GLOBAL UNIQUENESS IN THE IMPEDANCE IMAGING PROBLEM FOR LESS REGULAR CONDUCTIVITIES^{∗}

RUSSELL BROWN DEPARTMENT OF MATHEMATICS UNIVERSITY OF KENTUCKY LEXINGTON, KY 40506-0027 U.S.A.

                                   RBROWN@MS.UKY.EDU

Abstract. If L_{γ} = div γ∇ is an elliptic operator with scalar coefficient γ, we show that we can recover the coefficient γ from the Dirichlet to Neumann map under the assumption that γ has only 3/2 + ϵ derivatives. Previously, the best result required γ to have two derivatives.

Key words. Inverse problem, Dirichlet to Neumann map, Impedance imaging, Besov space

AMS subject classifications. 35R30

Let Ω ⊂ R^{n}, n ≥ 3, be a bounded open set and let L_{γ} = div γ∇ be an elliptic operator on Ω with scalar coefficient γ. We let Λ_{γ} denote the Dirichlet to Neumann map Λ_{γ}f = γ∂u/∂ν where u is the solution to the Dirichlet problem L_{γ}u = 0 in Ω, u = f on ∂Ω. In 1987, Sylvester and Uhlmann [10] showed that if we restrict attention to γ which are sufficiently smooth, then the map γ → Λ_{γ} is injective. Nachman, Sylvester and Uhlmann [8], showed that injectivity continues to hold if γ has two bounded derivatives. Extensions to slightly less smooth conductivities or the related Schrödinger equation are given in Chanillo [3] and Ramm [9]. Isakov [5] has established injectivity for conductivities with jump discontinuities.

Since the only smoothness assumption needed to define Λ_{γ} is that γ be measurable, it is reasonable to ask if the map γ → Λ_{γ} is injective under less restrictive

hypotheses on γ. In this paper, we show that γ need have only \frac{3}{2} +ϵ derivatives. There is no reason to believe that the result in this paper is optimal. I conjecture that the right smoothness assumption is that γ have one derivative. However, the methods presented here do not give this. To state our main result, we recall the standard space of Hölder continuous functions C^{α}(Ω̄) = {f : f : Ω → R and |f(x) − f(y)| ≤ M|x − y|^{α} for some M > 0}.

Theorem 0.1. Let Ω ⊂ R^{n}, n ≥ 3 be a bounded, Lipschitz domain. Then the map γ → Λ_{γ} is injective on the set {γ : γ > 0 in Ω̄, ∇γ ∈ ∪_{ϵ>0} C^{1/2+ϵ}(Ω̄)}.

The outline of our argument is the same as in [10]. We construct special solutions of L_{γ}u = 0 by studying a Schrödinger operator Δ − q. The innovation here is that we consider potentials q which lie in a Besov space of negative order.

We begin by recalling the Besov spaces and some of their simple properties. We will use the monograph of Bergh and Löfstrom [2] as our reference for these spaces. For s ∈ R and 1 ≤ p, q ≤ ∞, we let B_{p,q}^{s} denote the Besov space of distributions. Roughly speaking, a distribution in B_{p,q}^{s} has s derivatives in L^{p}. We recall that if 0 < s < 1, 1 ≤ p, q < ∞, then f ∈ B_{p,q}^{s} if and only if

∥f∥_{L^{p}} + (∫_{R^{n}} (∫_{R^{n}} |f(x + h) − f(x)|^{p} dx)^{q/p} |h|^{−n−sq} dh)^{1/q} (1)

∗ Research supported by NSF grant DMS 9305753 and by the NSF and the Commonwealth of Kentucky through NSF-EPsCOR.

is finite. Furthermore, the expression in (1) gives a norm on B_{p,q}^{s}. When p = q = ∞, the limiting version of (1) is

∥f∥_{L^{∞}} + sup_{x∈R^{n}, |h|≠0} |h|^{−s}|f(x + h) − f(x)|.

This provides a norm for B_{∞,∞}^{s} and thus, for 0 < s < 1, B_{∞,∞}^{s} = C^{s}(R^{n}). We also consider a scale of weighted Besov spaces B_{p,q}^{s,δ} defined for δ ∈ R by

B_{p,q}^{s,δ} = {f : (1 + |x|^{2})^{δ/2}f ∈ B_{p,q}^{s}}

with the norm

∥f∥_{B_{p,q}^{s,δ}} = ∥(1 + |x|^{2})^{δ/2}f∥_{B_{p,q}^{s}}.

We will use B_{p,q}^{s,c} to denote the distributions in B_{p,q}^{s} which are compactly supported and

B_{p,q}^{s,loc} = {f : ψf ∈ B_{p,q}^{s} for each ψ ∈ C_{0}^{∞}(R^{n})}.

We recall that B_{2,2}^{0} is the usual Lebesgue space L^{2} on R^{n}. If follows that

B_{2,2}^{0,δ} = {f : (1 + |x|^{2})^{δ/2}f ∈ L^{2}}

is the weighted Lebesgue space L_{δ}^{2} used by Sylvester and Uhlmann. We also have that B_{2,2}^{1} is the Sobolev space of functions having one derivative in L^{2} and that

B_{2,2}^{1,δ} = {f : f, ∇f ∈ L_{δ}^{2}}. (2)

Next, we note that since B_{2,2}^{s} and B_{2,2}^{s,δ} are isomorphic, we may identify the complex interpolation spaces

[B_{2,2}^{s_{0},δ}B_{2,2}^{s_{1},δ}]_{θ} = B_{2,2}^{s_{θ},δ}, 0 < θ < 1

where s_{0}, s_{1} ∈ R, s_{θ} = (1 − θ)s_{0} + θs_{1} (see [2, Theorem 6.4.5]).

The reason for introducing the Besov spaces to be able to define products of (certain) distributions as bilinear maps between Besov spaces. This depends on the following elementary result regarding multiplication in Besov spaces.

Proposition 0.2. a) If ∥ϕ∥_{∞} + ∥∇ϕ∥_{∞} ≤ M, then for 0 < s < 1, δ ∈ R, 1 ≤ p, q ≤ ∞

∥ψu∥_{B_{p,q}^{s,δ}} ≤ C∥u∥_{B_{p,q}^{s,δ}}M

where C = C(n, p, q).

b) For 0 < s < 1,

∥uv∥_{B_{1,2}^{s}} ≤ C∥u∥_{B_{2,2}^{s}}∥v∥_{B_{2,2}^{s}}

where C = C(s, n).

We do not prove this Proposition, but note that each result follows easily from the norm for B_{p,q}^{s} given in (1). I thank Mike Frazier for telling me of part b) of the above Proposition.

Next, we give estimates for the operator G_{ζ} which is the solution operator to the equation

Δu + 2ζ · ∇u = f

where ζ ∈ C^{n}.

We observe that G_{ζ} defined by

G_{ζ}f = (\frac{f̂}{−|ξ|^{2} + 2iζ · ξ} )^{∨} (3)

maps from S to S′. Here we are using the Fourier transform defined by f̂(ξ) = ∫_{R^{n}} e^{−ix·ξ} dx. In [10], it is shown that if ζ ·ζ = 0, then G_{ζ} : L_{δ+1}^{2} → L_{δ}^{2}, −1 < δ < 0, with the bound

∥G_{ζ}f∥_{L_{δ}^{2}} ≤ \frac{C}{|ζ|}∥f∥_{L_{δ+1}^{2}}, |ζ| > 1 and − 1 < δ < 0. (4)

We give a simple extension of this result to obtain mapping properties of G_{ζ} on B_{2,2}^{s,δ}. Shortly before this paper was written, A. Nachman established related estimates for the operator G_{ζ} in two dimensions [7, Lemma 1.3].

Theorem 0.3. Let ζ ∈ C^{n} satisfy ζ · ζ = 0 and |ζ| > 1. Then for −1 < δ < 0 and 0 ≤ s ≤ 1/2, the map G_{ζ} defined by (3) satisfies

∥G_{ζ}f∥_{B_{2,2}^{s,δ}} ≤ \frac{C}{|ζ|^{1−2s}} ∥f∥_{B_{2,2}^{−s,δ+1}}.

where C = C(n, s, δ).

Proof. We choose a function ϕ satisfying ϕ = 1 on {ξ : |ξ| ≤ 4|ζ|}, supp ϕ ⊂ {ξ : |ξ| < 8|ζ|} and |∇ϕ| ≤ C/|ζ|. For u ∈ L_{δ}^{2}, we define

T u = ∇[(ϕû)^{∨}].

We claim that

∥T u∥_{L_{δ}^{2}} ≤ C|ζ| ∥u∥_{L_{δ}^{2}}, −1 ≤ δ ≤ 1 (5)

When δ = 0, this is elementary since T is a multiplier operator whose symbol is bounded by C|ζ|. To obtain (5) when δ = 1, note that

∥û∥_{L^{2}} + ∥∇û∥_{L^{2}}

gives an equivalent norm on the weighted Lebesgue space L_{1}^{2}. Now

∇T̂ u = iξϕ∇û + û∇(iξϕ)

and hence

∥∇T̂ u∥_{L^{2}} ≤ C(|ζ| + 1) ∥u∥_{L_{1}^{2}}.

If we recall that |ζ| ≥ 1, then (5) follows for δ = 1. The estimate (5) follows by duality when δ = −1 and for the remaining values of δ, by interpolation.

Next, define on operator S by

\begin{aligned}(Sf)^{∧}(ξ) & = \frac{iξ(1 − ϕ)f̂}{(−|ξ|^{2} + 2iξ · ζ)} \\ & = \frac{1}{|ξ|}ψf̂ \end{aligned}

where ψ(ξ) = iξ|ξ|(1 − ϕ)/(−|ξ|^{2} + 2iξ · ζ). The argument used to treat T shows that

f → (ψf̂)^{∨}

is bounded on L_{δ}^{2}, −1 ≤ δ ≤ 1, and the norm of this operator is bounded for |ζ| ≥ 1.

The fractional integral f → (|ξ|^{−1}f̂)^{∨} maps L_{δ+1}^{2} to L_{δ}^{2}, −1 < δ < 0, by the argument in Lemma 3.1, [10]. This gives

∥Sf∥_{L_{δ}^{2}} ≤ C(n, δ)∥f∥_{L_{δ+1}^{2}}, −1 < δ < 0.

Summarizing, we have ∇G_{ζ}f = T (G_{ζ}f) + Sf and hence

\begin{aligned}∥∇G_{ζ}f∥_{L_{δ}^{2}} & ≤ C|ζ| ∥G_{ζ}f∥_{L_{δ}^{2}} + C∥f∥_{L_{δ+1}^{2}} \\ & ≤ C∥f∥_{L_{δ+1}^{2}} \end{aligned}

where the second inequality is (4).

Combining this with (5) and the characterization of B_{2,2}^{1,δ} in (2) gives

∥G_{ζ}f∥_{B_{2,2}^{1,δ}} ≤ C∥f∥_{B_{2,2}^{0,δ+1}}, −1 < δ < 0.

By duality, we have

∥G_{ζ}f∥_{B_{2,2}^{0,δ}} ≤ C∥f∥_{B_{2,2}^{−1,δ+1}}, −1 < δ < 0.

Interpolating between these estimates and (4) gives

∥G_{ζ}f∥_{B_{2,2}^{s,δ}} ≤ \frac{C}{|ζ|^{1−s}} ∥f∥_{B_{2,2}^{0,δ+1}} (6)

and

∥G_{ζ}f∥_{B_{2,2}^{0,δ}} ≤ \frac{C}{|ζ|^{1−s}} ∥f∥_{B_{2,2}^{−s,δ+1}} (7)

where each inequality holds for 0 ≤ s ≤ 1 and −1 < δ < 0. Finally interpolating between (6) and (7) gives the estimate of the Theorem.

If g is a function on R^{n} satisfying λ^{−1} < g < λ (8) for some λ > 0 and ∇g is bounded and compactly supported, then for u ∈ C^{∞}(R^{n}), we may define a distribution m_{q}(u) by

m_{q}(u)(v) = − ∫_{R^{n}} ∇g · ∇ (\frac{1}{g}uv) dx. (9)

Formally, q = g^{−1}Δg will be the potential in our Schrödinger operator and m_{q}(u) is the product qu. Our main result on m_{q} is:

Theorem 0.4. Suppose that g is defined on R^{n}, satisfies (8) and for some s, 0 < s < 1, and M > 0, satisfies

∥∇g∥_{B_{∞,2}^{1−s}} ≤ M (10)

supp ∇g ⊂ {x : |x| < M}. (11)

Then there exists C = C(M, λ, s) so that the map m_{q} satisfies

∥m_{q}(u)∥_{B_{2,2}^{−s,δ+1}} ≤ C∥u∥_{B_{2,2}^{s,δ}}.

Before presenting the proof of this theorem, we note that if ψ ∈ C_{0}^{∞}(R^{n}), then

∥ψu∥_{B_{p,q}^{s}} ≤ C(ψ, δ, s, p, q)∥u∥_{B_{p,q}^{s,δ}} (12)

and if u ∈ B_{p,q}^{s,c}, with supp u ⊂ {x : |x| < R}, then

∥u∥_{B_{2,2}^{s,δ}} ≤ C(R, δ)∥u∥_{B_{2,2}^{s}}. (13)

In each case, the stated inequality follows by observing that if ψ ∈ C_{0}^{∞}(R^{n}) and r ∈ R, then u → (1 + |x|^{2})^{r}ψu is bounded on each Besov space.

Proof of Theorem 4. We prove the estimate of the theorem for u smooth and then we may extend m_{q} to B_{2,2}^{s,δ} by density. Let ψ = 1 on supp ∇g with ψ ∈ C_{0}^{∞}(R^{n}). Then we write

\begin{aligned}|m_{q}(u)(ϕ)| & = |∫ ψ∇g · ∇ (ψ^{2} \frac{uϕ}{g}) dx| \\ & ≤ ∥ψ∇g∥_{B_{∞,2}^{1−s}}∥∇(ψ^{2}g^{−1}uϕ)∥_{B_{1,2}^{s−1}} \end{aligned} (14)

We use that ∂/∂x_{i} : B_{2,2}^{s} → B_{2,2}^{s−1}, Proposition 2 (b) and then (13) to obtain

\begin{aligned}∥∇(ψ^{2}g^{−1}uϕ)∥_{B_{1,2}^{s−1}} & ≤ C∥ψ^{2}uϕ∥_{B_{1,2}^{s}} \\ & ≤ C∥ψu∥_{B_{2,2}^{s}}∥ψϕ∥_{B_{2,2}^{s}} \\ & ≤ C∥u∥_{B_{2,2}^{s,δ}}∥ϕ∥_{B_{2,2}^{s,−δ−1}} \end{aligned} (15)

Using (12) and (15) in (14) gives that

|m_{q}(u)(ϕ)| ≤ C∥∇g∥_{B_{∞,2}^{1−s}}∥u∥_{B_{2,2}^{s,δ}}∥ϕ∥_{B_{2,2}^{s,−δ−1}}

or that m_{q}(u) is in the dual of B_{2,2}^{s,−δ−1}, (B_{2,2}^{s,−δ−1})′ = B_{2,2}^{−s,δ+1} (see [2, Corollary 6.2.8] for the duals of unweighted Besov spaces).

Remark: An examination of the above proof shows that in fact we have m_{q} : B_{2,2}^{s,loc} → B_{2,2}^{s,c}. We will use this in Corollary 6 to define m_{q}(1).

Our next theorem considers solutions to the equation

Δψ + 2ζ · ∇ψ − m_{q}(ψ) = f.

Theorem 0.5. Let g satisfy (8), (10) and (11) and let ζ ∈ C^{n} satisfy ζ · ζ = 0. If 0 < s < 1/2, −1 < δ < 0 and f ∈ B_{2,2}^{−s,δ+1}, then there exists C_{0} = C_{0}(λ, M, s, δ, n) so that for |ζ| > C_{0}, there exists a unique solution to

Δψ + 2ζ · ∇ψ − m_{q}(ψ) = f, ψ ∈ B_{2,2}^{s,δ} (16)

and this solution satisfies

∥ψ∥_{B_{2,2}^{s,δ}} ≤ \frac{C}{|ζ|^{1−2s}} ∥f∥_{B_{2,2}^{−s,δ+1}}

where C = C(n, s, δ, M, λ).

Proof. Consider the map ψ → G_{ζ}(m_{q}(ψ)). By Theorems 3 and 4, we have

∥G_{ζ}(m_{q}(ψ))∥_{B_{2,2}^{s,δ}} ≤ \frac{C}{|ζ|^{1−s}} ∥ψ∥_{B_{2,2}^{s,δ}}

Hence, if |ζ| is sufficiently large, then this map is a contraction on B_{2,2}^{s,δ}.

From the uniqueness of solutions to Δψ + 2ζ · ∇ψ = 0, ψ ∈ L_{δ}^{2} (see [10, Cor. 3.4], [4, Theorem 7.1.27]), ψ satisfies (16) if and only if

ψ = G_{ζ}(f) + G_{ζ}(m_{q}(ψ)). (17)

and by Theorem 3, G_{ζ}(f) ∈ B_{2,2}^{s,δ}. Thus the contraction mapping principle implies solutions to (17) exist and are unique in B_{2,2}^{s,δ}.

Now we are ready to return to the study of L_{γ} = div γ∇. It will be convenient to assume that γ is defined in all of R^{n} and satisfies for some 1/2 > ϵ > 0 and R > 0, λ > 1,

λ^{−1} < γ < λ (18)

∇γ ∈ B_{∞,∞}^{1/2+2ϵ,c} ⊂ B_{∞,2}^{1/2+ϵ,c} for some ϵ > 0 (19)

γ(x) = 1, if |x| > R (20)

The embedding in (19) follows easily from the definition of the B_{p,q}^{s}-norm [2, Definition 6.2.2]. Thus if g = \sqrt{γ}, g satisfies the hypotheses of Theorem 5 with s = \frac{1}{2} − ϵ.

Corollary 0.6. Suppose that γ satisfies (18)-(20) and ζ ∈ C^{n} satisfies ζ · ζ = 0 and |ζ| > C_{0}. Then there exists a solution to L_{γ}u = 0 of the form

u(x) = γ(x)^{−1/2}(1 + ψ(x))e^{x·ζ}, ψ ∈ B_{2,2}^{1/2−ϵ,δ}.

Furthermore, D^{2}u ∈ L_{loc}^{2}(R^{n}).

Proof. Given γ, we construct m_{q} as in (9), with g = \sqrt{γ}. We let ψ be the solution of

Δψ + 2ζ · ∇ψ − m_{q}(ψ) = m_{q}(1).

from Theorem 5. Since v = e^{x·ζ}(1 + ψ) solves Δv − m_{q}(v) = 0 in S′ and m_{q}(v) ∈ B_{2,2}^{ϵ−1/2,c}, regularity theory for Δ implies v ∈ B_{2,2}^{ϵ+3/2,loc} and in particular, ∇v ∈ L_{loc}^{2}. Then a calculation shows that u = γ^{−1/2}v solves L_{γ}u = 0, ∇u ∈ L_{loc}^{2}. Finally, since γ is C^{1}, regularity theory for L_{γ} implies ∇^{2}u ∈ L_{loc}^{2}.

Theorem 0.7. Suppose that ∂Ω is Lipschitz, and Λ_{γ_{1}} = Λ_{γ_{2}}. If ∇γ_{i} ∈ C^{1/2+2ϵ}(Ω̄) for some ϵ > 0, then there exist extensions of γ_{i} to R^{n} so that, with g_{i} = \sqrt{γ}_{i},

∫_{R^{n}} ∇g_{1} · ∇(g_{1}^{−1}ϕ) = ∫_{R^{n}} ∇g_{2} · ∇(g_{2}^{−1}ϕ), ϕ ∈ C_{0}^{∞}(R^{n}).

Proof. We begin by observing that since Λ_{γ_{1}} = Λ_{γ_{2}} and ∂Ω is Lipschitz, we have γ_{1} = γ_{2} and ∇γ_{1} = ∇γ_{2} on ∂Ω. This result was proven for smooth conductivities in [6] and for C^{1} conductivities in Lipschitz domains by [1]. Thus we may extend γ_{1} and γ_{2} to R^{n} so that γ_{1} = γ_{2} in R^{n} \ Ω̄ and satisfies (18)-(20).

We let u_{1} and u_{2}, be solutions of L_{γ_{i}} u_{i} = 0, ∇u_{i} ∈ L^{2}(Ω), i = 1, 2. We let v_{i} = γ_{i}^{1/2}u_{i} and obtain

\begin{aligned}_{∫∂Ω}u_{2}Λ_{γ_{1}}u_{1} & = _{∫Ω} γ_{1}∇(γ_{1}^{−1/2}v_{1}) · ∇(γ_{1}^{−1/2}v_{2}) dx \\ & = _{∫Ω} − ∇γ_{1}^{1/2} · ∇(γ_{1}^{−1/2}v_{1}v_{2}) + ∇v_{1} · ∇v_{2} dx \end{aligned}

where have used that γ_{1} = γ_{2} on ∂Ω and the second equality depends on the product rule.

Reversing the roles of u_{1} and u_{2} gives

∫_{∂Ω} u_{1}Λ_{γ_{2}}u_{2} = ∫_{Ω} −∇γ_{2}^{1/2} · ∇(γ_{2}^{−1/2}v_{1}v_{2}) + ∇v_{1} · ∇v_{2} dx.

If we subtract there expressions and use that Λ_{γ_{2}} is a symmetric operator, we have

\begin{aligned}_{∫∂Ω} u_{1}(Λ_{γ_{1}} − Λ_{γ_{2}})u_{2} & = _{∫Ω} − ∇γ_{1}^{1/2} · ∇(γ_{1}^{−1/2}v_{1}v_{2}) \\ & +∇γ_{2}^{1/2} · ∇(γ_{1}^{−1/2}v_{1}v_{2}) dx. \end{aligned} (21)

If we assume that u_{1} and u_{2} and defined in all of R^{n}, then (21), our assumption that Λ_{γ_{1}} = Λ_{γ_{2}} and that γ_{1} = γ_{2} in R^{n} \ Ω̄ give

0 = ∫_{R^{n}} −∇γ_{1}^{1/2} · ∇(γ_{1}^{−1/2}v_{1}v_{2}) + ∇γ_{2}^{1/2} · ∇(γ_{2}^{−1/2}v_{1}v_{2}) dx.

To choose u_{1} and u_{2}, we fix k ∈ R^{n} and then note that the argument in [10, p. 157] and the estimate of Theorem 5 allow us to construct sequences u_{1}^{(n)}, u_{2}^{(n)} so that L_{γ_{i}}u_{i}^{(n)} = 0 and v_{1}^{(n)} · v_{2}^{(n)} → e^{ix·k} in B_{1,2}^{1/2−ϵ,loc} as n → ∞. Hence we conclude that

∫ ∇γ_{1}^{1/2} · ∇(γ_{1}^{−1/2}e^{ix·k}) = ∫ ∇γ_{2}^{1/2} · ∇(γ_{2}^{−1/2}e^{ix·k}), k ∈ R^{n}.

This implies the conclusion of the Theorem.

Proposition 0.8. If the conclusion of Theorem 7 holds, then g_{1} = g_{2}.

Proof. We have

∫ ∇g_{1} · ∇ (\frac{1}{g_{1}} ϕ) = ∫ ∇g_{2} · ∇ (\frac{1}{g_{2}} ϕ)

for all ϕ ∈ C_{0}^{1}(R^{n}). Replace ϕ by g_{1}g_{2}ψ and observe that this gives

∫ g_{1}g_{2}∇(log g_{1} − log g_{2}) · ∇ψ = 0.

In particular if ψ = log \frac{g_{1}}{g_{2}} then log \frac{g_{1}}{g_{2}} = 0 in R^{n}.

This proposition amounts to observing that the equality g_{1}^{−1}Δg_{1} = g_{2}^{−1}Δg_{2} implies that div g_{1}g_{2}∇ log(g_{1}/g_{2}) = 0. I thank J. Tolle for showing me this argument, which is due to G. Alessandrini.

Proof of Theorem 1. Suppose that γ_{1} and γ_{2} are as in the Theorem and that Λ_{γ_{1}} = Λ_{γ_{2}}. Then we conclude that \sqrt{γ}_{1} = \sqrt{γ}_{2} from Theorem 7 and Proposition 8.

REFERENCES

[1] G. Alessandrini, Singular solutions of elliptic equations and the determination of conductivity by boundary measurements, J. Diff. Equations, 84 (1990), pp. 252–272.

[2] J. Bergh and J. Löfström, Interpolation spaces, an introduction, Springer-Verlag, 1976.

[3] S. Chanillo, A problem in electrical prospection and a n-dimensional Borg-Levinson theorem, Proc. Amer. Math. Soc., (1990), pp. 761–767.

[4] L. Hörmander, The analysis of linear partial differential operators, I, Springer-Verlag, 1983.

[5] V. Isakov, On uniqueness of recovery of a discontinuous conductivity coefficient, Comm. Pure Appl. Math., 41 (1988), pp. 865–977.

[6] R. Kohn and M. Vogelius, Determining conductivity by boundary measurements, Comm. Pure Appl. Math., 37 (1984), pp. 289–298.

[7] A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem. To appear, Annals of Math.

[8] A. I. Nachman, J. Sylvester, and G. Uhlmann, An n-dimensional Borg-Levinson theorem, Comm. Math. Physics, 115 (1988), pp. 595–605.

[9] A. G. Ramm, Completeness of products of solutions to pde and uniqueness theorems in inverse scattering, Inverse Problems, 3 (1987), pp. L77–L82.

[10] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Annals of Math., 125 (1987), pp. 153–169.

August 16, 1996