Uniqueness in the inverse conductivity problem for conductivites with 3/2 derivatives in L^{p}, p > 2n

Russell M. Brown^{∗} Department of Mathematics University of Kentucky Lexington, Kentucky

Rodolfo H. Torres Department of Mathematics University of Kansas Lawrence, Kansas

The purpose of this note is to establish a small extension of a result of Panchenko, Päivärinta and Uhlmann [14]. These authors recently showed that we have uniqueness in the inverse conductivity problem for conductivities which are in the class C^{3/2} in three dimensions and higher. This built on earlier work of one the authors, Brown [3]. In this note, we relax this condition to conductivities which have 3/2 derivatives in L^{p} for p > 2n. We will obtain an end-point result with p = 2n for a related problem for Schrödinger equations. However, the problem of defining the trace on the boundary prevents us from obtaining uniqueness for the inverse conductivity problem at the end-point. Our results are in R^{n} with n ≥ 3. Better results are available in R^{2} in the work of Brown and Uhlmann [4].

The inverse conductivity problem is the problem of determining the coefficient γ in the elliptic operator divγ∇ from information about solutions to this operator at the boundary. This problem was posed in the mathematics community by A.P. Calderón [6]. As has been known for some time [8, 12, 16], the key technique in establishing such uniqueness results is the construction of solutions which are asymptotic to harmonic exponentials at infinity. To construct these solutions, we use the standard relationship between an elliptic operator of the form divγ∇ and a Schrödinger operator. Of course, the coefficient γ is a scalar valued function. The Schrödinger operators we consider are of the form

Δ − q

where the potential q = Δ\sqrt{γ}/\sqrt{γ} may be a distribution if γ is not smooth. We look for solutions v of the Schrödinger equation Δv − qv = 0 which are asymptotic to e^{x·ζ} where ζ is in C^{n} and satisfies ζ · ζ = ∑_{j} ζ_{j}^{2} = 0. A simple calculation shows that if we look for solutions which are of the form

v(x, ζ) = e^{x·ζ}(1 + ψ(x, ζ))

∗Research supported, in part, by the National Science Foundation.

then the function ψ must satisfy

Δψ + 2ζ · ∇ψ − qψ = q.

And, at least formally, the solution ψ can be constructed as the series

ψ(x, ζ) = ∑_{j=0}^{∞}(G_{ζ}m_{q})^{j}G_{ζ}(q).  (1)

In this expression, m_{q} is the multiplication operator given by q

m_{q}(ϕ) = qϕ

and G_{ζ} is the inverse of Δ + 2ζ · ∇ defined by

Ĝ_{ζ}(f)(ξ) = \frac{f̂(ξ)}{−|ξ|^{2} + 2iζ · ξ} .

Since the symbol −|ξ|^{2} + 2iζ · ξ = 0 vanishes to first order on a sphere of codimension 2, the right-hand side of the definition of G_{ζ} will be a tempered distribution if f is in the Schwartz class. The Fourier transform is normalized by f̂(ξ) = ∫ f(x)e^{−ix·ξ} dx. To make sense of the infinite sum (1), we need to make estimates for the operator G_{ζ} and the multiplication operator m_{q} acting on Banach spaces. The contribution of this note beyond previous work is to provide an improved estimate for m_{q}. To be precise, we will show that if q is in the Sobolev space W^{−s,n/s}, then the multiplication operator m_{q} (which is defined, say, as a map from S to S′) also maps W^{s,2} to W^{−s,2}. The case s = 1 is a consequence of such familiar results as the product rule and the Sobolev embedding theorem. To see this, suppose u, v ∈ W^{1,2}, then we have that uv lies not only in W^{1,1}, but we also have ∇(uv) ∈ L^{n/(n−1)}. This follows from the product rule and Sobolev embedding: u∇v + v∇u is in L^{n/(n−1)} since ∇v, ∇u ∈ L^{2} and u, v ∈ L^{2n/(n−2)}. Thus, we have

|⟨\frac{∂p}{∂x_{j}} , uv⟩| = |∫ p\frac{∂uv}{∂x_{j}} dx| ≤ C∥p∥_{L^{n}}∥u∥_{W^{1,2}}∥v∥_{W^{1,2}}

from which it follows that the map u → u\frac{∂p}{∂x_{j}} maps W^{1,2} to the dual space W^{−1,2} when p is in L^{n}. Here, we use ⟨·, ·⟩ to denote the bilinear pairing between distributions and functions.

In this paper, we contribute nothing to the analysis of G_{ζ}. The estimates used are from the paper of Sylvester and Uhlmannn [16]. It is possible that some improvement can be made here. We expect that one should be able to prove uniqueness for conductivities which have 3/2 derivatives in L^{p} with p > 2n/3. However, the straightforward generalization of the argument presented below would require that f → ∇G_{ζ}f map L^{p} functions which are compactly supported to functions which are locally in L^{r} with p and r satisfying 1/p − 1/r = 1/n. Many such estimates fail, see [2] for further discussion.

Finally, let us mention that one of us (Brown) conjectures that uniqueness should hold when the conductivity γ has one derivative in L^{p} with p > n. The methods presented here do not come close to addressing this conjecture.

We begin the formal development by stating a theorem on products of functions in Sobolev spaces W^{s,p}(R^{n}). There does not seem to be a standard notation for these spaces. For s ∈ R and 1 < p < ∞, we will use W^{s,p} to denote the space of distributions (or functions, if s ≥ 0) which are defined by the Bessel potential operator. Thus u ∈ W^{s,p} if and only if J_{−s}u ∈ L^{p} where J_{s} is the operator given by

f → ((1 + |ξ|^{2})^{−s/2}f̂(ξ))ˇ .

We recall that if s = 0 and 1 < p < ∞, then this space coincides with L^{p}. If s is a positive integer and 1 < p < ∞, then the space W^{s,p} is precisely the functions with s derivatives in L^{p}. For 1 < p < ∞, this space is also known as the Triebel-Lizorckin space F_{2}^{s,p}. We refer to the monograph of Triebel [18] for properties of these spaces.

Theorem 1 Let u ∈ W^{s,p} and let v ∈ W^{s,q}, with 1 < p, q < ∞, 1/p + 1/q ≤ 1, and 0 ≤ s < n min(1/p, 1/q). Then uv ∈ W^{s,r^{∗}} where 1/r^{∗} = 1/p + 1/q − s/n.

This result is well-known, see the monograph of Runst and Sickel [15, p. 177]. This theorem immediately implies the following Corollary.

Corollary 2 If q ∈ W^{−1/2,2n} with n ≥ 3, then the operator m_{q} given by m_{q}(u) = qu maps W^{1/2,2} → W^{−1/2,2} and satisfies

∥m_{q}(u)∥_{W^{−1/2,2}} ≤ C∥u∥_{W^{1/2,2}}∥q∥_{W^{−1/2,2n}}.

Proof. To be precise, m_{q}(u) is defined by ⟨m_{q}(u), v⟩ = ⟨q, uv⟩. Using the duality between W^{s,n/(n−s)} and W^{−s,n/s} which is valid (at least) for 1 < n/s < ∞, and Theorem 1 we conclude that

|⟨q, vu⟩| ≤ C∥q∥_{W^{−1/2,2n}}∥uv∥_{W^{1/2,2n/(2n−1)}} ≤ C∥q∥_{W^{−1/2,2n}}∥u∥_{W^{1/2,2}}∥v∥_{W^{1/2,2}}.

This inequality implies the Corollary.

Now, we recall standard estimates for the operator G_{ζ}. In order to simplify what follows, we will not use the weighted estimates of Sylvester and Uhlmann, but a simple consequence of these estimates. This will allow us to avoid mention of weighted Sobolev spaces. A weighted version of the previous Corollary is undoubtedly true, but it might be a chore to chase down the proof. We will restrict our attention to compactly supported potentials. This restriction is acceptable for our application to the inverse conductivity problem. However, there may be interest in Schrödinger operators where the potential is not compactly supported. If X is a space of distributions, we will fix R_{0} > 1 and let X_{c} denote the subspace of X which is supported in B̄_{R_{0}}(0). Our notation obscures the dependence of the space X_{c} on R_{0}, however we will give the dependence of the constants in our estimates on R_{0}. Also, we are using B_{s}(x) to denote the ball of radius s and center x. The following result is due to Sylvester and Uhlmann [16]. The spaces we use are similar to those used by Agmon and Hörmander (see [10, p. 227]), except that we require smoothness of order s. We fix η an infinitely differentiable function which is supported in a ball of radius 1 and is identically one in a neighborhood of B̄_{1/2}(0) and set η_{R}(x) = η(x/R). With this function η, we define the space B_{s}^{∗} to be the space of distributions for which the norm

∥f∥_{B_{s}^{∗}} = sup_{R>1} R^{−1/2}∥η_{R}f∥_{W^{s,2}}

is finite. We leave it as an exercise to see that different choices of η, subject to the above conditions, give equivalent norms on B_{s}^{∗}.

Theorem 3 If n ≥ 3, |ζ| ≥ 1, ζ · ζ = 0 and 0 ≤ s ≤ 1, then

∥G_{ζ}f∥_{B_{1/2}^{∗}} ≤ \frac{CR_{0}^{1/2}}{|ζ|^{s}} ∥f∥_{W_{c}^{−1/2+s,2}}.

Proof. From the main estimate in [16] (see also [3] for the gradient estimate), we find that

∫_{R^{n}}(1 + |x|^{2})^{−1/2}(|G_{ζ}f(x)|^{2}|ζ|^{2} + |∇G_{ζ}f(x)|^{2}) dx ≤ C ∫_{R^{n}}(1 + |x|^{2})^{1/2}|f(x)|^{2} dx. (2)

On the ball B_{R}(0), (with R > 1) the weight on the left-hand side of (2) is bounded below by 2R and the weight on the right-hand side is bounded above by 2R. This gives the estimate

R^{−1/2}∥G_{ζ}f∥_{L^{2}(B_{R}(0))} ≤ \frac{C}{|ζ|}R_{0}^{1/2}∥f∥_{L_{c}^{2}}.  (3)

Also, since G_{ζ} and ∇ commute, we obtain that for f compactly supported in B_{2R_{0}}

R^{−1/2}∥∇G_{ζ}f∥_{L^{2}(B_{R}(0))} ≤ \frac{C}{|ζ|}R_{0}^{1/2}∥∇f∥_{L_{c}^{2}}.  (4)

We let η be a smooth function as in the definition of the B_{.}^{∗}-spaces. The inequalities (3) and (4) together imply that for s = 0, 1, we have

R^{−1/2}∥η_{R}G_{ζ}η_{2R_{0}}f∥_{W^{s,2}} ≤ \frac{C}{|ζ|}R_{0}^{1/2}∥f∥_{W^{s,2}}.  (5)

Complex interpolation implies that the inequality (5) continues to hold for 0 < s < 1. Finally, if we use the estimate for the gradient in (2), it is not hard to see that for R, R_{0} > 1, we have

R^{−1/2}∥η_{R}G_{ζ}η_{2R_{0}}f∥_{W^{s+1,2}} ≤ CR_{0}^{1/2}∥f∥_{W^{s,2}}  (6)

for s = 0. By duality (and interchanging R and 2R_{0}), we obtain (6) with s = −1 also. Then, interpolating gives (6) with s = −1/2. Interpolating between (6) with s = −1/2 and (5) with s = 1/2, and taking the supremum in R gives the conclusion of the theorem. Notice that if f is in W_{c}^{s,2} and η = 1 in a neighborhood of B̄_{1/2}(0), then η_{2R_{0}}f = f, hence the estimates for G_{ζ} on W_{c}^{s,2} follow from estimates for the map f → G_{ζ}(η_{2R_{0}}f).

Now, we consider a potential q in W_{c}^{−1/2,2n} and consider the equation

(Δ + 2ζ · ∇)ψ − qψ = f.  (7)

In what follows, we will use G_{ζ,q} to denote the inverse of the operator Δ + 2ζ · ∇ − q. We prove the existence of this map by showing that it maps from W_{c}^{−1/2,2} to the space B_{1/2}^{∗}.

Theorem 4 Let q ∈ W_{c}^{−1/2,2n}, then there exists C = C_{0}(q) so that for |ζ| ≥ C_{0}(q), we may find a unique solution, ψ = G_{ζ,q}f, to (7) in B_{1/2}^{∗}. This map satisfies

∥G_{ζ,q}f∥_{B_{1/2}^{∗}} ≤ A∥f∥_{W_{c}^{−1/2,2}}.

The constant A = A(R_{0}) in the previous estimate is independent of q and ζ.

Furthermore, we have

lim_{|ζ|→∞} ∥G_{ζ,q}f∥_{B_{1/2}^{∗}} = 0.

To carry out the proof of the Theorem, we begin with the case of smooth potentials. The key point is that smooth potentials are dense in W^{−1/2,2n} and that for smooth potentials, the norm of the inverse does not depend on the size of q. The size of q only enters in determining how large ζ must be.

Lemma 5 Suppose q ∈ C_{c}^{∞}(R^{n}) and that supp q ⊂ B_{R_{0}}(0). Then there is a constant C_{0} = C_{0}(q) so that for |ζ| > C_{0}(q) and 0 ≤ s ≤ 1, G_{ζ,q} satisfies

∥G_{ζ,q}f∥_{B_{1/2}^{∗}} ≤ \frac{A}{|ζ|^{s}} ∥f∥_{W_{c}^{−1/2+s,2}}.  (8)

The constant A = A(R_{0}) is independent of ζ and q.

Proof. We write

G_{ζ,q}(f) = ∑_{j=0}^{∞}(G_{ζ}m_{q})^{j}G_{ζ}f.

According to Theorem 3, we have

∥G_{ζ}f∥_{B_{1/2}^{∗}} ≤ \frac{CR_{0}^{1/2}}{|ζ|^{s}} ∥f∥_{W^{−1/2+s,2}}

for 0 ≤ s ≤ 1. Because, q is smooth and η = 1 in a neighborhood of B_{1/2}(0), it is clear that

∥m_{q}f∥_{W_{c}^{1/2,2}} ≤ C(q)∥η_{2R_{0}}f∥_{W^{1/2,2}} ≤ C(q)R_{0}^{1/2}∥f∥_{B_{1/2}^{∗}}.

Using this and the estimate in Theorem 3 with s = 1 , we obtain

∥G_{ζ}m_{q}f∥_{B_{1/2}^{∗}} ≤ \frac{C(q)R_{0}}{|ζ|} ∥f∥_{B_{1/2}^{∗}}.

Now, if we require |ζ| to be sufficiently large, the norm of G_{ζ} ∘ m_{q} on B_{1/2}^{∗} will be bounded by 1/2 and hence the norm of ∑_{j}(G_{ζ} ∘ m_{q})^{j} as an operator on B_{1/2}^{∗} will be at most 2. This proves the Lemma.

Proof of Theorem 4. We suppose now that q is in W^{−1/2,2n} and is supported, say, in B_{R_{0}/2}(0). We let ϵ > 0, be a number to be determined later. We can write q = q_{s} + q_{r} where the smooth part of q, q_{s}, is in C_{0}^{∞}(R^{n}) and is supported in B_{R_{0}}(0) and the remainder, the rough part, q_{r} is also supported in B_{R_{0}}(0) and satisfies ∥q_{r}∥_{W^{−1/2,2n}} < ϵ. We suppose f ∈ B_{1/2}^{∗} and observe that with η as in the definition of B_{1/2}^{∗}, we have that m_{q_{r}}(f) = m_{q_{r}}(η_{2R_{0}}f) because η_{2R_{0}} = 1 on the support of q_{r}. We use the estimate for G_{ζ,q_{s}} on W_{c}^{−1/2,2} in Lemma 5 and the estimate for m_{q_{r}} in Corollary 2 to obtain

∥G_{ζ,q_{s}}m_{q_{r}}f∥_{B_{1/2}^{∗}} ≤ A∥m_{q_{r}}f∥_{W_{c}^{−1/2,2}}
≤ CA∥q_{r}∥_{W^{2n,−1/2}}∥η_{2R_{0}}f∥_{W^{1/2,2}}  (9)
≤ CAϵ\sqrt{2R_{0}}∥f∥_{B_{1/2}^{∗}}.

We search for u which satisfies Δu + 2ζ · ∇u − q_{s}u − q_{r}u = f and if we let G_{ζ,q_{s}} be an inverse to Δu + 2ζ · ∇ − q_{s}, then we can write

G_{ζ,q}f = ∑_{j=0}^{∞}(G_{ζ,q_{s}}m_{q_{r}})^{j}G_{ζ,q_{s}}f.  (10)

If we choose ϵ so that CAϵ\sqrt{2R_{0}} = 1/2, then the series in (10) will converge thanks to the estimate in (9). According to Lemma 5, the maps G_{ζ,q_{s}} will exist for |ζ| sufficiently large depending on q_{s} and hence on q.

To obtain uniqueness, we suppose that u ∈ B_{1/2}^{∗} is a distribution solution of Δu + 2ζ · ∇u − qu = 0 in R^{n}. Note that u must have half of a derivative for the product qu to be defined. Again, we split q = q_{r} + q_{s} and by Proposition 2.1 in Sylvester and Uhlmann [16], see also Hörmander [9, Theorem 7.1.27], there is a unique solution of Δu + 2ζ · ∇u − q_{s}u = q_{r}u, namely u = G_{ζ,q_{s}}(q_{r}u). By the estimate of Lemma 5, we obtain

∥u∥_{B_{1/2}^{∗}} ≤ A∥q_{r}∥_{W^{−1/2,2n}}∥u∥_{B_{1/2}^{∗}},

at least for |ζ| sufficiently large. Since the norm of q_{r} may be arbitrarily small, it follows that we have u = 0 for ζ large.

Finally, we observe that if we fix f ∈ W^{−1/2,2}, then we have

lim_{|ζ|→∞} ∥η_{R}G_{ζ,q}f∥_{B_{1/2}^{∗}} = 0.

To see this, we let ϵ > 0 and split f = f_{s} + f_{r} with ∥f_{r}∥_{W^{−1/2,2}} < ϵ. Let q_{s} and q_{r} be the splitting used above in constructing G_{ζ,q}. Using the representation for G_{ζ,q}, (10), and the estimate for G_{ζ,q_{s}}, (8), we have

∥G_{ζ,q}f_{s}∥_{B_{1/2}^{∗}} ≤ C(f_{s}, q)/|ζ|.  (11)

While the estimate of this theorem, which we have already proved, shows that for |ζ| sufficiently large, we have

∥G_{ζ,q}f_{r}∥_{B_{1/2}^{∗}} ≤ Aϵ.  (12)

Together these observations imply that lim sup_{|ζ|→∞} ∥G_{ζ,q}f∥_{B_{1/2}^{∗}} < Aϵ and since ϵ is an arbitrary positive number, the limit is zero.

Corollary 6 If q ∈ W_{c}^{−1/2,2n}, then there is a constant C_{0}(q) so that if ζ satisfies |ζ| > C_{0}(q) and ζ · ζ = 0, we can construct solutions v of Δv − qv = 0 of the form v(x, ζ) = e^{x·ζ}(1 + ψ(x, ζ)) with

lim_{|ζ|→∞} ∥ψ∥_{B_{1/2}^{∗}} = 0

Furthermore, ψ is the unique function in B_{1/2}^{∗} and so that v = e^{x·ζ}(1+ψ(x, ζ)) satisfies Δv − qv = 0.

Proof. We construct ψ = ψ(·, ζ) = G_{ζ,q}(q) where G_{ζ,q} was constructed in Theorem 4. The existence and the limiting behavior of ψ follow from Theorem 4. It is a simple calculation to see that the ψ and then the v constructed in this way have the desired properties.

If we have two such ψ, call them ψ_{1} and ψ_{2}, then their difference is a solution of the homogeneous equation

Δ(ψ_{1} − ψ_{2}) + 2ζ · ∇(ψ_{1} − ψ_{2}) − q(ψ_{1} − ψ_{2}) = 0.

According to Theorem 4, 0 is the only solution of this equation in B_{1/2}^{∗}.

In what follows, we will also need function spaces on domains. For s ≥ 0, we define the space W^{s,p}(Ω) as the image of the space W^{s,p}(R^{n}) under the map u → u|_{Ω}. Thus elements in the space W^{s,p}(Ω) automatically have extensions to all of R^{n}. We also recall that the space W_{0}^{s,p}(Ω) denotes the closure of C_{0}^{∞}(Ω) in the space W^{s,p}(Ω).

Next, we recall the definition of the Dirichlet to Neumann map, which we denote Λ_{γ}, for the elliptic operator divγ∇. We suppose that we have a bounded domain with Lipschitz boundary. We define the space W^{1/2,2}(∂Ω) as the quotient space W^{1,2}(Ω)/W_{0}^{1,2}(Ω). It is well-known that this space can be identified with the Besov space B_{2}^{1/2,2} on the boundary. The coefficient γ in the operator divγ∇ will lie in L^{∞}(Ω) and satisfy

δ ≤ γ ≤ 1/δ  (13)

for some δ > 0. If f is in W^{1,2}(Ω), we can solve the Dirichlet problem

\{ \begin{matrix}divγ∇u = 0 & in Ω \\ u = f & on ∂Ω. \end{matrix}

This solution u is, of course, independent of the particular representative f of an equivalence class in W^{1/2,2}(∂Ω). Given the solution u of the Dirichlet problem with data f, we may define a map Λ_{γ} : W^{1/2,2}(∂Ω) → W^{−1/2,2}(∂Ω) by

Λ_{γ}(f)(g) = ∫_{Ω} γ∇u · ∇g dx.

Since u is a solution, the expression on the right does not change if we add a function in W_{0}^{1,2}(Ω) to g. Thus, we may define Λ_{γ}(f) as an element of the dual of W^{1/2,2}(∂Ω).

Now, we quote the following result on determining the coefficient at the boundary. This will be an important step in connecting the Dirichlet to Neumann map in Ω to a problem in all of space. In our argument below, we will work with two conductivities γ_{1} and γ_{2} with \sqrt{γ_{j}} in W^{3/2,2n+ϵ}(Ω) and try to extend them to preserve smoothness and so that the extensions are equal in the complement of Ω. The next two results explain when this can be done.

Proposition 7 Assume that ∂Ω is Lipschitz and assume that for some ϵ > 0, γ_{1} and γ_{2} are in W^{3/2,2n+ϵ} and that Λ_{γ_{1}} = Λ_{γ_{2}}. Then we have γ_{1} = γ_{2} and ∇γ_{1} = ∇γ_{2} on ∂Ω.

Proof. The hypothesis that γ_{j} is in W^{3/2,2n+ϵ}(Ω) implies, by the Sobolev embedding theorem, that there is an ϵ′ > 0 so that ∇γ_{j} is in C^{1+ϵ′}(Ω̄) for j = 1, 2. It has been proven many times that the boundary values of a conductivity and its derivatives are determined by the Dirichlet to Neumann map. The result stated here for C^{1+ϵ′} conductivities may be found in Alessandrini [1].

Corollary 8 If, for some ϵ > 0, \sqrt{γ_{1}} and \sqrt{γ_{2}} are in W^{3/2,2n+ϵ}(Ω), and Λ_{γ_{1}} = Λ_{γ_{2}}, then we may extend γ_{1} and γ_{2} to all of R^{n} so that \sqrt{γ_{j}} − 1 ∈ W^{3/2,2n+ϵ} and γ_{1} = γ_{2} in R^{n} \ Ω.

Proof. According to the previous proposition, γ_{1} = γ_{2} on ∂Ω and also ∇γ_{1} = ∇γ_{2} on ∂Ω. By our definition of the space W^{s,p}(Ω), it is immediate that there are extensions of \sqrt{γ_{1}} and \sqrt{γ_{2}} to all of R^{n}. Using smooth cutoff functions, we can arrange that the extensions (which we still denote by \sqrt{γ_{j}}) have \sqrt{γ_{j}} − 1 in W_{c}^{3/2,2n+ϵ}(R^{n}) for j = 1, 2. We claim that the function

β = \{ \begin{matrix}\sqrt{γ_{2}} − \sqrt{γ_{1}}, & in Ω, \\ 0, & in Ω̄^{c}, \end{matrix}

lies in W^{3/2,2n+ϵ}(R^{n}). This will suffice to prove our Corollary since \sqrt{γ_{1}} and \sqrt{γ_{1}} + β provide the two extensions which agree in the complement of Ω.

Thus, we must establish the claim. Of course, this depends on the hypothesis that γ_{1} and γ_{2} agree to first order on the boundary of Ω. In fact, this claim is a consequence of Corollary 2.11 in Triebel’s monograph [18, p. 210]. Alert readers will note that this monograph assumes that the domain is smooth. However, that assumption is not needed when p > 1. The key step is to show that the characteristic function of Ω is multiplier on W^{s,p}(R^{n}) when −1 + 1/p < s < 1/p. The proof proceeds by changing variables to flatten the boundary. As the discussion on p. 172 of Triebel indicates, when p > 1, the needed results for Lipschitz change of variables can be proven for s = 0, 1 using the chain rule (for s = 1) and change of variables. Then, we interpolate to obtain results for 0 < s < 1.

The next proposition uses the equality of the Dirichlet to Neumann maps to deduce the equality of expressions involving the solutions v_{j} constructed above. In this proposition and below, we will use a Schrödinger operator with a potential q_{j} which is defined by Δ\sqrt{γ_{j}}/\sqrt{γ_{j}}. Since, in general, the coefficient γ_{j} does not have two derivatives, we define the Laplacian in a weak sense. To be precise, we will define the pairing between q_{j} and a function ϕ with one derivative by

⟨q_{j}, ϕ⟩ = − ∫_{R^{n}} ∇\sqrt{γ_{j}} · ∇\frac{ϕ}{\sqrt{γ_{j}}} dx.  (14)

Proposition 9 Suppose n ≥ 3. Let γ_{1} and γ_{2} satisfy (13) and suppose that ∇\sqrt{γ_{j}} is in L_{c}^{n}. Suppose further that γ_{1} = γ_{2} outside Ω and that Λ_{γ_{1}} = Λ_{γ_{2}}. We let q_{j} = Δ\sqrt{γ_{j}}/\sqrt{γ_{j}}. If for j = 1, 2, v_{j} is a solution of Δv_{j} − q_{j}v_{j} = 0 which lies in W_{loc}^{1,2}(R^{n}), then

⟨q_{1}, v_{1}v_{2}⟩ = ⟨q_{2}, v_{1}v_{2}⟩.

Proof. We first observe that the multiplication operator given by Δ\sqrt{γ_{j}}/\sqrt{γ}_{j} maps W^{1,2}(R^{n}) to W^{−1,2}(R^{n}). To see this, we use the definition of the distribution Δ\sqrt{γ_{j}}/\sqrt{γ_{j}} in (14) to obtain

⟨\frac{Δ\sqrt{γ_{j}}}{\sqrt{γ_{j}}} , uv⟩ = − ∫_{R^{n}} ∇\sqrt{γ_{j}} · ∇\frac{uv}{\sqrt{γ}_{j}} dx.  (15)

The product rule, Hölder’s inequality, and then the Sobolev inequality (which requires n ≥ 3), imply that

|∫ ∇\sqrt{γ_{j}} · ∇\frac{uv}{\sqrt{γ_{j}}} dx| ≤ ∥u∥_{W^{1,2}B_{R_{0}}}∥v∥_{W^{1,2}B_{R_{0}}}(∥∇ log \sqrt{γ_{j}}∥_{L^{n}} + ∥∇ log \sqrt{γ_{j}}∥_{L^{n}}^{2}).

Thus the expressions in (15) are defined, at least when n ≥ 3.

Now, we turn to the proof of the equality of the theorem. First, we consider the integral outside of Ω. Since we have \sqrt{γ_{1}} = \sqrt{γ_{2}} outside Ω, it follows immediately that

∫_{R^{n}\Ω} ∇\sqrt{γ_{1}} · ∇\frac{v_{1}v_{2}}{\sqrt{γ_{1}}} dx = ∫_{R^{n}\Ω} ∇\sqrt{γ_{2}} · ∇\frac{v_{1}v_{2}}{\sqrt{γ_{2}}} dx.  (16)

To study the integral inside Ω, we observe that if we define u_{j} = v_{j}/\sqrt{γ_{j}}, then we have that divγ_{j}∇u_{j} = 0. This follows from a well-known calculation which we omit. Next, we claim that

∫_{Ω} ∇\sqrt{γ}_{1} · ∇(\frac{1}{\sqrt{γ_{1}}} v_{1}v_{2}) − ∇v_{1} · ∇v_{2} dx = −Λ_{γ_{1}}(u_{1})(u_{2}).  (17)

Now, if we interchange the indices 1 and 2 in (17), subtract the result from (17), use that Λ_{γ_{j}}(f)(g) = Λ_{γ_{j}}(g)(f), and then our assumption that Λ_{γ_{1}} = Λ_{γ_{2}} we obtain that

∫_{Ω} ∇\sqrt{γ}_{1} · ∇\frac{v_{1}v_{2}}{\sqrt{γ_{1}}} dx = ∫_{Ω} ∇\sqrt{γ_{2}} · ∇\frac{v_{1}v_{2}}{\sqrt{γ_{2}}} dx.  (18)

If we add (16) and (18), we obtain the conclusion of the theorem.

Thus, we turn to the proof of (17) and for this we will need an additional function ũ_{2} = v_{2}/\sqrt{γ_{1}}. By the definitions of u_{1} and ũ_{2} and the product rule, we obtain

∫_{Ω} ∇\sqrt{γ_{1}} · ∇(\frac{1}{\sqrt{γ_{1}}} v_{1}v_{2}) − ∇v_{1} · ∇v_{2} dx

= ∫_{Ω} ∇\sqrt{γ_{1}} · ∇(\sqrt{γ_{1}}u_{1}ũ_{2}) − u_{1}∇\sqrt{γ_{1}} · ∇(\sqrt{γ_{1}}ũ_{2})
−\sqrt{γ_{1}}∇u_{1} · (ũ_{2}∇\sqrt{γ_{1}}) − γ_{1}∇u_{1} · ∇ũ_{2} dx
= −Λ_{γ_{1}}(u_{1})(u_{2}).

In the last equality, we use that u_{2} − ũ_{2} is in W_{0}^{1,2}(Ω) and hence they restrict to the same element in W^{1/2,2}(∂Ω). This is a consequence of Lemma 10 below. This completes the proof of (17) and hence the Proposition.

Lemma 10 Assume ∂Ω is Lipschitz and that Ω is bounded. If β is C(Ω̄) ∩ W^{1,n}(Ω) and β(x) = 0 for x ∈ ∂Ω, then the map u → βu maps W^{1,2}(Ω) to W_{0}^{1,2}(Ω).

Proof. We let ϵ > 0 and construct a cutoff function η_{ϵ}(x) where η_{ϵ}(x) = 1 if δ(x) > 2ϵ and η_{ϵ}(x) = 0 if δ(x) < ϵ. Here, we are using δ(x) to denote the distance from x to ∂Ω. This function may be constructed to satisfy |∇η_{ϵ}| ≤ C/ϵ. We consider ∇(η_{ϵ}β) − ∇β = (η_{ϵ} − 1)∇β + β∇(η_{ϵ} − 1). By the dominated convergence theorem, we have lim_{ϵ→0^{+}} ∫_{Ω} |∇β|^{n}|(1 − η_{ϵ})|^{n} = 0. While Hardy’s inequality (see [7, p. 28], for example) implies that

∫_{Ω} |β|^{n}|∇η_{ϵ}|^{n} dx ≤ ∫_{{x:δ(x)<Cϵ}} |∇β|^{n} dx.

Hence, we can approximate β in the norm of W^{1,n}(Ω) by βη_{ϵ}. Furthermore, by regularizing, we can find a sequence of functions {β_{j}} which are infinitely differentiable, converge to β in W_{0}^{1,n}(Ω), the sequence is bounded in L^{∞}(Ω) and converges pointwise to β. Now, we claim that β_{j}u converges to βu in W^{1,2}(Ω). The convergence of β_{j}u to βu in L^{2} follows from the dominated convergence theorem. Now consider ∇(β_{j}u) = u∇β_{j}+β_{j}∇u. The convergence of β_{j}∇u to β∇u follows from the dominated convergence theorem. We have

∥u(∇β_{j} − ∇β)∥_{L^{2}(Ω)} ≤ ∥∇β_{j} − ∇β∥_{L^{n}(Ω)}∥u∥_{L^{2n/(n−2)}(Ω)}.  (19)

From the inequality (19), we see that the convergence of ∇β_{j} in L^{n} implies the convergence of u∇β_{j} to u∇β in L^{2}. We have established that ∥∇(β_{j}u) − ∇(βu)∥_{L^{n}(Ω)} goes to zero as j → ∞. The term ∥u∥_{L^{2n/(n−2)}(Ω)} is finite because of the Sobolev embedding theorem. Hence, the right-hand side of (19) tends to zero. Because the sequence of functions β_{j}u tend to βu in W^{1,2}(Ω) we can conclude that βu is in W_{0}^{1,2}(Ω).

Finally, we give the proof of our main theorem. Given the above, the proof of the main result follows familiar lines.

Theorem 11 Suppose n ≥ 3, ∂Ω is Lipschitz, γ_{j} ∈ W^{3/2,2n+ϵ}(Ω) for some ϵ > 0 and that Λ_{γ_{1}} = Λ_{γ_{2}}, then we have that γ_{1} = γ_{2}.

Proof. Because of the equality of the Dirichlet to Neumann maps, we may use the result of Corollary 8 to extend γ_{1} and γ_{2} to all of R^{n} so that γ_{1} = γ_{2} in R^{n} \ Ω. Thus, according to Proposition 9 we have

⟨(q_{1} − q_{2}), v_{1}v_{2}⟩ = 0  (20)

where q_{j} are the potentials, Δ\sqrt{γ_{j}}/\sqrt{γ_{j}} as above and v_{j} are solutions of Δv_{j}−q_{j}v_{j} = 0. Now, we fix ξ ∈ R^{n}, let R > 0 be large and construct ζ_{1} and ζ_{2} in C^{n} satisfying

ζ_{j} · ζ_{j} = 0,  j = 1, 2  (21)
ζ_{1} + ζ_{2} = −iξ,  j = 1, 2  (22)
|ζ_{j}| > R,  j = 1, 2.  (23)

We recall the standard construction of ζ_{j}. Choose e_{1} and e_{2} unit vectors in R^{n} and so that e_{1}, e_{2} and ξ are mutually orthogonal and then put ζ_{1} = −Re_{1}−ie_{2}\sqrt{R^{2} − |ξ|^{2}/4}−

iξ/2 and ζ_{2} = Re_{1} + ie_{2}\sqrt{R^{2} − |ξ|^{2}/4} − iξ/2. We construct the solutions v_{j} of Δv_{j} − q_{j}v_{j} = 0 corresponding to ζ_{j} as given by Corollary 6. Note that the solutions given by this Corollary must have 3/2 derivatives L_{loc}^{2} and thus they satisfy the hypotheses of Proposition 9. To see this, observe that the right-hand side of Δv = qv lies W_{loc}^{−1/2,2} and thus by elliptic regularity, v has two more derivatives. We substitute e^{x·ζ_{j}}(1+ψ_{j}) for v_{j} in (20) and obtain

0 = ⟨q_{1} − q_{2}, e^{−ix·ξ}(1 + ψ_{1} + ψ_{2} + ψ_{1}ψ_{2})⟩.

Because the functions ψ_{j} tend to zero in B_{1/2}^{∗} as R → ∞, passing to the limit in the previous equation gives q̂_{1} = q̂_{2}. Here, we must use Theorem 1 and our estimate for ψ_{1}, ψ_{2} in B_{1/2}^{∗} to conclude that the product ψ_{1}ψ_{2} goes to zero in W_{loc}^{1/2,2n/(2n−1)}. Hence, it follows that Δ\sqrt{γ_{1}}/\sqrt{γ_{1}} = Δ\sqrt{γ_{2}}/\sqrt{γ_{2}}. As in [3, p. 1056], this implies that log(γ_{1}/γ_{2}) solves div\sqrt{γ_{1}γ_{2}}∇ log(γ_{1}/γ_{2}) = 0 and is compactly supported and hence vanishes by the weak maximum principle.

We close with several questions motivated by the above work.

1. Can we obtain uniqueness in the inverse conductivity problem at the endpoint p = 2n? The above argument requires p > 2n in order to carry out the extension in Corollary 8.

2. Can we lower the L^{p}-space from p = 2n to p = 2n/3 in the uniqueness result for conductivities with 3/2 derivatives?

3. Can we lower the smoothness index to 1 and obtain a uniqueness result?

4. The result for uniqueness at the boundary in Proposition 7 that we quote from Alessandrini [1] does not seem sharp. There is a boundary uniqueness result for the gradient of the conductivity which requires that the coefficient be continuously differentiable in the work of Sylvester and Uhlmann [17], however the domain must be smooth. The work of Brown [5] gives a way of reconstructing the boundary values for continuous (and some discontinuous) conductivities in Lipschitz domains. However, this work does not discuss the gradient of the conductivity. The work Tanuma and Nakamura [13] and Kang and Yun [11] gives boundary identifiability of the gradient of a C^{1+ϵ} conductivity in a C^{2+ϵ} domain. Results of Nachman [12] give boundary identifiability for the gradient of C^{1,1} conductivities in domains with C^{1,1} boundaries. A reasonable conjecture is that we can determine the gradient of continuously differentiable conductivities at the boundary in a Lipschitz domain. This result does not seem to be in the literature.

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January 22, 2002