Estimate of an inclusion in a body with discontinuous conductivity∗

Estimate of an inclusion in a body with discontinuous conductivity ^∗

Tu Nguyen

Jenn-Nan Wang

Abstract

We study the problem of estimating the size of an inclusion embed- ded inside a two dimensional body with discontinuous conductivity by one voltage-current measurement. This problem is practically impor- tant because the conductivity of a human body is discontinuous. The proofs rely on quantitative uniqueness estimates for the conductivity equation with discontinuous coefficients.

1 Introduction

An important clinical problem is to estimate the size of a cancerous tumor inside an organ by noninvasive methods. In this paper, we study this problem by the method of electrical impedance tomography (EIT) with one measure- ment. Previous works on this problem assumed that the conductivity of the studied body is Lipschitz continuous (see, for example, [5, 6]). However, this is not guaranteed in reality, for example, the conductivities of heart, liver, intestines are 0.70 (S/m), 0.10 (S/m), 0.03 (S/m), respectively. In this paper, we show that in the two dimensional case, the assumption on the regularity of the conductivity can be weaken.

We briefly outline the framework, following [6]. Let Ω ⊂ R² be an open bounded domain with Lipschitz boundary. Assume that the background

∗This work is dedicated to Professor Neil Trudinger for his 70th birthday.

†Department of Mathematics, National Taiwan University, Taipei 106, Taiwan. Email:

anhtu@uw.edu

‡Institute of Applied Mathematical Sciences, National Taiwan University, NCTS (Taipei), Taipei 106, Taiwan. Email: jnwang@ntu.edu.tw

conductivity σ(x) is elliptic, i.e. for some λ > 0,

λ⁻¹|y|² ≤ hσ(x)y, yi ≤ λ |y|², ∀y ∈ R², a.e. x ∈ Ω. (1.1) Let D be a subdomain of Ω and ˜σ be a matrix-valued function on D with bounded measurable coefficients, representing the conductivity of the inclu- sion. Let v be the electric potential with boundary value φ, i.e.

(div((σ(x)χ_{Ω\ ¯D}+ ˜σ(x)χ_D)∇v) = 0 in Ω,

v = φ on ∂Ω. (1.2)

The energy required to maintain voltage potential φ on ∂Ω is W :=

ˆ

∂Ω

φ hσ∇v, νi ds.

Let u be the electric potential with the same boundary value when there is no inclusion, i.e.

(div(σ(x)∇u) = 0 in Ω,

u = φ on ∂Ω. (1.3)

Similarly, we define the energy W0 :=

ˆ

∂Ω

φ hσ∇u, νi ds.

In [6], it is shown that if σ is Lipschitz continuous and for some ζ, η > 0 either

(1 + η)σ ≤ ˜σ ≤ ζσ a.e. in Ω, (1.4) or

ζσ ≤ ˜σ ≤ (1 − η)σ a.e. in Ω, (1.5) then the size of D can be estimated using the normalized power gap

W −W0

W0

  . More precisely, the following estimate holds

K₁    

W − W₀ W0

≤ |D| ≤ K₂    

W − W₀ W0

1 p

, (1.6)

where p > 1, K₁and K₂are constants depending on a priori data. If moreover D satisfies the fatness condition (4.3), then a better estimate holds

K₁    

W − W₀ W₀

≤ |D| ≤ K₂    

W − W₀ W₀

. (1.7)

We will show that in two dimension, the method of [6] works even when σ is only piecewise H¨older continuous. Essentially, this is because in two dimension, the three-ball and doubling inequalities for solutions of (1.3) hold for bounded σ; and a gradient estimate needed in proving the propagation of smallness for ∇u was proved in [15] for piecewise H¨older σ (in any dimension).

We would like to mention that size estimates have also been derived for other systems, for example, [2] for the isotropic elasticity, [16, 17, 18] for the isotropic/anisotropic thin plate, [11, 10] for the shallow shell.

The paper is organized as follows. In next section, we define several notations and list several assumptions used in the paper. In Section 3, we prove some quantitative estimates for solutions of (1.3). In Section 4, we prove (1.6) and (1.7).

2 Notations and assumptions

Definition 2.1. Let Ω be an open bounded domain of R². Given 0 < α < 1, we say that ∂Ω is of class C^1,α with parameters r0, M0, if for any P ∈ ∂Ω, there exists a rigid coordinates transform under which P = 0 and

Ω ∩ B_r0(0) = {z = (z₁, z₂) ∈ B_r0(0) : z₂ > ψ(z₁)}, where ψ(z₁) ∈ C^1,α(−r₀, r₀) satisfying ψ(0) = 0 and ∇ψ(0) = 0 and

kψk_C^1,α_(−r0,r0) ≤ M₀. Recall that

kψk_C^1,α_(−r0,r0) = kψk_L^∞_(−r0,r0)+k∇ψk_L^∞_(−r0,r0)+ sup

x,y∈(−r0,r0)

|∇ψ(x) − ∇ψ(y)|

|x − y|^α . We now state the assumptions used in the paper.

Assumptions

• Ω ⊂ R² is an open bounded C^1,α domain with parameters r₀ and M₀.

• There exist disjoint C^1,α domains Ωj ⊂ Ω, 1 ≤ j ≤ m such that Ω =

∪^m_j=1Ω_j and for some µ > 0, we have σ_j(x) := σ(x)χ_Ωj ∈ C^0,µ(Ω_j), 1 ≤ j ≤ m. For α⁰ = min{µ,_3(α+1)^α }, let M₁ = sup_jkσ_jk_C0,α0(Ωj).

• For any x ∈ Ω, there exist r > 0 and an appropriate rotation of co- ordinates such that the set (∪^m_j=1∂Ω_j) ∩ B_r(x) consists of the graphs of `(x, r) functions of class C^1,α, whose C^1,α norms are bounded by L(x, r). We assume that

L := sup

x∈Ω r>0inf



L(x, r) + `(x, r) + 1 r



< ∞.

• d = dist(D, ∂Ω) > 0.

• For some Γ ⊂ ∂Ω of positive measure, φ|_Γ = 0.

Remark 2.2. The boundaries of subdomains may touch each other. The inclusion D is only required to stay away from the boundary ∂Ω, it may intersect ∂Ω_j’s (see Figure 2.1).

Figure 2.1: Ω_j’s may touch each other and D is allowed to intersect the interfaces.

We also define for h > 0,

Ω_h = {x ∈ Ω | dist(x, ∂Ω) > h}.

3 Quantitative uniqueness estimates

In this section, we prove quantitative uniqueness estimates for solutions of (1.3) that will be used in the next section. We first recall the three ball inequality of [4].

Lemma 3.1. [4, Theorem 3.11] For all 0 < r₁ < r₂ < r₃, there exist con- stants C > 0 and 0 < τ < 1 depending only on λ, ^r_r¹

3, and ^r_r²

3 such that for any solution of (1.3) in B_r3(x), we have

kuk_L²_(Br2(x))≤ Ckuk^τ_L2(B_r1(x))kuk^1−τ_L2(B_r3(x)). (3.1) Using this three-ball inequality, we can prove

Lemma 3.2. (propagation of smallness) Assume that the assumptions in Section 2 holds. Let u ∈ H¹(Ω) be the solution of (1.3). For any ρ > 0 and every x ∈ Ω4ρ, we have

ˆ

Bρ(x)

|∇u|² ≥ C ˆ

Ω

|∇u|², (3.2)

where C depends on Ω, Γ, λ, α, µ, r₀, M₀, M₁, L, ρ, and kφk_H²_(∂Ω)/kφk_H1/2(∂Ω). Proof. We follow the arguments presented in [6, Lemma 2.2]. We first ob- serve that it suffices to consider the case ρ is small, so we can assume that Ω_ρis connected. Using Caccioppoli and Poincar´e inequalities, we can deduce from Lemma 3.1 that

k∇uk_L²_(B3r(x)) ≤ Ck∇uk^τ_L2(Br(x))k∇uk^1−τ_L2(B4r(x)). (3.3) Given x, y ∈ Ω_4ρ, let γ be a curve in Ω_4ρjoining x and y. We define a sequence x_k’s as follows: Let x₁ = x. For k > 1, let x_k = γ(t_k) where t_k = max{t :

|γ(t) − x_k−1| = 2ρ} if |x_k− y| > 2ρ; otherwise let x_k = y, N = k and stop the process. Note that since the balls B_ρ(x_k) are disjoint, N ≤ N₀ = _πρ^|Ω|2. Using (3.3), noting that B_ρ(x_k+1) ⊂ B_3ρ(x_k) because |x_k+1− x_k| ≤ 2ρ, we can deduce that

k∇uk_L²_(Bρ(xk+1)) k∇uk_L²_(Ω) ≤ C

k∇uk_L²_(Bρ(xk)) k∇uk_L²_(Ω)

τ

. By induction, we obtain

k∇uk_L²_(Bρ(y))

k∇uk_L²_(Ω) ≤ C^{1/(1−τ )}

k∇uk_L²_(Bρ(x)) k∇uk_L²_(Ω)

τ^N

. (3.4)

Since we can cover Ω_5ρ by no more than _2ρ^|Ω|2 balls of radius ρ, we obtain k∇uk_L²_(Ω5ρ)

k∇uk_L²_(Ω) ≤ C

k∇uk_L²_(Bρ(x)) k∇uk_L²_(Ω)

τ^N0

, (3.5)

where C depends on λ, |Ω|, and ρ.

By Corollary 1.3 in [15], k∇uk²_L∞(Ω) ≤ Ckφk²_C1,1/2(Ω), hence by the em- bedding H²(∂Ω) ,→ C^1,1/2(∂Ω), we get

ˆ

Ω\Ω5ρ

|∇u|² ≤ C|Ω \ Ω_5ρ|kφk²_{C1,α0(∂Ω)} ≤ Cρkφk²_H2(∂Ω). (3.6)

Here we have used |Ω\Ω_5ρ| . ρ since ∂Ω is Lipschitz.

Using the Poincar´e inequality of [9, Theorem 6.1-8 (b)], recalling that ϕ|Γ= 0, we have

kφk²_H1/2(∂Ω) ≤ Ckuk²_H1(Ω) ≤ Ck∇uk²_L2(Ω). (3.7) Combining this and (3.6), we see that if ρ is small enough depending on Ω, Γ, λ, r₀, M₀, M₁, α, µ, L, and kφk_H²_(∂Ω)/kφk_H1/2(∂Ω),

k∇uk²_L2(Ω5ρ)

k∇uk²_L2(Ω)

≥ 1 2. The lemma follows from this and (3.5).

Next, we derive a local doubling inequality for solutions of (1.3).

Lemma 3.3. For any ρ > 0, there exist δ = δ(ρ, λ) ∈ (0, ρ) and a constant C = C(ρ, λ) such that for all x ∈ Ω_ρ and r ∈ (0, δ) and any non-trivial solution u of (1.3), we have

kuk_L2(B4r(x))

kuk_L2(Br(x))

≤ Ckuk_L∞(Bρ(x))

kuk_L∞(Bδ(x))

. (3.8)

Proof. The proof, using the theory of quasiconformal maps, follows the ideas of the proof of Proposition 2 in [1]. We first note that it suffices to consider the case u is real-valued. Let v ∈ H_loc¹ (Ω) be a σ-harmonic conjugate of u, i.e.

∇v = Jσ∇u where J = 0 −1

1 0



. Then f = u + iv satisfies

∂zf = ν1∂zf + ν2∂zf , where

ν₁ = bc − ad + 1 + i(b − c)

(a + 1)(d + 1) − bc , ν₂ = d − a + i(b + c) (a + 1)(d + 1) − bc.

It is easy to check that |ν₁| + |ν₂| ≤ κ < 1. Here κ is a constant depending only on λ.

By Bers-Nirenberg representation theorem (see [8], p. 259), there exists a quasiconformal map χ : Ω → χ(Ω) and an analytic function h : χ(Ω) → C

such that f = h ◦ χ. Furthermore, there exist K, α > 1 depending on κ such that

K⁻¹|x − y|^α ≤ |χ(x) − χ(y)| ≤ K |x − y|^α1 , ∀x, y ∈ Ω.

Let δ = (10K²)^−αρ^α2 and R = (10K)⁻¹ρ^α, then we have χ(B_δ(x)) ⊂ B_R(χ(x)) and B_10R(χ(x)) ⊂ Ω.

By Theorem 3.6.2 in [7], there exists an increasing function γ depending only on κ with γ(0) = 0 such that if x₁, x₂, x₃ ∈ B(x, δ) then

|χ(x₁) − χ(x₂)|

|χ(x₁) − χ(x₃)| ≤ γ |x₁− x₂|

|x₁− x₃|

 .

Let c = γ(8) > 1 then for any x ∈ Ω_ρ and r ∈ (0, δ), there exists s ∈ (0, R/c) such that if y = χ(x) then

B_s(y) ⊂ χ(B_r/2(x)) and χ(B_4r(x)) ⊂ B_cs(y). (3.9) Since ˜u = <h is harmonic on χ(Ω), by Hadamard’s three-circle theorem, there exists an absolute constant C such that

k˜uk_L∞(Bcs(y))

k˜uk_L∞(Bs(y))

≤ Ck˜uk_L∞(B4R(y))

k˜uk_L∞(B3R(y))

.

By Theorem 3.1.2 in [7], |E| = 0 iff |χ(E)| = 0, hence (3.9) implies kuk_L∞(B4r(x))

kuk_L∞(B_r/2(x))

≤ Ckuk_L∞(B_ρ2(x))

kuk_L∞(B_ρ1(x))

.

Here

ρ1 = (3R/K)^α = 3^αδ > δ, ρ2 = (4KR)^α1 = (2/5)^1/αρ < ρ.

Using well-known estimates for elliptic equations with measurable coeffi- cients, we have

kuk_L2(B4r(x))

kuk_L2(Br(x))

≤ C kuk_L∞(B4r(x))

kuk_L∞(Br/2(x))

≤ Ckuk_L∞(Bρ(x))

kuk_L∞(Bδ(x))

.

4 Size estimates

To begin, we recall the following energy inequalities proved in [6].

Lemma 4.1. [6, Lemma 2.1] Assume that σ satisfies the ellipticity condition (1.1). If either (1.4) or (1.5) holds, then

C₁ ˆ

D

|∇u|²dx ≤ |W₀− W | ≤ C₂ ˆ

D

|∇u|²dx, (4.1) where C₁, C₂ are constants depending only on λ, η, and ζ.

We now state and prove the main theorem.

Theorem 4.2. i/ Suppose that the assumptions in Section 2 hold. Then there exist constants K₁, K₂ > 0 and p > 1 depending only on Ω, Γ, λ, α, µ, r₀, M₀, M₁, L, d, η, ζ, ρ, and kφk_H²_(∂Ω)/kφk_H1/2(∂Ω) such that

K₁    

W₀ − W W0

≤ |D| ≤ K₂    

W₀− W W0

1 p

. (4.2)

ii/ If moreover, there exists h > 0 such that

|D_h| ≥ 1

2|D| (fatness condition). (4.3) then

K₁    

W0− W W₀

≤ |D| ≤ K₂    

W0− W W₀

, (4.4)

where K₁ and K₂ depend on the various constants as in i/ and also on h.

Proof. The proof closely follows the arguments of [6].

We first establish the lower bound. Let c = ¹

|^Ωd/4|

´

Ω_d/4u. By the gradient estimate of [15, Theorem 1.1], the interior estimate of [14, Theorem 8.17] and the Poincar´e inequality for the domain Ω_d/4, we have

k∇uk_L^∞_(Ωd/2) ≤ Cku − ck_L^∞_(Ωd/3) ≤ Cku − ck_L²_(Ωd/4)≤ Ck∇uk_L²_(Ω). From this, the trivial estimate k∇uk²_L2(D) ≤ C|D|k∇uk²_L∞(Ωd/2) and Lemma 4.1, the lower bound follows.

Next, we establish the upper bounds.

i/ We will first establish that |∇u|² is an A_p-weight, following the proof of Theorem 1.1 in [13]. Let ρ = d/5 and δ be the constant appears in Lemma 3.3. By Cacciopolli inequality and (3.2), for any x ∈ Ω5ρ we have

ku − ck_L∞(Bδ(x))≥ C ku − ck_L2(Bδ(x)) ≥ C k∇uk_L2(Bδ/2(x)) ≥ C k∇uk_L2(Ω). (Note that C depends also on δ). By interior estimate, we have

ku − ck_L∞(Bρ(x)) ≤ 2 kuk_L∞(Bρ(x))≤ C kϕk_H1/2(∂Ω).

For r ∈ (0, δ), applying the doubling inequality of 3.3 to u − c where c =

1

|B_r|

´

Br(x)u, we get ku − ck_L2(B2r(x))

ku − ck_L2(Br(x))

≤ Cku − ck_L∞(Bρ(x))

ku − ck_L∞(Bδ(x))

≤ C kϕk_H1/2(∂Ω)

k∇uk_L2(Ω)

≤ C.

At the last inequality we have used (3.8). We note that the constant C depends on various constants, including kϕk_H2(∂Ω)/ kϕk_H1/2(∂Ω) but is inde- pendent of r.

This and the Caccioppoli inequality give

r⁻¹k∇uk_L2(Br(x))≤ C ku − ck_L2(B2r(x))≤ C ku − ck_L2(Br(x)). Combining this with the Poincar´e inequality

 1

|B_r(x)|

ˆ

Br(x)

|u − c|²

¹₂

≤ Cr⁻¹

 1

|B_r(x)|

ˆ

Br(x)

|∇u|³²

²₃ , we get

 1

|B_r(x)|

ˆ

Br(x)

|∇u|²

¹₂

≤ C

 1

|B_r(x)|

ˆ

Br(x)

|∇u|³²

²₃ .

This reverse H¨older inequality shows that |∇u|² is an A_p-weight for some p > 1 (see [12, Chapter 7]).

We cover D with internally nonoverlapping closed squares Q_k, 1 ≤ k ≤ I, with side length 2ρ. Since |∇u|² is and A_p-weight, by [12, (7.2)], we have

|D ∩ Q_k|

|Q_k| ≤ C ´

D∩Qk|∇u|²

´

Qk|∇u|²

!1/p

.

Summing over k and using (3.2), we get

|D| ≤ C

´

D|∇u|² min_k´

Qk|∇u|²

!1/p

≤ C

´

D|∇u|²

´

Ω|∇u|²

!1/p

.

The upper bound of |D| now follows from (4.1).

ii/. Let ρ = ¹₄min{d, h} and cover D_h with internally nonoverlapping closed squares {Q_k}^J_k=1 of side length 2ρ. It is clear that Q_k ⊂ D, hence

ˆ

D

|∇u|²dx ≥ ˆ

∪^J_k=1Qk

|∇u|²dx ≥ |D_h| ρ² min

k

ˆ

Qk

|∇u|²dx.

≥ C|D|

ρ² ˆ

Ω

|∇u|²dx.

Here we have used Lemma 3.2 and the fatness condition at the last inequality.

The upper bound of |D| follows from this and Lemma 4.1.

Acknowledgements

The authors were supported in part by the National Science Council of Tai- wan grant 99-2115-M-002-006-MY3. We are grateful to Sergio Vessella for carefully reading this paper and for many insightful comments.

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