CONDUCTIVITY INTERFACE PROBLEMS. PART I - CiteSeerX

CONDUCTIVITY INTERFACE PROBLEMS. PART I: SMALL PERTURBATIONS OF AN INTERFACE HABIB AMMARI, HYEONBAE KANG, MIKYOUNG LIM, AND HABIB ZRIBI Abstract. We derive high-order terms in the asymptotic expansions of the boundary perturbations of steady-state voltage potentials resulting from small perturbations of the shape of a conductivity inclusion with C 2 -boundary. Our derivation is rigorous and based on layer potential techniques. The asymptotic expansion in this paper is valid for C 1 -perturbations and inclusions with extreme conductivities. It extends those already derived for small volume conductivity inclusions and leads us to very effective algorithms, aimed at determining certain properties of the shape of a conductivity inclusion based on boundary measurements.

1. Introduction The main objective of this paper is to present a schematic way based on layer potential techniques to derive high-order terms in the asymptotic expansions of the boundary perturbations of steady-state voltage potentials resulting from small perturbations of the shape of a conductivity inclusion. We then use this formula to design algorithms to recover certain properties of the perturbation of the shape. More precisely, consider a homogeneous conducting object occupying a bounded domain Ω ⊂ R2 , with a connected C 2 -boundary ∂Ω. We assume, for the sake of simplicity, that its conductivity is equal to 1. Let D with C 2 -boundary be a conductivity inclusion inside Ω of conductivity equal to some positive constant k 6= 1. We assume that dist(D, ∂Ω) ≥ C > 0. The voltage potential in the presence of the inclusion D is denoted by u. It is the solution to ³ ´    ∇· 1 + (k − 1)χD ∇u = 0 in Ω, Z (1.1) ∂u ¯¯   = g, u = 0, ¯ ∂ν ∂Ω ∂Ω where χD is the indicator function of D. Here ν denotes the unit outward normal to the domain Ω andR g represents the applied boundary current; it belongs to the set L20 (∂Ω) = {f ∈ L2 (∂Ω), ∂Ω f = 0}. Let D² be an ²-perturbation of D, i.e., there is h ∈ C 1 (∂D) such that ∂D² is given by (1.2)

∂D² = { x ˜:x ˜ = x + ²h(x)ν(x), x ∈ ∂D }.

2000 Mathematics Subject Classification. 35B30. Key words and phrases. Small perturbations, interface problem, full-asymptotic expansions, boundary integral method. 1

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AMMARI ET AL.

Let u² be the solution to

(1.3)

³ ´    ∇· 1 + (k − 1)χD² ∇u² = 0 Z ∂u² ¯¯   u² = 0. ¯ = g, ∂ν ∂Ω ∂Ω

in Ω,

The main achievement of this paper is a rigorous derivation, based on layer potential techniques, of high-order terms in the asymptotic expansion of (u² − u)|∂Ω as ² → 0. The solution u² to (1.3) can be represented using integral operators (see formula (3.1)), and hence derivation of asymptotic formula for u² is reduced to that of the integral operator ∗ defined by KD ² Z 1 h˜ x − y˜, ν˜(˜ x)i ∗ ˜ x) = KD² [ϕ](˜ p.v. ϕ(˜ ˜ y )dσ² (˜ y ), 2 2π |˜ x − y˜| ∂D² ∗ where p.v. stands for the Cauchy principal value. The operator KD is a singular integral ² 2 ∗ operator and known to be bounded on L (∂D² ) [8, 7]. It was proved in [14, 9] that KD ² ∗ converges to the operator KD on the non-perturbed domain D, defined for a density ϕ ∈ L2 (∂D), by Z 1 hx − y, ν(x)i ∗ KD [ϕ](x) = ϕ(y)dσ(y). 2π ∂D |x − y|2

Note that we dropped the notation p.v. in the above; it is because ∂D is C 2 . In this paper ∗ we will derive a complete asymptotic expansion of the singular integral operator KD on ² 2 L (∂D² ) in terms of ². This asymptotic expansion yields an expansion of u² − u which extends those already derived for small volume inclusions [1, 2, 3, 4, 5, 10, 16, 19]. The asymptotic formula derived in the paper is of significant interest from an “imaging point of view”. For instance, if one has a detailed knowledge of the “boundary signatures” of conductivity inclusions, then it becomes possible to design effective algorithms to identify certain properties of their shapes. Since it carries precise information on the shape of the inclusion, we will show how it can be efficiently exploited for designing significantly better algorithms. In connection with reconstruction of interfaces, we refer readers to [6, 12, 17].

∗ 2. High-order terms in the expansion of KD ²

Let a, b ∈ R, with a < b, and let X(t) : [a, b] → R2 be the arclength parametrization of ∂D, namely, X is a C 2 -function satisfying |X 0 (t)| = 1 for all t ∈ [a, b] and ∂D := {x = X(t), t ∈ [a, b]}. Then the outward unit normal to ∂D, ν(x), is given by ν(x) = R− π2 X 0 (t), where R− π2 is the rotation by −π/2, the tangential vector at x, T (x) = X 0 (t), and X 0 (t) ⊥ X 00 (t). Set the curvature τ (x) to be defined by X 00 (t) = τ (x)ν(x). We will sometimes use h(t) for h(X(t)) and h0 (t) for the tangential derivative of h(x).

SMALL PERTURBATIONS OF AN INTERFACE

3

˜ Then, X(t) = X(t) + ²h(t)ν(x) = X(t) + ²h(t)R− π2 X 0 (t) is a parametrization of ∂D² . By ν˜(˜ x), we denote the outward unit normal to ∂D² at x ˜. Then, we have ˜ 0 (t) R− π2 X ˜ 0 (t)| |X ³ ´ 1 − ²h(t)τ (x) ν(x) − ²h0 (t)X 0 (t) = r ³ ´

ν˜(˜ x) =

2

2

²2 h0 (t) + 1 − ²h(t)τ (x) ³ ´ 1 − ²h(t)τ (x) ν(x) − ²h0 (t)T (x) = r ³ ´2 , 2 2 0 ² h (t) + 1 − ²h(t)τ (x)

(2.1)

and hence ν˜(˜ x) can be expanded uniformly as (2.2)

ν˜(˜ x) =

∞ X

²n ν (n) (x),

x ∈ ∂D,

n=0

where the vector-valued functions ν (n) are uniformly bounded regardless of n. In particular, the first two terms are given by ν (0) (x) = ν(x),

ν (1) (x) = −h0 (t)T (x).

Likewise, we get a uniformly convergent expansion for the length element dσ² (˜ y ): ˜ 0 (s)|ds = dσ² (˜ y ) = |X

(2.3)

p

(1 − ²τ (s)h(s))2 + ²2 h02 (s)ds =

∞ X

²n σ (n) (y)dσ(y),

n=0

where σ (n) are functions bounded independently of n, and σ (0) (y) = 1,

(2.4) Set

σ (1) (y) = −τ (y)h(y).

˜ x = X(t), x ˜ = X(t) = x + ²h(t)R− π2 X 0 (t), ˜ y = X(s), y˜ = X(s) = y + ²h(s)R− π2 X 0 (s).

Since

³ ´ x ˜ − y˜ = x − y + ² h(t)ν(x) − h(s)ν(y) ,

(2.5) we get

|˜ x − y˜|2 = |x − y|2 + 2²hx − y, h(t)ν(x) − h(s)ν(y)i + ²2 |h(t)ν(x) − h(s)ν(y)|2 , and hence (2.6)

1 1 1 = , |˜ x − y˜|2 |x − y|2 1 + 2²F (x, y) + ²2 G(x, y)

where F (x, y) =

hx − y, h(t)ν(x) − h(s)ν(y)i , |x − y|2

and G(x, y) =

|h(t)ν(x) − h(s)ν(y)|2 . |x − y|2

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AMMARI ET AL.

One can easily see that 1

|F (x, y)| + |G(x, y)| 2 ≤ CkXkC 2 khkC 1

for all x, y ∈ ∂D.

It follows from (2.1), (2.3), (2.5), and (2.6) that ³ hx − y, ν(x)i h hh(t)ν(x) − h(s)ν(y), ν(x)i h˜ x − y˜, ν˜(˜ x)i dσ (˜ y ) = + ² ² |˜ x − y˜|2 |x − y|2 |x − y|2 0 hx − y, τ (x)h(t)ν(x) + h (t)T (x)i i − |x − y|2 hh(t)ν(x) − h(s)ν(y), τ (x)h(t)ν(x) + h0 (t)T (x)i ´ − ²2 |x − y|2 p (1 − ²τ (y)h(s))2 + ²2 h02 (s) 1 p × dσ(y) 1 + 2²F (x, y) + ²2 G(x, y) (1 − ²τ (x)h(t))2 + ²2 h02 (t) ³ ´ : = K0 (x, y) + ²K1 (x, y) + ²2 K2 (x, y) p (1 − ²τ (y)h(s))2 + ²2 h02 (s) 1 p dσ(y). × 2 1 + 2²F (x, y) + ² G(x, y) (1 − ²τ (x)h(t))2 + ²2 h02 (t) Let

p ∞ (1 − ²τ (y)h(s))2 + ²2 h02 (s) X n 1 p = ² Fn (x, y), 1 + 2²F (x, y) + ²2 G(x, y) (1 − ²τ (x)h(t))2 + ²2 h02 (t) n=0

where the series converges absolutely and uniformly. In particular, we can easily see that F0 (x, y) = 1,

F1 (x, y) = −2F (x, y) + τ (x)h(x) − τ (y)h(y).

Then we now have ³ ´ h˜ x − y˜, ν˜(˜ x)i hx − y, ν(x)i dσ² (˜ y) = dσ(y) + ² K0 (x, y)F1 (x, y) + K1 (x, y) dσ(y) 2 2 |˜ x − y˜| |x − y| ∞ ³ ´ X + ²2 ²n Fn+2 (x, y)K0 (x, y) + Fn+1 (x, y)K1 (x, y) + Fn (x, y)K2 (x, y) dσ(y). n=0

Therefore, we obtain that ∞ X h˜ x − y˜, ν˜(˜ x)i dσ (˜ y ) = ²n kn (x, y)dσ(y), ² |˜ x − y˜|2 n=0

where k0 (x, y) =

hx − y, ν(x)i , |x − y|2

k1 (x, y) = K0 (x, y)F1 (x, y) + K1 (x, y),

and for any n ≥ 2, kn (x, y) = Fn (x, y)K0 (x, y) + Fn−1 (x, y)K1 (x, y) + Fn−2 (x, y)K2 (x, y). (n)

Introduce a sequence of integral operators (KD )n∈N , defined for any φ ∈ L2 (∂D) by: Z (n) KD φ(x) = kn (x, y)φ(y)dσ(y) for n ≥ 0. ∂D

(0) KD

∗ KD .

Note that = Observe that the same operator with the kernel kn (x, y) replaced with Kj (x, y), j = 0, 1, 2, is bounded on L2 (∂D). In fact, it is an immediate consequence of

SMALL PERTURBATIONS OF AN INTERFACE

5 (n)

the celebrated theorem of Coifman-McIntosh-Meyer [7]. Therefore each KD is bounded on L2 (∂D). Let Ψ² be the diffeomorphism from ∂D onto ∂D² given by Ψ² (x) = x + ²h(t)ν(x), where x = X(t). The following theorem holds. Theorem 2.1. Let N ∈ N. There exists C depending only on N , kXkC 2 , and khkC 1 such that for any φ˜ ∈ L2 (∂D² ), ° ° N ° ° X ° ° ∗ ˜ ∗ n (n) ≤ C²N +1 ||φ||L2 (∂D) , ² KD [φ]° (2.7) °KD² [φ] ◦ Ψ² − KD [φ] − ° 2 ° n=1

L (∂D)

where φ := φ˜ ◦ Ψ² . 3. Derivation of the full asymptotic formula for the steady-state voltage potentials In this section we derive high-order terms in the asymptotic expansion of (u² − u)|∂Ω as ² → 0. k+1 Suppose that the conductivity of D is k. Let λ := 2(k−1) . Define the background voltage potential, U , to be the unique solution to   ∆U = 0 in Ω, Z ∂U ¯¯  = g, U = 0.  ∂ν ∂Ω ∂Ω Let N (x, y) be the Neumann function for ∆ in Ω corresponding to a Dirac mass at y, that is, N is the solution to  ∆x N (x, y) = −δy in Ω,      1  ∂N ¯¯ =− , ∂ν ∂Ω |∂Ω|  Z     N (x, y)dσ(x) = 0 for y ∈ Ω.  ∂Ω

Let ND be defined by

Z ND [ϕ](x) :=

N (x, y)ϕ(y)dσ(y),

x ∈ ∂Ω,

∂D

for ϕ ∈ L20 (∂D). Let u² be the solution to (1.3). Then the following representation formula holds [1]: (3.1)

u² (x) = U (x) − ND² [φ˜² ](x),

x ∈ ∂Ω

where φ˜² ∈ L20 (∂D² ) is given by (3.2)

H² (x) := DΩ [u² |∂Ω ](x) − SΩ [g](x),

x ∈ Ω,

and ∂H² ∗ (λI − KD )[φ˜² ] = ² ∂ ν˜

on ∂D² .

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Here and throughout this paper SΩ and DΩ denote the single and double layer potential on ∂Ω: Z 1 hy − x, ν(y)i DΩ [ϕ](x) = ϕ(y)dσ(y), x ∈ R2 \ ∂Ω, 2π ∂Ω |x − y|2 Z 1 SΩ [ϕ](x) = log |x − y|ϕ(y)dσ(y), x ∈ R2 . 2π ∂Ω Likewise the solution u to (1.1) has the representation u(x) = U (x) − ND [φ](x),

x ∈ ∂Ω,

where φ ∈ L20 (∂D) is given by (3.3)

H(x) := DΩ [u|∂Ω ](x) − SΩ [g](x),

x ∈ Ω,

and ∗ (λI − KD )[φ] =

(3.4)

∂H ∂ν

on ∂D.

We then get u² (x) − u(x) = −ND² [φ˜² ](x) + ND [φ](x),

(3.5)

x ∈ ∂Ω.

We now investigate the asymptotic behavior of ND² [φ˜² ] as ² → 0. After the change of variables y˜ = Ψ² (y), we get from (2.3) and the Taylor expansion of N (x, y˜) in y ∈ ∂D for each fixed x ∈ ∂Ω that Z ³X ∞ X n ∞ ´ ³X ´ ² α α ˜ ND² [φ² ](x) = (h(y)ν(y)) ∂y N (x, y) φ² (y) ²n σ (n) (y) dσ(y), α! ∂D n=0 n=0 |α|=n

where φ² = φ˜² ◦ Ψ² . One can see from Theorem 2.1 that, for each integer N , φ² satisfies µ ¶ N X (n) ∗ λI − KD − ²n KD [φ² ] + O(²N +1 ) = (∇H² ◦ Ψ² ) · 9˜ ν ◦ Ψ² )

on ∂D.

n=1

We obtain from (2.2) that ∞ X n ∞ ³X ´ ³X ´ ² α α (∇H² ◦ Ψ² )(y) · (˜ ν ◦ Ψ² )(y) = (∇∂ H² )(y)(h(y)ν(y)) · ²n ν (n) (y) α! n=0 n=0 |α|=n

:=

(3.6)

∞ X

²n Gn (y).

n=0

Note that (3.7)

G0 (y) =

∂H² (y), ∂ν

­ ® G1 (y) = h(y) D2 H² (y)ν(y), ν(y) − h0 (y)h∇H² , T (y)i,

where D2 H² is the Hessian of H² . Therefore, we obtain the following integral equation to solve: µ ¶ N +∞ X X (n) ∗ (3.8) λI − KD − ²n KD [φ² ] + O(²N +1 ) = ²n Gn on ∂D. n=1

n=0

SMALL PERTURBATIONS OF AN INTERFACE

7

The equation (3.8) can be solved recursively in the following way: Define h ¯ i ∗ −1 ∗ −1 ∂H² ¯ (3.9) φ(0) = (λI − KD ) [G0 ] = (λI − KD ) , ∂ν ∂D and for 1 ≤ n ≤ N , n−1 i h X (n−p) (n) ∗ −1 (3.10) φ = (λI − KD ) Gn + KD φ(p) . p=0

We obtain the following lemma. Lemma 3.1. Let N ∈ N. There exists C depending only on N , the C 2 -norm of X, and the C 1 -norm of h such that N X ||φ² − ²n φ(n) ||L2 (∂D) ≤ C²N +1 , n=0

where φ(n) are defined by the recursive relation (3.10). Define, for n ∈ N and for x ∈ ∂Ω, ´ X Z ³X 1 (h(y)ν(y))α ∂yα N (x, y) σ (j) (y)φ(k) (y)dσ(y). (3.11) vn (x) := α! ∂D i+j+k=n

|α|=i

It then follows from (3.7) and (3.11) that N h ¯ i X ∗ −1 ∂H² ¯ ˜ ND² [φ² ](x) = ND (λI − KD ) + ²n vn (x) + O(²N +1 ). ∂ν ∂D n=1

Hence we get from (3.5) that u² (x) − u(x) = −ND (λI −

∗ −1 KD )

N h ∂H ¯ ∂H ¯¯ i X n ²¯ − − ² vn (x) + O(²N +1 ), ∂ν ∂D ∂ν ∂D n=1

x ∈ ∂Ω.

Observe from (3.2) and (3.3) that (3.12)

H² (x) − H(x) = DΩ [u² |∂Ω − u|∂Ω ](x),

If we define the operator E on

L20 (∂Ω)

x ∈ Ω.

by h∂ ¯ i (DΩ v)¯∂D (x), ∂ν

x ∈ ∂Ω,

²n vn (x) + O(²N +1 ),

x ∈ ∂Ω.

∗ −1 E[v](x) := ND (λI − KD )

then it follows that (3.13)

(I + E)[u² − u](x) = −

N X n=1

We need the following lemma. Lemma 3.2. The operator I + E is invertible on L20 (∂Ω). Let us continue derivation of asymptotic expansion of (u² − u)|∂Ω leaving the proof of Lemma 3.2 at the end of this section. We get from (3.13) that (3.14)

u² (x) − u(x) = −

N X n=1

²n (I + E)−1 [vn ](x) + O(²N +1 ),

x ∈ ∂Ω.

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Observe that the function vn is still depending on ² since Gn in (3.6) is defined by H² and hence φ(n) depends on ². We can remove this dependence on ² from the asymptotic formula in an iterative way. Observe from (3.14) that (u² − u)|∂Ω = O(²), and hence, by (3.12), H² (x) − H(x) = O(²). (n)

Thus if we define G1n , n ∈ N, by (3.6) with H² replaced with H, and define φ1 (3.9), (3.10), and (3.11), then vn − vn1 = O(²). Therefore we get (3.15)

u² (x) − u(x) = −²1 (I + E)−1 [v11 ](x) + O(²2 ),

and vn1 by

x ∈ ∂Ω.

Repeat the same procedure with H − ²DΩ (I + E)−1 [vn1 ] instead of H to get vn2 . Then vn − vn2 = O(²2 ) and hence u² (x) − u(x) = −

2 X

²n (I + E)−1 [vn2 ](x) + O(²3 ),

x ∈ ∂Ω.

n=1

Repeating the same procedure until we get vnN , and we obtain the following theorem. Theorem 3.3. Let vnN , n = 1, . . . , N , be the functions obtained by the above procedure. Then the following formula holds uniformly for x ∈ ∂Ω: u² (x) − u(x) = −

N X

²n (I + E)−1 [vnN ](x) + O(²N +1 ).

n=1

The remainder O(²N +1 ) depends only on N , Ω, the C 2 -norm of X, the C 1 -norm of h, and dist(D, ∂Ω). (0)

Let us compute the first order approximation of (u² − u)|∂Ω explicitly. Note that φ1 = φ where φ is defined by (3.4), and h ­ i ® (1) (1) ∗ −1 ) h (D2 H)ν, ν − h0 h∇H, T i + KD φ . φ1 = (λI − KD Therefore, by (2.4) and (3.11), v11 takes the form Z v11 (x) = ∇y N (x, y) · ν(y)h(y)φ(y)dσ(y) ∂D Z Z − N (x, y)τ (y)h(y)φ(y)dσ(y) + ∂D

∂D

(1)

N (x, y)φ1 (y)dσ(y).

Using this formula and (3.15) we find the first-order term in the asymptotic expansion of u² − u on ∂Ω. The first term in the asymptotic expansions is exactly the domain derivative of the solution derived in [11, Theorem 1]. To see that, it suffices to prove that (3.16)

(I + E)[w] = −

1 v1 k−1 1

on ∂Ω,

SMALL PERTURBATIONS OF AN INTERFACE

9

where w is the solution of  ∆w = 0 in (Ω \ D) ∪ D,    ¯    ∂u ¯¯   w|+ − w|− = −h ¯ on ∂D,   ∂ν −  ¯ ¯ (3.17) ∂w ¯¯ ∂w ¯¯ ∂ ³ ∂u ´   h on ∂D, − k = −   ∂ν ¯+ ∂ν ¯− ∂T ∂T        ∂w = 0 on ∂Ω. ∂ν It is easy to see that h ∂u ¯¯ i w = DΩ [w|∂Ω ] + DD h ¯¯ + SD [θ], x ∈ Ω, ∂ν −

where the density θ on ∂D is given by ¯ ¸ · ¯ 1 ∂ ∂u ∂ ∂ ∂u ¯¯ ¯¯ ∗ −1 ¯ θ = (λI − KD ) − ( h )+ (DΩ [w]) ∂D + (DD [h ¯ ]) ∂D . k − 1 ∂T ∂T ∂ν ∂ν ∂ν − Thus, for x ∈ ∂Ω,

¯ Z h∂ ¯ i ∂N ∂u ¯¯ ¯ w(x) + ND (λI − (DΩ w) ∂D (x) = − (x, y)h(y) ¯ (y)dσ(y) ∂ν ∂ν − ∂D ∂ν(y) Z ∂ ∂u 1 ∗ −1 N (x, y)(λI − KD ) [ h ](y)dσ(y) k − 1 ∂D ∂T ∂T ¯ Z ∂u ¯¯ ¯¯ ∗ −1 ∂ − N (x, y)(λI − KD ) [ (DD (h ¯ )] ∂D )dσ(y). ∂ν ∂ν ∗ −1 KD )

∂D

Since

and

−

¯ ∂u ¯¯ 1 φ = ∂ν ¯− k−1 µ ¶ ­ ® ∂ ∂u ∂H ∂H ∂ ∂ h = h − (D2 H)ν, ν − τ + h0 + h SD [φ], ∂T ∂T ∂ν ∂T ∂T ∂T

where φ is defined by (3.4), it is not difficult to see that (3.16) holds by using the expression (1) of KD . Proof of Lemma 3.2. Since E is a compact operator, we can apply the Fredholm alternative. Suppose that (I +E)[v] = 0. Then, first of all, v is smooth on ∂Ω. Since (− 12 I +KΩ )ND = SD on L20 (∂D) as was proved in [1, 2, 3], we get h ³ 1 ´ ¯ i ∗ −1 ∂ − I + KΩ [v] + SD (λI − KD ) (DΩ v)¯∂D = 0 on ∂Ω. 2 ∂ν Since ´ ³1 I + KΩ [f ](x), x ∈ ∂Ω, DΩ [f ]|− (x) = 2 where the subscript − denotes the limit from the inside of Ω [18], we get h ∂ ¯ i ∗ −1 ( DΩ [v])¯∂D (x), x ∈ ∂Ω. v(x) = DΩ [v]|− (x) + SD (λI − KD ) ∂ν

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AMMARI ET AL.

Thus v can be extended to whole Ω to satisfy h ∂ ¯ i ∗ −1 v(x) = DΩ [v](x) + SD (λI − KD ) ( DΩ [v])¯∂D (x), x ∈ Ω. ∂ν 1,2 Let the space W (Ω) be the set of functions f ∈ L2 (Ω) such that ∇f ∈ L2 (Ω). We now recall the following facts from [13, 15]: The W 1,2 -solution u to (1.1) has the representation (3.18)

(3.19)

u(x) = H(x) + SD [φ](x),

x ∈ Ω,

where the Harmonic function H ∈ W 1,2 (Ω) and φ ∈ L20 (∂D) is given by (3.3) and (3.4). Moreover the harmonic function H and φ are unique. Observe that v given in (3.18) takes exactly the form in (3.19) with H = DΩ [v]. By (3.3) ∂v ∂v |∂Ω ] = 0 in Ω. It then follows that ∂ν |∂Ω = 0 on ∂Ω, and uniqueness of H, we have SΩ [ ∂ν 2 and hence v is constant in Ω. Since v ∈ L0 (∂Ω), we get v = 0. So, I + E is injective, and hence invertible. This completes the proof. ¤ 4. Reconstruction of the interface deformation We finally give a reconstruction formula of the shape deformation h from measurements of boundary perturbations u² − u on ∂Ω. Recall from the previous section that (4.1)

(u² − u)(x) = ²(k − 1)w(x) + O(²2 ),

For f ∈ L20 (∂Ω), let v be the solution to ³ ´    ∇· 1 + (k − 1)χD ∇v = 0 Z ∂v ¯¯   v = 0. ¯ = f, ∂ν ∂Ω ∂Ω

x ∈ ∂Ω.

in Ω,

In view of (3.17), multiplying (4.1) by f and integrating over ∂Ω yields · ¸ Z Z ∂v ∂u ∂v ¯¯ ∂u ¯¯ (4.2) f (u² − u) dσ = ²(1 − k) h +k ¯ ¯ dσ + O(²2 ), ∂T ∂T ∂ν − ∂ν − ∂Ω ∂D where u is the solution to (1.1). Formula (4.2) can be used in order to reconstruct an approximation of h by choosing v appropriately. Higher-order terms given in Theorem 3.3 can be used to reconstruct a better approximation of h. To illustrate this, let us consider D to be the disk centered at the origin with radius ρ. Set ( −|n| |n| inθ cn ρ r e , r < ρ, ´ (4.3) un (r, θ) = cn ³ −|n| |n| |n| −|n| inθ ρ (k + 1)r − ρ (k − 1)r e , r ≥ ρ, 2 where 1 2 cn = , for |n| ≥ 1. |n| ρ−|n| (k + 1) + ρ|n| (k − 1) Now, define the boundary currents as ∂un ¯¯ ∂um ¯¯ g(θ) = ¯ , f (θ) = ¯ , |m|, |n| ≥ 1. ∂ν ∂Ω ∂ν ∂Ω Since un satisfies ∇ · (1 + (k − 1)χD )∇un = 0, we have u = un and v = um .

SMALL PERTURBATIONS OF AN INTERFACE

11

Therefore, Z (4.4)

Z hei(n+m)θ dθ + O(²2 ),

f (u² − u) dσ = ²cn,m (k) ∂Ω

∂D

where cn,m (k) = ρ−1 cn cm (1 − k)(k|n||m| − nm). This implies that the Fourier coefficient hp can be determined from measurements of u² −u on ∂Ω provided that the order of magnitude of |hp | is much larger than ². To reconstruct higher Fourier coefficients of h or more accurately the first ones, the high-order asymptotic expansions derived in this paper should be used. The idea is quite simple. We first use the leading-order term to determine the first Fourier coefficients of h, say hp , for |p| ≤ L with a C certain L. Since h is C 1 , |hp | ≤ |p| for some constant C, and hence we may choose L so that L = O(1/²). Then, we plug in these coefficients into the second-order term by approximating P h by |p|≤L hp eipθ and use the second-order asymptotic expansion in Theorem 3.3 to both determine further Fourier coefficients and get more accurate approximations of (hp )|p|≤L . This procedure can be repeated recursively. 5. Numerical Results We present several examples of interface reconstructions. The inclusion’s conductivity is fixed to be k = 4, and the background domain Ω is assumed to be an ellipse given by the following equation: y2 (x + 0.3)2 + ≤ 1, a = 1.2, b = 1. a2 b2 From (4.4), the Fourier coefficients h(p) of h is obtained as Z 1 f (u² − u)dσ, h(p) = 2π²cn,m (k) ∂Ω where

  for p = 0, (−1, 1), (n, m) = (−1, −p + 1), for p ≤ −1,   (1, −p − 1), for p ≥ 1.

Solving Forward Problem. To have the boundary data u² |∂Ω , we solve the forward problem using the boundary integral representation of u² . Using the jump formula of single and double layer potential, it can be formulated (5.1)

u(x) = SD ϕ(x) + SΩ ψ(x) + C,

x ∈ Ω,

where (ϕ, ψ) ∈ L20 (∂D) × L20 (∂Ω) is a solutio to ¶µ µ ¶ µ ¶ ∂ k+1 ∗ I − KD )|∂D − ∂ν (SΩ ·)|∂D ( 2(k−1) ϕ 0 (5.2) = , ∂ ∗ ψ g (− 12 I + KΩ )|∂Ω ∂ν (SD ·)|∂Ω R and C is a constant chosen to make ∂Ω u = 0. We discretize (5.2) and solve the obtained linear system to have (ϕ, ψ). Then, the boundary voltage u² |∂Ω is calculated from (5.1).

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Example 1. We presents the reconstruction results of the inclusion when it’s perturbation is given by h(θ) = 1 + 2 cos(jθ), j = 0, 3, 6, 9 and ² = 0.03, 0.05. Since cn,m in (4.2) increases as |n + m| increases, a higher frequency perturbation is less accurate than a lower P P one. The perturbation function h is approximated by |p|≤6 hp eipθ for j = 0, 3, 6, and |p|≤9 hp eipθ for j = 9. 1

1

0

0

−1 −1.5

0

1

−1 −1.5

1

1

0

0

−1 −1.5

0

1

−1 −1.5

1

1

0

0

−1 −1.5

0

1

−1 −1.5

1

1

0

0

−1 −1.5

0

1

−1 −1.5

0

1

0

1

0

1

0

1

Figure 1. The solid blue line represents the inclusion, which is a perturbation of disk, a dashed line. The red one represents the reconstructed inclusion. The perturbation is given by ²h, where ² = 0.03, the left column, and ² = 0.05, the right column.

SMALL PERTURBATIONS OF AN INTERFACE

13

Example 2. We present the reconstruction results for a shifted inclusion. The initial circular D is shifted by 0.1 or 0.2. The perturbation function h is approximated by P inclusion ipθ h e . p |p|≤6 1

1

0

0

−1 −1.5

0

1

−1 −1.5

0

1

Figure 2. Reconstruction result when the inclusion is shifted by 0.1, the left figure, or 0.2, the right figure. 2

x Example 3. We present the reconstruction when the inclusion is an ellipse given by (0.4+a) 2+ P 2 y ipθ . |p|≤6 hp e 0.42 = 1, a = 0.2, 0.3. The perturbation function h is approximated by 1

1

0

0

−1 −1.5

0

1

−1 −1.5

0

1

Figure 3. Reconstruction result when the inclusion is an ellipse. Acknowledgements This research was partly supported by CNRS-KOSEF grant No. 14889, Korea Science and Engineering Foundation grant R01-2006-000-10002-0, ACI Nouvelles Interfaces des Math´ematiques No. 171, and the ANR project EchoScan. References [1] H. Ammari and H. Kang, High-order terms in the asymptotic expansions of the steady-state voltage potentials in the presence of conductivity inhomogeneities of small diameter, SIAM J. Math. Anal., 34 (2003), 1152–1166. [2] —————–, Reconstruction of Small Inhomogeneities from Boundary Measurements, Lecture Notes in Mathematics, Vol. 1846, Springer-Verlag, Berlin, 2004. [3] —————–, Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory, Applied Mathematical Sciences, Vol. 162, Springer-Verlag, New York, 2007. [4] Y. Capdeboscq and M.S. Vogelius, A general representation formula for the boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction, Math. Modelling Num. Anal., 37 (2003), 159–173.

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[5] D.J. Cedio-Fengya, S. Moskow, and M. Vogelius, Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction, Inverse Problems 14 (1998), 553–595. [6] T. Chan and X.-C. Tai, Level set and total variation regularization for elliptic inverse problems with discontinuous coefficients, J. Comput. Phys. 193 (2003), 40–66. [7] R.R. Coifman, A. McIntosh, and Y. Meyer, L’int´ egrale de Cauchy d´ efinit un op´ erateur bourn´ e sur L2 pour les courbes lipschitziennes, Ann. Math., 116 (1982), 361–387. [8] E.B. Fabes, M. Jodeit, and N.M. Rivi´ ere, Potential techniques for boundary value problems on C 1 domains, Acta Math., 141 (1978), 165–186. [9] E. Fabes, H. Kang, and J.K. Seo, Inverse conductivity problem: error estimates and approximate identification for perturbed disks, SIAM J. Math. Anal., 30 (4) (1999), 699–720. [10] A. Friedman and M. Vogelius, Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence, Arch. Rat. Mech. Anal., 105 (1989), 563–579. [11] F. Hettlich and W. Rundell, The determination of a discontinuity in a conductivity from a single boundary measurement, Inverse Problems, 14 (1998), 67–82. [12] K. Ito, K. Kunish, and Z. Li, Level-set function approach to an inverse interface problem, Inverse Problems 17 (2001), 1225–1242. [13] H. Kang and J.K. Seo, Layer potential technique for the inverse conductivity problem, Inverse Problems 12 (1996), 267–278. [14] —————, On stability of a transmission problem, Jour. Korean Math. Soc., 34 (1997), 695-706. [15] —————, Recent progress in the inverse conductivity problem with single measurement, in Inverse Problems and Related Fields, CRC Press, 2000, 69–80. [16] O. Kwon, J.K. Seo, and J.R. Yoon, A real-time algorithm for the location search of discontinuous conductivities with one measurement, Commun. Pure Appl. Math. LV (2002), 1–29. [17] C.F. Tolmasky and A. Wiegmann, Recovery of small perturbations of an interface for an elliptic inverse problem via linearization, Inverse Problems 15 (1999), 465–487. [18] G.C. Verchota, Layer potentials and boundary value problems for Laplace’s equation in Lipschitz domains, J. Funct. Anal., 59 (1984), 572–611. [19] M. Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities, Math. Model. Numer. Anal. 34 (2000), 723–748. ´matiques Applique ´es, CNRS UMR 7641 and Ecole Polytechnique, 91128 Palaiseau Centre de Mathe Cedex, France E-mail address: [email protected] Department of Mathematical Sciences and RIM, Seoul National University, Seoul 151-747, Korea E-mail address: [email protected] Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA E-mail address: [email protected] ´matiques Applique ´es, CNRS UMR 7641 and Ecole Polytechnique, 91128 Palaiseau Centre de Mathe Cedex, France E-mail address: [email protected]