2 Statement of the result

Uniqueness for the two dimensional Calder´ on’s problem with unbounded conductivites

C˘ at˘ alin I. Cˆ arstea

Jenn-Nan Wang

Abstract

In this work we consider the Calder´on problem in two dimensions with conductivity γ ∈ W^1,2(Ω). This condition allows for the conductivity to be unbounded. We prove a uniqueness result when ||∇ log γ||_L2

is bounded by a fixed constant depending on the domain Ω.

1 Introduction

The inverse conductivity problem, first discussed by Calder´on in [8], consists of determining the conductivity of the interior of an object from measurements of electrical potential and current taken on the boundary.

One aspect of this problem is weather or not two different conductivity functions might give rise to the same set of boundary measurements. This question has been considered for different spatial dimensions and under various assumptions on the regularity of the conductivity function. An early result in dimension greater than two and for smooth conductivities was obtained by Sylvester and Uhlmann in their seminal paper [18].

Recently, Haberman and Tataru have shown in [12], also in dimensions higher than two, that uniqueness holds for Lipschitz conductivities γ such that ||∇ log γ||L^∞ is small. In [11], Haberman has further improved on this result in dimensions 3 ≤ n ≤ 6. In dimensions n = 3, 4 he proves uniqueness for conductivities in W^1,n. We want to point out that in [11] the conductivity γ satisfies c ≤ γ ≤ c⁻¹, in particular, γ ∈ L^∞. In dimension two, Nachman proved in [14] uniqueness for conductivities in W^2,p for some p > 1. Brown and Uhlmann in [7] established the uniqueness for conductivities in W^1,pwith p > 2. In [2], Astala and P¨aiv¨arinta proved the uniqueness for conductivities in L^∞. In the recent paper [3], Astala, Lassas, and P¨aiv¨arinta have proved a uniqueness result (see their Theorem 1.9) that also applies to some unbounded conductivities.

In this paper we will study the two-dimensional Calder´on’s problem in a different class of singular con- ductivities. Precisely, we prove that uniqueness holds for the strictly positive conductivity γ ∈ W^1,2, with and additional constraint that ||∇ log γ||_L2(Ω) < C with C = C(Ω). It is important to emphasize that we do not assume γ ∈ L^∞. For example, γ(x) = log |log |ax|| or γ(x) = (log |ax|)^s, 0 < s < ¹₂, have finite

||∇ log γ||_L2(B 1 2|a|

), thus can be used to construct examples. The second of these two examples also falls outside the conditions required by Theorem 1.9 of [3]. Therefore, our result is not a consequence of previous work. In view of the unique continuation property for the linear convection equation with L² coefficients in R²[13], the assumption of γ ∈ W^1,2 is most likely optimal for the two dimensional Calder´on’s problem. It is not yet certain that the restriction on the L² norm of ∇ log γ is necessary.

This work is inspired by the paper of Cheng and Yamamoto in [9] where they prove a uniqueness result for the equation

4u + ~b · ∇u = 0

with ~b ∈ L^p, p > 2. Here we push their result to the optimal case p = 2 (with an added requirement that the L² norm of ~b be bounded by a constant) and obtain the result for Calder´on’s problem as a corollary.

The proof relies on reducing the second order equation to a first order one in the complex plane (see [19], [1]

for background). It also employs an inverse scattering method developed by Beals and Coifman in [4] and by Sung in [15], [16], and [17]. Along the way we also prove a version of Brown’s result in [6] regarding the recovery of the boundary values of the conductivity from the knowledge of the Dirichlet–Neumann map.

∗National Center for Theoretical Sciences Mathematics Division, Taipei 106, Taiwan. Email: cicarstea@ncts.ntu.edu.tw

†Institute of Applied Mathematical Sciences, NCTS, National Taiwan University, Taipei 106, Taiwan. Email: jn- wang@math.ntu.edu.tw

2 Statement of the result

We consider a domain Ω ⊂ R² which is open, bounded, simply connected, and with sufficiently smooth boundary. For a conductivity γ : Ω → R positive and bounded away from zero, we will consider the boundary value problem

( ∇(γ∇u)(x) = 0, x ∈ Ω,

u|∂Ω= ω. (1)

The Dirichlet–Neumann map for the classical Calderon problem is then Λγ(ω) := γ|∂Ω

∂u

∂ν,

where u is the solution with boundary value ω. Equation (1) can be put in non-divergence form as ( 4u(x) + ~b(x) · ∇u(x) = 0, x ∈ Ω,

u|∂Ω= ω,

(2)

where

~b := ∇ log γ.

For this equation, we consider the Dirichlet–Neumann map Λ~b(ω) := ∂u

∂ν.

Throughout, we assume γ ∈ W^1,2(Ω) or ~b ∈ L²_R(Ω), where L²_R(Ω) denotes the space of L² real-valued vectors in Ω. For both forms of the equation, we consider strong solutions u ∈ W^2,p(Ω), for some 1 < p < 2.

The Dirichlet boundary data then belong to W^2−1p,p(∂Ω). In the appendix we give a proof of existence and uniqueness of solutions for the case when ||~b||L²(Ω)< C, with C = C(p, Ω).

Our main result is the following

Theorem 2.1. There exists a constant C = C(Ω) such that if γ₁, γ₂∈ W^1,2(Ω), ||∇ log γ_1,2||L²(Ω)< C and Λ_γ1 = Λ_γ2, then γ₁= γ₂.

This follows immediately from Theorem 2.2 and Proposition 2.1 stated below.

Theorem 2.2. There exists a constant C = C(Ω) such that if ~b1,~b2∈ L²_R(Ω), ||~b1,2||_L2(Ω)≤ C and Λ_~b

1 = Λ_~b

2, then ~b1= ~b2.

Note that this second theorem is more general than the first. It is this result that we will prove below.

We also prove the following boundary determination result.

Proposition 2.1. If γ ∈ W^1,2(Ω), the trace γ|∂Ω can be determined from the Dirichlet-to-Neumann map Λ_γ.

Since the proof of Proposition 2.1 is of a different nature from that of Theorem 2.2, the principal result of this paper, we have provided it as an appendix.

3 Complex variable notation and the first order form of the equa- tion

We will identify R²with C and use the usual notation dz = dx1+i dx₂, d¯z = dx₁−i dx₂, ∂_z =¹₂(∂_x1−i∂_x2),

∂_z¯ =¹₂(∂_x1+i∂_x2). If u ∈ W^2,p(Ω) is a solution of (2), then w := ∂_zu ∈ W^1,p(Ω) is a solution of the equation

∂z¯w + Bw + ¯B ¯w = 0, (3)

where B(z) := ¹₄(b1+ ib2). We can also go the other way:

Lemma 3.1. (e.g. see [19]) Given w ∈ W^1,p(Ω) a solution to (3), then there exists a solution u ∈ W^2,p(Ω) of (2) such that w = ∂_zu.

For Dirichlet boundary conditions, as we have in (2), w must satisfy 2Re ( ˙zw)|∂Ω= ∂ω

∂t (the tangential derivative along ∂Ω). (4) Here ˙z is the derivative of a parametrization z(t) of the boundary ∂Ω. In the case of Neumann boundary conditions, we would instead need

2Im ( ˙zw)|∂Ω= ∂u

∂ν|∂Ω. (5)

For this see [1], section 16.5. Note that the relations (4) and (5) are valid for real solutions u of (2). Since ~b is a real vector, these relations remain true for complex-valued solutions of (2).

We define the Cauchy transform

(Cf )(z) := −1 π

Z

Ω

f (ζ) ζ − zd²ζ.

It maps C : L^q(Ω) → W^1,q(Ω) continuously for all 1 < q < ∞ (again, see [1]). The operator C is an “inverse”

to the differential operator ∂z¯. For f ∈ C^∞(Ω) ∩ C( ¯Ω)

(C∂¯zf )(z) = −1 πlim



Z

Ω−B(z,)

∂ζ¯

f (ζ) ζ − z

d ¯ζ ∧ dζ 2i

= 1 2πi lim



Z

∂B(z,)

f (ζ) − f (z)

ζ − z dζ + 2πif (z) − Z

∂Ω

f (ζ) ζ − zdζ

! , so

(C∂¯zf )(z) = f (z) − 1 2πi

Z

∂Ω

f (ζ)

ζ − zdζ. (6)

Also, defining hϕ, f i :=R

Ωϕf , for any ϕ ∈ C_c^∞(Ω)

hϕ, ∂z¯Cf i = hC∂z¯ϕ, f i = hϕ, f i and so we have

∂_z¯Cf = f. (7)

Given the mapping properties of the Cauchy transform, the formulas (6), (7) extend to any f ∈ W^1,q(Ω), 1 < q < ∞.

The Cauchy transform may be used to turn (3) into an integral equation.

Lemma 3.2. A function w ∈ W^1,p(Ω) that satisfies (3) will also satisfy the integral equation w(z) + C(Bw + ¯B ¯w)(z) = 1

2πi Z

∂Ω

w(ζ) ζ − zdζ.

Also, if Φ ∈ Hol(Ω) ∩ W^1,p(Ω) and w ∈ W^1,p(Ω) satisfies

w(z) + C(Bw + ¯B ¯w)(z) = Φ(z), (8)

then w satisfies (3) and

1 2πi

Z

∂Ω

w(ζ)

ζ − zdζ = Φ(z).

Next we would like to show the existence of solutions to (8), with suitable Φ. A contraction principle argument of the same form as the one in the proof of Proposition A.1 would work. Instead, we present a different type of argument which doesn’t require the norm of the coefficients to be small. Testing the equation (8) against a test function φ we get

hφ −

B + ¯Bw¯ w

C(φ), wi = hφ, Φi. (9)

Let A := B + ¯B ¯w/w ∈ L²(Ω), ||A||_L2(Ω) ≤ 2||B||_L2(Ω), and ψ := C(φ). Let 1 < p < 2 and p^∗= 2p/(2 − p).

In order to obtain an a priori estimate for the L^p∗ norm of w, we would like that

∂z¯ψ − Aψ = |w|^p∗−2w.¯ (10)

It turns out that one solution of (10) is explicitly expressed by ψ = C

|w|^p∗−2w e¯ ^−CA e^CA.

The left hand side of (9) is just ||w||^p_L^∗_p∗(Ω). The right hand side of (9) consists of two terms:

h|w|^p∗−2w, Φi + hAψ, Φi =: I + II.¯ H¨older’s inequality implies

I ≤ ||w||^p_L^∗_p∗⁻¹_(Ω)||Φ||_Lp∗(Ω).

To deal with the second term we begin by noticing that |w|^p∗−2w ∈ L¯ ^{p∗/(p∗−1)}(Ω). With the help of Trudinger’s inequality, we can see that e^CA, e^−CA∈ L^q(Ω) for any q < ∞ (for example, see the computation in [13]). It follows that ψ is in all L^r(Ω) with r < 2p^∗/(p^∗− 2). We can then bound the second term

II ≤ ||A||_L2(Ω)||ψ||_Lr(Ω)||Φ||_Lp∗ +(Ω), r = 2(p^∗+ ) p^∗+  − 2 with  > 0. Since ||ψ||_Lr(Ω)≤ C||w||^p_L^∗_p∗⁻¹_(Ω), we get

II ≤ C||w||^p_L^∗_p∗⁻¹_(Ω)||A||_L2(Ω)||Φ||_Lp∗ +(Ω). Putting these together implies that

kwk_Lp∗(Ω)≤ C(1 + kAk_L2(Ω))kΦk_Lp∗ +(Ω)

and

kwk_W1,p(Ω)≤ C(1 + kAk_L2(Ω))kΦk_Lp∗ +(Ω).

If Φ ∈ W^1,p+(Ω) for some  > 0 and B ∈ C₀^∞(Ω), then a solution to (8) is known to exits (see, for example, the proof of [9, Lemma 3.3]). Given B ∈ L²(Ω) let B_n ∈ C₀^∞(Ω) be such that ||B_n− B||L² → 0, and let w_n be the solutions of

w_n+ C(B_nw_n+ ¯B_nw¯_n) = Φ.

Then

||wn||_W1,p+/2(Ω)≤ C||Φ||W^1,p+(Ω)(1 + ||B||L²).

Also

wn− wm+ C(Bn(wn− wm) + Bn(wn− wm)) = −C((Bn− Bm)wm+ (Bn− Bm)wm).

According to the observation above, the right had side is in W^1,p+/2(Ω) and

||RHS||_W1,p+/2(Ω)≤ C||Bn− Bm||L²||Φ||_W1,p+(Ω)(1 + ||B||L²).

Applying the a priori estimate to the difference w_n− w_m we get

||wn− wm||_W1,p(Ω)≤ C||Bn− Bm||_L2||Φ||_W1,p+(Ω)(1 + ||B||_L2)².

This proves that the sequence of approximate solutions is Cauchy in W^1,p(Ω). That the limit is a solution to the equation follows from the continuity of C and the continuity of products w.r.t. strong convergence.

We have therefore proven:

Lemma 3.3. If 1 < p < 2, Φ ∈ W^1,p+(Ω),  > 0 with p +  < 2, then the equation w + C(Bw + ¯B ¯w) = Φ

has a unique solution w ∈ W^1,p+/2(Ω) and the following estimate holds

kwk_W1,p+/2(Ω)≤ CkΦk_W1,p+(Ω), (11)

where C = C(kAk_L2(Ω), p, , Ω).

4 CGO solutions (introduction of the parameter k)

In this section, we want to discuss complex geometrical optics (CGO) solutions of (3). These special solutions are useful in Calder´on’s problem. Let k = kx+ iky ∈ C. If w is a solution to (3) then αk(z) := w(z)e^−2ikz¯ satisfies the equation

∂_¯zα_k(z) + B(z)α_k(z) + e^{−2i(¯kz+k ¯z)}B(z)α_k(z) = 0 in Ω. (12) We will use the notation

ek(z) = exp



−i

2(¯kz + k ¯z)



= e^i(kxx+kyy). Clearly, |ek(z)| = 1. Analogously to the previous section, we have

Lemma 4.1. If α_k∈ W^1,p(Ω), 1 < p < 2, satisfies (12), then it also satisfies the integral equation αk(z) + C(Bαk+ ekB ¯¯αk)(z) = 1

2πi Z

∂Ω

αk(ζ) ζ − z dζ.

Conversely, the equation

α_k(z) + C(Bα_k+ e_kB ¯¯α_k)(z) = 1 (13) has a unique solution αk ∈ W^1,p(Ω) and

||αk||W^1,p(Ω)≤ C(||~b||L²(Ω), Ω, p).

This solution will also satisfy (12) and 1 2πi

Z

∂Ω

α_k(ζ)

ζ − z dζ = 1.

4.1 Differentiability in k

In order to investigate the differentiability with respect to k of α_k , for some fixed k = k_x+ ik_y we introduce the notations

δκα = 1

κ(αk+κ(z) − αk(z)) and δκe = 1

κ(ek+κ(z) − ek(z)).

Note that this quantity satisfies the integral equation

δκα + C Bδκα + ek+κBδ¯ κα + δ¯ κe ¯B ¯αk
= 0.

We will only consider real valued κ (the case of imaginary κ will only different by a minus sign). In this case δκα + C Bδκα + ek+κBδ¯ κα
= −C (δκe) ¯B ¯αk
. (14) Let  > 0 be such that p +  < 2, then it follows from (11) that

||δκα||_W1,p+/2(Ω)≤ C(||~b||_L2(Ω), p, )||δκe||_L∞(Ω). In order to prove that δκα is Cauchy, we need to estimate the quantity

∆κ,κ⁰α := δκα − δκ⁰α.

Note that ∆_κ,κ⁰e → 0 in C( ¯Ω). It is clear that ∆_κ,κ⁰α satisfies the integral equation

∆_κ,κ⁰α + C B∆_κ,κ⁰α + e_k+κ⁰B∆¯ _κ,κ⁰α

= −C (∆_κ,κ⁰e) ¯B ¯α_k+ (e_k+κ⁰− e_k+κ) ¯Bδ_κα
. Applying the a priori estimate (11) again we obtain

||∆κ,κ⁰α||_W1,p(Ω)≤ C(||~b||_L2(Ω), p, ) ||∆κ,κ⁰e||_L∞(Ω)+ κ||δκe||_L∞(Ω)+ κ⁰||δκ⁰e||_L∞(Ω)
.

We have thus shown that δκα is Cauchy in W^1,p(Ω) as κ → 0 for real κ. Taking the limit in (14) we see that

∂k_xα ∈ W^1,p(Ω) satisfies

∂k_xα + C B∂k_xα + ekB∂¯ k_xα
= −C (ixek) ¯B ¯αk
.

We can again apply (11) to conclude that ∂k_xα is bounded in W^1,p(Ω) uniformly in k. The case of imaginary κ is almost identical. We conclude that

Proposition 4.1. The partial derivatives of α_k with respect to k exist and ∂_kxα_k, ∂_kyα_k ∈ L^∞(Ck; W^1,p(Ω)).

4.2 Behavior as k → ∞

Because the αkare bounded in W^1,p(Ω) uniformly in k, we can extract a subsequence, also denoted αk, such that αk * α0 in L^p∗ and αk → α0 in L^q for any 1 ≤ q < p^∗. Of course, α0∈ W^1,p(Ω).

To find the equation satisfied by α0 we are going to integrate (13) against a test function ϕ ∈ C_c^∞(Ω), then for the first term we have

hϕ, αki → hϕ, α0i.

For the second term, since Cϕ ∈ L^∞(Ω) and B ∈ L²(Ω),

hϕ, C(Bαk)i = −hBCϕ, α_ki → −hBCϕ, α0i = hϕ, C(Bα0)i.

To handle the third term, we write

hϕ, C(ekB ¯¯αk)i = −hekBCϕ, ¯¯ α0i − hekBCϕ, ¯¯ αk− ¯α0i.

Since ¯αk→ ¯α0 strongly in L²(Ω),

hekBCϕ, ¯¯ αk− ¯α0i → 0.

As ¯α0BCϕ ∈ L¯ ^p(Ω), the Riemann-Lebesgue lemma implies hekBCϕ, ¯¯ α0i → 0.

Putting these together we get that

hϕ, α0+ C(Bα₀) − 1i = 0, for any ϕ ∈ C_c^∞(Ω) and so

α₀(z) + C(Bα₀)(z) = 1, ∀z ∈ Ω.

Of course it then follows that

∂¯zα0+ Bα0= 0, and applying back the Cauchy transform we need to have

1 2πi

Z

∂Ω

α0(ζ)

ζ − z dζ = 1.

We have thus proved

Lemma 4.2. There exists a subsequence of solutions αk of (13) such that αk * α0 in W^1,p(Ω) ⊂ L^p∗ and αk→ α0 in L^q for any 1 ≤ q < p^∗. The limit satisfies the integral equation

α0(z) + C(Bα0)(z) = 1, ∀z ∈ Ω,

and 1

2πi Z

∂Ω

α₀(ζ)

ζ − z dζ = 1.

In fact

α0= e^−C(B). (15)

Proof. We only have to show that the representation formula (15) holds. First not that extending B to the whole plane such that B = BχΩand using the equation to extend α0 we have that α0∈ 1 + W^1,p(C).

Define h := α0e^C(B)and note that ∂z¯h = 0 in C, so it is holomorphic on C. Since B is supported within the bonded domain Ω, C(B) and C(Bα0) are both holomorphic on C − ¯Ω. Furthermore, they both decay as 1/z at infinity. It follows then that lim_z→∞h = 1 and by Liouville’s theorem this implies that h ≡ 1.

5 Cauchy transforms of B

and B

In this section, we would like to show that under the assumptions of Theorem 2.2, the Cauchy transforms of B1 and B2 are identical outside of the domain Ω. Let α1,k, α2,k∈ W^1,p(Ω) be the solutions of the integral equations

αj,k+ C(Bjαj,k+ ekB¯jα¯j,k) = 1, j = 1, 2.

First note that, according to Lemma 4.1, 1 2πi

Z

∂Ω

α_1,k(ζ)

ζ − z dζ = 1, ∀z ∈ Ω.

Define w₁(z, k) := e^i2¯kzα_1,k(z), w₁(·, k) ∈ W^1,p(Ω). It satisfies

∂¯zw1+ B1w1+ ¯B1w¯1= 0.

By Lemma 3.1, there exists a u1(·, k) ∈ W^2,p(Ω) such that w1= ∂zu1and 4u₁+ ~b₁· ∇u₁= 0.

Denote ω := u₁|∂Ω∈ W^2−1p,p(∂Ω) and let u₂∈ W^2,p(Ω) be the solution of ( 4u2+ ~b2· ∇u2= 0,

u2|∂Ω= ω.

The existence of u2 with boundary condition ω is guaranteed by choosing the constant C in the statement of Theorem 2.2 such that Proposition A.1 applies. Then w2:= ∂zu2will satisfy

∂¯zw2+ B2w2+ ¯B2w¯2= 0.

Since u₁and u₂share the same Dirichlet data, Re ( ˙zw₁) = Re ( ˙zw₂). Since we are assuming the Dirichlet–

Neumann maps produced by ~b1 and ~b2 are identical, the Neumann data of u1 must be the same as u2, so Im( ˙zw1) = Im ( ˙zw2). It follows then that

w₁|∂Ω= w₂|∂Ω. Define α⁰_2,k(z) := w₂(z, k)e^−i2¯kz. It satisfies the differential equation

∂z¯α⁰_2,k+ B2α⁰_2,k+ ekB¯2α¯⁰_2,k= 0 and, as α1,k|∂Ω= α⁰_2,k|∂Ω, we see that α⁰_2,ksatisfies the integral equation

α⁰_2,k+ C(B2α⁰_2,k+ ekB¯2α¯⁰_2,k) = 1.

It follows the uniqueness of the solution that α⁰_2,k= α2,k. Consequently, we proved that Lemma 5.1. α_1,k|_∂Ω= α_2,k|_∂Ω.

We know from Lemma 4.2 that we can find a subsequence such that αj,k * αj,0 in W^1,p(Ω), j = 1, 2, and

αj,0+ C(Bjαj,0) = 1.

The trace operator τ maps W^1,p(Ω) to W^1−1p,p(∂Ω) ⊂ L^3−p2p (∂Ω) continuously and is compact from W^1,p(Ω) to L^s(∂Ω) for 1 < s < _3−p^2p . Therefore

α_j,k|∂Ω→ αj,0|∂Ω, in L^s(∂Ω).

Lemma 5.1 them implies

α1,0|∂Ω= α2,0|∂Ω. We can now prove

Γ

Ω

z

R

Figure 1: Contour for the determination of e^−C(Bj)(z).

Lemma 5.2. If Λ~b₁= Λ~b₂, then C(B1)(z) = C(B2)(z) for any z ∈ C − Ω.

Proof. By equation (15), we know that e^−C(B1)|∂Ω= e^−C(B2)|∂Ω.

Let z ∈ C − ¯Ω. Since e^−C(Bj) is holomorphic in C − ¯Ω, using the integration contour from the figure, we can write

e^−C(Bj)(z) = 1 2πi

Z

∂Ω

e^−C(Bj)(ζ)

ζ − z dζ − 1 2πi

Z

|ζ|=R

e^−C(Bj)(ζ) ζ − z dζ.

as ζ → ∞, e^−C(Bj)= 1 + O(¹_ζ) so

lim

R→∞

1 2πi

Z

|ζ|=R

e^−C(Bj)(ζ)

ζ − z dζ = 1.

It then follows that

e^−C(B1)(z) = e^−C(B2)(z),

for any z ∈ C − ¯Ω. This can happen only if C(B1)(z) = C(B2)(z) + 2πni. Since both Cauchy transforms vanish at infinity we must have n = 0.

6 A related first order system

To proceed further, we now would like to apply an inverse scattering method for a related first order system based on [4], [15], [16], and [17]. A similar idea was also used in [7] and [9]. We define

Cj:= e^CBje^−CBjB¯j, j = 1, 2.

Note that, since CBj− CBj = 2iIm (CBj), Cj ∈ L²(Ω) and ||Cj||L²(Ω) = ||Bj||L²(Ω). We also define the matrix

Qj:=

 0 −Cj

−Cj 0

 .

Let µj(z) be the 2 × 2 matrix valued function that satisfies the integral equation µj(z, k) = I + C (ekQjµ¯j) = I − 1

π Z

Ω

e_k(ζ)

ζ − zQj(ζ)µj(ζ, k) d²ζ. (16) Consider the orthogonal matrix

R := 1

√2

1 1 1 −1



Conjugating Qj by R we get

RQjR^t=−C_j 0 0 C_j



Thus, conjugating the integral equation (16) by R, we obtain a decoupled system of four scalar integral equations. We then can apply the same method we have used to prove the existence of solutions to the equation (8) to show (16) has unique solutions µj(·, k) ∈ W^1,p(Ω). These solutions satisfy the differential equation

∂z¯µj(z, k) − ek(z)Qj(z)µj(z, k) = 0 and the equality

I = 1 2πi

Z

∂Ω

µj(ζ, k)

ζ − z dζ, ∀z ∈ Ω, k ∈ C.

Also, just like the CGO solutions αk, µj are differentiable in k and ∂k_xµj, ∂k_yµj ∈ W^1,p(Ω).

Define

η1(z, k) := e^i2¯kzµ1(z, k).

This new matrix-valued function satisfies the differential equation

∂z¯η1= Q1η¯1. Let

v1:=1 1 i −i

 η1, then

∂z¯v1=1 2

1 1 i −i

 Q1

1 i 1 −i



¯

v1= −C1v¯1. Finally let

w1:= e^−CB1v1

and we get that w1 satisfies the matrix differential equation

∂_z¯w₁(z, k) + B₁(z)w_j(z, k) + B₁(z)w₁(z, k) = 0, ∀z ∈ Ω, k ∈ C.

Applying Lemma 3.1 to the components of w₁, we obtain that there is a matrix-valued function u₁ ∈ W^2,p(Ω) such that w₁= ∂_zu₁ and

4u₁+ ~b₁· ∇u₁= 0.

Let ω := u1|∂Ω∈ W^2−1p,p(∂Ω) and let u2∈ W^2,p(Ω) be the 2 × 2 matrix valued solution of the equation ( 4u2+ ~b2· ∇u2= 0,

u₂|_∂Ω= ω.

Just as before, we know that both the Dirichlet and the Neumann data of u₁ and u₂ coincide. Then w₂:= ∂_zu₂ satisfies

∂_¯zw₂+ B₂w₂+ ¯B₂w¯₂= 0 and

w1|∂Ω= w2|∂Ω. Also v2:= e^CB2w2 satisfies

∂z¯v2= −C2¯v2

and since, by Lemma 5.2, e^CB1|∂Ω= e^CB2|∂Ωwe have

v2|∂Ω= v1|∂Ω. Let

η2:=1 2

1 −i 1 i



v2, i.e., v2:=1 1 i −i

 η2, and finally

µ⁰₂:= e^−2i¯kzη₂.

This last matrix-valued function satisfies the differential equation

∂_¯zµ⁰₂= e_kQ₂µ¯⁰₂ and

µ1|∂Ω= µ⁰₂|∂Ω. Applying the Cauchy transform to the differential equation we get

µ⁰₂= Φ2+ C(ekQ2µ¯⁰₂), where

Φ₂(z) = 1 2πi

Z

∂Ω

µ⁰₂(ζ)

ζ − z dζ = 1 2πi

Z

∂Ω

µ₁(ζ)

ζ − z dζ = I.

So µ⁰₂≡ µ2and we have Lemma 6.1. µ1|∂Ω= µ2|∂Ω.

7 The ∂

equation

We now extend µj(z, k), j = 1, 2, to z ∈ C \ Ω by setting µj(z, k) = I + C (ekQjµ¯j) = I − 1

π Z

Ω

ek(ζ)

ζ − zQj(ζ)µj(ζ, k) d²ζ, ∀ z ∈ C \ Ω.

Abusing notation a little, if we write Qj= QjχΩ, then µj(z, k) satisfies µj(z, k) = I + C (ekQjµ¯j) = I − 1

π Z

C

ek(ζ)

ζ − zQj(ζ)µj(ζ, k) d²ζ, ∀ z ∈ C, ∀ k ∈ C.

We can easily see that

lim

|z|→∞µ_j(z, k) = I, that

µj(·, k) ∈ W^1,p(Ω), and µj(z, k) − I decays like ¹_z as z → ∞.

To simplify the notations, we suppress the index j. In other words, the computations below are carried out for µ_j, j = 1, 2, respectively. Define the matrix

ν(z, k) := µ₁₁(z, k) e_−k(z)µ₁₂(z, k) e_−kµ₂₁(z, k) µ₂₂(z, k)

! .

Note that

ν(z, k) = I + 1 π

Z

Ω





C(ζ)

ζ−¯¯ zν21(ζ, k) ^{ek(ζ−z)C(ζ)}_ζ−z ν22(ζ, k)

e_k(ζ−z)C(ζ)

ζ−z ν₁₁(ζ, k) ^C(ζ)_{ζ−¯¯ z}ν₁₂(ζ, k)



 d²ζ

and ν is the unique solution of this integral equation. Repeating the proof of Proposition 4.1, we can show that ∂k¯ν(z, k) exists in W^1,p(Ω) and it satisfies

∂¯kν = e−k(z) 2πi

Z

Ω

0 ek(ζ)C(ζ)µ22(ζ, k) ek(ζ)C(ζ)µ11(ζ, k) 0

! d²ζ

+e_−k(z) π

Z

Ω





C(ζ)

ζ−¯¯ ze_k(z)∂¯kν₂₁(ζ, k) ^ek(ζ)C(ζ)_ζ−z ∂k¯ν₂₂(ζ, k)

ek(ζ)C(ζ)

ζ−z ∂¯kν11(ζ, k) ^C(ζ)_{ζ−¯¯ z}ek(z)∂k¯ν12(ζ, k)



 d²ζ.

This can be rewritten as e_k(z)∂¯kν = 0 T₁₂(k)

T21(k) 0

!

+1 π

Z

Ω





e_k(ζ−z)C(ζ)

ζ−¯¯ z ek(ζ)∂¯kν21(ζ, k) ^C(ζ)_ζ−zek(ζ)∂k¯ν22(ζ, k)

C(ζ)

ζ−zek(ζ)∂k¯ν11(ζ, k) ^{ek(ζ−z)C(ζ)}_{ζ−¯¯ z} ek(ζ)∂¯kν12(ζ, k)



 d²ζ, (17) where we define

T₁₂(k) = 1 2πi

Z

Ω

e_k(ζ)C(ζ)µ₂₂(ζ, k) d²ζ, T21(k) = 1

2πi Z

Ω

ek(ζ)C(ζ)µ11(ζ, k) d²ζ.

(18)

Comparing the integral equation (17) with

ν(z, k) 0 T₁₂(k) T21(k) 0

!

=

= T21(k)ν12(z, k) T12(k)ν11(z, k) T21(k)ν22(z, k) T12(k)ν21(z, k)

!

= 0 T12(k) T₂₁(k) 0

!

+ 1 π

Z

C





e_k(ζ−z)C(ζ)

ζ−¯¯ z T₂₁(k)ν₂₂(ζ, k) ^C(ζ)_ζ−zT₁₂(k)ν₂₁(ζ, k)

C(ζ)

ζ−zT21(k)ν12(ζ, k) ^{ek(ζ−z)C(ζ)}_{ζ−¯¯ z} T12(k)ν11(ζ, k)



 d²ζ, we deduce from the uniqueness of integral equation (17) that

∂k¯ν(z, k) = e_−k(z)ν(z, k) 0 T12(k) T₂₁(k) 0

! . Since

µ11= 1 − C(ekC) + C(ekC ¯C(e_−kC ¯¯µ11)), µ₂₂= 1 − C(e_kC) + C(e_kC ¯C(e−kC ¯¯µ₂₂)), it follows that µ11= µ22, so also T12= T21. Now let us denote

T (k) =

 0 T12(k) T12(k) 0



and specify Tj(k) be as T above defined for Cj, j = 1, 2. At this point we can also note that

∂k¯ν(z, k) = e_−k(z)ν(z, k)T (k) = e_−k(z)T (k)ν(z, k).

We have

Lemma 7.1. T₁(k) = T₂(k) for all k.

The proof of this is identical to the argument given for the corresponding statement of [9, (4.25)] as part of the proof of their Lemma 4.4. The result of our Lemma 6.1 is needed in the proof.

We can now use a known result of Brown on the mapping properties of the scattering map to conclude that

Lemma 7.2. C₁= C₂.

Proof. Note that the off-diagonal entries of Q_jare zero. We use Theorem 2 of [5]. There it is shown that the mapping Q → T is invertible, hence injective, provided kQk_L2 <√

2. This can be arranged by choosing the constant C in the statement of Theorem 2.2 appropriately. Since we have already determined that T₁= T₂, it follows that Q₁= Q₂.

8 Conclusion of the argument

By Lemma 7.2 we have that

B₁e^CB1−CB1 = B₂e^CB2−CB2, ∀ z ∈ Ω, and hence

B1e^CB1−CB1 = B2e^CB2−CB2, ∀ z ∈ C

since B1 = B2 = 0 in C \ Ω. In view of Lemma 5.2, if we let Z := C(B²− B1), then Z ∈ W^1,2(C) and supp(Z) ⊆ Ω. Denote

κ := e^Z−Z¯ . Then we obtain

B1= κB2

and

∂z¯Z = (1 − κ)B2, z ∈ C.

By an easy computation (see [9, P. 1389]), we can deduce that

|κ − 1| ≤ 2|Z|

and thus

|∂_z¯Z| ≤ 2|B₂| |Z|

holds. Applying the generalized Liouville theorem of [7, Corollary 3.11] (or [1, Theorem 5.8.3]), it follows then that Z ≡ 0 and so B1≡ B2. This ends the proof of Theorem 2.2.

A Existence of strong solutions for coefficients with small norm

In this appendix, we will show that if k~bkL²(Ω)is not too large, then (2) has a unique solution for any Dirichlet condition in W^2−p1,p(∂Ω). The proof is based on a contraction mapping principle argument.

Proposition A.1. There exists a constant c(p, Ω) > 0 such that if ||~b||L²(Ω)< c(p, Ω), then (2) has a unique solution u ∈ W^2,p(Ω).

Proof. Let H ∈ W^2,p(Ω) be the unique solution to the Dirichlet boundary value problem:

( 4H(x) = 0, x ∈ Ω, H|_∂Ω= ω.

Defining u₀:= u − H, the equation becomes

4u₀(x) + ~b(x) · ∇u₀(x) = −~b · ∇H(x) (19) with u0∈ W₀^1,p(Ω). Let L(f ) be the solution of the inhomogeneous problem

( 4U (x) = f, x ∈ Ω, U |_∂Ω= 0.

Then L : L^p(Ω) → W^2,p(Ω) ∩ W₀^1,p(Ω) is continuous (e.g. Theorem 9.15 of [10]). If (19) can be solved, then the solution should satisfy

u₀= −L~b · ∇u₀+ ~b · ∇H . Seeing this, we define the operator T for functions v ∈ W^2,p(Ω) by

T v := −L~b · ∇v +~b · ∇H .

Since ∇v ∈ W^1,p(Ω) ⊂ L^p∗(Ω), _p¹∗ = ¹_p−¹₂, it follows that ~b · ∇v ∈ L^p(Ω). The same is true of the second term in the definition of T . Putting these together we get that there is a constant C(p, Ω) > 0 such that

||T v||W^2,p(Ω)≤ C(p, Ω)



||~b||L²(Ω)||v||W^2,p(Ω)+ ||~b||_L2(Ω)||ω||

W^{2− 1p,p}(∂Ω)

 . So T : W^2,p(Ω) → W^2,p(Ω). Moreover, for two v1, v2∈ W^2,p(Ω),

||T v1− T v2||_W2,p(Ω)≤ C(p, Ω)||~b||_L2(Ω)||v1− v2||_W2,p(Ω).

If ||~b||L²(Ω)< _C(p,Ω)¹ , the operator T is a contraction hence it will have a fixed point: the solution of (19).

Uniqueness follows easily from these same estimates.

B Recovering the conductivity at the boundary

Since γ ∈ W^1,2(Ω) we can consider its trace γ|_∂Ω ∈ W^12,2(∂Ω). In this section, following the method of Brown in [6], we will prove that we can recover γ|_∂Ω from the Dirichlet-to-Neumann map. We cannot quote the result of [6] directly since there the conductivity is assumed to be bounded.

First we need to straighten out the boundary. If P ∈ ∂Ω, then there is a ball B(P, R) and a diffeomorphism φ : Ω → ˜Ω such that φ(Ω ∩ B(P, R)) ⊂ {y₂> 0} and 0 ∈ φ(∂Ω ∩ B(P, R)) ⊂ {y₂= 0}. In the new coordinates the equation (1), denoting F := φ⁻¹ and v := u ◦ F , becomes

∂_k(a_kj∂_jv) = 0, where

akj(y) := γ(F (y))∂iφ^k(F (y))∂iφ^j(F (y)) det(DF )(y).

Hereafter, we adopt the summation convention. We can make sure that Dφ ∈ C¹∩ L^∞, DF ∈ C¹∩ L^∞, so a_kj∈ W_loc^1,2.

Choose η : R → [0, ∞), supp η ⊂ [−1, 1], η|[−¹₂,¹₂]≡ 1, smooth. Also choose α ∈ R² such that

∂iφ^j(P )αj∂iφ^k(P )αk = ∂iφ^j(P )δj2∂iφ^k(P )δk2,

∂iφ^j(P )αj∂iφ^k(P )δk2= 0.

In other words, R² 3 α satisfies |Dφ(p)α| = |Dφ(P )e₂| and Dφ(P )α ⊥ Dφ(P )e₂. Define µ := Dφ(P )(iα − e₂) ∈ C². Then µ · µ = 0, µ · ¯µ = 2|Dφ(P )e₂|²> 0. With these notations we introduce the functions

vN(y) := η(N^1/2y1)η(N^1/2y2)e^{N (iα−e2)·y}=: ψ(N^1/2y)E(N y), where N ∈ N.

Lemma B.1. There is a non-zero constant A such that Z

Ω˜

a_ij(y)∂_iv_N(y)∂_jv¯_N(y) dy = N¹²γ(F (0))A + o(N¹²), as N → ∞.

Proof. Let ˜γ(y) := γ(F (y)) det(DF )(y). We compute Z

Ω˜

aij(y)∂ivN(y)∂j¯vN(y) dy

= N² Z

Ω˜

˜

γ(0)µ · ¯µψ(N^1/2y)²e^{−2N y2} dy + N²

Z

Ω˜

(˜γ(y) − ˜γ(0)) µ · ¯µψ(N^1/2y)²e^{−2N y2} dy + N

Z

Ω˜

aij(y)(∂iψ)(N^1/2y)(∂jψ)(N^1/2y)e^{−2N y2} dy

− N^3/2 Z

Ω˜

aij(y)δj2(∂iψ)(N^1/2y)ψ(N^1/2y)e^{−2N y2} dy

=: I1+ I2+ I3+ I4. (20)

We begin with the estimate of I₄. Note that

N⁻³²|I₄| ≤ sup

i,j

Z N^{− 1}2

−N^{− 1}2

Z N^{− 1}2

0

|a_ij(0)|e^{−2N y2} dy₂dy₁

+ sup

i,j

Z N^{− 1}2

−N^{− 12}

Z N^{− 1}2

0

|aij(y1, 0) − aij(0)|e^{−2N y2} dy2dy1

+ sup

i,j

Z N^{− 1}2

−N^{− 1}2

Z N^{− 1}2

0

|aij(y1, y2) − aij(y1, 0)|

y2

y2e^{−2N y2} dy2dy1

=: J₁+ J₂+ J₃. By direct computation, we have that

J1= γ(F (0))CN⁻³² for some constant C. If 0 is a Lebesgue point for aij|_{∂ ˜Ω} then

N¹² Z N^{− 1}2

−N^{− 1}2

|a_ij(y₁, 0) − a_ij(0)| dy₁→ 0, which gives that also J₂= o(N⁻³²). And finally Hardy’s inequality implies

J₃≤ C sup

i,j

||∇aij||L²N⁻⁷⁴.

Hence I4= O(1). The same type of estimates give I3= O(N⁻¹²).

The first term of (20) is easily seen by direct computation to satisfy I₁= γ(F (0))AN¹² + o(N¹²).

Finally, arguing in the same way as in the case of the terms J2and J3we have N⁻²I2=

Z

Ω˜

(γ(F (y1, 0)) − γ(F (0))) µ · ¯µψ(N^1/2y)²e^{−2N y2} dy +

Z

Ω˜

(γ(F (y1, y2)) − γ(F (y1, 0))) µ · ¯µψ(N^1/2y)²e^{−2N y2} dy

= o(N⁻³²) + O(N⁻⁷⁴), which gives I₂= o(N¹²).

Lemma B.2. ||∂i(aij∂jvN)||_H−1( ˜Ω) = o(N¹⁴).

Proof. We begin by splitting the left hand side into several terms

∂_i(a_ij∂_jv_N)(y) = a_ij(0)N^3/2(∂_iψ)(N^1/2y)(iα − e₂)_jE(N y)

+ aij(0)N (∂i∂jψ)(N^1/2y)E(N y) + ∂_i

(a_ij(y) − a_ij(0))∂_j(ψ(N^1/2y)E(N y))

= I1+ I2+ I3. Let ϕ ∈ H₀¹( ˜Ω) (we use the norm ||ϕ||_H1

0 = ||∇ϕ||_L2) and denote δ(y) the distance between y and ∂ ˜Ω.

N⁻³²     Z

Ω˜

ϕ(y)I1(y) dy    

≤ |aij(0)|

Z

Ω˜

ϕ(y)

δ(y)δ(y)(∂iψ)(N^1/2y)|(iα − e2)j|e^{−N y2} dy

≤ sup

ij

|aij(0)|

Z

Ω˜

ϕ(y)² δ(y)² dy

¹₂



 Z N^{− 1}2

−N^{− 12}

Z N^{− 1}2

0

y₂²e^{−2N y2} dy





1 2

≤ C||ϕ||_H1 0N⁻⁷⁴,

hence

||I₁||_H−1( ˜Ω)= O(N⁻¹⁴).

In the same way we get

||I2||_H−1( ˜Ω)= O(N⁻³⁴).

For the third term, we have 

   Z

Ω˜

ϕ(y)I₃(y) dy    

≤ C||∇ϕ||L²

×





N^1/2sup

i



 Z N^{− 1}2

−N^{− 12}

Z N^{− 1}2

0

(a_ij(y) − a_ij(0))²(∂_jψ)²(N^1/2y)e^{−2N y2} dy





1 2

+ N sup

ij



 Z N^{− 1}2

−N^{− 12}

Z N^{− 1}2

0

(aij(y) − aij(0))²ψ²(N^1/2y)e^{−2N y2} dy





1 2





≤ C||∇ϕ||L²

h

N^1/2J1+ N J2

i . We first look at J1,

J1≤ sup

ij



 Z N^{− 1}2

−N^{− 12}

Z N^{− 1}2

0

(aij(y1, y2) − aij(y1, 0))²

y²₂ y²₂e^{−2N y2} dy





1 2

+ sup

ij



 Z N^{− 1}2

−N^{− 12}

Z N^{− 1}2

0

(aij(y1, 0) − aij(0))²e^{−2N y2} dy





1 2

= K₁+ K₂. Using the inequality

s²e^{−2N s}≤ N⁻²e⁻², ∀s ∈ [0, ∞),

we get K1 = O(N⁻¹). Using the same methods as in previous estimates, we also have K2 = o(N⁻³⁴). This gives J1= o(N⁻³⁴) and similarly also J2= o(N⁻³⁴). Then we have

||I₃||_H−1( ˜Ω)= o(N¹⁴) and the lemma is proved.

Lemma B.3. The boundary value problem

∂i(aij∂jwN) = ∂i(aij∂jvN), wN|_{∂ ˜Ω}≡ 0, has solutions that satisfy ||wN||_H1

0( ˜Ω)= o(N¹⁴).

Proof. Existence of weak solutions follows from applying the standard Lax-Milgram argument in the energy space

Ha :=



v : ˜Ω → C : Z

Ω˜

aij(y)∂iv(y)∂jv(y) dy < ∞, v|_{∂ ˜Ω}= 0

 .

Recall that γ is positive and bounded away from zero. So aij is a positive-definite matrix function and we can see that H_a ⊂ H₀¹( ˜Ω). Since H⁻¹ ⊂ (H_a)⁰ the right hand side of the equation the usual method can indeed be applied. Integrating the equation against w_N we obtain

||∇wN||²_L2 ≤ ||∇wN||_L2||∂i(aij∂jvN)||_H−1

and the result follows.