A CALDERÓN PROBLEM WITH FREQUENCY-DIFFERENTIAL DATA IN DISPERSIVE MEDIA. has a unique solution. The Dirichlet-to-Neumann map

A CALDERÓN PROBLEM WITH FREQUENCY-DIFFERENTIAL DATA IN DISPERSIVE MEDIA SUNGWHAN KIM AND ALEXANDRU TAMASAN ABSTRACT. We consider the problem of identifying a complex valued coefficient γ(x, ω) in the conductivity equation γ(, ω) u(, ω) = 0 from knowledge of the frequency differentials of the Dirichlet-to-Neumann map. For a frequency analytic γ(, ω) = k=0 (σ k + iϵ k )ω k, in three dimensions and higher, we show that d j dω j Λ γ(,ω) ω=0 for j = 0, 1,..., N recovers σ 0,...σ N and ϵ 1,,,, ϵ N. This problem arises in frequency differential electrical impedance tomography of dispersive media. 1. INTRODUCTION Let R n, n 2, be a bounded domain with Lipschitz boundary. For some ω 0 > 0 and c > 0, let γ C([0, ω 0 ]; L ()) be complex valued satisfying (1) Re(γ(x, ω)) c > 0, (x, ω) [0, ω 0 ]. For each fixed ω and f H 1/2 () the Dirichlet problem (2) γ(, ω) u(, ω) = 0 in, u(, ω) = f, has a unique solution. The Dirichlet-to-Neumann map is a bounded operator defined by Λ γ(,ω) : H 1/2 () H 1/2 () (3) Λ γ(,ω) (f) := n γ(, ω) u(, ω), where n is the outer normal to the boundary. Moreover the map ω Λ γ(,ω) C([0, ω 0 ]; L(H 1 2 (), H 1 2 ())), where L(H 1 2 (), H 1 2 ()) denotes the space of linear operators endowed with the strong topology. The Calderón problem refers to the determination of γ(, ω) from Λ γ(,ω), for each fixed frequency ω. At ω = 0 this is the original question in 2000 Mathematics Subject Classification. 35R30, 35J65, 65N21. Key words and phrases. Calderón problem, frequency differential electrical impedance tomography, complex geometrical optics. This author was supported in part by the NSF Grant DMS 1312883. 1

2 SUNGWHAN KIM AND ALEXANDRU TAMASAN [6], which has been mostly settled in the affirmative: In dimensions n 3, of direct relevance here, we mention the breakthrough result in [22], where Λ γ(,0) is shown to uniquely determine γ(, 0), and the reconstruction method in [16]. Although not stated, these results extend to ω 0, case in which the coefficient γ(, ω) becomes complex valued. The analogous two dimensional problem (also at ω = 0) was cracked in [17], refined in [3] (with reconstruction in [14]) and completely settled in [1]. The breakthrough result in [5] settles in two dimensions the uniqueness part: Λ γ(,ω) uniquely determines γ(, ω) for each fixed ω (an earlier result [9] assumed γ = σ 0 (x) + iωϵ 0 (x) with sufficiently small ωϵ 0 ). For an understanding on the breadth of development of the Calderón problem we refer to the reviews [7], [4], [10], [2], and [23]. Recent biomedical advances [11, 12, 13, 18, 19] allow to measure discrepancies in boundary data at different frequencies, and produce images based on such differential data. Given that biological materials have frequency dependent electric-magnetic properties (they are dispersive) [8] and only one image is produced from multiple frequencies, the following questions arise: What can be determined from the frequency derivative d k dω Λ k γ(,ω) ω=ω0 for a fixed order k 0, or for several orders? Can one expect quantitative information from multi-frequency data? What do the images represent? We address these questions when the coefficient γ(, ω) depends analytically in ω in dimensions three or higher. In [21] the authors considered the case k = 1 and the ansatz γ = σ 0 (x) + iωϵ 1 (x) specific to media with frequency-independent electrical properties (i.e., non-dispersive). For some r, c > 0, let us define the class A r,c of functions γ(x, ω) such that (4) Re(γ(x, ω)) = σ n (x)ω n, Im(γ(x, ω)) = ϵ n (x)ω n, n=0 where σ 0 C 1,1 () is real valued and constant near the boundary with (5) 0 < c 1 σ 0 (x) c, and σ k, ϵ k C 1,1 0 () are real valued and compactly supported in with σ k r, ϵ k (6) r, k 1. σ 0 Note that ω γ(, ω) is analytic in ω < 1/r with values in L (), We prove the following: σ 0 n=1

A CALDERÓN PROBLEM FOR DISPERSIVE MEDIA 3 Theorem 1.1. Let n 3 and R n be open with C 1,1 boundary. Assume that γ A r,c, for some r, c > 0 and N 1 be fixed. Then dk ω=0 Λ dω k γ(,ω) for k = 0,..., N, uniquely determine σ 0, σ 1,..., σ N, and ϵ 1,..., ϵ N inside. Consequently, we also obtain: Corollary 1.1. Let n 3 and R n be open with C 1,1 boundary. Assume γ A r,c, for some r, c > 0. Then the map ω Λ γ(,ω) given in an arbitrarily small neighborhood of some fixed ω 0 [0, (1 + 2r) 1 ) recovers γ on [0, r 1 ). The proof of Theorem 1.1 is based on the complex geometrical optic solutions of Sylvester-Uhlmann in [22], whose existence is recalled below to establish notations. For k, η, l R n with k η = k l = k η = 0, and η 2 = k 2 + l 2, consider the vectors 4 ( ) ( ) k k (7) ξ 1 (η, k, l) := η i 2 + l, ξ 2 (η, k, l) := η i 2 l, ( which satisfy ξ 1 ξ 1 = ξ 2 ξ 2 = 0, ξ 1 2 = ξ 2 2 k = 2 2 + l ), 2 and 2 ξ 1 + ξ 2 = ik. The result is stated in the equivalent form: Existence of CGOs (Theorem 2.3, [22]): Let n 3, and σ 0 C 1,1 (R n ) be constant outside. For 1 < δ < 0 there are two constants R, C > 0 dependent only on δ, σ 0 / σ 0 L (), and such that the following holds: For ξ j C n as in (7) with l > R, there exist w j := w(, ξ j ) Hloc 1 (Rn ) solutions in R n of of the form σ 0 w j = 0 (8) w j (x) = e x ξ j σ 1/2 0 (1 + ψ j (x)), with (9) ψ j L 2 δ C, ξ j j = 1, 2, where f 2 := f(x) 2 (1 + x 2 ) δ dx. L 2 δ R n The assumptions on the coefficients may be relaxed as follows. The boundary values σ 0 and σ 0 can be recovered from Λ γ(,0) as shown in [16] for this regularity (and earlier in [15] and [22] for C -regular). Then σ 0 can be extended outside with preserved regularity while making it constant near the boundary of a neighborhood of the original domain. Similarly, the compact support assumption on σ n, ϵ n, for n = 1, 2,... can σ be replaced by the knowledge of the boundary values of σ n,ϵ n, n, and ϵ n. With the same proof as in [17], the Dirichlet-to-Neumann map

4 SUNGWHAN KIM AND ALEXANDRU TAMASAN Λ γ(,ω) transfers from to and we can consider the problem on the larger domain. 2. A FREQUENCY RANGE FOR ANALYTIC DEPENDENCE In this section we show analytic dependence of the map ω Λ γ(,ω) in the space of linear operators from H 1/2 () to H 1/2 () endowed with the strong topology, and introduce two operators needed in the proof of Theorem 1.1. Existence and uniqueness of solution for the conductivity problem (2) with a complex coefficient has been known for some time; e.g., in [20] the Lax-Milgram lemma is employed. Below we provide a solution which is frequency-analytic. Since the solution of (2) is unique, the result below also shows the analytic dependence of the solution of (2) for a frequencyanalytic coefficient. Theorem 2.1. Let R n, n 2, be a domain with Lipschitz boundary. Consider the Dirichlet problem (2) for f H 1/2 () real valued and (10) γ(x, ω) := σ 0 (x) + (σ k (x) + iϵ k (x))ω n, k=1 where σ 0 is real valued satisfying (5) for some c > 0, and σ k, ϵ k L (), k 1, are real valued satisfying (6) for some r > 0. Then, for (11) ω < 1 1 + 2r, the Dirichlet problem (2) has a unique solution u(, ω) H 1 () with the series representation (12) Re{u(x, ω)} = v k (x)ω k, Im{u(x, ω)} = h k (x)ω k k=0 absolutely convergent in H 1 (), where h 0 0, v 0 solves (13) σ 0 v 0 = 0 in, v 0 = f, and, for k = 1, 2,..., the following recurrence holds: (14) (15) σ 0 v k = σ 0 h k = k=0 (σ j v k j ϵ j h k j ) in, v k = 0, (σ j h k j + ϵ j v k j ) in, h k = 0.

A CALDERÓN PROBLEM FOR DISPERSIVE MEDIA 5 We stress here that the recurrence (14) and (15) uses the reality of the Dirichlet boundary data f. The proof of the theorem is a consequence of the estimate in Lemma 2.1 below. We start by a simple remark, which can be justified by induction. Remark 2.1. Let {a k } k=0 be a sequence of non-negatives and m > 0 such that a 1 a 0 and Then a k a 0 + m(a 1 +... + a k 1 ), k 2. a k (1 + m) k a 0, k 0. We denote by the L 2 ()-norm, and by 1 the H 1 ()-norm. Lemma 2.1. Let v 0 satisfy (13) and assume the sequences {v n } n=1, {h n } n=1 in H 1 () satisfy the corresponding problems (14) and (15). Then, for k = 1, 2,... we have (16) v k 1 + h k 1 c(1 + 2r) k v 0. Proof. Let k be a fixed positive index. Since v k = 0, by applying the divergence theorem to (14) we estimate σ 0 v k 2 = (σ j v k j ϵ j h k j ) v k dx r r σ 0 v k σ 0 v k j v k + σ 0 h k j v k dx ( σ 0 v k j + σ 0 h k j ), where the first inequality uses (6). Thus σ 0 v k r ( σ 0 v k j + σ 0 h k j ). Similarly, using h k = 0 and the divergence theorem in (15) we obtain σ 0 h k r ( σ 0 v k j + σ 0 h k j ). Now apply Remark 2.1 for a k = σ 0 v k + σ 0 h k and m = 2r to obtain

6 SUNGWHAN KIM AND ALEXANDRU TAMASAN (17) σ 0 v k + σ 0 h k (1 + 2r) k σ 0 v 0. Since v k, h k H 1 0() the estimate (16) follows from the lower and upper bounds on σ 0 in (5). Proof. [of Theorem 2.1] We seek the solution in the ansatz (18) u(x, ω) := v 0 (x) + (v n (x) + ih n (x))ω n n=1 and then the recurrence in (14) and (15) is obtained by identifying the same order coefficients in a formal series. It is in here that the reality of the Dirichlet boundary data f (hence of v 0 ) is used. We construct the sequences {v n }, {h n } inductively by solving the Dirichlet problems (14) and (15). Part of the inductive step, the right hand hand sides (in divergence form) of the equations in (14) and (15) belong to H 1 (). Thus the recurrence is well defined to construct v 0 H 1 () and the two sequences {v n } n=1 and {h n } n=1 in H 1 0(). The estimate (16) and ω < 1/(1 + 2r) shows (19) u 1 c v 0 1 n=0 (1 + 2r) k ω k <. Therefore u(, ω) defined in (18) is in H 1 (). A direct calculation involving (14) and (15) shows that it solves the Dirichlet problem (2) for γ(x, ω) := σ 0 (x) + (σ k (x) + iϵ k (x))ω n. k=1 As a consequence of the theorem above we can explicit the real and imaginary part of the operators dk dω k Λ γ(,ω) ω=0 (f) for a real valued f as follows. Corollary 2.1. Let γ(x, ω) be as in the Theorem 2.1 and f H 1/2 () be real valued. Then, with h 0 0, v 0 solution of (2), and v k and h k, k = 1, 2,... given by the recurrences (14) and (15), we obtain (20) (21) { } 1 d k k! Re dω Λ k γ(,ω) (f) = ω=0 { } 1 d k k! Im dω Λ k γ(,ω) (f) = ω=0 ( σ j v k j ( σ j h k j ϵ j + ϵ j h k j ), ) v k j.

A CALDERÓN PROBLEM FOR DISPERSIVE MEDIA 7 Proof. For ω < (1 + 2r) 1 < r 1 the series in (10) defining γ is absolutely convergent in L (), and the series (18) defining u is absolutely convergent in H 1 (). Therefore the two series below u(, ω) γ(, ω) = k=0 + i ω k k=0 ω k ( σ j v k j ϵ j ( σ j h k j ) h k j + ϵ j ) v k j are absolutely convergent in H 1/2 (). The formulas (20) and (21) now follow by differentiation in ω. The proof of Theorem 1.1 requires applications of dk dω k Λ γ(,ω) ω=0 to traces of the complex (valued) geometrical optics solutions. However, if the Dirichlet data in (13) is complex valued, the formulas (20) and (21) no longer hold for v k and h k obtained by the recurrence (14) and (15). In order to preserve this compatibility we extend the real and imaginary part of d k dω k Λ γ(,ω) ω=0 by complex linearity: For k 0 and real valued f, g H 1/2 () we define R (k), I (k) : H 1/2 () H 1/2 () by (22) R (k) (f + ig) := 1 { } d k k! Re Λ γ(,ω) dω k (f) ω=0 (23) I (k) (f + ig) := 1 { } d k k! Im Λ γ(,ω) dω k (f) ω=0 + i k! Re { d k Λ γ(,ω) dω k + i k! Im { d k Λ γ(,ω) dω k } (g), ω=0 } (g). ω=0 Proposition 2.1. Let R (k), I (k) : H 1/2 () H 1/2 () be as defined in (22) and (23). Then ( ) R (k) v k j (f + ig) = σ j ϵ h k j (24) j (25) I (k) (f + ig) = where h 0 0, v 0 solves σ 0 v 0 = 0, ( σ j h k j + ϵ j v 0 = f + ig ) v k j, and h k and v k are defined inductively by the recurrence (14) and (15) for k 1.

8 SUNGWHAN KIM AND ALEXANDRU TAMASAN Proof. The identities hold when g 0 by Corollary 2.1. The left hand sides of (24) and (25) are complex linear by definition. But the right hand side is also complex linear since the sequences{h k } and {v k } produced by the recurrence (14) and (15) are complex linear with the Dirichlet data. 3. PROOF OF THEOREM 1.1 AND ITS COROLLARY In this section we assume the hypotheses in Theorem 1.1. The proof is inductive and relies on the basic identities in the lemma below which relate the quantities (26) (27) Q r N := ( σ N σ N ln σ 0 ) σ 0, Q i N := ( ϵ N ϵ N ln σ 0 ) σ 0, for N 1, with the boundary operator R (N) in (22), respectively I (N) in (23), and some interior data previously determined in the inductive step. In the following lemma σ j, ϵ j for j = 1,..., N, need not be compactly supported in, and σ 0 need not be constant near the boundary, but rather its normal derivative be vanishing at the boundary. Lemma 3.1. Let w 1, w 2 H 1 () be σ 0 -harmonic. Then ( σ 1 σ 1 ln σ 0 )w 1 w 2 dx = 2 R (1) [w 1 ]w 2 ds (w 1 w 2 ) (σ 1 σ 1 (28) w 1w 2 )ds, and, for N 2, ( σ N σ N ln σ 0 )w 1 w 2 dx = 2 R (N) (w 1 w 2 ) [w 1 ]w 2 ds (σ N σ N w 1w 2 )ds N 1 (29) 2 (σ j v N j ϵ j h N j ) w 2 dx. Also (30) ( ϵ 1 ϵ 1 ln σ 0 )w 1 w 2 dx = 2 I (1) [w 1 ]w 2 ds (w 1 w 2 ) (ϵ 1 ϵ 1 w 1w 2 )ds,

A CALDERÓN PROBLEM FOR DISPERSIVE MEDIA 9 and, for N 2, ( ϵ N ϵ N ln σ 0 )w 1 w 2 dx = 2 I (N) [w 1 ]w 2 ds ( (w 1 w 2 ) ϵ N ϵ ) N w 1w 2 ds N 1 (31) 2 (σ j h N j + ϵ j v N j ) w 2 dx. Proof. From Proposition 2.1, (32) R (N) [w 1 ] = N ( σ j v N j ϵ j ) h N j where v 0 = w 1, h 0 = ϵ 0 = 0, and v j, h j, j = 1,..., N are defined recursively in (14) and (15). Upon multiplication of (32) by w 2 and integration by parts we obtain (33) N ( R (N) v N j [w 1 ]w 2 ds = σ j ϵ h N j j ( N ) = w 2 (σ j v N j ϵ j h N j ) dx + = = N + N 1 (σ j v N j ϵ j h N j ) w 2 dx σ 0 v N w 2 dx + σ N w 1 w 2 dx N 1 ) w 2 ds (σ j v N j ϵ j h N j ) w 2 dx (σ j v N j dx ϵ j h N j ) w 2 dx + σ N w 1 w 2 dx, where in the third equality we use the recurrence (14), and for the last equality we use the facts that w 2 is σ 0 -harmonic and v N H0(). 1 Note that for N = 1 the identity (33) reduces to (34) R (1) [w 1 ]w 2 ds = σ 1 w 1 w 2 dx.

10 SUNGWHAN KIM AND ALEXANDRU TAMASAN (35) Since σ 0 C 1,1 (), w j also solves w j = ln σ 0 w j, j = 1, 2. By using the general identity 2 w 1 w 2 = (w 1 w 2 ) w 1 w 2 w 2 w 1, and (35) we get 2 σ N w 1 w 2 dx = σ N ( (w 1 w 2 ) w 1 w 2 w 2 w 1 )dx = σ N ( (w 1 w 2 ) + ln σ 0 (w 1 w 2 ))dx ( (w 1 w 2 ) = σ N σ ) N w 1w 2 ds + ( σ N )w 1 w 2 dx + σ N ln σ 0 (w 1 w 2 )dx ( (w 1 w 2 ) = σ N σ ) N w ln σ 0 1w 2 ds + σ N w 1w 2 ds + ( σ N σ N ln σ 0 )w 1 w 2 dx ( (w 1 w 2 ) = σ N σ ) N w 1w 2 ds (36) + ( σ N σ N ln σ 0 )w 1 w 2 dx, where, for the last equality, the vanishing of the second boundary integral accounts for σ 0 being constant near the boundary. The use of (34), respectively (36), in (33) proves the identity (28), respectively (29). The identity (31) follows similarly: An integration by parts and use of (15) yield (37) I (N) [w 1 ]w 2 ds = N 1 (σ j h N j + ϵ j v N j ) w 2 dx + and the calculation in (36) with ϵ N replacing σ N shows ( (w 1 w 2 ) 2 ϵ N w 1 w 2 dx = ϵ N ϵ ) N w 1w 2 ds (38) + ( ϵ N ϵ N ln σ 0 )w 1 w 2 dx,. By using (38) in (37) we obtain (31). ϵ N w 1 w 2 dx,

A CALDERÓN PROBLEM FOR DISPERSIVE MEDIA 11 We prove Theorem 1.1 by induction on the order of derivative in frequency. We will employ the basic identities in the lemma above with w 1, w 2 being the CGO solutions in (8). Recall the dependence of w 1 and w 2 on the vectors η, k, l, which we do not made explicit to keep the formulas readable. Case N = 0: Assume that Λ γ(,0) is known, then we appeal to [16, Lemmae 2.7 and 2.12 b)] to conclude that Λ γ(,0) recovers the traces w j for j = 1, 2, and to [16, Theorem 5.1] to recover coefficient σ 0 in. Case N = 1: Assume that R (1) and I (1) are known. By using the CGO solutions w 1, w 2 in (30), and accounting for the compact support of σ 1, ϵ 1, we get (39) e ix k Q r 1(x)(1 + ψ 1 (x))(1 + ψ 2 (x))dx = R n R (1) [w 1 ]w 2 ds. Due to the decay estimate (9), the left hand side in (39) has a limit with l, thus so does the right hand side, and the Fourier transform F[Q r 1] is determined. The Fourier inversion yields [ Q r 1 = F 1 lim l ] R (1) [w 1 ]w 2 ds, Starting with (30) and proceeding similarly with the boundary data I (1), we find by Fourier inversion the L ()-function [ Q i 1 = F 1 lim l ] I (1) [w 1 ]w 2 ds. d In the case N = 1 we note that dω ω=0 Λ γ(,ω) uniquely determines the quantities Q r 1 and Q i 1, independent of knowledge of the DtN map Λ γ(,0). However, since σ 0 is known, we can also recover σ 1 and ϵ 1 as the unique solutions of the respective Dirichlet problems: (40) (41) σ 1 σ 1 ln σ 0 σ 1 ln σ 0 = σ 0 Q r 1 in, σ 1 = 0, ϵ 1 ϵ 1 ln σ 0 ϵ 1 ln σ 0 = σ 0 Q i 1 in, ϵ 1 = 0. The inductive step: Assume that σ 0, and σ j, ϵ j, j = 1,..., N 1 have been determined, and R (N), I (N) are known. Below we reconstruct σ N and ϵ N. By solving the recurrence (14) and (15) starting with v 0 = w 1, we can determine v j, h j H0() 1 for j = 1,..., N 1 (recall v 0 = w 1 and h 0 = 0).

12 SUNGWHAN KIM AND ALEXANDRU TAMASAN Since σ N is assumed compactly supported in, the identity (29) yields e ix k Q r N(x)(1 + ψ 1 (x))(1 + ψ 2 (x))dx R n N 1 (42) = 2 R (N) [w 1 ]w 2 ds 2 (σ j v N j ϵ j h N j ) w 2 dx. Note that the right hand side of (42) is known. Due to the decay estimate (9) the left hand side of (42) has a limit with l, thus so does the right hand side, and the Fourier transform F[Q r N ] is recovered from { F [Q r N] (k) = lim R (N) [w 1 ]w 2 ds l (43) 2 N 1 (σ j v N j ϵ j h N j ) w 2 dx By Fourier inversion in (42), we recovered Q r N L 0 (). By a similar reasoning starting from (31) with w 1, w 2 being the CGO solutions in (8), we determine Q i N L 0 (). Finally, the coefficients σ N and ϵ N, are recovered as the unique solution to the respective Dirichlet problems: (44) (45) σ N σ N ln σ 0 σ N ln σ 0 = σ 0 Q r N in, σ N = 0, ϵ N ϵ N ln σ 0 ϵ N ln σ 0 = σ 0 Q i N in, ϵ N = 0. This finishes the proof of Theorem 1.1. For the proof of the corollary, we note that the analyticity of the map ω Λ γ(,ω) allows for each derivative dk dω k Λ γ(,ω) at ω = 0 to be determined from the derivatives dj dω j Λ γ(,ω) ω0, j 0, taken at any other frequency ω 0 within the domain of analyticity [0, (1 + 2r) 1 ). 4. ACKNOWLEDGMENT The authors would like to thank the anonymous referees for the constructive criticism, which uncover a gap in a previous version. REFERENCES [1] K. Astala and L. Päivärinta, Calderón s inverse conductivity problem in the plane, Ann. of Math. 163(2006), 265 299 [2] R.H. Bayford, Bioimpedance Tomography (Electrical impedance tomography), Annu. Rev. Biomed. Eng., 8(2006), 63 91. [3] R. M. Brown and G. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions. Comm. Partial Differential Equations 22 (1997), no. 5-6, 10091027. }.

A CALDERÓN PROBLEM FOR DISPERSIVE MEDIA 13 [4] L. Borcea, EIT Electrical impedance tomography, Inverse Problems, 18(2002), R99 R136. [5] A.L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inv. Ill-posed Problems 16(2008), 19 33 [6] A.P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasiliera de Matematica, Rio de Janeiro, 1980 [7] M. Cheney, D. Isaacson, and J. C. Newell, Electrical impedance tomography, SIAM Review, 41(1999), 85 101. [8] K.R. Foster and H.P. Schwan, Dielectric properties of tissues and biological materials, a critical review, Crit. Rev. Biomed. Eng. 17 (1989), no. 1, 25 104. [9] E. Francini, Recovering a complex coefficient in a planar domain from the Dirichletto-Neumann map, Inverse Problems 16 107, doi:10.1088/0266-5611/16/1/309 [10] D. Holder, Electrical Impedance Tomography: Methods, History and Applications (Bristol, UK: IOP Publishing), 2005. [11] S. Kim, J. Lee, J. K. Seo, E. J. Woo and H. Zribi, Multifrequency trans-admittance scanner: mathematical framework and feasibility, SIAM J. Appl. Math., 69(2008), no. 1, 22-36. [12] S. Kim, J.K. Seo, and T. Ha, A nondestructive evaluation method for concrete voids: frequency differential electrical impedance scanning,siam J. Appl. Math., 69(2009), no. 6, 1759-1771. [13] S. Kim, E.J. Lee, E.J. Woo, and J.K. Seo, Asymptotic analysis of the membrane structure to sensitivity of frequency-difference electrical impedance tomography, Inverse problems, 28(2012), 075004 (17pp). [14] K. Knudsen and A. Tamasan, Reconstruction of less regular conductivities in the plane, Comm. Partial Differential Equations 29 (2004), no. 3-4, 361381. [15] R.V. Kohn and M. Vogelius, Determining conductivity by boundary measurements, Commun. Pure Appl. Math., 37(1984), 113-123. [16] A. Nachman, Reconstructions from boundary measurements, Ann. Math., 128(1988), 531-576. - [17] A. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. Math. 143(1996), 71 96 [18] J.K. Seo, J. Lee, S.W. Kim, H. Zribi, and E.J. Woo, Frequency-difference electrical impedance tomography (fdeit): algorithm development and feasibility study, Inverse Problems, 29(2008), 929-944. [19] J.K. Seo, B. Harrach, and E.J. Woo, Recent progress on frequency difference electrical impedance tomography,esaim: PROCEEDINGS, 26(2009), 150-161. [20] E. Somersalo, M. Cheney, and D. Isaacson, Existence and Uniqueness for an Electrode Model for ECCT, SIAM J. App. Math. 52(1992), 1023 1040. [21] K. Sungwhan and A. Tamasan, On a Calderón problem in frequency differential electrical impedance tomography, SIAM J. Math. Anal. 45 (2013), no. 5, 2700-2709. [22] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. Math., 125(1987), 153-169. [23] G. Uhlmann, Electrical impedance tomography and Calderón s problem, Inverse Problems, 25(2009), 1-39.

14 SUNGWHAN KIM AND ALEXANDRU TAMASAN DIVISION OF LIBERAL ARTS, HANBAT NATIONAL UNIVERSITY, KOREA E-mail address: sungwhan@hanbat.ac.kr DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CENTRAL FLORIDA, ORLANDO, FLORIDA 32816 E-mail address: tamasan@math.ucf.edu