INCREASING STABILITY FOR THE CONDUCTIVITY AND ATTENUATION COEFFICIENTS

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ATTENUATION COEFFICIENTS

VICTOR ISAKOV, RU-YU LAI, AND JENN-NAN WANG

Abstract. In this work we consider stability of recovery of the conductivity and attenuation coefficients of the stationary Maxwell and Schr¨odinger equations from a complete set of (Cauchy) boundary data. By using complex geometrical optics solutions we derive some bounds which can be viewed as an evidence of increasing stability in these inverse problems when frequency is growing.

1. Introduction

The main goal of this paper is to demonstrate increasing stability of recovery of the conduc- tivity coefficient from results of all possible electromagnetic boundary measurements when frequency of the stationary waves is growing. When the frequency is large the Maxwell system can not be reduced to a scalar conductivity equation, so we have to handle the full Maxwell system. We do it by reducing the first order system to a vectorial Schr¨odinger type equation containing conductivity coefficient in matrix potential coefficient as in [3], [13] and use ideas in [7], [9] to derive increasing stability for potential coefficient involving variable attenuation.

Use of the full system seems to be crucial, since increasing stability for the conductivity in the scalar equation does not seems to be true. We first consider a simpler case of scalar equation and extend the first and third author’s result [9] on the increasing stability estimate in the case of a constant attenuation to a variable one. Observe, that the method used in [9] works only for the constant attenuation. Extending this method to the case with variable attenuation is not an obvious routine, so the result for the scalar equation is of independent interest.

The problem of recovering the conductivity from boundary measurements has been well studied since the 1980s. The uniqueness issue was first settled by Sylvester and Uhlmann in [15] where they constructed complex geometrical optics (CGO) solutions and proved global uniqueness of potential in the Schr¨odinger equation. A logarithmic stability estimate was obtained by Alessandrini [1]. Later, the log-type stability was shown to be optimal by Man- dache [11]. The logarithmic stability makes it impossible to design reconstruction algorithms with high resolution in practice since small errors in measurements will result in exponen- tially large errors in the reconstruction of the target material parameters. Nonetheless, in some cases, it has been observed numerically that the stability increases if one increases the frequency in the equation. The study of the increasing stability phenomenon has attracted a lot of attention recently. There are several results [7], [8], [9] and [12] which rigorously demonstrated the increasing stability behaviours in different settings.

For the Schr¨odinger equation without attenuation, the first author in [7] derived some bounds in different ranges of frequency which can be viewed as an evidence of increasing stability behaviour. The idea in [7] is to use complex- and real-valued geometrical optics

The first and the second author were supported in part by the National Science Foundation.

The third author was supported in part by the MOST102-2115-M-002-009-MY3.

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solutions to control low and high frequency ranges, respectively. The proof was simplified in [8] by using only complex-valued geometrical optics solutions. Recently, the first and third author showed in [9] that the increasing stability also holds for the the same problem with constant attenuation. The proofs in [9] use complex and bounded geometrical optics solutions.

Continuing the research in [9] we show in this work even in the case of variable attenuation, the stability is improving when we increase the frequency. More precisely, the stability estimate will consist of two parts, one is the logarithmic part and the other one is the Lipschitz part (see Theorem 2.1). In the high frequency regime, the logarithmic part becomes weaker and the stability behaves like a Lipschitz continuity. Moreover, the constant of the Lipschitz part grows only polynomially in terms of the frequency. We would like to point out that in [12] an increasing stability phenomenon was proved for the acoustic equation in which the constant associated with the Lipschitz part grows exponentially in the frequency. In order to obtain polynomially growing constants as in [7], [8] and [9], it seems like the smallness assumption on the attenuation is needed.

The main result of the paper is a proof of the increasing stability for the Maxwell equations in a conductive medium. The first global uniqueness result for all three electromagnetic parameters of an isotropic medium was proved in [14]. We refer the reader to the latest results in [2] where the uniqueness and stability (log-type) were established using local data. Since our aim here is to demonstrate the increasing stability for the conductivity, we will assume that both the electric permittivity and the magnetic permeability are known constants. As in the previous results on the increasing stability, our proof here relies on CGO solutions. Here we will use CGO solutions constructed in [3]. Theorem 2.2, shows that the logarithmic part becomes weaker and the Lipschitz part becomes dominant when the frequency is increasing.

In order to derive a polynomially frequency-dependent constant in the Lipschitz part, we impose the smallness assumption on the conductivity. Of course, it is an interesting question whether one can remove the smallness assumption and still obtain a polynomially frequency- dependent constant in the Lipschitz part.

The paper is structured as follows. Section 2 introduces the mathematical settings for the Schr¨odinger and the Maxwell equations and presents our main results. The proof of stability estimates for the Schr¨odinger equation is carried out in section 3. In section 4 we derive the stability estimate for the Maxwell equations.

2. Preliminaries and main results

Let Ω be a sub domain of the unit ball B in R3, with Lipschitz boundary ∂Ω. By Hs(Ω) we denote Sobolev spaces of functions on Ω with the standard norm k · k(s)(Ω). These notations also hold for surfaces instead of Ω and for vector and matrix valued functions.

2.1. Schr¨odinger equation with attenuation. We consider the Schr¨odinger equation with attenuation

−∆u − ω2u + qu = 0 in Ω, q = iωσ + c, (2.1) where σ, c are real-valued bounded measurable functions in Ω. Since the boundary value problem for (2.1) does not necessarily has a unique solution, for the study of the inverse problem we consider the Cauchy data set defined by

C(q) =n

(u, ∂νu)|∂Ω∈ H1/2(∂Ω) × H−1/2(∂Ω) : u is a H1(Ω)-solution to (2.1)o .

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Hereafter, ν is the unit outer normal vector to ∂Ω. Let qj = iωσj + cj. To measure the distance between two Cauchy data sets, we define

dist(C(q1), C(q2)) = sup

j6=k

sup

(fj,gj)∈Cj

inf

(fk,gk)∈Ck

k(fj, gj) − (fk, gk)k(1/2,−1/2)(∂Ω) k(fj, gj)k(1/2,−1/2)(∂Ω) ,

where (fj, gj) = (uj, ∂νuj), j = 1, 2, are the Cauchy data for solutions uj to the equation (2.1) with q = qj and

k(f, g)k(1/2,−1/2)(∂Ω) =

kf k2(1/2)(∂Ω) + kgk2(−1/2)(∂Ω)1/2

. Denote

 = dist(C1, C2), E = − log .

With these notations, the bound which indicates the increasing stability phenomenon for the Schr¨odinger equation with attenuation can now be stated as follows.

Theorem 2.1. Let s > 3/2 and qj = iωσj+ cj, j = 1, 2. Suppose that supp(q1 − q2) ⊂ Ω and cj, σj ∈ H2s(Ω) satisfy kcjk(2s)(Ω) ≤ M for some constant M > 0. Let ω > 1, E > 1.

Then there exist constants m > 0 depending on s, M and C depending on s, Ω, M , such that if

jk(2s)(Ω) ≤ m, then

kq1− q2k(−s)(Ω) ≤ C(ω2 + ω(ω + E )−(2s−3)/2). (2.2) It is an immediate consequence of Theorem 2.1 that if σ1 = σ2 = b in Ω where b is a fixed constant, |b| ≤ m, then one gets a bound similar to [9].

When 52 < s the logarithmic term is decreasing with respect to ω, so the bound (2.2) implies improving (better than logarithmic) stability of recovery of q from complete boundary data.

We discuss it in detail.

Obviously,

ω2 + ω(ω + E )−(2s−3)/2 ≤ ω2 + (ω + E )−(2s−5)/2≤ ω2 + (ω2+ E2)−(2s−5)/4.

The function F (t) = t + (t + E2)−(2s−5)/4 with the derivative F0(t) =  − (2s − 5)/4(t + E2)−(2s−1)/4 is increasing on the interval [1, +∞) if a) (2s − 5)/4(1 + E2)−(2s−1)/4 ≤  and has minimum when (2s − 5)/4(t + E2)−(2s−1)/4 =  if b)  < (2s − 5)/4(1 + E2)−(2s−1)/4. In the exceptional case a) the minimal value of F on [1, +∞) is  + (1 + E2)−(2s−5)/4

 + (4/(2s − 5))(2s−5)/(2s−1). In the case b) the minimal value is

((4/(2s − 5))−4/(2s−1)− E2) + (4/(2s − 5))(2s−5)/(2s−1)

((4/(2s − 5))−4/(2s−1)) + (4/(2s − 5))(2s−5)/(2s−1) = ((4/(2s − 5))−4/(2s−1)+ (4/(2s − 5))(2s−5)/(2s−1))(2s−5)/(2s−1).

In the both cases, given error level  one can choose frequency ω to guarantee at least H¨older stability for q which is far better than logarithmic stability. Moreover, this analysis of the stability estimate (2.2) suggests that for given  there is an optimal choice of ω for the best reconstruction of q.

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2.2. Maxwell equations. Let E and H denote the electric and magnetic vector fields in the medium with the electric permittivity ε, µ is the magnetic permeability µ, and the electric conductivity σ. The Maxwell equations at frequency ω are

 ∇ × H + iωγE = 0,

∇ × E − iωµH = 0, γ = ε + iσ/ω. (2.3)

We define the function space

H(curl; Ω) = {v ∈ H0(Ω) : ∇ × v ∈ H0(Ω)}, with the norm

kvkH(curl;Ω)= kvk(0)(Ω) + k∇ × vk(0)(Ω).

For any v ∈ H(curl; Ω) the tangential trace of v, ν × v, can be defined as an element of the space H−1/2(∂Ω). Namely, for any w∈ H1(Ω),

hν × v, wi = Z

(∇ × v) · w − Z

v · (∇ × w), (see [3]). We define the space

T H(∂Ω) = {w ∈ H−1/2(∂Ω) : there exists v ∈ H(curl; Ω), ν × v = w}

with the norm

kwkT H(∂Ω)= inf{kvkH(curl;Ω) : ν × v = w}.

The Cauchy data set corresponding to (2.3) is defined by C = {(ν × E|∂Ω,ν × H|∂Ω) ∈ (T H(∂Ω))2:

(E, H) is (H(curl; Ω))2− solution of (2.3)}.

In this work we assume that ε = µ = 1. Suppose that C1 and C2 are two Cauchy data of (2.3) corresponding to conductivities σ1 and σ2, respectively. We measure the distance between two Cauchy data C1, C2 of the Maxwell equations as follows:

dist(C1, C2) = sup

j6=k

sup

(F (j), G(j)) ∈ Cj,

inf

(F (k),G(k))∈Ck

k(F (j), G(j)) − (F (k), G(k))k(T H(∂Ω))2

kF (j)kT H(∂Ω) , where F (j) = ν × E(j), G(j) = ν × H(j) on ∂Ω and E(j), H(j) is a H(curl; Ω) solution to (2.3) with σ = σj. Likewise, we denote

 = dist(C1, C2), E = − log .

Our main result is the following stability estimate for the Maxwell equations.

Theorem 2.2. Let s > 3/2 be an integer. Suppose that supp(σ1 − σ2) ⊂ Ω. Assume that E > c (depending on s, Ω) and ω > 1.

Then there exist constants m and C depending on s, Ω such that if

jk(2s+2)(Ω) ≤ m (2.4)

for j = 1, 2, then

1− σ2k(−s)(Ω) ≤ Cω−12+ E2)3/21/2+ C(ω + E )−(2s−3)/2+ C(ω + E )−1. (2.5) As above, Theorem 2.2 implies increasing stability for recovery of the conductivity coeffi- cient.

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3. Stability estimates for Schr¨odinger equation

In this section we prove Theorem 2.1. We will use some ideas in [8]. C will stay for a generic constant depending only on s, Ω, M . First, following the arguments in [5], it is not hard to see that the following inequality holds.

Proposition 3.1. Let u1 and u2 be the solutions of (2.1) corresponding to the coefficients (σ1, c1) and (σ2, c2), respectively. Then

Z

(q1− q2)u1u2

≤ k(u1, ∂νu1)k(1/2,−1/2)(∂Ω) k(u2, ∂νu2)k(1/2,−1/2)(∂Ω). (3.1) We use CGO solutions for (2.1).

Proposition 3.2. Let ζ ∈ C3 satisfy ζ · ζ = ω2. Let s > 3/2. Then there exist constants C0

and C1, depending on s and Ω, such that if |ζ| > C0kqk(2s)(Ω), then there exists a solution u(x) = eiζ·x(1 + ψ(x))

of the equation (2.1) with

kψk(2s)(Ω) ≤ C1

|ζ|kqk(2s)(Ω).

Proof. By extension theorems for Sobolev spaces there is an extension of q from Lipschitz Ω onto the unit ball B (denoted also q) with kqk(2s)(B) ≤ C(Ω)kqk(2s)(Ω). We can assume that q is compactly supported in B.

We observe that since e−iζ·x(−∆)(eiζ·x(1 + ψ(x))) = (−∆ − 2iζ · ∇ + ζ · ζ)(1 + ψ(x)), if u = eiζ·x(1 + ψ) is a solution of (2.1) if and only if the reminder ψ solves

(−∆ − 2iζ · ∇)ψ = −q(1 + ψ) in Ω. (3.2)

As known [15], there are constants C0, C1, depending on s, such that if C0kqk(2s)(B) < |ζ|, there exists a solution ψ to (3.2) with

kψk(2s)(B) ≤ C1

|ζ|kqk(2s)(B).

 Let ξ ∈ R3. We select e(j) ∈ R3 satisfying |e(1)| = |e(2)| = 1 and e(j) · ξ = e(1) · e(2) = 0 for j = 1, 2. Let R > 0. We choose

ζ(1) = −1

2ξ + i R2 2

1/2

e(1) +



ω2+R2 2 − |ξ|2

4

1/2

e(2),

ζ(2) = −1

2ξ − i R2 2

1/2

e(1) −



ω2+R2 2 − |ξ|2

4

1/2

e(2), (3.3)

provided

ω2+R2 2 ≥ |ξ|2

4 . It is clear that

ζ(1) + ζ(2) = −ξ, ζ(j) · ζ(j) = ω2, |ζ(j)|2= R2+ ω2, for j = 1, 2, which implies that √

2|ζ(j)| ≥ R + ω. We observe that if R + ω ≥

2C0(mω + M ), (3.4)

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then one has

|ζ(j)| ≥ C0(mω + M ) > C0kiωσj+ cjk(2s)(Ω).

Note that when

R ≥√

2C0M and √

2C0m ≤ 1, (3.5)

the inequality (3.4) is clearly satisfied. So from Proposition 3.2, there exist CGO solutions uj(x) = eiζ(j)·x(1 + ψj(x)), kψjk(2s)(Ω) ≤ C1

|ζ(j)|kiωσj+ cjk(2s)(Ω). (3.6) To derive the stability, we would like to estimate the Fourier transform of q1− q2. Before doing so, we need the following estimate on the Cauchy data of CGO solutions uj, j = 1, 2, constructed above.

Lemma 3.3. If uj are the solutions (3.6), then

k(uj, ∂νuj)k(1/2,−1/2)(∂Ω) ≤ CeR2p

R2+ ω2. (3.7)

Proof. Recall that s > 3/2. Thus, by Sobolev embedding theorems kψjk(Ω) ≤ Ckψjk(2s)(Ω) ≤ CC1

|ζ(j)|kiωσj+ cjk(2s)(Ω) ≤ CC1 C0

≤ C and

|uj(x)| = |eiζ(j)·x(1 + ψj(x))| ≤ e|=ζ(j)|(1 + |ψj(x)|) ≤ Ce

R 2. Moreover,

|∇uj(x)| ≤ |iζ(j)eiζ(j)·x(1 + ψj)| + |eiζ(j)·x∇ψj| ≤ CeR2|ζ(j)|(1 + |∇ψ(x)|), hence

kujk(1)(Ω) ≤ CeR2(1 + |ζ(j)|)(1 + kψjk(1)(Ω) ≤ CeR2p

R2+ ω2 and thus by Trace Theorems

k(uj, ∂νuj)k(1/2,−1/2)(∂Ω) ≤ Ce

R

2|ζ(j)| = CeR2p

R2+ ω2.

 We denote ˆf the Fourier transform of the function f . Let ˜q = q1− q2 . Lemma 3.3 is used to bound the low frequency of ˆq as the following lemma shows.˜

Lemma 3.4. Let s > 3/2 and ξ = re with r ≥ 0, |e| = 1. Assume that (3.5) is satisfied.

There exists a constant C, such that

|ˆq(re)| ≤ Cφ(R, ω)Q(ξ) + Ce˜

2R(R2+ ω2), (3.8) provided ω2+ R2/2 > r2/4. Here

Q(ξ)2 = Z

hxi−4s Z

h−η + ξ − xi−2s|ˆq(η)|˜ 2dηdx and

φ(R, ω) = mω + M R + ω .

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Proof. Substituting CGO solutions (3.6) into (3.1) yields

Z

˜

q(x)e−iξ·x(1 + ψ1(x))(1 + ψ2(x))dx

≤ k(u1, ∂νu1)k(1/2,−1/2)(∂Ω) k(u2, ∂νu2)k(1/2,−1/2)(∂Ω), which leads to

Z

˜

qe−iξ·xdx

≤ Z

˜

q(x)e−iξ·xχ(x)Ψ(x)dx

+ k(u1, ∂νu1)k(1/2,−1/2)(∂Ω) k(u2, ∂νu2)k(1/2,−1/2)(∂Ω), (3.9) where Ψ(x) = ψ1(x) + ψ2(x) + ψ1(x)ψ2(x), χ ∈ C0(Ω), and χ = 1 in supp(˜q).

Since R f gdx = (2π)−3R f ˆˆgdη and cf g = (2π)−3f ∗ ˆˆ g, one has Z

˜

q(x)e−iξ·xχ(x)Ψ(x)dx

= (2π)−3 Z

bq(η)( \˜ e−iξ·xχΨ)(−η)dη = (2π)−6 Z

q(η)( \e−iξ·xχ ∗ bΨ)(−η)dη.

Denote χξ= e−iξ·xχ and hξi = (1 + |ξ|2)1/2. By the H¨older’s inequality, we obtain

| Z

ˆ˜

q(η)( \e−iξ·xχ ∗ bΨ)(−η)dη|

≤ Z

|ˆq(η)||(˜ cχξ∗ ˆΨ)(−η)|dη

≤ Z

|ˆq(η)||˜ Z

ξ(−η − x) ˆΨ(x)dx|dη

≤ Z Z

|ˆq(η)||˜ χcξ(−η − x)|dη| ˆΨ(x)|dx

≤ Z

hxi−4s

Z

|ˆq(η)||˜ χcξ(−η − x)|dη

2

dx

!1/2

kΨk(2s)

≤ Z

hxi−4s

Z

h−η + ξ − xi−s|ˆq(η)|h−η + ξ − xi˜ s| ˆχ(−η + ξ − x)|dη

2

dx

!1/2

kΨk(2s)

Z

hxi−4s

Z

h−η + ξ − xi−2s|ˆq(η)|˜ 2

 dx

1/2

kχk(s)kΨk(2s). (3.10) From (3.6) and a priori assumptions, we have

kΨk(2s)(Ω) ≤ Cmω + M

R + ω . (3.11)

Therefore, the estimate (3.8) follows from (3.7), (3.9), (3.10) and (3.11).  The following lemma is an immediate consequence of Lemma 3.4.

Lemma 3.5. Let

R >√

2C0M and √

2C0m ≤ 1, (3.12)

where C0 is the constant given in Proposition 3.2.

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If 0 ≤ r ≤ ω + R, then

|ˆq(re)| ≤ Cφ(R˜ , ω)Q(ξ) + C(ω2+ R∗2)e

2R; (3.13)

if r ≥ ω + R, then

|ˆq(re)| ≤ Cφ(r, ω)Q(ξ) + C(ω˜ 2+ r2)e

2r. (3.14)

Proof. It is enough to take R = R when 0 ≤ r ≤ ω + R and take R = r when r ≥ ω + R

in Lemma 3.4. 

With the help of Lemma 3.5 we can prove

Lemma 3.6. Let the conditions of Lemma 3.5 be satisfied. Then for any T ≥ ω + R, we have

k˜qk2(−s)(Ω) ≤ Cφ(R, ω)2k˜qk2(−s)(Ω) + C(ω2+ T2)2(e2

2R+ χ(T )e2

2T)2+ C(mω + M )2T−(2s−3), (3.15) where χ(T ) ≤ 1 and χ(T ) = 0 if T = ω + R.

Proof. Using polar coordinates, we obtain k˜qk2(−s)(Ω) =

Z 0

Z

|e|=1

|ˆq(re)|˜ 2(1 + r2)−sr2dedr

=

Z ω+R

0

Z

|e|=1

|ˆq(re)|˜ 2(1 + r2)−sr2dedr

+ Z T

ω+R

Z

|e|=1

|ˆq(re)|˜ 2(1 + r2)−sr2dedr +

Z T

Z

|e|=1

|ˆq(re)|˜ 2(1 + r2)−sr2dedr

=: I1+ I2+ I3. (3.16)

We begin with bounding I3. Since supp ˜q ⊂ Ω by the H¨older’s inequality |ˆq(ξ)| ≤˜ Ck˜qk(0)(Ω) and so

I3≤ Ck˜qk2(0)(Ω) Z

T

Z

|e|=1

(1 + r2)−sr2dedr

≤ Ck˜qk2(0)(Ω)T−(2s−3) ≤ C(ωm + M )2T−(2s−3). (3.17) Before evaluating I1 and I2 terms, we need the following estimate. Let A = {η; | − η + ξ − x| ≤ |η|/2}. By direct computation, we have

Z hξi−2s

Z

hxi−4s

Z

Ac

h−η + ξ − xi−2s|ˆq(η)|˜ 2

 dxdξ

≤ C Z

hξi−2s Z

hxi−4s

Z

Ac

hηi−2s|ˆq(η)|˜ 2

 dxdξ

≤ Ck˜qk2(−s),

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with C depends on s. On the other hand, by using the fact that A ⊂ {η; 23|ξ − x| ≤ |η| ≤ 2|ξ − x|} and hxi−2shξi−2s≤ 2shx − ξi−2s, one has

Z hξi−2s

Z

hxi−4s

Z

A

h−η + ξ − xi−2s|ˆq(η)|˜ 2

 dxdξ

≤ Z

hξi−2s Z

hxi−4s Z

2

3|ξ−x|≤|η|≤2|ξ−x|

h−η + ξ − xi−2s|ˆq(η)|˜ 2dηdxdξ

≤ C Z

hxi−2s Z Z

2

3|ξ−x|≤|η|≤2|ξ−x|

hx − ξi−2sh−η + ξ − xi−2s|ˆq(η)|˜ 2dηdξdx

≤ C Z

hxi−2s Z Z

2

3|ξ−x|≤|η|≤2|ξ−x|

hηi−2sh−η + ξ − xi−2s|ˆq(η)|˜ 2dηdξdx

≤ C

Z Z Z

hxi−2sh−η + ξ − xi−2sdξdx



|ˆq(η)|˜ 2hηi−2s

≤ Ck˜qk2(−s).

Therefore, we can deduce that Z

hξi−2sQ(ξ)2dξ ≤ Ck˜qk2(−s). (3.18) Next, by using (3.13) and (3.18), we yield

I1

Z ω+R 0

C φ(R, ω)Q(ξ) + (ω2+ R∗2)e

2R2

(1 + r2)−sr2dr

≤ C(φ(R, ω)2k˜qk2(−s)(Ω) + (ω2+ R∗2)2e

2R2)

Z ω+R 0

(1 + r2)−sr2dr

≤ Cφ(R, ω)2k˜qk2(−s)(Ω) + C(ω2+ R∗2)2e

2R∗2, (3.19)

since due to 2s > 3 we have Z ω+R

0

(1 + r2)−sr2dr ≤ Z

0

(1 + r2)−sr2dr = C.

In the same way, the estimate (3.14) implies that I2 ≤ C φ(R, ω)2k˜qk2(−s)(Ω) + (ω2+ T2)2e

2T2

Z T ω+R

(1 + r2)−sr2dr

≤ Cφ(R, ω)2k˜qk2(−s)(Ω) + C(ω2+ T2)2e2

2T2. (3.20)

Combining (3.16)-(3.20) and using that I2 = 0 when T = ω + R, we complete the proof.

 We are now ready to prove Theorem 2.1.

Proof of Theorem 2.1. We will use the estimate (3.15). Obviously, φ(R, ω) = mω + M

R+ ω ≤ m + M R. Therefore, we can choose m < 1/(√

2C0) and R>√

2C0M so that Cφ(R, ω)2< 1

2.

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The choice of m depends only on s, Ω and of R only depends on s, Ω, M . Thus, the first term on the right hand side of (3.15) can be absorbed by the left side, and we obtain

k˜qk2(−s)(Ω) ≤ C(ω2+ T2)2(e2

2R+ χ(T )e2

2T)2+ Cω2T−(2s−3). (3.21)

We consider the following two cases:

(i) ω + R≤ 1

2E and (ii) ω + R≥ 1 2E.

In the case (i) we choose T = E2. We want to show that there exists C2 such that (ω2+ T2)2e2

2T2≤ C2ω2(ω + E )−(2s−3) (3.22) and

ω2T−(2s−3) ≤ C2ω2(ω + E )−(2s−3). (3.23) It is clear that (2.2) is a consequence of (3.21), (3.22), and (3.23).

Due to our choice of T , (3.22) is satisfied if (ω + E )42−

2 ≤ C(ω + E)−2s+3, or

(ω + E )4+2s−32−

2 ≤ C, which follows from the elementary inequality (E )4+2s−32−

2 ≤ C.

Note that (3.23) is equivalent to

C2−1/(2s−3)(ω + E ) ≤ T = E

2. (3.24)

Obviously,

ω + E ≤ ω + R+ E ≤ 3E 2 ,

according to (i). Now we see that (3.24) holds whenever C2−1/(2s−3)≤ 1/3.

Next we consider case (ii). We choose T = ω + R. Then from (3.21) we have k˜qk2H(−s)(Ω) ≤ C(ω4+ R∗4)e2

2R2+ Cω2(ω + R)−(2s−3). Since R< C and in the case (ii) ω + Rω2 +14E, the estimate (2.2) follows.

The proof now is complete.

4. Stability estimates for Maxwell equations

In this section we will prove Theorem 2.2. We first discuss a reduction that transforms the Maxwell equations to the vectorial Schr¨odinger equation by adapting Section 1 in [4] to the special case of ε = µ = 1. We then use arguments similar to the previous section to derive a bound of the difference of the conductivities.

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4.1. Reduction to the Schr¨odinger equation. Let Ω be a bounded domain in R3 with smooth boundary. We consider the time-harmonic Maxwell equations (2.3) with ε = µ = 1,

 ∇ × H + iωγE = 0,

∇ × E − iωH = 0, (4.1)

where γ = 1 + iσ/ω, σ ∈ C2(Ω) and σ > 0 in Ω. From (4.1), one has the following compatibility conditions for E and H

 ∇ · (γE) = 0,

∇ · H = 0. (4.2)

Let α = log γ (the principal branch). Combining (4.1) and (4.2) gives the following eight equations





∇ · E + ∇α · E = 0,

∇ × E − iωH = 0,

∇ · H = 0,

∇ × H + iωγE = 0, which leads to the 8 × 8 system

∗ 0 ∗ D·

∗ 0 ∗ −D×

∗ D· ∗ 0

∗ D× ∗ 0

 +

∗ 0 ∗ Dα·

∗ ωI3 ∗ 0

∗ 0 ∗ 0

∗ 0 ∗ ωγI3

 0 H

0 E

= 0.

Here Ij is the (j × j)-identity matrix,

D· = −i(∂123), D = −i(∂123)t and

D× = −i

0 −∂32

3 0 −∂1

−∂21 0

.

∗ means that we obtain the same equations regardless of values of ∗.

Define the operators P+(D) =

 0 D·

D D×



, P(D) =

 0 D·

D −D×



acting on 4-vectors. It is easy to check that

P+(D)P(D) = P(D)P+(D) = −∆I4 and

P+(D)= P(D), P(D) = P+(D).

Also, we can see that the 8 × 8 operator P =

 0 P(D)

P+(D) 0



is a self-adjoint, i.e., P= P . For ζ ∈ C3, we set P (ζ) = −i

 0 P(ζ) P+(ζ) 0

 .

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Let X = (Φ1, Ht, Φ2, Et)tbe 8-vectors, where Φ1 and Φ2 are additional scalar fields. The augmented Maxwell’s system is

(P + V )X = 0 in Ω, where

V =

ω 0 0 Dα·

0 ωI3 Dα 0

0 0 ωγ 0

0 0 0 ωγI3

 .

Note that when Φj = 0 the solution X corresponds to the solution of the original Maxwell’s system in Ω. Now we want to rescale the augmented system. Let

Y =

 I4 0 0 γ1/2I4

 X, then

(P + V )X =

 γ−1/2I4 0 0 I4



(P + W )Y, where

W = κI8+1 2

0 0 0 Dα·

0 0 Dα D × α

0 0 0 0

0 0 0 0

 with κ = ωγ1/2. Hence

(P + V )X = 0 if and only if (P + W )Y = 0.

The advantage of rescaling is that no first order terms appear in (4.3)-(4.5). The proof of next lemma can be found in [4].

Lemma 4.1.

(P + W )(P − Wt) = −∆I8+ Q, (4.3)

(P − Wt)(P + W ) = −∆I8+ Q(1), (4.4) (P + W)(P − W ) = −∆I8+ Q(2), (4.5) where the matrix potentials are given by

Q = 1 2

∆α

2∇2α − ∆αI3

− κ2I8

1

4(Dα · Dα)I4 2Dκ·

2Dκ 2Dκ·

2Dκ

, (4.6)

Q(1) = −1 2

 ∆α

2∇2α − ∆αI3

− κ2I8

2Dκ×

−2Dκ×

1

4(Dα · Dα)I4

 ,

and

Q(2) = −1 2

 ∆α

2∇2α − ∆αI3

− κ2I8

−2Dκ×

2Dκ×

1

4(Dα · Dα)I4

 .

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Here ∇2f = (∂ij2f ) is the Hessian of f and only non zero elements are shown in 8×8 matrices Q, Q(1), and Q(2). Moreover, Wt denotes the transpose of W and W = Wt.

4.2. Construction of CGO solutions. We recall that Ω ⊂ B. We extend σ to a function in H2s+2(R3), still denoted by σ, so that supp σ in B. Therefore, ω2I8+ Q, ω2I8+ Q(1) and ω2I8+ Q(2) are compactly supported in B.

Let Y = (f1, (u1)t, f2, (u2)t)t∈ H2s(B). We recall the following construction of CGO solutions for Maxwell equations (4.1) in [3].

Proposition 4.2. [3, Proposition 9] Let s > 3/2 be an integer. Assume that a, b, ζ ∈ C3 and ζ · ζ = ω2. Then there exists a constant C0 depending on s, such that if

|ζ| > C0

 X

j=1,2

2+ qjk(2s)(B) + kω2I8+ Qk(2s)(B)

, where

q1 = −κ2, q2= −1

2∆α − κ2− 1

4(Dα · Dα), then there exists a solution

Z = eiζ·x(A + Ψ) of (−∆I8+ Q)Z = 0 with

A = 1

|ζ|

 ζ · a

ωb ζ · b

ωa

 ,

satisfying

kΨk(2s)(Ω) ≤ C0

|ζ||A|kω2I8+ Qk(2s)(B).

Moreover, Y = (P − Wt)Z is a solution of (P + W )Y = 0 and Y = (0, Ht, 0, γ1/2Et)t where E, H solve (4.1).

Proof. The proposition can be proved following [3, Proposition 9]. The only difference is that we use the estimate from Hδ+12s to Hδ2s with −1 < δ < 0, that is,

kGζf kH2s

δ ≤ C(δ)

|ζ| kf kH2s

δ+1, (4.7)

instead of [3, (22)] where the estimate is from L2δ+1 to L2δ, where Gζ is the convolution operator with the fundamental solution of (−∆ − 2iζ · ∇). This estimate (4.7) holds due to the easy fact (−∆ − 2iζ · ∇) and (I − ∆)2s commute. Due to the fact that ω2I8+ Q is compactly supported in B, one can replace the Hδ+12s norm on the right hand side of (4.7) by H2s(B). Recall that H2s(B) is a Banach algebra for s > 3/4. The rest of the proof follows

[3, Proposition 9]. 

We also need CGO solutions for the rescaled system. These solutions can be constructed as in [3, Lemma 10, Proposition 11].

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Proposition 4.3. [3, Lemma 10, Proposition 11] Let a, b, ζ ∈ C3 and ζ · ζ = ω2. Then there exists a constant C0 depending on s, such that if

|ζ| > C02I8+ Q(2)k(2s)(B), then there is a solution

Y = eiζ·x(A+ Ψ) of (P + W)Y = 0, where

A = 1

|ζ|

ζ · a

−ζ × a ζ · b ζ × b

and Ψ= P Ψ+ iP (ζ)Ψ− W A− W Ψ with A = |ζ|1 (0, b∗t, 0, a∗t)t, kΨk(2s)(Ω) ≤ C0

|ζ||A|kω2I8+ Q(2)k(2s)(B) (4.8) and

kP Ψk(2s)(Ω) ≤ C0(|A| + kΨk(2s)(Ω))kω2I8+ Q(2)k(2s)(B) (4.9) Moreover,

k(2s)(Ω) ≤ C0|A|



1 +kσk(2s+2)(B)ω

|ζ|



2I8+ Q(2)k(2s)(B) + kW k(2s)(B) . (4.10) 4.3. Stability estimates. Suppose that ε = µ = 1, kσjk(2s+2)(B) ≤ m < 1 and supp(σ1− σ2) ⊂ Ω. Hence we have supp(γ1− γ2) ⊂ Ω. We first prove an inequality that connects the unknowns and the boundary measurements.

In the proofs C denote generic constants depending on s, Ω.

Theorem 4.4. Assume that σ1 and σ2 belong to H2s+2(Ω) and supp(σ1− σ2) ⊂ Ω. There exists a constant C dependent on Ω, s such that, for any Z1 ∈ H2s(Ω) satisfying Y1 = (P − W1t)Z1= (0, H1t, 0, γ11/2E1t)t with (E1, H1) solution to (4.1) in Ω with coefficient σ1, and any H2s(Ω) solution Y2 = (f1, (u1)t, f2, (u2)t)t of (P + W2)Y2 = 0, one has

|((Q1− Q2)Z1, Y2)| ≤ C dist(C1, C2)kY1k(1)(Ω)kY2k(1)(Ω). (4.11) Note that Qj is defined in (4.6) corresponding to σj for j = 1, 2. The matrix-valued functions W1, W2 are defined similarly.

Proof. Using supp(σ1− σ2) ⊂ Ω, from the first identity in the proof of [3, Proposition 7] one has

((Q1− Q2)Z1, Y2)= (Y1, P Y2)− (P Y1, Y2). (4.12) Moreover, using the estimate of the right hand of (4.12) derived in [3] (see the inequality before [3, Proposition 7]) and the support assumption of σ1− σ2, we can obtain that

|(Y1, P Y2)− (P Y1, Y2)|

≤ C(kν × E1− ν × E2kT H(∂Ω)+ kν × H1− ν × H2kT H(∂Ω))kY2k(1)(Ω)

≤ C dist(C1, C2)kν × E1kT H(∂Ω)kY2k(1)(Ω)

≤ C dist(C1, C2)kY1k(1)(Ω)kY2k(1)(Ω), (4.13) where (ν × E2, ν × H2) ∈ C2. The required estimate follows from (4.12) and (4.13). 

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Let ξ ∈ R3. We select e(j) ∈ R3 satisfying |e(1)| = |e(2)| = 1 and e(j) · ξ = e(1) · e(2) = 0 for j = 1, 2. Let R > 0. We choose

ζ(1) = −1

2ξ + i R2 2

1/2

e(1) +



ω2+R2 2 − |ξ|2

4

1/2

e(2),

ζ(2) = 1

2ξ − i R2 2

1/2

e(1) +



ω2+R2 2 −|ξ|2

4

1/2

e(2), where we assume

ω2+R2 2 ≥ |ξ|2

4 . Then

ζ(1) − ζ(2) = −ξ, ζ(j) · ζ(j) = ω2, |ζ(j)|2= R2+ ω2, for j = 1, 2.

To construct CGO solutions of Proposition 4.2 with Q = Q1 corresponding to σ1, we choose a(1) = 0 and

b(1) = −ie(1)

2 +e(2)

√ 2 . By direct calculation, we can see that

2+ qjk(2s)(B) + kω2I8+ Q1k(2s)(B) ≤ C21k(2s+2)(B)ω + C3, where C2, C3 are independent of ω. If we take

R + ω > 21/2(C0C21k(2s+2)(B)ω + C0C3), (4.14) then

|ζ(1)| ≥ 2−1/2(R + ω) > C02+ qjk(2s)(B) + kω2I8+ Q1k(2s)(B) . By Proposition 4.2, there exist solutions

Z1 = eiζ(1)·x(A(1) + Ψ1) of (−∆I8+ Q1)Z1 = 0 with

A(1) = 1

|ζ(1)|

 0 ωb(1) ζ(1) · b(1)

0

 and

1k(2s)(Ω) ≤ C0

|ζ(1)||A(1)|kω2I8+ Q(1)k(2s)(B).

Likewise, if

R + ω > 21/2(C0C22k(2s+2)(B)ω + C0C3), (4.15) then one has

|ζ(2)| > C0k(ω2I8+ Q(2)2k(2s)(B),

where Q(2)2 is the matrix-valued function Q(2) corresponding to σ = σ2. By choosing a = 0 and

b = b(2) = ie(1)

√2 +e(2)

√2

in Proposition 4.3 with Q(2) corresponding to σ2, there exist solutions Y2= eiζ(2)·x(A(2) + Ψ2)

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of (P + W2)Y2= 0 with

A(2) = 1

|ζ(2)|

 0 0 ζ(2) · b(2) ζ(2) × b(2)

and Ψ2 = P Ψ2∗+ iP (ζ(2))Ψ2∗− W2A2∗− W2Ψ2∗, where A2∗ = |ζ(2)|−1(0, b(2), 0, 0)t and Ψ2∗ satisfies (4.8), (4.9). Moreover,

2k(2s)(Ω) ≤ C0

 1

|ζ(2)| +kσ2k(2s+2)(B)ω

|ζ(2)|2



2I8+ Q(2)2k(2s)(B) + C0

 1

|ζ(2)| +kσ2k(2s+2)(B)ω

|ζ(2)|2



kW2k(2s)(B) (4.16) (see (4.10)).

For Y2, we can obtain the following estimate.

Lemma 4.5. Let Y2 be the CGO solution of (P + W2)Y2 = 0 as above. Then for ω ≥ 1 there exists a constant C depending on s, Ω such that

kY2k(1)(Ω) ≤ C (1 + |ζ(2)|) e2−1/2R. (4.17) Proof. Since ∇(eiζ(2)·xA(2)) = iζ(2)eiζ(2)·xA(2) and |A(2)| ≤ 21/2, one gets

keiζ(2)·xA(2)k(1)(Ω) ≤ C(e2−1/2R|A(2)| + |ζ(2)|e2−1/2R|A(2)|) ≤ C(1 + |ζ(2)|)e2−1/2R. By direct calculation,

2I8+ Q(2)2k(2s)(B) ≤ Cω, kW2k(2s)(B) ≤ Cω,

where we used the assumption kσ2k(2s+2)(Ω) ≤ m < 1. Thus, from (4.16) we have kΨ2k(1)(Ω) ≤ C|ζ(2)|ω which implies that

keiζ(2)·xΨ2k(1)(Ω) ≤ keiζ(2)·xΨ2k(0)(Ω) + kiζ(2)eiζ(2)·xΨ2k(0)(Ω) + keiζ(2)·x∇Ψ2k(0)(Ω)

≤ C(1 + |ζ(2)|) ω

|ζ(2)|e2−1/2R≤ C(1 + |ζ(2)|)e2−1/2R.

The proof is complete. 

Similarly, we can prove that

Lemma 4.6. Let Z1 be the CGO solution of (−∆I8+ Q1)Z1 = 0 and Y1 = (P − W1t)Z1 as above. Then for ω ≥ 1 there exists a constant C depending on s, Ω such that

kY1k(1)(Ω) ≤ C (1 + |ζ(1)|)2e2−1/2R. (4.18) Proof. Substituting Z1= eiζ(1)·x(A(1) + Ψ1) gives

Y1 = (P − W1t)Z1= (P − W1t)(eiζ(1)·x(A(1) + Ψ1)).

We evaluate

k(P − W1t)(eiζ(1)·xA(1))k(1)(Ω)

≤ C|ζ(1)|keiζ(1)·xA(1)k(1)(Ω) + CkW1teiζ(1)·xA(1)k(1)(Ω)

≤ C(1 + |ζ(1)|)2e2−1/2R

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and

k(P − W1t)(eiζ(1)·xΨ1)k(1)(Ω)

≤ C|ζ(1)|keiζ(1)·xΨ1k(1)(Ω) + CkW1teiζ(1)·xΨ1k(1)(Ω) + Ckeiζ(1)·xP Ψ1k(1)(Ω)

≤ C(1 + |ζ(1)|)2e2−1/2R,

which completes the proof. 

We now choose m < 1 satisfying 21/2C0C2m < 1 and R > 21/2C0C3, i.e., (4.14) and (4.15) hold. Combining (4.11), (4.17) and (4.18) yields that

|((Q1− Q2)Z1, Y2)| ≤ C dist(C1, C2)kY1k(1)(Ω)kY2k(1)(Ω)

≤ C(1 + |ζ(2)|)3e21/2Rdist(C1, C2). (4.19) Lemma 4.7. Let s > 3/2 and ξ = re with r ≥ 0, |e| = 1. Under the assumptions of Theorem 4.4, there exist a constant C depending on Ω only and a constant C depending on s, Ω such that

|(ˆσ1− ˆσ2)(re)| ≤ C+ Cm

β(ξ)2 G(∇2σ)(ξ) +˜ Cβ(ξ) + Cm(ω + β(ξ))

β(ξ)2 G(∇˜σ)(ξ) + Cω−1β(ξ) + Cm(ω2+ ωβ(ξ))

β(ξ)2 G(˜σ)(ξ) + C(ω2+ R2)7/2

ωβ(ξ)2 e21/2Rdist(C1, C2), (4.20) where we denote

β(ξ) = R +



2+ R2− |ξ|2 2

1/2

,

˜

σ = σ1− σ2, and

G(f )(ξ)2 = Z

hxi−4s Z

h−η + ξ − xi−2s| ˆf (η)|2dηdx.

Proof. Using Z1= eiζ(1)·x(A(1) + Ψ1), Y2 = eiζ(2)·x(A(2) + Ψ2) and the definition of Q1, Q2, we have

((Q1− Q2)Z1, Y2)= Z

e−iξ·x(A(2) + Ψ2)t(Q1− Q2)(A(1) + Ψ1)dx

= Z

e−iξ·xA(2)t(Q1− Q2)A(1)dx + Z

e−iξ·xΨ2t(Q1− Q2)A(1)dx +

Z

e−iξ·x(A(2) + Ψ2)t(Q1− Q21dx

=: I1+ I2+ I3. (4.21)

We will evaluate each term in (4.21). We begin with I1. Since A(1) = |ζ(1)|−1(0, ωb(1), ζ(1)·

b(1), 0)t and A(2) = |ζ(2)|−1(0, 0, ζ(2) · b(2), ζ(2) × b(2))t, we deduce that I1 = (ζ(1) · b(1))(ζ(2) · b(2))

|ζ(1)||ζ(2)|

Z

e−iξ·x22− κ21)dx

 +

2ω ζ(2) · b(2)

|ζ(1)||ζ(2)|

Z

e−iξ·xD(κ2− κ1) · b(1)dx



. (4.22)

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To bound I2, we recall that Ψ2 = P Ψ2∗+ iP (ζ(2))Ψ2∗− W2A2∗− W2Ψ2∗, then I2 =

Z

e−iξ·x(P Ψ2∗)t(Q1− Q2)A(1)dx + Z

e−iξ·x(iP (ζ(2))Ψ2∗)t(Q1− Q2)A(1)dx +

Z

e−iξ·x(−W2A2∗)t(Q1− Q2)A(1)dx + Z

e−iξ·x(−W2Ψ2∗)t(Q1− Q2)A(1)dx

=: J1+ J2+ J3+ J4.

We now bound Jk , k = 1, 2, 3, 4. Applying (4.8), (4.9) and using that m < 1, ω ≤ |ζ(j)|, we can deduce that

2∗k(2s)(Ω) ≤ C mω

|ζ(2)|2 , kP Ψ2∗k(2s)(Ω) ≤ C mω

|ζ(2)|. (4.23)

By an argument similar to Lemma 3.4, we have

|J1| ≤ CQ(ξ)kP Ψ2∗k(2s)(Ω)

≤ C mω

|ζ(2)|Q(ξ), where

Q(ξ) =

 1

|ζ(1)|G(∇2σ)(ξ) +˜ ω + ζ(1) · b(1)

|ζ(1)| G(∇˜σ)(ξ) + ω2+ ω(ζ(1) · b(1))

|ζ(1)| G(˜σ)(ξ)

 . Using the bound kP (ζ(2))Ψ2∗k(2s)(Ω) ≤ Cmω|ζ(2)|−1 and following a similar argument in Lemma 3.4, we can derive

|J2| ≤ C mω

|ζ(2)|Q(ξ).

For J4, it follows from the definition of W2 that kW2k(2s)(Ω) ≤ Cω, then applying (4.23) and the proof of Lemma 3.4, we yield

|J4| ≤ C mω

|ζ(2)|Q(ξ).

Finally, we would like to estimate J3. Note that

W2A2∗= 1

|ζ(2)|

κ2I8+ 1 2

0 0 0 Dα2· 0 0 Dα2 D × α2

0 0 0 0

0 0 0 0

 0 b(2)

0 0

= ωγ21/2

|ζ(2)|

 0 b(2)

0 0

 .

(4.24) In view of the definition of Qj, direct calculations show that

(Q1− Q2)A(1)

= ω

2|ζ(1)|

0

2(∇21− α2))b(1) − (∆(α1− α2))b(1) 0

0

−iω(σ1− σ2)

|ζ(1)|

 0 ωb(1) ζ(1) · b(1)

0

− 1

|ζ(1)|

0

1

4ω(Dα1· Dα1− Dα2· Dα2)b(1) + 2(ζ(1) · b(1))D(κ1− κ2) 2ωb(1) · D(κ1− κ2)

0

. (4.25)

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