OPTIMAL STABILITY ESTIMATE OF THE INVERSE BOUNDARY VALUE PROBLEM BY PARTIAL MEASUREMENTS

arXiv:0708.3289v1 [math.AP] 24 Aug 2007

BOUNDARY VALUE PROBLEM BY PARTIAL MEASUREMENTS

HORST HECK AND JENN-NAN WANG

Abstract. In this work we establish log type stability estimates for the inverse potential and conductivity problems with partial Dirichlet-to-Neumann map, where the Dirichlet data is homoge- neous on the inaccessible part. This result, to some extent, im- proves our former result on the partial data problem [HW06] in which log-log type estimates were derived.

1. Introduction

In this paper we study the stability question of the inverse bound- ary value problem for the Schr¨odinger equation with a potential and the conductivity equation by partial Cauchy data. This type of in- verse problem with full data, i.e., Dirichlet-to-Neumann map, were first proposed by Calder´on [Ca80]. For three or higher dimensions, the uniqueness issue was settled by Sylvester and Uhlmann [SU87] and a reconstruction procedure was given by Nachman [Na88]. For two di- mensions, Calder´on’s problem was solved by Nachman [Na96] for W^2,p conductivities and by Astala and P¨aiv¨arinta [AP06] for L^∞ conductivi- ties. This inverse problem is known to be ill-posed. A log-type stability estimate was derived by Alessandrini [Al88]. On the other hand, it was shown by Mandache [Ma01] that the log-type estimate is optimal.

All results mentioned above are concerned with the full data. Re- cently, the inverse problem with partial data has received lots of at- tentions [GU01], [IU04], [BU02], [KSU05], [FKSU07], [Is07]. A log-log type stability estimate for the inverse problem with partial data was derived by the authors in [HW06]. The method in [HW06] was based on [BU02] and a stability estimate for the analytic continuation proved in [Ve99]. We believe that the log type estimate should be the right estimate for the inverse boundary problem, even with partial data. In this paper, motivated by the uniqueness proof in Isakov’s work [Is07], we prove a log type estimate for the inverse boundary value problem

The second author was supported in part by the National Science Council of Taiwan (NSC 95-2115-M-002-003).

1

under the same a priori assumption on the boundary as given in [Is07].

Precisely, the inaccessible part of the boundary is either a part of a sphere or a plane. Also, one is able to use zero data on the inaccessible part of the boundary. The strategy of the proof in [Is07] follows the framework in [SU87] where complex geometrical optics solutions are key elements. A key observation in [Is07] is that when Γ0 is a part of a sphere or a plane, we are able to use a reflection argument to guarantee that complex geometrical optics solutions have homogeneous data on Γ0.

Let n ≥ 3 and Ω ⊂ Rⁿ be an open domain with smooth boundary

∂Ω. Given q ∈ L^∞(Ω), we consider the boundary value problem:

(∆ − q)u = 0 in Ω

u = f on ∂Ω, (1.1)

where f ∈ H^1/2(∂Ω). Assume that 0 is not a Dirichlet eigenvalue of

∆ − q on Ω. Then (1.1) has a unique solution u ∈ H¹(Ω). The usual definition of the Dirichlet-to-Neumann map is given by

Λqf = ∂νu|∂Ω

where ∂_νu = ∇u · ν and ν is the unit outer normal of ∂Ω.

Let Γ0 ⊂ ∂Ω be an open part of the boundary of Ω. We set Γ =

∂Ω \ Γ0. We further set H₀^1/2(Γ) := {f ∈ H^1/2(∂Ω) : supp f ⊂ Γ}

and H^−1/2(Γ) the dual space of H₀^1/2(Γ). Then the partial Dirichlet- to-Neumann map Λ_q,Γ is defined as

Λq,Γf := ∂νu|Γ∈ H^−1/2(Γ)

where u is the unique weak solution of (1.1) with Dirichlet Data f ∈ H₀^1/2(Γ). In what follows, we denote the operator norm by

kΛq,Γk_∗ := kΛq,Γk_H^1/2

0 (Γ)→H^−1/2(Γ)

We consider two types of domains in this paper: (a) Ω is a bounded domain in {xn< 0} and Γ0 = ∂Ω ∩ {xn < 0}; (b) Ω is a subdomain of B(a, R) and Γ0 = ∂B(a, R) ∩ ∂Ω with Γ0 6= ∂B(a, R), where B(a, R) is a ball centered at a with radius R. Denote by ˆq the zero extension of the function q defined on Ω to Rⁿ. The main result of the paper reads as follows:

Theorem 1.1. Assume that Ω is given as in either (a) or (b). Let N > 0, s > ⁿ₂ and qj ∈ H^s(Ω) such that

kq_jk_H^s_(Ω) ≤ N (1.2)

for j = 1, 2, and 0 is not a Dirichlet eigenvalue of ∆ − qj for j = 1, 2.

Then there exist constants C > 0 and σ > 0 such that kq₁− q₂k_L^∞_(Ω) ≤ C

log kΛ_q1,Γ− Λ_q2,Γk_∗

^−σ (1.3)

where C depends on Ω, N, n, s and σ depends on n and s.

Theorem 1.1 can be generalized to the conductivity equation. Let γ ∈ H^s(Ω) with s > 3 + ⁿ₂ be a strictly positive function on Ω. The equation for the electrical potential in the interior without sinks or sources is

div(γ∇u) = 0 in Ω u = f on ∂Ω.

As above, we take f ∈ H₀^1/2(Γ). The partial Dirichlet-to-Neumann map defined in this case is

Λγ,Γ: f 7→ γ∂νu|Γ.

Corollary 1.2. Let the domain Ω satisfy (a) or (b). Assume that γj ≥ N⁻¹> 0, s > ⁿ₂, and

kγ_jk_H^s+3_(Ω)≤ N (1.4)

for j = 1, 2, and

∂_ν^βγ1|Γ = ∂_ν^βγ2|Γ on ∂Ω, ∀ 0 ≤ β ≤ 1. (1.5) Then there exist constants C > 0 and σ > 0 such that

kγ1− γ2kL^∞(Ω) ≤ C

log kΛγ1,Γ− Λγ2,Γk_∗

^−σ (1.6)

where C depend on Ω, N, n, s and σ depend on n, s.

Remark 1.3. For the sake of simplicity, we impose the boundary iden- tification condition (1.5) on conductivities. However, using the argu- ments in [Al88] (also see [HW06]), this condition can be removed. The resulting estimate is still in the form of (1.6) with possible different constant C and σ.

2. Preliminaries

We first prove an estimate of the Riemann-Lebesgue lemma for a certain class of functions. Let us define

g(y) = kf (· − y) − f (·)k_L¹_(Rⁿ₎ for any f ∈ L¹(Rⁿ). It is known that lim_|y|→0g(y) = 0.

Lemma 2.1. Assume that f ∈ L¹(Rⁿ) and there exist δ > 0, C0 > 0, and α ∈ (0, 1) such that

g(y) ≤ C0|y|^α (2.1)

whenever |y| < δ. Then there exists a constant C > 0 and ε0 > 0 such that for any 0 < ε < ε0 the inequality

|F f (ξ)| ≤ C(exp(−πε²|ξ|²) + ε^α) (2.2)

holds with C = C(C0, kf k_L¹, n, δ, α).

Proof. Let G(x) := exp(−π|x|²) and set Gε(x) := ε⁻ⁿG(^x_ε). Then we define fε := f ∗ Gε. Next we write

|F f (ξ)| ≤ |F fε(ξ)| + |F (fε− f )(ξ)|.

For the first term on the right hand side we get

|F f_ε(ξ)| ≤ |F f (ξ)| · |F G_ε(ξ)|

≤ kf k1|ε⁻ⁿεⁿF G(εξ)|

≤ kf k1exp(−πε²|ξ|²).

(2.3)

To estimate the second term, we use the assumption (2.1) and derive

|F (fε− f )(ξ)| ≤ kfε− f k1

≤ Z

Rⁿ

Z

Rⁿ

|f (x − y) − f (x)|G_ε(y) dy dx

= Z

|y|<δ

Z

Rⁿ

|f (x − y) − f (x)|Gε(y) dx dy +

Z

|y|≥δ

Z

Rⁿ

|f (x − y) − f (x)|Gε(y) dx dy

= I + II.

In view of (2.1) we can estimate I =

Z

|y|<δ

g(y)Gε(y) dy

≤ C₀ Z

|y|<δ

|y|^αG_ε(y) dy

= C₀ Z

Sⁿ⁻¹

Z δ 0

r^αε⁻ⁿexp(−πε⁻²r²)rⁿ⁻¹dr dψ

= C₁ Z δ

0

ε^αu^αε⁻ⁿexp(−u²)εⁿ⁻¹u^u−1ε du

= C2ε^α Z δ

0

u^n+α−1exp(−u²) du = C3ε^α, where C₃ = C₃(C₀, n, δ, α).

As for II, we obtain that for ε sufficiently small II =

Z

|y|≥δ

g(y)Gε(y) dy

≤ 2kf k_L¹ Z

|y|≥δ

Gε(y) dy

≤ C4kf k1

Z _∞

δ

ε⁻ⁿexp(−πε⁻²r²)rⁿ⁻¹dr

= C4kf k1

Z _∞

δε⁻¹

uⁿ⁻¹exp(−πu²) du

≤ C₄kf k₁ Z _∞

δε⁻¹

exp(−πu) du

≤ C4kf k1

1

πexp(−πδε⁻¹) ≤ C5ε^α,

where C5 = C5(kf kL¹, n, δ, α). Combining the estimates for I, II, and

(2.3), we immediately get (2.2). 

We now provide a sufficient condition on f , defined on Ω, such that (2.1) in the previous lemma holds.

Lemma 2.2. Let Ω ⊂ Rⁿ be a bounded domain with C¹ boundary. Let f ∈ C^α(Ω) for some α ∈ (0, 1) and denote by ˆf the zero extension of f to Rⁿ. Then there exists δ > 0 and C > 0 such that

k ˆf(· − y) − ˆf (·)k_L¹_(Rⁿ₎ ≤ C|y|^α for any y ∈ Rⁿ with |y| ≤ δ.

Proof. Since Ω is bounded and of class C¹, there exist a finite number of balls, say m ∈ N, B_i(x_i) with center x_i ∈ ∂Ω, i = 1, . . . , m and associated C¹-diffeomorphisms ϕ_i : B_i(x_i) → Q where Q = {x^′ ∈ Rⁿ⁻¹ : kx^′k ≤ 1} × (−1, 1). Set d = dist (∂Ω, ∂(Sm

i=1B_i(x_i))) > 0 and Ω˜_ε =S

x∈∂ΩB(x, ε), where B(x, ε) denotes the ball with center x and radius ε > 0. Obviously, for ε < d, it holds that ˜Ω_ε⊂Sm

i=1B_i(x_i). Let x ∈ ∂Ω and 0 < |y| < δ ≤ d, then for any z₁, z₂ ∈ B(x, |y|) ∩ B_i(x_i) we get that

|ϕi(z1) − ϕi(z2)| ≤ k∇ϕikL^∞|z1− z2| ≤ C|y|

for some constant C > 0. Therefore, ϕi( ˜Ω_|y|∩ Bi(xi)) ⊂ {x^′ ∈ Rⁿ⁻¹ : kx^′k ≤ 1} × (−C|y|, C|y|). By the transformation formula this yields vol( ˜Ω_|y|) ≤ C|y|.

Since |y| < δ we have ˆf (x − y) − ˆf (x) = 0 for x 6∈ Ω ∪ ˜Ω_|y|. Now we write

k ˆf (· − y) − ˆfk_L¹_(Rⁿ₎ = Z

Ω\ ˜Ω_|y|

| ˆf (x − y) − ˆf (x)| dx

+ Z

Ω˜_|y|

| ˆf(x − y) − ˆf (x)| dx

≤ C vol(Ω)|y|^α+ 2kf kL^∞vol( ˜Ω_|y|)

≤ C(|y|^α+ |y|) ≤ C|y|^α

for δ ≤ 1. 

Now let q1 and q2 be two potentials and their corresponding partial Dirichlet-to-Neumann maps are denoted by Λ1,Γ and Λ2,Γ, respectively.

The following identity plays a key role in the derivation of the stability estimate.

Lemma 2.3. Let vj solve (1.1) with q = qj for j = 1, 2. Further assume that v1 = v2 = 0 on Γ0. Then

Z

Ω

(q₁− q₂)v₁v₂dx = h(Λ_1,Γ− Λ_2,Γ)v₁, v₂i

Proof. Let u₂ denote the solution of (1.1) with q = q₂ and u₂ = v₁ on

∂Ω. Therefore Z

Ω

∇v1· ∇v2+ q1v1v2dx = h∂νv1, v2i Z

Ω

∇u2· ∇v2+ q2u2v2dx = h∂νu2, v2i.

Setting v := v1 − u2 and q0 = q1 − q2 we get after subtracting these identities

Z

Ω

∇v · ∇v2+ q2vv2+ q0v1v2 = h(Λ1− Λ2)v1, v2i.

Since v2 solves (∆ − q2)v2 = 0, v = 0 on ∂Ω and v2 = 0 on Γ0, we have Z

Ω

∇v · ∇v2+ q2vv2 = 0,

h(Λ1− Λ2)v1, v2i = h(Λ1,Γ− Λ2,Γ)v1, v2i,

and the assertion follows. 

In treating inverse boundary value problems, complex geometrical optics solutions play a very important role. We now describe the com- plex geometrical optics solutions we are going to use in our proofs.

We will follow the idea in [Is07]. Assume that q1, q2 ∈ L^∞(Rⁿ) are compactly supported and are even in xn, i.e.

q₁^∗(x₁, · · · , x_n−1, x_n) = q₁(x₁, · · · , x_n−1, x_n) and

q₂^∗(x1, · · · , x_n−1, xn) = q2(x1, · · · , x_n−1, xn).

Hereafter, we denote

h^∗(x1, · · · , x_n−1, xn) = h(x1, · · · , x_n−1, −xn).

Given ξ = (ξ₁, · · · , ξ_n) ∈ Rⁿ. Let us first introduce new coordinates obtained by rotating the standard Euclidean coordinates around the xn

axis such that the representation of ξ in the new coordinates, denoted by ˜ξ, satisfies ˜ξ = ( ˜ξ1, 0, · · · , 0, ˜ξn) with ˜ξ1 = pξ₁²+ · · · + ξ_n−1² and ξ˜n = ξn. In the following we also denote by ˜x the representation of x in the new coordinates. Then we define for τ > 0

˜

ρ1 := (ξ˜1

2 − τ ˜ξn, i|ξ|(1

4+ τ²)^1/2, 0, · · · , 0,ξ˜n

2 + τ ˜ξ1),

˜

ρ2 := (ξ˜1

2 + τ ˜ξn, −i| ˜ξ|(1

4+ τ²)^1/2, 0, · · · , 0,ξ˜n

2 − τ ˜ξ1),

(2.4)

and let ρ1 and ρ2 be representations of ˜ρ1 and ˜ρ2 in the original coordi- nates. Note that xn = ˜xn and Pn

i=1xiyi =Pn

i=1x˜iy˜i. It is clear that, for j = 1, 2, ρj · ρj = 0 as well as ρ^∗_j · ρ^∗_j = 0 hold.

The construction given in [SU87] ensures that there are complex geometrical optics solutions u_j = e^iρj·x(1 + w_j) of (∆ − q_j)u_j = 0 in

Rⁿ, j = 1, 2, and the functions wj satisfy kwjkL²(K) ≤ CKτ⁻¹ for any compact set K ⊂ Rⁿ. We then set

v1(x) = e^iρ1·x(1 + w1) − e^iρ∗1·x(1 + w₁^∗)

v2(x) = e^−iρ2·x(1 + w2) − e^{−iρ∗2·x}(1 + w₂^∗). (2.5) From this definition it is clear that these functions are solutions of (∆ − qj)vj = 0 in Rⁿ₊ with vj = 0 on xn = 0.

3. Stability estimate for the potential

Now we are in the position to prove Theorem 1.1. We first consider the case (a) where Γ₀ is a part of a hyperplane. To construct the special solutions described in the previous section, we first perform zero extension of q₁ and q₂ to R⁺_n and then even extension to the whole Rⁿ. As in the last section, we can construct special geometrical optics solutions vj of the form (2.5) to (∆ − qj)vj = 0 in Ω for j = 1, 2.

Note that v1 = v2 = 0 on Γ0. We now plug in these solutions into the identity (2.3) and write q0 = q1− q2. This gives

h(Λ1,Γ− Λ2,Γ)v1, v2i

= Z

Ω

q0v1v2dx

= Z

Ω

q0(x)

e^{i(ρ1+ρ2)·x}(1 + w1)(1 + w2) + e^{i(ρ∗1+ρ∗2)·x}(1 + w₁^∗)(1 + w₂^∗)

− e^{i(ρ1+ρ∗2)·x}(1 + w₁)(1 + w₂^∗) − e^{i(ρ∗1+ρ2)·x}(1 + w^∗₁)(1 + w₂) dx

= Z

Ω

q0(x)(e^iξ·x+ e^iξ∗·x) dx + Z

Ω

q0(x)f (x, w1, w2, w₁^∗, w₂^∗) dx

− Z

Ω

q0(x) e^{i(ρ1+ρ∗2)·x}+ e^{i(ρ∗1+ρ2)·x}
dx,

(3.1) where

f = e^iξ·x(w1 + w2+ w1w2) + e^iξ∗·x(w₁^∗+ w^∗₂+ w^∗₁w^∗₂)

− e^{i(ρ∗1+ρ2)·x}(w₁^∗+ w2+ w₁^∗w2) − e^{i(ρ1+ρ∗2)·x}(w1+ w₂^∗+ w1w₂^∗).

The first term on the right hand side of (3.1) is equal to Z

Rⁿ

q0(x)e^iξ·xdx = F q0(ξ)

because q₀ is even in x_n. For the second term, we use the estimate kw₁k₂+ kw₁^∗k₂+ kw₂k₂+ kw^∗₂k₂ ≤ Cτ⁻¹

to obtain  

Z

Ω

q0f (x, w1, w2, w^∗₁, w^∗₂) dx

≤ Ckq0k2τ⁻¹. (3.2) As for the last term on the right hand side of (3.1), we first observe that

(ρ1+ ρ^∗₂) · x = (˜ρ1+ ˜ρ^∗₂) · ˜x = ˜ξ1x˜1+ 2τ ˜ξ1x˜n= ξ^′· x^′+ 2τ |ξ^′|xn

and

(ρ^∗₁+ ρ₂) · x = (˜ρ₁ + ˜ρ^∗₂) · ˜x = ˜ξ₁x˜₁ + 2τ ˜ξ₁x˜_n = ξ^′· x^′− 2τ |ξ^′|x_n, where ξ^′ = (ξ₁, · · · , ξ_n−1) and x^′ = (x₁, · · · , x_n−1). Therefore, we can write

Z

Ω

q0(x)e^{i(ρ1+ρ∗2)·x}dx = F q0(ξ^′, 2τ |ξ^′|) as well as

Z

Ω

q0(x)e^{i(ρ∗1+ρ2)·x}dx = F q0(ξ^′, −2τ |ξ^′|).

The Sobolev embedding and the assumptions on q_j ensure that q₀ ∈ C^α(Ω) for α = s − ⁿ₂ and therefore q0 satisfies the assumption of Lemma 2.2. Applying Lemma 2.1 to q0 yields that for ε < ε0

|F q0(ξ^′, 2τ |ξ^′|)| + |F q0(ξ^′, −2τ |ξ^′|)| ≤ C(exp(−πε²(1 + 4τ²)|ξ^′|²) + ε^α).

(3.3) Finally, we estimate the boundary integral

    Z

Γ

(Λ1,Γ− Λ2,Γ)v1· v2dσ    

≤ kΛ1,Γ− Λ2,Γk_∗kv1k

H¹²(Γ)kv2k

H¹²(Γ)

≤ kΛ1,Γ− Λ2,Γk_∗kv1kH¹(Ω)kv2kH¹(Ω)

≤ C exp(|ξ|τ )kΛ1− Λ2k_∗.

(3.4)

Combining (3.1), (3.2), (3.3), and (3.4) leads to the inequality

|F q₀(ξ)| ≤ C{exp(|ξ|τ )kΛ₁− Λ₂k_∗+ exp(−πε²(1 + 4τ²)|ξ^′|²) + ε^α+1 τ} (3.5) for all ξ ∈ Rⁿ and ε < ε0, where C only depends on a priori data on the potentials.

Next we would like to estimate the norm of q₀in H⁻¹. As usual, other estimates of q₀ in more regular norms can be obtained by interpolation.

To begin, we set Z_R = {ξ ∈ Rⁿ : |ξ_n| < R and |ξ^′| < R}. Note that

B(0, R) ⊂ ZR ⊂ B(0, cR) for some c > 0. Now we use the a priori assumption on potentials and (3.5) and calculate

kq0k²_H⁻¹ ≤ Z

ZR

|F q0(ξ)|²(1 + |ξ|²)⁻¹dξ + Z

ZRc|F q0(ξ)|²(1 + |ξ|²)⁻¹dξ

≤ Z

ZR

|F q0(ξ)|²(1 + |ξ|²)⁻¹dξ + CR⁻²

≤ C{Rⁿexp(cRτ )kΛ1− Λ2k²_∗+ Rⁿε^2α + Rⁿτ⁻²+ R⁻² +

Z R

−R

Z

B^′(0,R)

exp(−2πε²(1 + 4τ²)|ξ^′|²) dξ^′dξ_n},

(3.6) here B^′(x^′, R) denotes the ball in Rⁿ⁻¹ with center x^′ and radius R >

0. For the second term on the right hand side of (3.6), we choose ε = (1 + 4τ²)^−1/4 with τ ≥ τ0 ≫ 1 and integrate

Z R

−R

Z

B^′(0,R)

exp(−2πε²(1 + 4τ²)|ξ^′|²) dξ^′dξn

= 2R Z

B^′(0,R)

exp(−2π(1 + 4τ²)^1/2|ξ^′|²) dξ^′

= 2R Z

Sⁿ⁻²

Z R 0

rⁿ⁻²exp(−2π((1 + 4τ²)^1/4r)²) dr dω

≤ CR(1 + 4τ²)^−(n−1)/4 Z _∞

0

uⁿ⁻²exp(−2πu²) du

≤ CRτ^−(n−1)/2.

(3.7)

Plugging (3.7) into (3.6) with the choice of ε = (1 + 4τ²)^−1/4 we get for R > 1

kq0k²_H−1 ≤ C{Rⁿexp(cRτ )kΛ1− Λ2k²_∗ + Rⁿτ^−α+ Rτ^−(n−1)/2+ R⁻²}

≤ C{Rⁿexp(cRτ )kΛ₁− Λ₂k²_∗ + Rⁿτ^−˜α+ R⁻²},

(3.8) where ˜α = min{α, (n − 1)/2}.

Observing from (3.8), we now choose τ such that Rⁿτ^−˜α = R⁻², namely, τ = R^{(n+2)/ ˜α}. Substituting such τ back to (3.8) yields

kq0k²_H−1 ≤ C{Rⁿexp(cR^{n+2α˜ +1})kΛ1− Λ2k²_∗+ R⁻²}. (3.9) Finally, we choose a suitable R so that

Rⁿexp(cR^{n+2α˜ +1})kΛ₁− Λ₂k²_∗ = R⁻²,

i.e., R =

log kΛ1− Λ2k_∗ 

γ for some 0 < γ = γ(n, ˜α). Thus, we obtain from (3.9) that

kq1− q2k_H⁻¹_(Ω)≤ C

log kΛ1− Λ2k_∗

^−γ. (3.10) The derivation of (3.10) is legitimate under the assumption that τ is large. To make sure that it is true, we need to take R sufficiently large, i.e. R > R0 for some large R0. Consequently, there exists ˜δ > 0 such that if kΛ1− Λ2k_∗ < ˜δ then (3.10) holds. For kΛ1 − Λ2k_∗ ≥ ˜δ, (3.10) is automatically true with a suitable constant C when we take into account the a priori bound (1.2).

The estimate (1.3) is now an easy consequence of the interpola- tion theorem. Precisely, let ǫ > 0 such that s = ⁿ₂ + 2ǫ. Using that [H^t0(Ω), H^t1(Ω)]β = H^t(Ω) with t = (1 − β)t0 + βt1 (see e.g.

[Tr95, Theorem 1 in 4.3.1]) and the Sobolev embedding theorem, we get kq1−q2kL^∞ ≤ Ckq1−q2k_Hⁿ₂+ǫ ≤ Ckq1−q2k^(1−β)_Ht0 kq1−q2k^β_Ht1. Setting t0 = −1 and t1 = s we end up with

kq1− q2kL^∞(Ω) ≤ Ckq1− q2k

s+1−ǫ s+1

H⁻¹(Ω)

which yields the desired estimate (1.3) with σ = γ^s+1−ǫ_s+1 .

We now turn to case (b). With a suitable translation and rotation, it suffices to assume a = (0, · · · , 0, R) and 0 /∈ Ω. As in [Is07], we shall use Kelvin’s transform:

y = 2R

|x|

2

x and x = 2R

|y|

2

y. (3.11)

Let

˜

u(y) =  2R

|y|

_n−2

u(x(y)), then

 |y|

2R

n+2

∆yu(y) = ∆˜ xu(x).

Denote by ˜Ω the transformed domain of Ω. In view of this transform, Γ₀ now becomes ˜Γ₀ ⊂ {y_n = 2R} and Γ is transformed to ˜Γ and Γ = ∂ ˜˜ Ω∩{yn > 2R}. On the other hand, if u(x) satisfies ∆u−q(x)u = 0 in Ω, then ˜u satisfies

∆˜u − ˜q˜u = 0 in Ω,˜ (3.12) where

˜

q(y) = 2R

|y|

4

q(x(y)).

Therefore, for (3.12) we can define the partial Dirichlet-to-Neumann map ˜Λ_q,˜˜Γ acting boundary functions with homogeneous data on ˜Γ0.

We now want to find the relation between Λq,Γ and ˜Λ_q,˜˜Γ. It is easy to see that for f, g ∈ H₀^1/2(Γ)

hΛq,Γf, gi = Z

Ω

(∇u · ∇v + quv) dx, where u solves

∆u − qu = 0 in Ω, u = f on ∂Γ and v ∈ H¹(Ω) with v|_∂Ω= g. Defining

f =˜  2R

|y|

_n−2  

∂ ˜Ωf, g =˜  2R

|y|

_n−2   ∂ ˜Ωg, and

˜

v(y) = 2R

|y|

_n−2

v(x(y)).

Then we have ˜f , ˜g ∈ H₀^1/2(˜Γ) and

hΛq,Γf, gi = h˜Λ_q,˜˜Γf, ˜˜ gi, in particular,

h(Λ_q1,Γ− Λ_q2,Γ)f, gi = h(˜Λ_q˜1,˜Γ− ˜Λ_q˜2,˜Γ) ˜f , ˜gi. (3.13) With the assumption 0 /∈Ω, the change of coordinates x → y by (3.11) is a diffeomorphism from Ω onto ˜Ω. Note that (2R/|y|)ⁿ⁻²is a positive smooth function on ∂ ˜Ω. Recall a fundamental fact from Functional Analysis:

kΛq1,Γ− Λq2,Γk_∗ = sup

(|h(Λq1,Γ− Λq2,Γ)f, gi|

kf k_H^1/2

0 (Γ)kgk_H^1/2

0 (Γ)

: f, g ∈ H₀^1/2(Γ) )

. (3.14) The same formula holds for k˜Λ_q˜1,˜Γ − ˜Λ_q˜2,˜Γk_∗. On the other hand, it is not difficult to check that kf k_H¹/2

0 (Γ) and k ˜f k_H¹/2

0 (˜Γ), kgk_H¹/2

0 (Γ) and k˜gk_H^1/2

0 (˜Γ)are equivalent, namely, there exists C depending on ∂Ω such that

1

Ckf k_H¹/2

0 (Γ) ≤ k ˜fk_H¹/2

0 (˜Γ)≤ Ckf k_H¹/2 0 (Γ), 1

Ckgk_H^1/2

0 (Γ) ≤ k˜gk_H^1/2

0 (˜Γ)≤ Ckgk_H^1/2

0 (Γ).

(3.15)

Putting together (3.13), (3.14), and (3.15) leads to

k˜Λ_q˜1,˜Γ− ˜Λ_q˜2,˜Γk_∗ ≤ CkΛq1,Γ− Λq2,Γk_∗ (3.16) with C only depending on ∂Ω.

With all the preparations described above, we use case (a) for the domain ˜Ω with the partial Dirichlet-to-Neumann map ˜Λ_q,˜˜Γ. Therefore, we immediately obtain the estimate:

k˜q1− ˜q2k_L∞( ˜Ω) ≤ C

log k˜Λ_q˜1,˜Γ− ˜Λ_q˜2,˜Γk_∗ ^−σ. Finally, rewinding ˜q and using (3.16) yields the estimate (1.3).

4. Stability estimate for the conductivity

We aim to prove Corollary 1.2 in this section. We recall the following well-known relation: let q = ^∆√γ√γ then

Λq,Γ(f ) = γ^−1/2|ΓΛγ,Γ(γ^−1/2|Γf ) + 1

2(γ⁻¹∂νγ)|Γf.

In view of the a priori assumption (1.5), we have that

(Λq1,Γ− Λq2,Γ)(f ) = γ^−1/2|Γ(Λγ1,Γ− Λγ2,Γ)(γ^−1/2|Γf ) where γ^−1/2|Γ := γ₁^−1/2|Γ = γ₂^−1/2|Γ, which implies

kΛq1,Γ− Λq2,Γk_∗ ≤ CkΛγ1,Γ− Λγ2,Γk_∗ (4.1) for some C = C(N) > 0. Hereafter, we set qj = ^∆√γ_√γ ^j

j , j = 1, 2. The regularity assumption (1.4) and Sobolev’s embedding theorem imply that q1, q2 ∈ C¹(Ω). Using this and (1.5), we conclude that ˆq1 − ˆq2

satisfies the assumptions of Lemma 2.2 with α = 1. Therefore, Theo- rem 1.1 and (4.1) imply that

kq1− q2k_L^∞_(Ω) ≤ C

log kΛγ1,Γ− Λγ2,Γk_∗ 

−σ¹

(4.2) where C depend on Ω, N, n, s and σ₁ depend on n, s. Next, we recall from [Al88, (26) on page 168] that

kγ1− γ2kL^∞(Ω)≤ Ckq1− q2k^σ_L²∞(Ω) (4.3) for some 0 < σ2 < 1, where C = C(N, Ω) and σ2 = σ2(n, s). Finally, putting together (4.2) and (4.3) yields (1.6) with σ = σ1σ2 and the proof of Corollary 1.2 is complete.

References

[Al88] G. Alessandrini, Stable determination of conductivity by boundary mea- surements, Appl. Anal. 27 (1988), no. 1-3, 153-172.

[Al90] G. Alessandrini, Singular solutions of elliptic equations and the determina- tion of conductivity by boundary measurements, J. Differential Equations, 84(1990), 252-272.

[AP06] K. Astala and L. P¨aiv¨arinta, Calderon’s inverse conductivity problem in the plane, Ann. of Math. (2), 163 (2006), 265-299.

[BU02] A. L. Bukhgeim and G. Uhlmann, Recovering a potential from partial Cauchy data, Comm. Partial Differential Equations 27 (2002), 653-668.

[Ca80] A. Calder´on, On an inverse boundary value problem, Seminar on Numer- ical Analysis and its Applications to Continuum Physics, Soc. Brasileira de Matem´atica, R´ıo de Janeiro (1980), 65-73.

[FKSU07] D. Dos Santos Ferreira, C.E. Kenig, J. Sj¨ostrand, and G. Uhlmann, Determining the magnetic Schr¨odinger operator from partial Cauchy data, Comm. Math. Phys., to appear.

[GU01] A. Greenleaf and G. Uhlmann, Local uniqueness for the Dirichlet-to- Neumann map via the two-plane transform, Duke Math. J. 108 (2001), 599-617.

[HW06] H. Heck and J.-N. Wang, Stability estimates for the inverse boundary value problem by partial Cauchy data, Inverse Problems 22 (2006), 1787-1796.

[IU04] H. Isozaki and G. Uhlmann, Hyperbolic geometry and the local Dirichlet- to-Neumann map, Advances in Math., 188 (2004), 294-314.

[Is07] V. Isakov, On uniqueness in the inverse conductivity problem with local data, Inverse Problems and Imaging, 1 (2007), 95-105.

[KSU05] C.E. Kenig, J. Sj¨ostrand, and G. Uhlmann, The Calder´on problem with partial data, to appear in Ann. of Math.

[Ma01] N. Mandache, Exponential instability in an inverse problem for the Schrdinger equation, Inverse Problems 17 (2001), no. 5, 1435-1444.

[Na88] A. Nachman, Reconstructions from boundary measurements, Ann. of Math. (2) 128 (1988), no. 3, 531-576.

[Na96] , Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. (2), 143 (1996), 71-96.

[SU87] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2) 125 (1987), 153-169.

[SU88] J. Sylvester and G. Uhlmann, Inverse boundary value problems at the boundary—continuous dependence, Comm. Pure Appl. Math., 41 (1988), 197-219.

[Tr95] H. Triebel, Interpolation theory, function spaces, differential operators, 2nd ed., Johann Ambrosius Barth, Heidelberg, Leipzig, 1995.

[Ve99] S. Vessella, A continuous dependence result in the analytic continuation problem, Forum Math. 11 (1999), 695-703.

Technische Universit¨at Darmstadt, FB Mathematik, AG 4, Schloss- gartenstr. 7, D-64289 Darmstadt, Germany

E-mail address: heck@mathematik.tu-darmstadt.de

Department of Mathematics, National Taiwan University, Taipei 106, Taiwan

E-mail address: jnwang@math.ntu.edu.tw