Increasing stability in an inverse problem for the acoustic equation

Increasing stability in an inverse problem for the acoustic equation

Sei Nagayasu

Gunther Uhlmann

Jenn-Nan Wang

Abstract

In this work we study the inverse boundary value problem of determin- ing the refractive index in the acoustic equation. It is known that this inverse problem is ill-posed. Nonetheless, we show that the ill-posedness decreases when we increase the frequency and the stability estimate changes from log- arithmic type for low frequencies to a Lipschitz estimate for large frequen- cies.

1 Introduction

In this paper we study the issue of stability for determining the refractive index in the acoustic equation by boundary measurements. It is well known that this in- verse problem is ill-posed. However, one anticipates that the stability will increase if one increases the frequency. This phenomenon was observed numerically in the inverse obstacle scattering problem [5]. Several rigorous justifications of the in- creasing stability phenomena in different settings were obtained by Isakov et al [6, 7, 8, 10, 11]. Especially, in [8], Isakov considered the Helmholtz equation with a potential

−∆u − k²u + qu = 0 in Ω. (1.1)

∗Department of Mathematical Science, Graduate School of Material Science, University of Hyogo, 2167 Shosha, Himeji, Hyogo 671-2280, Japan. Email:[email protected]

†Department of Mathematics, University of Washington, Box 354305, Seattle, WA 98195- 4350 and Department of Mathematics, University of California, Irvine, CA 92697-3875, USA.

Email:[email protected]

‡Department of Mathematics, NCTS (Taipei), National Taiwan University, Taipei 106, Taiwan.

Email:[email protected]

He obtained stability estimates of determining q by the Dirichlet-to-Neumann map for different ranges of k’s. All of these results demonstrate the increasing stability phenomena in k. For the case of the inverse source problem for Helmholtz equa- tion and an homogeneous background it was shown in [3] that the ill-posedness of the inverse problem decreases as the frequency increases.

In this paper, we study the acoustic wave equation. Let Ω⊂ Rⁿbe a bounded domain, where n≥ 3. Let ∂Ω be smooth. We consider the equation

(∆ + k²q(x))

u(x) = 0 in Ω, (1.2)

where the real-valued q(x) is the refractive index. Assume that the kernel of the operator ∆ + k²q(x) on H₀¹(Ω) is trivial. Associated with (1.2), we define the Dirichlet-to-Neumann map (DN map) Λ : H^1/2(∂Ω)→ H^−1/2(∂Ω) by

Λf = ∂u

∂ν 

∂Ω

,

where u is the solution to (1.2) with the Dirichlet condition u = f on ∂Ω, and ν is the unit outer normal vector of ∂Ω. The uniqueness of this inverse problem is well known [13]. This inverse problem is notoriously ill-posed. For this aspect, Alessandrini proved that the stability estimate for this problem is of log type [1]

and Mandache showed that the log type stability is optimal [9]. In this paper, we would like to focus on how the stability behaves when the frequency k increases.

Now we state the main result.

Theorem 1.1. Assume that q₁(x) and q₂(x) are two sound speeds with associ- ated DN maps Λ1 and Λ2, respectively. Let s > (n/2) + 1, M > 0. Suppose

∥ql∥H^s(Ω) ≤ M (l = 1, 2) and supp(q1− q2) ⊂ Ω. Denote eq a zero extension of q₁− q2. Then there exists a constant C₁, depending only on n, s, and Ω, such that if k² ≥ 1/(C1M ) and∥Λ1− Λ2∥∗ ≤ 1/e then

∥eq∥H^−s(Rⁿ) ≤ C

k²exp(Ck²)∥Λ1− Λ2∥_∗+ C (

k² + log 1

∥Λ1− Λ2∥_∗

)_−(2s−n)

(1.3) holds, where C > 0 depends only on n, s, Ω, M and supp(q₁ − q2). Here∥·∥∗ is the operator norm from H^1/2(∂Ω) into H^−1/2(∂Ω).

Remark 1.2. 1. The estimate (1.3) consists two parts – Lipschitz and logarithmic estimates. As k increases, the logarithmic part decreases and the Lipschitz part

becomes dominated. In other words, the ill-posedness is alleviated when k is large.

2. We would like to remark on the constant C exp(Ck²)/k²appearing in the Lips- chitz part of (1.3). 1/k²comes from k²q in the equation, which appears naturally, while, exp(Ck²) is due to the fact that we use the complex geometrical optics so- lutions in the proof. Even so, we expect that the there is an exponential growth of the constant with frequency since we do not assume any geometrical restriction on q(x) other than regularity. For the wave equation it has been shown by Burq for the obstacle problem [4] that the local energy decay is log-slow and this is due to the presence of trapped rays. Notice that in our case we can have trapped rays.

For the case of simple sound speeds we expect that there is no exponential increase in the constant. In [12] a H¨older stability estimate was obtained for the hyperbolic DN map for generic simple metrics. For very general metrics there is not known modulus of continuity for the hyperbolic DN map, see [2] for convergence results.

However, in practice, k is fixed and so is the constant. Therefore, one should expect to obtain a better resolution of q from boundary measurements when the chosen k is large.

3. Unlike the result in [8, Theorem 2.1] (for equation (1.1)) where the stability estimates were derived in different ranges of k, estimate (1.3) is valid for all range of k provided k² ≥ 1/(C1M ) .

The proof of Theorem 1.1 makes use of Alessandrini’s arguments [1] and the CGO solutions constructed in [13]. The main task is to keep track of how k appears in the proof of the stability estimates.

2 Complex geometrical optics solutions

In this section, we construct CGO solutions to the equation (1.2) by using the idea in [13]. The main point is to express the dependence of constants on k explicitly.

We first state two easy consequences from the results in [13].

Lemma 2.1(see [13, Proposition 2.1 and Corollary 2.2]). Let s≥ 0 be an integer.

Let ε₀ > 0. Let ξ ∈ Cⁿ satisfy ξ· ξ = 0 and |ξ| ≥ ε0. Then for any f ∈ H^s(Ω) there exists w ∈ H^s(Ω) such that w is a solution to

∆w + ξ· ∇w = f in Ω

and satisfies the estimate

∥w∥H^s(Ω) ≤ C₀

|ξ|∥f∥H^s(Ω), where a positive constant C₀ depends only on n, s, ε₀and Ω.

By using this lemma, we can obtain a solution to the equation

∆ψ + ξ· ∇ψ + gψ = f (2.1)

satisfying some decaying property as in the following lemma.

Lemma 2.2 ([13, Theorem 2.3 and Corollary 2.4]). Let s > n/2 be an integer.

Let ε₀ > 0. Let ξ ∈ Cⁿsatisfy ξ· ξ = 0 and |ξ| ≥ ε0. Let f, g ∈ H^s(Ω). Then there exists C₁ > 0 depending only on n, s, ε₀ and Ω such that if

|ξ| ≥ C1∥g∥H^s(Ω)

then there exists a solution ψ ∈ H^s(Ω) to the equation (2.1) satisfying the estimate

∥ψ∥H^s(Ω) ≤ 2C₀

|ξ| ∥f∥H^s(Ω), where C0 is the positive constant in Lemma 2.1.

The needed CGO solutions are constructed as follows.

Proposition 2.3. Let s > n/2 be an integer. Let ε₀ > 0. Let ξ ∈ Cⁿ satisfy ξ· ξ = 0 and |ξ| ≥ ε0. Define the constants C₀and C₁ as in Lemma 2.2. Then if

|ξ| ≥ C1k²∥q∥H^s(Ω)

then there exists a solution u to the equation (1.2) with the form of u(x) = exp

(ξ 2 · x) (

1 + ψ(x))

, (2.2)

where ψ has the estimate

∥ψ∥H^s(Ω) ≤ 2C0k²

|ξ| ∥q∥H^s(Ω). Proof. Substituting (2.2) into (1.2), we have

∆ψ + ξ· ∇ψ + k²qψ =−k²q.

Then by Lemma 2.2, we obtain this proposition.

3 Proof of stability estimate

This section is devoted to the proof of Theorem 1.1. We begin with Alessandrini’s identity.

Proposition 3.1. Let ulbe a solution to (1.2) with q = q_l, then we have k²

∫

Ω

(q₂− q1)u₁u₂dx =⟨

(Λ₁− Λ2)u₁|∂Ω, u₂|∂Ω

⟩.

Now we would like to estimate the Fourier transform of the difference of two q’s. We denoteF(f) the Fourier transformation of a function f.

Lemma 3.2. Let s > (n/2) + 1 be an integer and M > 0. Assume ∥ql∥H^s(Ω) ≤ M , supp(q₁ − q2) ⊂ Ω and k² ≥ 1/C1M , where C₁ is the constant defined in Lemma 2.2 corresponding to ε₀ = 1. Let eq be a zero extension of q1 − q2 and a0 ≥ C1. Suppose that χ∈ C0^∞(Ω) satisfies χ≡ 1 near supp(q1− q2). Then for r ≥ 0 and η ∈ Rⁿwith|η| = 1 the following statements hold: if 0 ≤ r ≤ a0k²M then

|F eq(rη)| ≤ C∥χ∥H^s(Ω)

a₀ ∥eq∥H^−s(Rⁿ)+ C

k² exp(Ca₀k²M )∥Λ1− Λ2∥_∗ (3.1) holds; if r≥ C1k²M then

|F eq(rη)| ≤ CM k²∥χ∥H^s(Ω)

r ∥eq∥H^−s(Rⁿ)+ C

k² exp(Cr)∥Λ1− Λ2∥_∗ (3.2) holds, where C > 0 depends only on n, s and Ω.

Proof. In the following proof, the letter C stands for a general constant depending only on n, s and Ω. By Proposition 2.3, we can construct CGO solutions u_l(x) to the equation (1.2) with q = qlhaving the form of

u_l(x) = exp (ξ_l

2 · x) (

1 + ψ_l(x)) for l = 1, 2, and we have

∫

Ω

(q₂− q1) exp (1

2(ξ₁+ ξ₂)· x )

(1 + ψ₁+ ψ₂+ ψ₁ψ₂) dx

= 1 k²

⟨(Λ1− Λ2)u1|∂Ω, u2|∂Ω

⟩ (3.3)

from Proposition 3.1, where ψ_lsatisfies

∥ψl∥H^s(Ω)≤ Ck²

|ξl| ∥ql∥H^s(Ω)

if ξl ∈ Cⁿsatisfies ξl· ξl = 0,|ξl| ≥ 1 and

|ξl| ≥ C1k²∥ql∥H^s(Ω). (3.4) We remark that∥ψl∥H^s(Ω) ≤ C also holds. Indeed, we have

∥ψl∥H^s(Ω) ≤ Ck²∥ql∥H^s(Ω)

|ξl| ≤ Ck²∥ql∥H^s(Ω)

C₁k²∥ql∥H^s(Ω)

= C C₁ = C.

Now, let r≥ 0, and η ∈ Rⁿsatisfy|η| = 1. We assume that α, ζ ∈ Rⁿsatisfy α· η = α · ζ = η · ζ = 0 and |ζ|² =|α|²+ r². (3.5) Define ξ₁ and ξ₂as

ξ1 = ζ + iα− irη and ξ2 =−ζ − iα − irη.

Then we have

ξl· ξl= 0, |ξl|² =|ζ|²+|α|²+ r² = 2|ζ|²(l = 1, 2) and 1

2(ξ1+ ξ2) =−irη.

Hence by (3.3), we immediately obtain that F eq(rη) = −

∫

Ω

(q₂− q1) exp(−irη · x)(ψ1+ ψ₂+ ψ₁ψ₂) dx + 1

k²

⟨(Λ₁− Λ2)u₁|∂Ω, u₂|∂Ω

⟩ (3.6)

provided |ξl| ≥ 1 and (3.4) are satisfied. We first estimate the first term on the

right hand side of (3.6) by ∫

Ω

(q₂− q1) exp(−irη · x)(ψ1+ ψ₂+ ψ₁ψ₂) dx 

= ∫

Ω

(q2− q1) exp(−irη · x)χ(ψ1+ ψ2+ ψ1ψ2) dx 

≤ ∥q2− q1∥H^−s(Ω) χ(ψ₁+ ψ₂+ ψ₁ψ₂)

H^s(Ω)

≤ ∥eq∥H^−s(Rⁿ)∥χ∥H^s(Ω)

(∥ψ1∥H^s(Ω)+∥ψ2∥H^s(Ω)+∥ψ1∥H^s(Ω)∥ψ2∥H^s(Ω)

)

≤ ∥eq∥H^−s(Rⁿ)∥χ∥H^s(Ω)

( Ck²

√2|ζ| + Ck²

√2|ζ| + C Ck²

√2|ζ|

)∑2

l=1

∥ql∥H^s(Ω)

= Ck²∥χ∥H^s(Ω)

|ζ| ∥eq∥H^−s(Rⁿ)

∑2 l=1

∥ql∥H^s(Ω).

since χ(ψ₁+ ψ₂+ ψ₁ψ₂)∈ H₀^s(Ω) and s > n/2.

On the other hand, by taking R large enough such that Ω⊂ BR(0), we have ul|∂Ω

L²(∂Ω) ≤ |∂Ω|^1/2∥ul∥C⁰(Ω) ≤ |∂Ω|^1/2exp

(|Re ξl| 2 R) (

1 +∥ψl∥L^∞(Ω)

)

≤ C exp

(|Re ξl| 2 R) (

1 +∥ψl∥H^s(Ω)

)

≤ C exp

(|Re ξl|

2 R

)

(1 + C) = C exp (|ζ|

2 R )

. Likewise, we can get that

∇u_l|∂Ω

L²(∂Ω) = ξl

2u_l+ exp (ξl

2 · • )

(∇ψl) L²(∂Ω)

≤

√2|ζ|

2 C exp (|ζ|

2 R )

+|∂Ω|^1/2exp (|ζ|

2 R )

∥∇ψl∥C⁰(Ω)

≤ C|ζ| exp (|ζ|

2 R )

+ C exp (|ζ|

2 R )

∥∇ψl∥H^s−1(Ω)

≤ C|ζ| exp (|ζ|

2 R )

+ C exp (|ζ|

2 R )

∥ψl∥H^s(Ω)

≤ C exp(C|ζ|)

since s− 1 > n/2. Consequently, we have u_l|∂Ω

H^1/2(∂Ω) ≤ C exp(C|ζ|).

Therefore, we can estimate the second term of the right-hand side of (3.6) by ⟨(Λ₁− Λ2)u₁|∂Ω, u₂|∂Ω⟩ ≤ ∥Λ₁− Λ2∥_∗ u₁|∂Ω

H^1/2(∂Ω) u₂|∂Ω

H^1/2(∂Ω)

≤ C exp(C|ζ|)∥Λ1− Λ2∥∗.

Summing up, we have shown that for r > 0 and for η ∈ Rⁿ with|η| = 1 if we take α and ζ satisfying the conditions (3.5),|ζ| ≥ 2^−1/2and

|ζ| ≥ 2^−1/2C₁k²∥ql∥H^s(Ω) (3.7) then

|F eq(rη)| ≤ Ck²∥χ∥H^s(Ω)

|ζ| ∥eq∥H^−s(Rⁿ)

∑2 l=1

∥ql∥H^s(Ω)

+ C

k² exp(C|ζ|)∥Λ1− Λ2∥∗ (3.8) holds.

Now assume that∥ql∥H^s(Ω) ≤ M and k² ≥ 1/C1M . Thus if

|ζ| ≥ C1k²M (3.9)

holds, then (3.7) and |ζ| ≥ 2^−1/2 are satisfied. Pick a0 ≥ C1. We first consider the case where 0≤ r ≤ a0k²M . By choosing α and ζ satisfying

α· η = α · ζ = η · ζ = 0, |ζ| = a0k²M (≥ r) and |α| =√

(a0k²M )²− r² both (3.5) and (3.9) are then satisfied since a₀ ≥ C1. Hence we obtain (3.8), that is (3.1). On the other hand, when r ≥ C1k²M , we can choose α = 0, η · ζ = 0 and|ζ| = r. Then (3.5), (3.9) are satisfied and thus (3.8) holds and consequently (3.2) is valid.

Now we prove our main result.

Proof. As above, C denotes a general constant depending only on n, s and Ω.

Written in polar coordinates, we have

∥eq∥²_H−s(Rⁿ) = C

∫ _∞

0

∫

|η|=1|F eq(rη)|²(1 + r²)^−srⁿ⁻¹dη dr

= C

(∫ a0k²M 0

∫

|η|=1|F eq(rη)|²(1 + r²)^−srⁿ⁻¹dη dr +

∫ T a0k²M

∫

|η|=1|F eq(rη)|²(1 + r²)^−srⁿ⁻¹dη dr +

∫ _∞

T

∫

|η|=1|F eq(rη)|²(1 + r²)^−srⁿ⁻¹dη dr )

=: C(I₁+ I₂+ I₃), (3.10)

where a0 ≥ C1and T ≥ a0k²M are parameters which will be chosen later. Here C₁is the constant given in Lemma 3.2. From now on, we take k² ≥ 1/(C1M ).

Our task now is to estimate each integral separately. We begin with I₃. Since

|F eq(rη)| ≤ C∥q1− q2∥L²(Ω), q1− q2 ∈ H0^s(Ω), and s > n/2, we have that I₃ ≤ C

∫ _∞

T

∥q1− q2∥²_L²_(Ω)(1 + r²)^−srⁿ⁻¹dr ≤ CT^−m∥q1− q2∥²_L²_(Ω)

≤ CT^−m (

ε∥q1− q2∥²_H−s(Ω)+C

ε∥q1− q2∥²_H^s_(Ω) )

≤ CT^−m (

ε∥eq∥²H^−s(Rⁿ)+M² ε

)

(3.11) for ε > 0, where m := 2s− n.

On the other hand, by Lemma 3.2, we can estimate I₁ ≤ C

∫ _a0k²_M

0

(1 + r²)^−srⁿ⁻¹dr

×

[∥χ∥²_H^s_(Ω)

a²₀ ∥eq∥²_H−s(Rⁿ)+ exp(2Ca₀k²M )

k⁴ ∥Λ1− Λ2∥²_∗ ]

≤ C

∫ _∞

0

(1 + r²)^−srⁿ⁻¹dr [C_χ²

a²₀ ∥eq∥²H^−s(Rⁿ)+exp(Ca₀k²M )

k⁴ ∥Λ1− Λ2∥²_∗ ]

= CC_χ²

a²₀ ∥eq∥²H^−s(Rⁿ)+ C exp(Ca₀k²M )

k⁴ ∥Λ1− Λ2∥²_∗, (3.12)

where χ ∈ C0^∞(Ω) satisfies χ ≡ 1 near supp(q2 − q1) and C_χ := ∥χ∥H^s(Ω). In view of

∫ _T

a0k²M

(1 + r²)^−srⁿ⁻³dr≤

∫ _T

a0k²M

r^−2s+n−3dr≤ C(a0k²M )^−2s+n−2

≤ C(a0k²M )⁻²(C₁k²M )^−m≤ C a²₀k⁴M² and ∫ T

a0k²M

exp(Cr)(1 + r²)^−srⁿ⁻¹dr ≤ exp(CT )

∫ T a0k²M

(1 + r²)^−srⁿ⁻¹dr

≤ exp(CT )

∫ _∞

0

(1 + r²)^−srⁿ⁻¹dr

≤ C exp(CT ), we have that

I₂ ≤ CM²k⁴∥χ∥²H^s(Ω)∥eq∥²H^−s(Rⁿ)

∫ _T

a0k²M

(1 + r²)^−srⁿ⁻³dr + C

k⁴∥Λ1− Λ2∥²_∗

∫ T a0k²M

exp(Cr)(1 + r²)^−srⁿ⁻¹dr

≤ CC_χ²

a²₀ ∥eq∥²H^−s(Rⁿ)+ C

k⁴ exp(CT )∥Λ1− Λ2∥²_∗. (3.13) Combining (3.10)–(3.13) gives

∥eq∥²H^−s(Rⁿ)≤ C(I1+ I₂+ I₃)

≤ CC_χ²

a²₀ ∥eq∥²H^−s(Rⁿ)+ C exp(Ca₀k²M )

k⁴ ∥Λ1− Λ2∥²_∗ +CC_χ²

a²₀ ∥eq∥²H^−s(Rⁿ)+ C

k⁴ exp(CT )∥Λ1 − Λ2∥²_∗ + CT^−m

(

ε∥eq∥²H^−s(Rⁿ)+ M² ε

)

=

(C₂²C_χ²

a²₀ + C₃T^−mε )

∥eq∥²H^−s(Rⁿ)

+ C k⁴

(exp(Ca0k²M ) + exp(CT ))

∥Λ1− Λ2∥²_∗+CM² ε T^−m,

where positive constants C₂ and C₃ depend only on n, s and Ω.

Now we pick a₀ and ε as

a0 = 2C2Cχ≥ C1and ε = T^m 4C₃ (if needed, we take C₂large enough). We then obtain that

∥eq∥²_H−s(Rⁿ) ≤ C k⁴

[exp(2C₂CC_χk²M ) + exp(CT )]

∥Λ1− Λ2∥²_∗+ CT^−2mM²

= C

k⁴ exp(Cak²)A + CΦ(T ) (3.14)

for T ≥ a0k²M = 2C₂C_χk²M = ak², where Φ(T ) := 1

k⁴exp(C₄T )A + M²T^−2m,

A :=∥Λ1− Λ2∥²_∗, a := 2C2C_χM²and C4 > 0 depends only on n, s and Ω.

To continue, we consider two cases:

ak² ≤ p log 1

A (3.15)

and

ak² ≥ p log 1

A, (3.16)

where p will be determined later (see (3.24)).

For the first case (3.15), our aim is to show that there exists T ≥ ak²such that Φ(T )≤ 2C5

(

k²+ log 1 A

)_−2m

. (3.17)

Substituting (3.17) into (3.14) clearly implies (1.3). Now to derive (3.17), it is enough to prove that

1

k⁴ exp(C4T )A≤ C5

(

k²+ log 1 A

)_−2m

(3.18) and

M²T^−2m ≤ C5

(

k²+ log 1 A

)_−2m

. (3.19)

Remark that (3.19) in equivalent to T ≥ C5^−1/2mM^1/m

(

k²+ log 1 A

) , which holds if

T ≥ C5^−1/2mM^1/m (

1 + p a

) log 1

A (3.20)

because of (3.15). Setting T = p log(1/A) (≥ ak² by (3.15)), then (3.20) holds provided

p≥ C₅^−1/2mM^1/m (

1 + p a )

. (3.21)

Now we turn to (3.18). It is clear that (3.18) is equivalent to C₄p log 1

A ≤ log C5+ 2 log k²+ log 1

A − 2m log (

k²+ log 1 A

)

(3.22) since T = p log(1/A). It follows from (3.15) that

log (

k² + log 1 A

)

≤ log (p

alog 1

A + log 1 A

)

= log (p

a + 1 )

+ log log 1 A. Hence (3.22) is verified if we can show that

C₄p log 1

A ≤ log C5− 2 log(MC1) + log 1 A − 2m

( log

(p a + 1

)

+ log log 1 A

) ,

i.e.

(1− C4p) log 1

A− 2m log log 1

A + log C₅ − 2 log(MC1)− 2m log(p a + 1

)≥ 0 (3.23) for log(1/A) ≥ 1. Now we choose

p = 1

2C₄. (3.24)

Then (3.23) becomes log 1

A− 4m log log 1

A + 2 log C₅− 4 log(MC1)− 4m log(p a + 1

)≥ 0. (3.25)

Notice that

0<A≤1/einf (

log 1

A − 4m log log 1 A

)

= inf

z≥1(z− 4m log z)

≥ inf

z>0(z− 4m log z) = 4m log e 4m. Hence if we choose C₅ such that

C5 ≥ (MC1)² (p

a + 1 )2m(

4m e

)2m

(3.26) then (3.25) follows. Finally, we take

C₅ := max {

C₁² (4m

e )2m

, p^−2m }

M² (

1 + p a

)2m

,

which depends only on n, Ω, s, M and χ. With such choice of C₅, the conditions (3.26) and (3.21) hold, and thus estimate (3.17) is satisfied.

Next we consider the second case (3.16). By (3.14) with T = ak², we get that

∥eq∥²H^−s(Rⁿ) ≤ C

k⁴ exp(Cak²)A + C

k⁴ exp(C₄ak²)A + CM²(ak²)^−2m

≤ C

k⁴ exp(Cak²)A + CM²a^−2mk^−4m. Hence it remains to show that

k^−4m ≤ C6

(

k²+ log 1 A

)_−2m , i.e.

k² ≥ C6^−1/2m

(

k²+ log 1 A

)

. (3.27)

Since

k²+ log 1 A ≤

( 1 + a

p )

k² by (3.16), we have (3.27) if we take C₆ large enough so that

C₆ ≥ (

1 + a p

)2m

. The proof is completed.

Acknowledgements

Nagayasu was partially supported by Grant-in-Aid for Young Scientists (B). Uhlmann was partly supported by NSF and a Visiting Distinguished Rothschild Fellowship at the Isaac Newton Institute. Wang was partially supported by the National Sci- ence Council of Taiwan. We would also like to thank P. Stefanov for helpful discussions.

References

[1] G. Alessandrini, Stable determination of conductivity by boundary measure- ments, Appl. Anal., 27 (1988), 153-172.

[2] M. Anderson, A. Katsuda, Y. Kurylev, M. Lassas, M. Taylor, Boundary regu- larity for the Ricci equation, Geometric Convergence, and Gel’fand’s Inverse Boundary Problem, Inventiones Mathematicae, 158 (2004), 261-321.

[3] G. Bao, J. Lin and F. Triki, A multi-frequency inverse source problem, J. Diff.

Eq., 249, (2010), 3443-3465.

[4] N. Burg, Decay of the local energy of the wave equation for the exterior prob- lem and absence of resonance near the real axis, Acta Math., 180 (1998), 1-29.

[5] D. Colton, H. Haddar, and M. Piana, The linear sampling method in inverse electromagnetic scattering theory, Inverse Problems, 19 (2003), S105-S137.

[6] T. Hrycak and V. Isakov, Increased stability in the continuation of solutions to the Helmholtz equation, Inverse Problems, 20 (2004), 697-712.

[7] V. Isakov, Increased stability in the continuation for the Helmholtz equa- tion with variable coefficient, Control methods in PDE-dynamical systems, 255V267, Contemp. Math., 426, AMS, Providence, RI, 2007.

[8] V. Isakov, Increasing stability for the Schr¨odinger potential from the Dirichlet-to-Neumann map, DCDS-S, 4 (2011), 631-640.

[9] N. Mandache, Exponential instability in an inverse problem for the Schr¨odinger equation, Inverse Problems, 17 (2001), 1435-1444.

[10] D. A. Subbarayappa and V. Isakov, On increased stability in the continuation of the Helmholtz equation, Inverse Problems, 23 (2007), no. 4, 1689V1697.

[11] D. A. Subbarayappa and V. Isakov, Increasing stability of the continuation for the Maxwell system, Inverse Problems, 26 (2010), no. 7, 074005, 14 pp.

[12] P. Stefanov and G. Uhlmann, Stable determination of the hyperbolic Dirichlet-to-Neumann map for generic simple metrics, International Math Re- search Notices (IMRN), 17 (2005), 1047-1061.

[13] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 185 (1987), 153-169.