1.Introduction HorstHeck,Jenn-NanWang Optimalstabilityestimateoftheinverseboundaryvalueproblembypartialmeasurements

Optimal stability estimate of the inverse boundary value problem by

partial measurements

Horst Heck, Jenn-Nan Wang

We dedicate this work to Giovanni Alessandrini for his 60th birthday and for his pioneering contribution in the stability estimates of inverse problems.

Abstract. This manuscript was originally uploaded to arXiv in 2007 (arXiv:0708.3289v1). In the current version, we expand the Introduc- tion and the list of references which are related to the results of this pa- per after 2007. 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 homogeneous on the inac- cessible part. The proof is based on the uniqueness result of the inverse boundary value problem in Isakov’s work [16].

1. Introduction

In this paper we study the stability question of the inverse boundary value prob- lem for the Schr¨odinger equation with a potential and the conductivity equa- tion by partial Cauchy data. This type of inverse problem with full data, i.e., Dirichlet-to-Neumann map, were first proposed by Calder´on [6]. For three or higher dimensions, the uniqueness issue was settled by Sylvester and Uhlmann [27] and a reconstruction procedure was given by Nachman [25]. For two dimen- sions, Calder´on’s problem was solved by Nachman [26] for W^2,pconductivities and by Astala and P¨aiv¨arinta [3] for L^∞ conductivities. This inverse problem is known to be ill-posed. A log-type stability estimate was derived by Alessan- drini [1]. On the other hand, it was shown by Mandache [24] that the log-type estimate is optimal.

All results mentioned above are concerned with the full data. Over the last decade, the inverse problems with partial data have received a lot of attention.

We list several earlier results [14], [17], [5], [20], [12], [16], [23], [21], [18] [11]

and refer the reader to the survey article [19] for its detailed development and for related references. After the uniqueness proof comes stability estimates.

We summarize related results in the following.

• log log type: [15], [29], [7], [9], [8], [10], [22].

• log type: [7], [13], [4], [2].

The method in [15] was based on [5] and a stability estimate for the analytic continuation proved in [30]. 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 [16], we prove a log type estimate for the inverse boundary value problem under the same a priori assumption on the boundary as given in [16]. 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 [16] follows the framework in [27] where complex geometrical optics solutions are key elements. A key observation in [16] 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. Caro in [7] also used Isakov’s idea to derive a log type estimate for the Maxwell equations. The articles [13], [4], [2] have a common feature that the undetermined coefficients are known near the boundary.

Now we would like to describe the results in this work. 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)

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

Λ_q,Γf := ∂_νu|_Γ∈ H^−1/2(Γ)

where u is the unique weak solution of (1) with Dirichlet Data f ∈ H₀^1/2(Γ).

In what follows, we denote the operator norm by kΛq,Γk_∗:= kΛq,Γk_H1/2

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 Γ₀6= ∂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

kqjkH^s(Ω)≤ N (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− q2kL^∞(Ω)≤ C

log kΛq1,Γ− Λq2,Γk∗ 

−σ (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 γ^j≥ N⁻¹ >

0, s > ⁿ₂, and

kγjk_Hs+3(Ω)≤ N (4)

for j = 1, 2, and

∂^β_νγ₁|Γ= ∂_ν^βγ₂|Γ on ∂Ω, ∀ 0 ≤ β ≤ 1. (5) Then there exist constants C > 0 and σ > 0 such that

kγ1− γ2kL^∞(Ω)≤ C

log kΛγ₁,Γ− Λγ₂,Γk∗ 

−σ (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 identification condition (5) on conductivities. However, using the arguments in [1] (also see [15]), this condition can be removed. The resulting estimate is still in the form of (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 (7)

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) (8) 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 (f_ε− f )(ξ)|.

For the first term on the right hand side we get

|F fε(ξ)| ≤ |F f (ξ)| · |F Gε(ξ)|

≤ kf k1|ε⁻ⁿεⁿF G(εξ)|

≤ kf k1exp(−πε²|ξ|²).

(9)

To estimate the second term, we use the assumption (7) and derive

|F (fε− f )(ξ)| ≤ kfε− f k1

≤ Z

Rⁿ

Z

Rⁿ

|f (x − y) − f (x)|Gε(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 (7) we can estimate I =

Z

|y|<δ

g(y)G_ε(y) dy

≤ C0

Z

|y|<δ

|y|^αGε(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 k₁1

πexp(−πδε⁻¹) ≤ C₅ε^α,

where C5= C5(kf k_L1, n, δ, α). Combining the estimates for I, II, and (9), we immediately get (8).

We now provide a sufficient condition on f , defined on Ω, such that (7) 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, Bi(x_i) with center x_i∈ ∂Ω, i = 1, . . . , m and associated C¹- diffeomorphisms ϕi: Bi(xi) → Q where Q = {x⁰∈ Rⁿ⁻¹: kx⁰k ≤ 1} × (−1, 1).

Set d = dist (∂Ω, ∂(Sm

i=1Bi(xi))) > 0 and ˜Ωε=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 ˜Ωε⊂Sm

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

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

for some constant C > 0. Therefore, ϕ_i( ˜Ω_|y|∩ Bi(x_i)) ⊂ {x⁰ ∈ Rⁿ⁻¹: kx⁰k ≤ 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) − ˆf kL¹(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 q₁and q₂be two potentials and their corresponding partial Dirichlet- to-Neumann maps are denoted by Λ_1,Γ and Λ_2,Γ, respectively. The following identity plays a key role in the derivation of the stability estimate.

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

Z

Ω

(q1− q2)v1v2dx = h(Λ1,Γ− Λ2,Γ)v1, v2i

Proof. Let u2 denote the solution of (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 complex geometrical optics solutions we are going to use in our proofs. We will follow the idea in [16]. Assume that q1, q2 ∈ L^∞(Rⁿ) are compactly supported and are even in xn, i.e.

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

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

Hereafter, we denote

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

Given ξ = (ξ1, · · · , ξn) ∈ Rⁿ. Let us first introduce new coordinates ob- tained 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, ˜ξ_n) with ˜ξ₁=q

ξ²₁+ · · · + ξ²_n−1and ˜ξ_n= ξ_n. 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 ˜ξ_n, −i| ˜ξ|(1

4 + τ²)^1/2, 0, · · · , 0, ξ˜_n

2 − τ ˜ξ₁),

(10)

and let ρ1 and ρ2 be representations of ˜ρ1 and ˜ρ2 in the original coordinates.

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 [27] ensures that there are complex geometrical optics solutions uj= e^iρj·x(1 + wj) of (∆ − qj)uj= 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 + w^∗₂). (11) 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 Γ0is a part of a hyperplane. To construct the special solutions described in the previous section, we first perform zero extension of q1and 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 (11) 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 + w^∗₁)(1 + w^∗₂)

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

− Z

Ω

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

(12)

where

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

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

The first term on the right hand side of (12) is equal to Z

Rⁿ

q₀(x)e^iξ·xdx = F q₀(ξ)

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

  Z

Ω

q0f (x, w1, w2, w₁^∗, w^∗₂) dx

≤ Ckq0k2τ⁻¹. (13) As for the last term on the right hand side of (12), we first observe that

(ρ1+ ρ^∗₂) · x = ( ˜ρ1+ ˜ρ^∗₂) · ˜x = ˜ξ1x˜1+ 2τ ˜ξ1x˜n= ξ⁰· x⁰+ 2τ |ξ⁰|xn

and

(ρ^∗₁+ ρ2) · x = ( ˜ρ^∗₁+ ˜ρ2) · ˜x = ˜ξ1x˜1− 2τ ˜ξ1x˜n= ξ⁰· x⁰− 2τ |ξ⁰|xn,

where ξ⁰ = (ξ₁, · · · , ξ_n−1) and x⁰= (x₁, · · · , x_n−1). Therefore, we can write Z

Ω

q0(x)e^{i(ρ1+ρ∗2)·x}dx = F q0(ξ⁰, 2τ |ξ⁰|) as well as

Z

Ω

q0(x)e^{i(ρ∗1+ρ2)·x}dx = F q0(ξ⁰, −2τ |ξ⁰|).

The Sobolev embedding and the assumptions on qj 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(ξ⁰, 2τ |ξ⁰|)| + |F q0(ξ⁰, −2τ |ξ⁰|)| ≤ C(exp(−πε²(1 + 4τ²)|ξ⁰|²) + ε^α). (14) Finally, we estimate the boundary integral

    Z

Γ

(Λ_1,Γ− Λ_2,Γ)v₁· v₂dσ    

≤ kΛ1,Γ− Λ2,Γk_∗kv1k

H¹2(Γ)kv2k

H¹2(Γ)

≤ kΛ1,Γ− Λ2,Γk_∗kv1k_H1(Ω)kv2k_H1(Ω)

≤ C exp(|ξ|τ )kΛ₁− Λ₂k_∗.

(15)

Combining (12), (13), (14), and (15) leads to the inequality

|F q0(ξ)| ≤ C{exp(|ξ|τ )kΛ1− Λ2k_∗+ exp(−πε²(1 + 4τ²)|ξ⁰|²) + ε^α+1 τ} (16) 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 q0 in H⁻¹. As usual, other estimates of q0 in more regular norms can be obtained by interpolation. To begin, we set ZR = {ξ ∈ Rⁿ : |ξn| < R and |ξ⁰| < R}. Note that B(0, R) ⊂ ZR ⊂ B(0, cR) for some c > 0. Now we use the a priori assumption on potentials and (16) 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

Z_R

|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

B⁰(0,R)

exp(−2πε²(1 + 4τ²)|ξ⁰|²) dξ⁰dξ_n},

(17)

here B⁰(x⁰, R) denotes the ball in Rⁿ⁻¹with center x⁰ and radius R > 0. For the second term on the right hand side of (17), we choose ε = (1 + 4τ²)^−1/4 with τ ≥ τ0 1 and integrate

Z R

−R

Z

B⁰(0,R)

exp(−2πε²(1 + 4τ²)|ξ⁰|²) dξ⁰dξn

= 2R Z

B⁰(0,R)

exp(−2π(1 + 4τ²)^1/2|ξ⁰|²) dξ⁰

= 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.

(18)

Plugging (18) into (17) 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}, (19) where ˜α = min{α, (n − 1)/2}.

Observing from (19), we now choose τ such that Rⁿτ^{− ˜α} = R⁻², namely, τ = R^{(n+2)/ ˜α}. Substituting such τ back to (19) yields

kq0k²_H−1 ≤ C{Rⁿexp(cR^{n+2α˜ +1})kΛ1− Λ2k²_∗+ R⁻²}. (20) 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 (20) that

kq₁− q₂k_H−1(Ω)≤ C

log kΛ₁− Λ₂k_∗ 

−γ. (21)

The derivation of (21) 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 > R₀for some large R₀. Consequently, there exists ˜δ > 0 such that if kΛ₁− Λ₂k_∗ < ˜δ then (21) holds. For kΛ₁− Λ₂k_∗≥ ˜δ, (21) is automatically true with a suitable constant C when we take into account the a priori bound (2).

The estimate (3) is now an easy consequence of the interpolation theorem.

Precisely, let  > 0 such that s = ⁿ₂ + 2. Using that [H^t0(Ω), H^t1(Ω)]β = H^t(Ω) with t = (1 − β)t0+ βt1 (see e.g. [28, Theorem 1 in 4.3.1]) and the

Sobolev embedding theorem, we get kq₁− q₂k_L^∞ ≤ Ckq₁− q₂k_Hⁿ₂+≤ Ckq₁− q₂k^(1−β)_Ht0 kq1− q2k^β_Ht1. Setting t₀= −1 and t₁= s we end up with

kq1− q2k_L^∞_(Ω)≤ Ckq1− q2k

 s+1

H⁻¹(Ω)

which yields the desired estimate (3) with σ = γ_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 [16], we shall use Kelvin’s transform:

y = 2R

|x|

²

x and x = 2R

|y|

²

y. (22)

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⊂ {yn= 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,˜ (23)

where

˜

q(y) = 2R

|y|

4

q(x(y)).

Therefore, for (23) 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  ∂ ˜Ω

f, ˜g = 2R

|y|

ⁿ⁻²  ∂ ˜Ω

g,

and

˜

v(y) = 2R

|y|

n−2

v(x(y)).

Then we have ˜f , ˜g ∈ H₀^1/2(˜Γ) and

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

h(Λq1,Γ− Λq2,Γ)f, gi = h( ˜Λ_q˜

1,˜Γ− ˜Λ_q˜

2,˜Γ) ˜f , ˜gi. (24) With the assumption 0 /∈ Ω, the change of coordinates x → y by (22) is a diffeomorphism from Ω onto ˜Ω. Note that (2R/|y|)ⁿ⁻² is a positive smooth function on ∂ ˜Ω. Recall a fundamental fact from Functional Analysis:

kΛq1,Γ− Λq2,Γk∗= sup

(|h(Λ_q1,Γ− Λ_q2,Γ)f, gi|

kf k_H1/2

0 (Γ)kgk_H1/2 0 (Γ)

: f, g ∈ H₀^1/2(Γ) )

. (25)

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_H1/2 0 (˜Γ)

are equivalent, namely, there exists C depending on ∂Ω such that 1

Ckf k_H1/2

0 (Γ)≤ k ˜f k_H1/2

0 (˜Γ)≤ Ckf k_H1/2 0 (Γ), 1

Ckgk_H1/2

0 (Γ)≤ k˜gk

H₀^1/2(˜Γ)≤ Ckgk_H1/2 0 (Γ).

(26)

Putting together (24), (25), and (26) leads to k ˜Λ_q˜

1,˜Γ− ˜Λ_q˜

2,˜Γk∗≤ CkΛq₁,Γ− Λq₂,Γk∗ (27) 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 (27) yields the estimate (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 (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∗ (28) for some C = C(N ) > 0. Hereafter, we set q_j =^∆

√γj

√γ_j , j = 1, 2. The regularity assumption (4) and Sobolev’s embedding theorem imply that q1, q2 ∈ C¹(Ω).

Using this and (5), we conclude that ˆq1 − ˆq2 satisfies the assumptions of Lemma 2.2 with α = 1. Therefore, Theorem 1.1 and (28) imply that

kq1− q2k_L^∞_(Ω)≤ C

log kΛγ₁,Γ− Λγ₂,Γk_∗ 

−σ₁

(29) where C depend on Ω, N, n, s and σ1 depend on n, s. Next, we recall from [1, (26) on page 168] that

kγ₁− γ₂k_L∞(Ω)≤ Ckq₁− q₂k^σ_L²∞(Ω) (30) for some 0 < σ2< 1, where C = C(N, Ω) and σ2 = σ2(n, s). Finally, putting together (29) and (30) yields (6) with σ = σ1σ2 and the proof of Corollary 1.2 is complete.

References

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

[2] G. Alessandrini and K. Kim, Single-logarithmic stability for the calder´on prob- lem with local data, J. Inv. Ill-Posed Problems 20 (2012), 389–400.

[3] K. Astala and L. P¨aiv¨arinta, Calder´on’s inverse conductivity problem in the plane, Ann. of Math. 163 (2006), no. 1, 265–299.

[4] H. Ben Joud, A stability estimate for an inverse problem for the Schr¨odinger equation in a magnetic field from partial boundary measurements, Inverse Prob- lems 25 (2009), no. 4, 045012.

[5] A. L. Bukhgeim and G. Uhlmann, Recovering a potential from partial Cauchy data, Comm. Partial Differential Equations 27 (2002), no. 3-4, 653–668.

[6] A.-P. Calder´on, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), Soc.

Brasil. Mat., Rio de Janeiro, 1980, pp. 65–73.

[7] P. Caro, On an inverse problem in electromagnetism with local data: stability and uniqueness, Inverse Problems and Imaging 5 (2011), 297–322.

[8] P. Caro, D. Dos Santos Ferreira, and A. Ruiz, Stability estimates for the Calder´on problem with partial data, arXiv:1405.1217.

[9] P. Caro, D. Dos Santos Ferreira, and A. Ruiz, Stability estimates for the Radon transform with restricted data and applications, Advances in Mathematics 267 (2014), 523–564.

[10] P. Caro and M. Salo, Stability of the Calder´on problem in admissible geome- tries, Inverse Problems and Imaging 8 (2014), 939–957.

[11] F. J. Chung, P. Ola, M. Salo, and L. Tzou, Partial data inverse problems for Maxwell equations via Carleman estimates, arXiv:1502.01618, 2015.

[12] D. Dos Santos Ferreira, C. Kenig, J. Sj¨ostrand, and G. Uhlmann, De- termining the magnetic Schr¨odinger operator from partial Cauchy data, Comm.

Math. Physics 271 (2007), 467–488.

[13] I. K. Fathallah, Stability for the inverse potential problem by the local Dirichlet-to-Neumann map for the Schr¨odinger equation, Appl. Anal. 86 (2007), 899–914.

[14] 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.

[15] H. Heck and J.-N. Wang, Stability estimates for the inverse boundary value problem by partial Cauchy data, Inverse Problems 22 (2006), no. 5, 1787–1796.

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

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

[18] C. Kenig and M. Salo, The Calder´on problem with partial data on manifolds and applications, Analysis & PDE 6 (2013), no. 8, 2003–2048.

[19] C. Kenig and M. Salo, Recent progress in the Calder´on problem with partial data, Contemp. Math. 615 (2014), 193–222.

[20] C. Kenig, J. Sj¨ostrand, and G. Uhlmann, The Calder´on problem with partial data, Annals of Math. 165 (2007), 567–591.

[21] K. Krupchyk, M. Lassas, and G. Uhlmann, Inverse problems with partial data for a magnetic schr¨odinger operator in an infinite slab and on a bounded domain, Communications in Mathematical Physics 312 (2012), no. 1, 87–126.

[22] R.-Y. Lai, Stability estimates for the inverse boundary value problem by partial Cauchy data, Math. Meth. Appl. Sci. 38 (2015), no. 8, 1568–1581.

[23] X. Li and G. Uhlmann, Inverse problems with partial data in a slab, Inverse Problems and Imaging 4 (2010), no. 3, 449–462.

[24] N. Mandache, Exponential instability in an inverse problem for the Schr¨odinger equation, Inverse Problems 17 (2001), no. 5, 1435–1444.

[25] A. I. Nachman, Reconstructions from boundary measurements, Ann. of Math.

(2) 128 (1988), no. 3, 531–576.

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

[27] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2) 125 (1987), no. 1, 153–169.

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

[29] L. Tzou, Stability estimate for the coefficients of magnetic Schr¨odinger equation from full and partial boundary measurements, Comm. PDE 11 (2008), 1911–

1952.

[30] S. Vessella, A continuous dependence result in the analytic continuation prob- lem, Forum Math. 11 (1999), 695–703.

Authors’ addresses:

Horst Heck

Bern University of Applied Sciences Engineering and Information Technology Jlcoweg 1

CH-3400 Burgdorf

E-mail: horst.heck@bfh.ch Jenn-Nan Wang

Institute of Applied Mathematical Sciences, NCTS National Taiwan University

Taipei 106 Taiwan

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

The second author was supported in part by the MOST (NSC 102-2115-M-002-009-MY3).

Received