) : ∫_{∂D} fdσ = 0}. The following results are due to Verchota and Escauriaza-Fabes-Verchota.

Theorem 2.1 ([10], [25]) λI − K_{D}^{∗} is invertible on L_{0}^{∞}(∂D) if |λ| ≥ \frac{1}{2}, and for λ ∈ (−∞, −\frac{1}{2}] ∪ (\frac{1}{2}, ∞), λI − K_{D}^{∗} is invertible on L^{2}(∂D).

For proofs when ∂D is smooth, see [11], [13].

The following theorem was proved in [15, 16, 17].

Theorem 2.2 Suppose that D is a domain compactly contained in Ω with a connected Lipschitz boundary and conductivity k. Then the solution u of the problem

(2.6)  \{ \begin{matrix}∇ · ((1 + (k − 1)χ(D))∇u) = 0 in Ω, \\ \frac{∂u}{∂ν}|_{∂Ω} = g \end{matrix}

is represented as

(2.7)  u(x) = H(x) + S_{D}ϕ(x), x ∈ Ω,

where the harmonic function H is given by

(2.8)  H(x) = −S_{Ω}(g)(x) + D_{Ω}(f)(x), x ∈ Ω, f := u|_{∂Ω},

and ϕ ∈ L_{0}^{2}(∂D) satisfies the integral equation

(2.9)  (\frac{k + 1}{2(k − 1)}I − K_{D}^{∗})ϕ = \frac{∂H}{∂ν} |_{∂D} on ∂D.

Moreover, ∀ n ∈ ℕ, there exists a constant C_{n} = C(n, Ω, dist(D, ∂Ω)) independent of |D| and k such that

(2.10)  ∥H∥_{C^{n}(\overline{D})} ≤ C_{n}∥g∥_{L^{2}(∂Ω)}.

Proof. The representation formula (2.7) was proved in [15, 17]. (2.10) was proved in [16] for d = 2 and it is easily seen that the same proof works for d = 3. We only need to check carefully whether the constant C_{n} in the estimate (2.10) is independent of |D|. Before doing this, let us point out that the harmonic function H can be computed explicitly from the boundary measurements (\frac{∂u}{∂ν} |_{∂Ω}, u|_{∂Ω}) and the density ϕ is uniquely and explicitly determined by the domain D and the harmonic function H. The decomposition of the function u into a harmonic part H and a refraction part S_{D}ϕ is unique [15, 17]. The representation formula (2.7) seems to inherit geometric properties of D.

Suppose that dist(D, ∂Ω)) > 2c_{0} for some constant c_{0} > 0. From the definition of H in (2.8) it is easy to see that

(2.11)  ∥H∥_{C^{n}(\overline{D})} ≤ C_{n}(∥g∥_{L^{2}(∂Ω)} + ∥u|_{∂Ω}∥_{L^{2}(∂Ω)}),

where C_{n} depends only on n, ∂Ω, and c_{0}. Let α⃗ be a vector field supported in the set dist(x, ∂Ω) < 2c_{0} such that α⃗ · ν(x) ≥ δ for some δ > 0 for all x ∈ ∂Ω. Using the Rellich identity with this α⃗, we can show that

∥\frac{∂u}{∂T} ∥_{L^{2}(∂Ω)} ≤ C(∥g∥_{L^{2}(∂Ω)} + ∥∇u∥_{L^{2}(Ω\\overline{D})}),

where C depends only on ∂Ω and c_{0} and T (x) is the tangent vector to ∂Ω at x. (See the proof of Lemma 2.1 of [12] for the detail of the proof.) Observe that

∥∇u∥_{L^{2}(Ω\\overline{D})}^{2} ≤ C ∫_{Ω}(1 + (k − 1)χ(D))∇u · ∇u dx
= C ∫_{∂Ω} gu dσ
≤ C∥g∥_{L^{2}(∂Ω)}∥u|_{∂Ω}∥_{L^{2}(∂Ω)}.

Since ∫_{∂Ω} u dσ = 0, it follows from the Poincaré inequality that

∥u|_{∂Ω}∥_{L^{2}(∂Ω)} ≤ C∥\frac{∂u}{∂T} ∥_{L^{2}(∂Ω)}.

Thus we obtain

∥u|_{∂Ω}∥_{L^{2}(∂Ω)}^{2} ≤ C(∥g∥_{L^{2}(∂Ω)}^{2} + ∥g∥_{L^{2}(∂Ω)}∥u|_{∂Ω}∥_{L^{2}(∂Ω)}),

and hence

∥u|_{∂Ω}∥_{L^{2}(∂Ω)} ≤ C∥g∥_{L^{2}(∂Ω)}.

From (2.11) we finally obtain (2.10). □

Using above representation we can derive a formula similar to (1.4) which is potentially useful in detecting the inhomogeneities (see the remark at the end of this paper). However it uses the function H which depends on D and hence on ϵ. Thus in order to derive (1.4) we will transform it to representations using only background potentials.

Let N(x, y) be the Neumann function for Δ in Ω corresponding to a Dirac mass at z. That is, N is the solution to

\{ \begin{matrix}Δ_{x}N(x, z) = −δ_{z} in Ω, \\ \frac{∂N}{∂ν} |_{∂Ω} = −\frac{1}{|∂Ω|}. \end{matrix}

In addition, we assume that

(2.12)  ∫_{∂Ω} N(x, y)dσ(x) = 0 for y ∈ Ω.

Let us fix one more notation: For D, a subset of Ω, let

N_{D}f(x) := ∫_{∂D} N(x, y)f(y)dσ(y).

The following lemma relates the fundamental solution with the Neumann function.

Lemma 2.3 For z ∈ Ω and x ∈ ∂Ω, let Γ_{z}(x) := Γ(x − z) and N_{z}(x) := N(x, z). Then

(2.13)  (−\frac{1}{2}I + K_{Ω})(N_{z})(x) = Γ_{z}(x) modulo constants, x ∈ ∂Ω,

or to be more precise, for any simply connected Lipschitz domain D compactly contained in Ω and for any g ∈ L_{0}^{2}(∂D), we have

(2.14) ∫_{∂D}(−\frac{1}{2}I + K_{Ω})(N_{z})(x)g(z)dσ(z) = ∫_{∂D} Γ_{z}(x)g(z)dσ(z), ∀x ∈ ∂Ω.

Proof. Let f ∈ L_{0}^{2}(∂Ω) and define

u(z) := ⟨(−\frac{1}{2}I + K_{Ω})(N_{z}), f⟩_{∂Ω} z ∈ Ω.

Then

u(z) = ∫_{∂Ω} N(x, z)(−\frac{1}{2}I + K_{Ω}^{∗})f(z)dσ(z).

Therefore, Δu = 0 in Ω and \frac{∂u}{∂ν}|_{∂Ω} = (−\frac{1}{2}I + K_{Ω}^{∗})f. Thus by (2.4) we have

u(z) − S_{Ω}f(z) = constant, z ∈ Ω.

Thus if g ∈ L_{0}^{2}(∂Ω), then we obtain

\begin{matrix}∫_{∂Ω} ∫_{∂D}(−\frac{1}{2}I + K_{Ω})(N_{z})(x)g(z)dσ(z)f(x)dσ(x) \\ = ∫_{∂Ω} ∫_{∂D} Γ_{z}(x)g(z)dσ(z)f(x)dσ(x). \end{matrix}

Since f is arbitrary, we have (2.13) or equivalently, (2.14). This completes the proof. □

Let g ∈ L_{0}^{2}(∂Ω). Let U(y) := ∫_{∂Ω} N(x, y)g(x)dσ(x). Then U satisfies

(2.15)  \{ \begin{matrix}ΔU = 0 in Ω, \\ \frac{∂U}{∂ν} |_{∂Ω} = g ∈ L_{0}^{2}(∂Ω), \\ ∫_{∂Ω} U(x)dσ(x) = 0. \end{matrix}

Theorem 2.4 The solution u of (2.6) can be represented as

(2.16)  u(x) = U(x) − N_{D}ϕ(x), x ∈ ∂Ω

where ϕ is defined in (2.9).

Proof. By substituting (2.7) into the equation (2.8), we obtain

H(x) = −S_{Ω}(g)(x) + D_{Ω}(H|_{∂Ω} + (S_{D}ϕ)|_{∂Ω})(x), x ∈ Ω.

It then follows from (2.5) that

(2.17)  (\frac{1}{2}I − K_{Ω})(H|_{∂Ω}) = −(S_{Ω}g)|_{∂Ω} + (\frac{1}{2}I + K_{Ω})((S_{D}ϕ)|_{∂Ω}), on ∂Ω.

Since U = −S_{Ω}(g) + D_{Ω}(U|_{∂Ω}) in Ω, we have

(2.18)  (\frac{1}{2}I − K_{Ω})(U|_{∂Ω}) = −(S_{Ω}g)|_{∂Ω}.

Since ϕ ∈ L_{0}^{2}(∂D), it follows from (2.13) that

(2.19)  −(\frac{1}{2}I − K_{Ω})((N_{D}ϕ)|_{∂Ω}) = (S_{D}ϕ)|_{∂Ω}.

From (2.17), (2.18), and (2.19), we conclude that

(\frac{1}{2}I − K_{Ω})(H|_{∂Ω} − U|_{∂Ω} + (\frac{1}{2}I + K_{Ω})((N_{D}ϕ)|_{∂Ω})) = 0.

Therefore, we have

(2.20)  H|_{∂Ω} − U|_{∂Ω} + (\frac{1}{2}I + K_{Ω})((N_{D}ϕ)|_{∂Ω}) = C (constant).

Note that (\frac{1}{2}I + K_{Ω})((N_{D}ϕ)|_{∂Ω}) = (N_{D}ϕ)|_{∂Ω} + (S_{D}ϕ)|_{∂Ω}. Thus we get from (2.7) and (2.20)

(2.21)  u|_{∂Ω} = U|_{∂Ω} − (N_{D}ϕ)|_{∂Ω} + C.

Since all the functions entering in (2.21) belong to L_{0}^{2}(∂Ω), we conclude that C = 0, and the theorem is proved. □

We have a similar representation for solutions of the Dirichlet problem. Let G(x, y) be the Green function for the Dirichlet problem, i.e., the function V defined by V (x) := ∫_{∂Ω} \frac{∂G}{∂ν(y)}(x, y)f(y)dσ(y) is the solution of the problem ΔV = 0 in Ω and V |_{∂Ω} = f for any f ∈ L^{2}(∂Ω). Then we have the following representation theorem.

Theorem 2.5

(2.22)  (\frac{1}{2}I + K_{Ω}^{∗})^{−1}(\frac{∂Γ_{z}(y)}{∂ν(y)})(x) = \frac{∂G_{z}}{∂ν(x)}(x), x ∈ ∂Ω, z ∈ Ω.

Let u be the solution of (2.6) with the Neumann condition replaced by the Dirichlet condition u|_{∂Ω} = f. Then u can be represented as

(2.23)  \frac{∂u}{∂ν}(x) = \frac{∂V}{∂ν} (x) − G_{D}ϕ(x), x ∈ ∂Ω

where ϕ is defined in (2.9) and G_{D}ϕ(x) := ∫_{∂D} \frac{∂G}{∂ν(y)}(x, y)ϕ(y)dσ(y).

Theorem 2.5 can be proved in the same way as Theorem 2.4. In fact, it is simpler because of the solvability of the Dirichlet problem, or equivalently, the invertibility of (\frac{1}{2}I + K_{Ω}^{∗}). So we omit the proof.

3 Generalized Polarization Tensors

In this section we introduce the generalized polarization tensors (GPT) associated with a domain B and a conductivity k. These GPT are the basic building block for the asymptotic expansions in this paper.

Let B be a Lipschitz bounded domain in ℝ^{d} and the conductivity of B be k (k ≠ 1). The polarization tensor is M = (m_{ij}), 1 ≤ i, j ≤ d, is defined by

m_{ij} := (k − 1) [δ_{ij}|B| + ∫_{∂B} y_{i} \frac{∂}{∂ν^{+}} ψ_{j}(y)dσ(y)] ,

where ψ_{j} is the unique solution of the following transmission problem:

\begin{cases}Δψ_{j}(x) = 0, x ∈ B ∪ ℝ^{d} \ \overline{B}, \\ ψ_{j}|_{+} − ψ_{j}|_{−} = 0 on ∂B, \\ \frac{∂}{∂ν^{+}} ψ_{j} − k \frac{∂}{∂ν^{−}} ψ_{j} = ν_{j} on ∂B, \\ ψ_{j}(x) → 0 as |x| → ∞. \end{cases}

See [23], [7], and [14]. One can easily check using (2.4) that

ψ_{j} = S_{B}(λI − K_{B}^{∗})^{−1}(ν_{j}).

Using (2.4) again, we have

∫∂By_{i} \frac{∂}{∂ν^{+}} ψ_{j}(y)dσ(y)

= − ∫_{∂B} y_{i}(\frac{1}{2}I − K_{B}^{∗})(λI − K_{B}^{∗})^{−1}(ν_{j})(y)dσ(y)

= − ∫_{∂B} y_{i}ν_{j}dσ(y) + (λ − \frac{1}{2}) ∫_{∂B} y_{i}(λI − K_{B}^{∗})^{−1}(ν_{j})(y)dσ(y)

= −δ_{ij}|B| + \frac{1}{k − 1} ∫_{∂B} y_{i}(λI − K_{B}^{∗})^{−1}(ν_{j})(y)dσ(y).

Therefore we prove that the polarization tensor M associated with B and k is given by

(3.1)  m_{ij} = ∫_{∂B} y_{i}(λI − K_{B}^{∗})^{−1}(ν_{j})(y)dσ(y).

Recall λ := \frac{k+1}{2(k−1)}.

For a multi-index i = (i_{1}, …, i_{d}) ∈ ℕ^{d}, let ∂^{i}f = ∂_{1}^{i_{1}} … ∂_{d}^{i_{d}}f and x^{i} := x_{1}^{i_{1}} ⋯ x_{d}^{i_{d}}. For i, j ∈ ℕ^{d}, we define the generalized polarization tensor M_{ij} by

(3.2)  M_{ij} := ∫_{∂B} y^{j}ϕ_{i}(y)dσ(y),

where ϕ_{i} is defined by

ϕ_{i}(x) := (λI − K_{B}^{∗})^{−1}(\frac{1}{i!}ν_{y} · ∇y^{i})(x), x ∈ ∂B.

4 Derivation of the Full Asymptotic Formula

In this section we derive our asymptotic formula (1.4). As stated in the introduction, we restrict our derivation to the case of a single inhomogeneity (m = 1). We only give the details when considering the difference between the fields corresponding to one and zero inhomogeneities. In order to further simplify notation we assume that the single inhomogeneity D has the form D = ϵB + z, where z ∈ Ω and B is a bounded Lipschitz domain in ℝ^{d} containing the origin. Suppose that the conductivity of D is k. Let λ := \frac{k+1}{2(k−1)}. Then by (2.7) and (2.9), the solution u of (2.6) takes the form

u(x) = U(x) − N_{D}(λI − K_{D}^{∗})^{−1}(\frac{∂H}{∂ν} |_{∂D})(x), x ∈ ∂Ω,

where U is the background potential given in (2.15).

Define

H_{n}(x) := ∑_{|i|=0}^{n} \frac{1}{i!}(∂^{i}H)(z)(x − z)^{i}.

Then we have from (2.10) that

∥\frac{∂H}{∂ν} − \frac{∂H_{n}}{∂ν} ∥_{L^{2}(∂D)} ≤ sup_{x∈∂D} |∇H(x) − ∇H_{n}(x)||∂D|^{1/2}
≤ ∥H∥_{C^{n+1}(\overline{D})}|x − z|^{n}|∂D|^{1/2}
≤ C∥g∥_{L^{2}(∂Ω)}ϵ^{n}|∂D|^{1/2}.

Note that

(4.1)  if ∫_{∂D} hdσ = 0, then ∫_{∂D}(λI − K_{D}^{∗})^{−1}hdσ = 0.

If ∫_{∂D} hdσ = 0, then we have for x ∈ ∂Ω that

|N_{D}(λI − K_{D}^{∗})^{−1}h(x)| = |∫_{∂D}[N(x − y) − N(x − z)](λI − K_{D}^{∗})^{−1}h(y)dσ(y)|
≤ Cϵ|∂D|^{1/2} ∥h∥_{L^{2}(∂D)} .

It then follows that

|N_{D}(λI − K_{D}^{∗})^{−1}(\frac{∂H}{∂ν} |_{∂D} − \frac{∂H_{n}}{∂ν} |_{∂D})(x)| ≤ Cϵ|∂D|^{1/2} ∥\frac{∂H}{∂ν} − \frac{∂H_{n}}{∂ν} ∥_{L^{2}(∂D)}
≤ C∥g∥_{L^{2}(∂Ω)}ϵ^{d+n}.

Therefore, we have

(4.2)  u(x) = U(x) − N_{D}(λI − K_{D}^{∗})^{−1}(\frac{∂H_{n}}{∂ν} |_{∂D})(x) + O(ϵ^{d+n}), x ∈ ∂Ω

where O(ϵ^{d+n}) term is dominated by C∥g∥_{L^{2}(∂Ω)}ϵ^{d+n} for some C depending only on c_{0}. Note that

(λI − K_{D}^{∗})^{−1}(\frac{∂H_{n}}{∂ν} |_{∂D})(x) = ∑_{|i|=1}^{n}(∂^{i}H)(z)(λI − K_{D}^{∗})^{−1}(\frac{1}{i!}ν_{x} · ∇(x − z)^{i})(x).

Since D = ϵB + z, one can prove by using the change of variables y = \frac{x−z}{ϵ} that

(λI − K_{D}^{∗})^{−1}(\frac{1}{i!}ν_{x} · ∇(x − z)^{i})(x) = ϵ^{|i|−1}(λI − K_{B}^{∗})^{−1}(\frac{1}{i!}ν_{y} · ∇y^{i})(\frac{1}{ϵ}(x − z)).

Put

(4.3)  ϕ_{i}(x) := (λI − K_{B}^{∗})^{−1}(\frac{1}{i!}ν_{y} · ∇y^{i})(x), x ∈ ∂B.

Then we get

N_{D}(λI − K_{D}^{∗})^{−1}(\frac{∂H_{n}}{∂ν} |_{∂D})(x) (4.4)

= ∑_{|i|=1}^{n}(∂^{i}H)(z)ϵ^{|i|−1} ∫_{∂D} N(x, y)ϕ_{i}(ϵ^{−1}(y − z))dσ(y)

= ∑_{|i|=1}^{n}(∂^{i}H)(z)ϵ^{|i|+d−2} ∫_{∂B} N(x, ϵy + z)ϕ_{i}(y)dσ(y).

We now expand N(x, ϵy + z) asymptotically as ϵ → 0. By (2.13) we have the following relation:

(−\frac{1}{2}I + K_{Ω})[N(·, ϵy + z)](x) = Γ(x − z − ϵy) modulo constants, x ∈ ∂Ω.

Since

Γ(x − ϵy) = ∑_{|j|=0}^{+∞} \frac{1}{j!}ϵ^{|j|}∂^{j}(Γ(x))y^{j},

we obtain

(−\frac{1}{2}I + K_{Ω})[N(·, ϵy + z)](x) = ∑_{|j|=0}^{+∞} \frac{1}{j!}ϵ^{|j|}∂^{j}(Γ(x − z))y^{j}

= (−\frac{1}{2}I + K_{Ω}) [ ∑_{|j|=0}^{+∞} \frac{1}{j!}ϵ^{|j|}∂_{z}^{j}N(·, z)y^{j}] (x).

Since ∫_{∂Ω} N(x, w)dσ(x) = 0 for all w ∈ Ω, we have the following asymptotic expansion of the Neumann function which is of independent interest.

Lemma 4.1 For x ∈ ∂Ω, z ∈ Ω, and y ∈ ∂B, and ϵ → 0,

(4.5)  N(x, ϵy + z) = ∑_{|j|=0}^{+∞} \frac{1}{j!}ϵ^{|j|}∂_{z}^{j}N(x, z)y^{j}.

We now have from (4.4)

N_{D}(λI − K_{D}^{∗})^{−1}(\frac{∂H_{n}}{∂ν} |_{∂D})(x)

= ∑_{|i|=1}^{n}(∂^{i}H)(z)ϵ^{|i|+d−2} ∑_{|j|=0}^{+∞} \frac{1}{j!}ϵ^{|j|}∂_{z}^{j}N(x, z) ∫_{∂B} y^{j}ϕ_{i}(y)dσ(y).

Observe that

∑_{|i|=l} \frac{1}{i!}(∂^{i}H)(z)Δ(y^{i}) = Δ_{y} (∑_{|i|=l} \frac{1}{i!}(∂^{i}H)(z)y^{i}) = 0,

and therefore, by the Green’s theorem, it follows that

∫_{∂B} ∑_{|i|=l} \frac{1}{i!}(∂^{i}H)(z)∇(y^{i}) · ν(y) dσ(y) = 0.

Thus, in view of (4.3), the following identity holds by using observation (4.1)

(4.6)  ∑_{|i|=l}(∂^{i}H)(z) ∫_{∂B} ϕ_{i}(y)dσ(y) = 0, ∀ l ≥ 1.

In fact, this follows immediately from (4.1). Recall now that M_{ij} = ∫_{∂B} y^{j}ϕ_{i}(y)dσ(y)

is the generalized polarization tensor associated with the domain B and the conductivity k to obtain the following pointwise asymptotic formula: for x ∈ ∂Ω,

(4.7) u(x) = U(x) − ϵ^{d−2} ∑_{|i|=1}^{n} ∑_{|j|=1}^{n−|i|+1} \frac{1}{j!}ϵ^{|i|+|j|}(∂^{i}H)(z)M_{ij}∂_{z}^{j}N(x, z) + O(ϵ^{d+n}).

Observing that the formula (4.7) still contains ∂^{i}H factors, the remaining task is to convert (4.7) to a formula given solely by U and its derivatives.

As a simpliest case, let us now take n = 1 to find the leading order term in the asymptotic expansion of u|_{∂Ω} as ϵ → 0. From (2.7) and (2.16), we get

∥H − U∥_{L^{∞}(∂Ω)} ≤ Cϵ^{d}∥ϕ∥_{L^{2}(∂D)} ≤ Cϵ^{d}∥g∥_{L^{2}(∂Ω)}

for some C depending only on Ω and c_{0}. It then follows from the maximum principle that

∥H − U∥_{L^{∞}(Ω)} ≤ Cϵ^{d}∥g∥_{L^{2}(∂Ω)}.

Then, from the mean value property of harmonic functions, we obtain

|∇H(z) − ∇U(z)| ≤ Cϵ^{d}∥g∥_{L^{2}(∂Ω)}.

It thus follows from (4.7) that

(4.8) u(x) = U(x) − ϵ^{d} ∑_{|i|=1,|j|=1}(∂^{i}U)(z)M_{ij}∂^{j}N(x, z) + O(ϵ^{d+1}), x ∈ ∂Ω,

which is in view of (3.1) exactly the formula derived in [14] and [7] when D has C^{1,α} boundary.

We now return to (4.7). Substitution of (4.7) into (2.8) yields that, for any x ∈ Ω, (4.9)

H(x) = U(x) − ϵ^{d−2} ∑_{|i|=1}^{n} ∑_{|j|=1}^{n−|i|+1} \frac{1}{j!}ϵ^{|i|+|j|}(∂^{i}H)(z)M_{ij}D_{Ω}(∂_{z}^{j}N(·, z))(x) + O(ϵ^{d+n}).

In (4.9) the remainder O(ϵ^{d+n}) is uniform in the C^{n} norm on any compact subset of Ω for any n and therefore

(4.10) (∂^{l}H)(z) + ∑_{|i|=1}^{n} ϵ^{d−2} ∑_{|j|=1}^{n−|i|+1} ϵ^{|i|+|j|}(∂^{i}H)(z)P_{ijl} = (∂^{l}U)(z) + O(ϵ^{d+n}),

for all l ∈ ℕ^{d} with |l| ≤ n where

(4.11)  P_{ijl} = \frac{1}{j!}M_{ij}∂_{x}^{l}D_{Ω}(∂_{z}^{j}N(·, z))|_{x=z}.

Define the operator

P_{ϵ} : (v_{l})_{l∈ℕ^{d},|l|≤n} ↦(v_{l} + ϵ^{d−2} ∑_{|i|=1}^{n} ∑_{|j|=1}^{n−|i|+1} ϵ^{|i|+|j|}v_{i}P_{ijl})_{l∈ℕ^{d},|l|≤n}.

Observe that

P_{ϵ} = I + ϵ^{d}P_{1} + … + ϵ^{n+d−1}P_{n−1}.

Defining Q_{p}, p = 1, …, n − 1, by

(4.12) (I + ϵ^{d}P_{1} + … + ϵ^{n+d−1}P_{n−1})^{−1} = I + ϵ^{d}Q_{1} + … + ϵ^{n+d−1}Q_{n−1} + O(ϵ^{n+d}),

we finally obtain that

(4.13) ((∂^{i}H)(z))_{i∈ℕ^{d},|i|≤n} = (I + ∑_{p=1}^{n} ϵ^{d+p−1}Q_{p})((∂^{i}U)(z))_{i∈ℕ^{d},|i|≤n} + O(ϵ^{d+n})

which yields the main result of this paper stated in Theorem 1.1.

We also have a complete asymptotic expansion of the solutions of the Dirichlet problem:

Theorem 4.2 Suppose that the inhomogeneity consist of single component and let u be the solution of (1.2) with the Neumann condition replaced by the Dirichlet condition u|_{∂Ω} = f. Let V be the solution of ΔV = 0 in Ω with V |_{∂Ω} = f. The following pointwise asymptotic expansion on ∂Ω holds for d = 2, 3:

(4.14) \begin{aligned}\frac{∂u}{∂ν}(x) & = \frac{∂V}{∂ν} (x) − ϵ^{d−2} ∑_{|i|=1}^{n} ∑_{|j|=1}^{n−|i|+1} \frac{1}{j!}ϵ^{|i|+|j|}× \\ & [((I + ∑_{p=1}^{n+2−|i|−|j|−d} ϵ^{d+p−1}Q_{p})(∂^{l}U(z)))_{i}M_{ij}∂_{z}^{j} \frac{∂}{∂ν_{x}} G(x, z)] \\ & +O(ϵ^{d+n}), \end{aligned}

where the remainders O(ϵ^{d+n}) are is dominated by Cϵ^{d+n}∥g∥_{L^{2}(∂Ω)} for some C independent of x ∈ ∂Ω. Here G(x, z) is the Dirichlet Green function, M_{ij}, i, j ∈ ℕ^{d}, are the generalized polarization tensors, and Q_{p} is the operator defined in (4.12) where P_{ijk} is defined, in this case, by

(4.15)  P_{ijl} = \frac{1}{j!}M_{ij}∂_{x}^{l}S_{Ω}(∂_{z}^{j}(\frac{∂}{∂ν_{x}} G)(·, z))|_{x=z}.

Theorem 4.2 can be proved in the exactly same manner as Theorem 1.1. We begin with Lemma 2.5. Then the same arguments give us

u(x) = U(x) − ϵ^{d−2} ∑_{|i|=1}^{n} ∑_{|j|=1}^{n−|i|+1} \frac{1}{j!}ϵ^{|i|+|j|}(∂^{i}H)(z)M_{ij}∂_{z}^{j}G(x, z) + O(ϵ^{d+n}).

From this we can get (4.14) as before.

We conclude this paper by making a remark. The following formula is not exactly an asymptotic formula. However, since the formula is simple and has some potential applicability in solving the inverse conductivity problem, we make a record of it as a theorem.

Theorem 4.3 We have

(4.16) u(x) = H(x) + ϵ^{d−2} ∑_{|i|=1}^{n} ∑_{|j|=1}^{n−|i|+1} \frac{1}{j!}ϵ^{|i|+|j|}∂^{i}H(z)M_{ij}∂^{j}Γ(x − z) + O(ϵ^{d+n}),

where x ∈ Ω_{0} and O(ϵ^{d+n}) term is dominated by C∥g∥_{L^{2}(∂Ω)}ϵ^{d+n} for some C depending only on c_{0}, and H is given in (2.8).

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