On uniqueness in the inverse conductivity problem with local data

On uniqueness in the inverse conductivity problem with local data Victor Isakov June 21, 2006 1 Introduction The inverse condictivity problem with many boundary measurements consists of recovery of conductivity coefficient a (principal part) of an elliptic equation in a domain R n, n = 2, 3 from the Neumann data given for all Dirichlet data (Dirichlet-to-Neumann map). Calderon [5] proposed the idea of using complex exponential solutions to demonstrate uniqueness in the linearized inverse condictivity problem. Complex exponential solutions of elliptic equations have been introduced by Faddeev [7] for needs of inverse scattering theory. Sylvester and Uhlmann in their fundamental paper [19] attracted ideas from geometrical optics, constructed almost complex exponential solutions for the Schrödinger operator, and proved global uniqueness of a ( and of potential c in the Schrödinger equation) in the three-dimensional case. In the two-dimensional case the inverse conductivity problem is less overdetermined, and the Sylvester and Uhlmann method is not applicable, but one enjoys advantages of the methods of inverse scattering and of theory of complex variables. Using these methods Nachman [17] demonstrated uniqueness of a C 2 ( ) and Astala and Päivärinta [1] showed uniqueness of a L () which is a final result in the inverse conductivity problem in R 2 with many measurements from the whole boundary. There is a known hypothesis (see for example, [11], Problem 5.3, [14], [20]) that the Dirichlet-to-Neumann map given at any (nonvoid open) part Γ of the boundary also uniquely determines conductivity coefficient or potential in the Schrödinger equation. This local boundary measurements model important 1

applications, for example to geophysics or to semiconductors when collecting data from the whole boundary is either not possible or extremely expensive. Despite extended long term efforts this hypothesis remains not proven, altough there is some progress. Kohn and Vogelius [15] showed uniqueness of the boundary reconstruction (of all existing partial derivatives of a) and hence uniqueness of piecewise analytic a. When coefficients are known in a neighborhood of the boundary, then Runge type approximation argument reduces the partial Dirichlet-to-Neumann map to complete map [9], [11], Exercise 5.7.4, [15], and hence the hypothesis follows. In smooth case Bukhgeim and Uhlmann [4] made use of Carleman estimates (with linear phase function) to show that the Neumann data on a sufficiently large part Γ of the boundary given for all Dirichlet data on the whole boundary uniquely determine potential c in the three-dimensional Schrödinger equation. The most advanced and recent result is due to Kenig, Sjöstrand and Uhlmann [14]. They modified the scheme of [19], [4] by using quadratic phase function and demonstrated uniqueness of c from Neumann data on Γ for all Dirichlet data on a complementary part Γ 1. While Γ in [14] can be arbitrarely small one can not assume zero Dirichlet data on \ Γ (although they can be zero on \ Γ 1 ) and they need Γ and Γ 1 to have nonvoid intersection. Only result concering zero boundary data on part Γ 0 of the boundary is due to Hähner [8] who by explicit calculations proved completeness of products of harmonic functions which are zero on a spherical Γ 0. An inverse scattering in half-space was considered by Karamyan [13]. In this paper we give a complete proof of this hypothesis when the Dirichlet-to-Neumann map is given on arbitrary part Γ of while on the remaining part Γ 0 one has homogeneous Dirichlet or Neumann data. Our restrictive assumption is that Γ 0 is a part of a plane or of a sphere. In some applications this assumption is natural, but available uniqueness results [6], [16] require that coefficients of the differential equation are known near. An exception is the paper [13] where one is given scattering data in half-space. This assumption enables to reflect almost complex exponential solutions across Γ 0 and to avoid use of a special fundamental solution (Green s function) for the Schrödinger equation and of corresponding exponential weighted estimates with a large parameter in. Currently, such fundamental solution and estimates are available only when homogeneous boundary data are given at a part of the boundary (but not at the whole boundary) [4], [14]. It is not likely that such fundamenatl solutions and 2

estimates can be found when homogeneous Dirichlet (or Neumann) are prescribed at the whole boundary. We add new ingredients to the Sylvester- Uhlmann method. A crucial observation is that contributions of products of almost complex exponential solutions and of their reflections converge to zero when large parameter τ goes to. 2 Main results Let be a domain in R 3 with Lipschitz boundary. We consider the Schrödinger equation u + cu = 0 in (2.1) with the Dirichlet boundary data or the Neumann data u = g 0 on (2.2) ν u = g 1 on. (2.3) Let B 0 be some ball. We will assume that the (complex valued) potential c L (), c = k 2 on \ B 0. Let Γ 0 be an open bounded part of and Γ = \ Γ 0. We define the local Dirichlet-to-Neumann map Λ c (D, Γ) as Λ c (D, Γ)g 0 = ν u on Γ, g 0 H 1 2 ( ), g0 = 0 on Γ 0 and the local Neumann-to-Dirichlet map as Λ c (N, Γ)g 1 = u on Γ, g 1 H 1 2 ( ), g1 = 0 on Γ 0 (2.4) provided the Dirichlet or Neumann problems are uniquely solvable. In Theorems 2.1 and 2.2 we consider two cases and a) is a bounded subdomain of {x : x 3 < 0}, Γ = {x 3 < 0}; b) 0 < k, = {x : x 3 < 0} and Γ \ B. (2.5) It is well known that in case (2.5), a), the boundary value problems (2.1), (2.2) or (2.3) have unique solutions u H 1 () for any boundary data g 0 H 1 2 ( ), g 1 H 1 2 ( ) provided there is uniqueness of a solution. 3

Uniqueness is guaranteed by maximum principles or energy integrals when Ic = 0, 0 c on or when Ic 0 on a nonempty open subset of. In case (2.5), b), we will assume that Ic 0 on and we will augment the equation and the boundary condition by the Sommerfeld radiation condition lim r(σ u iku)(x) = 0, σ = r 1 x, as r = x. (2.6) By using integral equations or the Lax-Phillips method one can demonstrate uniqueness and existence of a solution u H 1 ( B) for any ball B to the scattering boundary value problem (2.1), (2.2) or (2.3), and (2.6) with compactly supported g 0 H 1 2 ( ), g 1 H 1 2 ( ) [11], [16]. Theorem 2.1 If or Λ c1 (D, Γ) = Λ c2 (D, Γ) (2.7) Λ c1 (N, Γ) = Λ c2 (N, Γ), (2.8) then c 1 = c 2 This result has immediate corollary for the conductivity equation div(a u) k 2 u = 0 in. (2.9) We will assume that a C 2 ( ), a = 1 on \ B 0, a > 0 on, and 0 < k in case (2.5), b). As for the Schrödinger equation, in case (2.5), a), the elliptic boundary value problems (2.9), (2.2) or (2.3) are uniquely solvable in H 1 () for all k except discrete set of eigenvalues. When k = 0, the Dirichlet problem is uniquely solvable. In case (2.5), b) the boundary value scattering scattering problems (2.9), (2.2) or (2.3), (2.6) are uniquely solvable in the same functional spaces as for the Schrödinger equation. We will assume the unique solvability condition and we define local Dirichlet-to-Neumann and Neumann-to-Dirichlet maps for the conductivity equation similarly to the Schrödinger equation and we will denote them by Λ(a; D, Γ), Λ(a; N, Γ). Theorem 2.2 If or Λ(a 1 ; D, Γ) = Λ(a 2 ; D, Γ) (2.10) Λ(a 1 ; N, Γ) = Λ(a 2 ; N, Γ) (2.11) 4

with the additional assumption then a 1 = a 2 on. 3 a 1 = 3 a 2 = 0 on Γ 0, (2.12) These results imply similar results for bounded domains when Γ 0 is a part of a sphere. In Theorems 2.3, 2.4 we use the following notation and assumptions. Let be a subdomain of B 0. Let Γ 0 = B 0 and Γ = \ Γ 0. We will assume that Γ 0 B 0. Theorem 2.3 The equality (2.7) implies that c 1 = c 2 Theorem 2.4 The equality (2.10) implies that a 1 = a 2 Finally we give available results the plane case and assume that is a bounded simply connected domain in R 2 with Lipschitz boundary. According to established theory of elliptic boundary value problems if a L (), k = 0, then the Dirichlet problem (2.9), (2.2) has a unique solution u H 1 () for any g 0 H 1 2 ( ), and the Neumann problem (2.9), (2.3) has a unique solution in the same space with normalization condition u = 0 provided g 1 = 0. We remind that the (conormal) derivative a ν u H 1 2 ( ) is defined as (a ν u)ϕ = a u ϕ, ϕ H 1 2 ( ), (2.13) where the integral on the left side is understood as dual pairing between H 1 2 ( ) and H 1 2 ( ). After these reminders we can define the partial Dirichlet-to-Neumann and Neumann-to-Dirichlet maps as above. Theorem 2.5 Let a 1, a 2 L (). Let Γ be any nonvoid open arc of. Then equalities (2.10) and (2.11) imply that a 1 = a 2 in. Theorem 2.5 was proven by Astala, Lassas, and Päivärinta [2], Theorem 2.3. Theorems 2.1-2.5 can be immediately generalized to semilinear Schrödinder equations and quasilinear conductivity equations as in [12], [18]. 5

3 Proofs for half-space In this section we consider the case of {x : x 3 < 0} when unobservable part Γ 0 of the boundary is contained in the plane {x 3 = 0}. We give proofs for Neumann boundary condition, because of its applied importance, and we will indicate how to modify them for Dirichlet condition. We start with a standard orthogonality relation. Lemma 3.1 Under condition (2.8) (c 1 c 2 )v 1 v 2 = 0 (3.14) for all functions v 1, v 2 H 1 ( B), for any ball B, satisfying v 1 + c 1 v 1 = 0 in, ν v 1 = 0 on Γ 0 (3.15) and v 2 + c 2 v 2 = 0 in, ν v 2 = 0 on Γ 0. (3.16) Proof: First we consider case a) of bounded domain. Let v 1 be any solution to (3.15). Let u 2 be the solution to the Schrödinger equation with c = c 2 and with the Neumann data ν u 2 = ν v 1 on. Subtracting the equations u 2 + c 2 u 2 = 0 and v 1 + c 1 v 1 = 0 and letting v = u 2 v 1 we yield v + c 2 v = (c 1 c 2 )v 1 on. (3.17) We have v = 0 on Γ by condition (2.8) and (2.4) and ν v = 0 (by definition of v) on. Multiplying equation (3.17) by solution v 2 to (3.16) and integrating by parts we yield (c 1 c 2 )v 1 v 2 = ( v + c 2 v)v 2 = v( v 2 + c 2 v 2 ) = 0 where we used the boundary conditions ν v = 0 = ν v 2 on Γ 0 and the equality v = ν v = 0 on Γ. Summing up we have the orthogonality relation (3.14). 6

The case of = {x 3 < 0} needs in addition a Runge type approximation argument Let u 1 be any solution to (3.15). Let u 2 be the solution to the Schrödinger equation with c = c 2 and with the Neumann data ν u 2 = ν u 1 on. Subtracting the equations u 2 +c 2 u 2 = 0 and u 1 +c 1 u 1 = 0 and letting v = u 2 u 1 we yield v + c 2 v = (c 1 c 2 )u 1 on. (3.18) Since v = 0 by condition (2.8) and ν v = 0 (by definition of v) on Γ we have v = 0 on \ B 0 due to uniqueness in the Cauchy problem for the Laplace equation. Multiplying equation (3.18) by v 2 and integrating by parts we yield (c 1 c 2 )u 1 v 2 = B 0 ( v + c 2 v)v 2 = B 0 v( v 2 + c 2 v 2 ) = 0 B 0 where we used the boundary conditions ν v = 0 = ν v 2 on Γ 0 and the equality v = 0 on \ B 0. Summing up we have the orthogonality relation (3.14) for v 1 = u 1. To complete the proof we will L 2 ( B 0 )-approximate arbitrary v 1 by u 1. Let us assume the opposite: the subspace {u 1 } is not dense in {v 1 }. Then by Hahn-Banach theorem there is f L 2 (), f = 0 outside B 0, such that for all u 1, but Let u 1(f) solve the Neumann problem From (3.19) and (3.21) we have fu 1 = 0 (3.19) fv 1 0 for some v 1. (3.20) u 1 + c 1 u 1 = f in, ν u 1 = 0 on. (3.21) 0 = ( u 1 + c 1 u 1)u 1 = Γ u 1 ν u 1 where we used the Green s formula and boundary conditions for u 1, u 1, Since ν u 1 can be arbitrary smooth function on Γ we conclude that u 1 is zero on Γ. Now u 1 solves the elliptic equation u 1 = 0 on \ B 0 and has zero 7

Cauchy data on Γ, so by uniqueness in the Cauchy problem u 1 = 0 on \B 0. Applying again the Green s formula in B 0 we yield which contradicts (3.20). The proof is complete. Lemma 3.2 Under condition (2.7) fv 1 = ( u 1 + c 1 u 1)v 1 = 0 B 0 (c 1 c 2 )v 1 v 2 = 0 for all functions v 1, v 2 H 1 ( B), for any ball B, satisfying v 1 + c 1 v 1 = 0 in, v 1 = 0 on Γ 0 and v 2 + c 2 v 2 = 0 in, v 2 = 0 on Γ 0 Proof is similar to the proof of Lemma 3.1. Proof of Theorem 2.1 First we remind known results about existence of special almost complex exponential solutions to the Schrödinger equation in R 3. Let ξ = (ξ 1, ξ 2, ξ 3 ), ξ = (ξ 1, ξ 2, ξ 3 ). We introduce e(1) = (ξ 2 1 + ξ 2 2) 1 2 (ξ1, ξ 2, 0), e(3) = (0, 0, 1), and the unit vector e(2) to get orthonormal basis e(1), e(2), e(3) in R 3. We denote the coordinates of x in this basis by (x 1e, x 2e, x 3e ) e. Observe that ξ = (ξ 1e, 0, ξ 3 ) e, ξ 1e = (ξ1 2 + ξ2) 2 1 2 and that in general x y = x 1 y 1 + x 2 y 2 + x 3 y 3 = x 1e y 1e + x 2e y 2e + x 3e y 3e. 8

We define ζ(1) = ( ξ 1e 2 τξ 3, i ξ ( 1 4 + τ 2 ) 1 2, ξ 3 2 + τξ 1e) e, ζ (1) = ( ξ 1e 2 τξ 3, i ξ ( 1 4 + τ 2 ) 1 2, ξ 3 2 τξ 1e) e, ζ(2) = ( ξ 1e 2 + τξ 3, i ξ ( 1 4 + τ 2 ) 1 2, ξ 3 2 τξ 1e) e, ζ (2) = ( ξ 1e 2 + τξ 3, i ξ ( 1 4 + τ 2 ) 1 ξ 3 2, 2 + τξ 1e) e, (3.22) where τ is a positive real number. By direct calculations we can see that ζ(1) ζ(1) = ζ (1) ζ (1) = ζ(2) ζ(2) = ζ (2) ζ (2) = 0. (3.23) Let us extend c 1, c 2 onto R 3 as even functions of x 3. Since (3.23) holds, it is known [11], section 5.3, [19], that there are almost exponential solutions to the equations with e iζ(1) x (1 + w 1 ), e iζ(2) x (1 + w 2 ) u 1 + c 1 u 1 = 0, u 2 + c 2 u 2 = 0 in R 3 (3.24) w 1 2 (B 0 ) + w 2 2 (B 0 ) 0 as τ, (3.25) where 2 (B) is the standard norm in L 2 (B). We define f (x 1, x 2, x 3 ) = f(x 1, x 2, x 3 ) and we let u 1 (x) = e iζ(1) x (1 + w 1 ) + e iζ (1) x (1 + w 1), u 2 (x) = e iζ(2) x (1 + w 2 ) + e iζ (2) x (1 + w 2). (3.26) It is obvious that u 1, u 2 H 2 ( B) for any B, solve the partial differential equations (3.24) and that ν u 1 = ν u 2 = 0 on Γ 0. (3.27) Let c = c 1 c 2. (3.28) 9

By (3.24), (3.27), (3.28), and Lemma 3.1 0 = cu 1 u 2 = c(x)(e i(ζ(1)+ζ(2)) x (1 + w 1 (x))(1 + w 2 (x))+ e i(ζ (1)+ζ(2)) x (1 + w 1(x))(1 + w 2 (x))+ e i(ζ(1)+ζ (2)) x (1 + w 1 (x))(1 + w 2(x))+ e i(ζ (1)+ζ (2)) x (1 + w 1(x))(1 + w 2(x)))dx due to (3.26). Using (3.22) we conclude that c(x)(e iξ x (1 + w 1 (x))(1 + w 2 (x))+ e i(ξ 1ex 1e 2τξ 1e x 3 ) (1 + w 1(x))(1 + w 2 (x))+ e i(ξ 1ex 1e +2τξ 1e x 3 ) (1 + w 1 (x))(1 + w 2(x))+ e iξ x (1 + w 1(x))(1 + w 2(x)))dx = 0. (3.29) Now we let τ. Observe that moduli of all exponents are bounded by 1. So due to (3.25) limits of all terms containing factors w j, wj are zero. By the Riemann-Lebesgue Lemma limits of c(x)e i(ξ 1ex 1e 2τξ 1e x 3 ), c(x)e i(ξ 1ex 1 +2τξ 1e x 3 ) dx as τ are also zero provided ξ 1e 0. Therefore from (3.29) we derive that c(x)(e iξ x + e iξ x )dx = 0 (3.30) for any ξ, ξ 1e 0. Since c j and hence c (given by (3.28) )are compactly supported, the right side in (3.30) is analytic with respect to ξ, so we have (3.30) for all ξ R 3. Since c is an even function of x 3, c(x)(e iξ x + e iξ x )dx = c(x)e iξ x dx. R 3 Hence from (3.30) R 3 c(x)e iξ x dx = 0 10

for any ξ R 3. By uniqueness of the inverse Fourier transformation c = 0, and hence c = 0 and c 1 = c 2. This completes the proof under condition (2.8). Now we will show how to adjust it to the case of Dirichlet boundary conditions. The argument until (3.25) is the same. Then we let u 1 (x) = e iζ(1) x (1 + w 1 (x)) e iζ (1) x (1 + w 1(x)), u 2 (x) = e iζ(2) x (1 + w 2 (x)) e iζ (2) x (1 + w 2(x)). (3.31) It is obvious that u 1, u 2 H 2 ( B 0 ), solve the partial differential equations (3.24) and that u 1 = u 2 = 0 on Γ 0. (3.32) By (3.24), (3.32) and Lemma 3.2 0 = cu 1 u 2 = c(x)(e i(ζ(1)+ζ(2)) x (1 + w 1 (x))(1 + w 2 (x)) e i(ζ (1)+ζ(2)) x (1 + w 1(x))(1 + w 2 (x)) e i(ζ(1)+ζ (2)) x (1 + w 1 (x))(1 + w 2(x))+ e i(ζ (1)+ζ (2)) x (1 + w 1(x))(1 + w 2(x)))dx due to (3.31). Using (3.22) we conclude that c(x)(e iξ x (1 + w 1 (x))(1 + w 2 (x)) e i(ξ 1ex 1e 2τξ 1e x 3 ) (1 + w 1(x))(1 + w 2 (x)) e i(ξ 1ex 1e +2τξ 1e x 3 ) (1 + w 1 (x))(1 + w 2(x))+ e iξ x (1 + w 1(x))(1 + w 2(x)))dx = 0. As above we let τ and repeating the argument after (3.29) we conclude that 0 = c(x)(e iξ x + e iξ x )dx = c(x)e iξ x dx. R 3 11

for any ξ R 3. By uniqueness of the inverse Fourier transformation c = 0, and hence c 1 = c 2. The proof is complete. Proof of Theorem 2.2 The well known substitution u = a 1 2 v (3.33) transforms the conductivity equation (2.9) into the Schrödinger equation (2.1) with c = a 1 1 2 a 2. (3.34) From (3.33) it follows that homogeneous boundary Dirichlet condition for u on Γ 0 implies the same condition for v and since 3 a = 0 on Γ 0 due to (2.12) the same is true for the Neumann condition. As known ([11]), section 5.2, ([15]) the local Dirichlet-to-Neumann (Neumann-to-Dirichlet) map on Γ uniquely determines a, 3 a on Γ. Hence again (3.33) implies that partial Dirichlet-to-Neumann map Λ(a; D, Γ) for the conductivity equation uniquely determines partial Dirichlet-to-Neumann map Λ c (D, Γ) for the Schrödinger equations. The same holds for the Neumann-to-Dirichlet maps. By Theorem 2.1 the potential c given by (3.34) is unique. (3.28) can be viewed as an linear elliptic equation a 1 2 + ca 1 2 = 0 in with respect to a 1 2. As mentioned Λ(a; D, Γ) or Λ(a; D, Γ) uniquely determine c on and a, 3 a on Γ. So due to uniqueness in the Cauchy problem for elliptic equations ([11], section 3.3) a is uniquely determined on as well. The proof is complete. We observe that Theorem 2.2 holds for complex valid a with Ra > 0 which are of importance in applications [11], p.7. The above proof is valid with the substitution (3.33) where a 1 is the principal branch which is well 2 defined on the half-plane Ra > 0. 4 Proofs for subsets of balls: use of the Kelvin transform We remind the definition of the Kelvin transform of a function in R 3. Let X(x) = x 2 x, x(x) = X 2 X (4.35) 12

and It is known [10] that U(X) = X 1 u(x(x)) (4.36) X U(X) = X 5 x u(x(x)) (4.37) Proof of Theorem 2.3 We can assume that B is the ball of radius 1 2 centered at x 0 = (0, 0, 1 2 ) and that the origin in R 3 does not belong to. Let us apply the Kelvin transform to the equation (2.1). Using (4.37) and (4.36) we yield X U + CU = 0, where C(X) = X 6 c(x(x)) The inversion (4.35) transforms the sphere { x x 0 = 1 } into the plane 2 {X 3 = 1}. Hence the domain in X-variables is a subdomain of the halfspace {X : 1 < X 3 }, and parts Γ 0, Γ of its boundary are correspondingly parts of the plane {X 3 = 1} and of the halfspace {1 < X 3 }. Due to (4.36) homogeneous Dirichlet data on Γ 0 are transformed into homogeneous Dirichlet data in new variables. Obviously, the Dirichlet-to-Neumann map in x-variables uniquely determines the Dirichlet-to-Neumann map on Γ in new variables. Applying Theorem 2.1 to the inverse problem in X variables we conclude that C(X) = X 6 c(x(x)) is uniquely determined. Hence c is uniquely determined on. The proof is complete. 5 Conclusion The main remaining open question is of course how obtain uniqueness from local Dirichlet-to-Neumann map when Γ 0 is an arbitrary surface. When this map is given at all k uniqueness follows from the results on inverse hyperbolic problems obtained by the methods of boundary control [3]. In particular, it is important to relax topological assumptions on and to consider case of several connected components of, especially when Γ 0 is an inner connected component of the boundary. For spherical Γ 0 one can most likely to use methods of this paper. In case of half-space by more careful study of bahavior at inifinity one can relax the assumtpion that c is compactly supported. We formulated results assuming that Dirichlet or Neumann problems are uniquely 13

solvable only to follow traditions. Arguments in the three-dimensional case will not change if we drop these assumptions and instead require equalities of Cauchy pairs (u, a ν u) on Γ (as in [20], section 3), under homogeneous Dirichlet or Neumann conditions on Γ 0. It is very interesting and probably more difficult to recover simulteneously a, Γ 0 and coefficient b of the boundary condition a ν u + bu = 0 on Γ 0. Again such results are available for hyperbolic equations. It is still not clear if it is possible to construct semilocalized (in weighted spaces) almost complex exponential solutions to the Schrödinger equation and to use them in uniqueness proofs and constructive methods of solution of the inverse problem with local data or with unknown multiple inclusions. Localization would most likely remove all geometrical assumptions and to obtain most general results. Due to substantial overdeterminancy of this inverse problem it is feasible that a completely different approach will work and resolve remaining questions. Aknowledgement: This research was in part supported by the NSF grant DMS 04-05976. References [1] Astala, K., Päivärinta, L. Calderon s inverse conductivity problem in the plane. Ann. Math.., 163 (2006), 265-299. [2] Astala, K., Lassa, M., Päivärinta, L. Calderon s inverse conductivity problem for anisotropic conductivity in the plane. Comm. Part. Diff. Equat., 30 (2005), 207-224. [3] Belishev, M. Boundary Control in Reconstruction of Manifolds and Metrics (the BC-method). Inverse Problems, 13 (1997), R1-R45. [4] Bukhgeim, A.L., Uhlmann, G. Recovering a Potential from Partial Cauchy Data. Comm. Part. Diff. Equat., 27 (2002), 653-668. 14

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[19] Sylvester, J., Uhlmann, G. Global uniqueness theorem for an inverse boundary value problem. Ann. Math., 125 (1987), 153-169. [20] Uhlmann, G. Developments in inverse problems since Calderon s fundamental paper. Harmonic Analysis and PDE, Univ. of Chicago Press. Victor Isakov Department of Mathematics and Statistics Wichita State University Wichita, KS 67260-0033, U.S.A. e-mail: victor.isakov@wichita.edu 16