Uniqueness in the inverse conductivity problem for conductivities with 3/2

derivatives in L^{p} for p > 2n.

Russell Brown Department of Mathematics

University of Kentucky Lexington, Kentucky

http://www.ms.uky.edu/∼rbrown

Rodolfo Torres Department of Mathematics

Kansas University Lawrence, Kansas

Ann Arbor, Michigan 3 March 2002 The inverse conductivity problem.

Let γ : Ω → (0, ∞) be a scalar-valued function such that divγ∇ is a strictly elliptic operator. Consider the Dirichlet problem for this operator in a domain Ω with reasonable boundary:

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

The inverse conductivity problem is the problem of recovering the coefficient γ from knowledge of the map Λ_{γ} which takes Dirichlet to Neumann data:

Λ_{γ}f = γ\frac{∂u}{∂ν}.

My particular interest is to find what regularity conditions the coefficient must satisfy in order to prove uniqueness. This may lead to interesting questions in harmonic analysis.

A related equation.

If we set v = √γu, then we have that v satisfies the Schrödinger equation,

Δv − qv = 0.

where the potential q is given by

q = \frac{Δ√γ}{√γ} .

We will see how to construct a large family of solutions to this equation.

Some history. Using this reduction to Schrödinger operators, a number of authors have proven that the map γ → Λ_{γ} is injective, provided, Sylvester and Uhlmann, 1987, n ≥ 3, γ is C^{2} Novikov, 1988, n ≥ 3, γ smooth. Nachman, 1988, n ≥ 3, γ C^{2}. Ramm, 1989, n = 3, γ with two derivatives in L^{2}. Chanillo, 1990, n ≥ 3, the analagous result for Schrödinger operators with potentials in the Fefferman-Phong class, F^{(n−1)/2} . Nachman, 1996, n = 2, provided γ has two derivatives in L^{p}, p > 1.

Brown, 1996, n ≥ 3, γ is of class C^{3/2+ϵ} . Brown and Uhlmann, 1997, n = 2, γ has one derivative in L^{p}, p > 2. Paivarinta, Panchenko, Uhlmann, 2000, n ≥ 3, γ is of class C^{3/2} . Lassas, Greenleaf, Uhlmann, 2002, n ≥ 3, γ smooth off of a hypersurface and globally C^{1+ϵ} .

The main technique.

It is comparatively easy to recover the boundary values of γ and derivatives of γ from the Dirichlet to Neumann map. Thus, if we have two coefficients γ_{1} and γ_{2}, with the same Dirichlet to Neumann map, we can extend γ_{j} to R^{n}, preserving smoothness and so that γ_{1} = γ_{2} outside of Ω.

We make the reduction to a Schrödinger operator, discussed above and then, based on an idea of Calderón, we try to construct solutions of the Schrödinger operator which approximate harmonic exponentials,

v(x) = e^{x·ζ}(1 + ψ(x)),

where ζ ∈ C^{n} satisfies ζ · ζ = 0.

Thus, the function ψ should satisfy

Δψ + 2ζ · ∇ψ − qψ = q.

Construction of exponentially growing solutions. We construct solutions of the equation:

Δψ + 2ζ · ∇ψ − qψ = q

in the form

ψ = ∞ ; ∑ ; j=0 (G_{ζ}m_{q})^{j}(1)

where m_{q} is the multiplication operator given by the distribution q.

Thus, we need:

1. Estimates for the operator G_{ζ} = (Δ + 2ζ · ∇)^{−1}.

2. Estimates for the bilinear map q, ψ → qψ.

The estimate for G_{ζ}, I.

The estimate we use is a simple consequence of the work of Sylvester and Uhlmann. We have a little bit new to say about the second point, where we essentially use the Sobolev embedding theorem to conclude that product is slightly better than one might first expect.

It is well-known (see Sylvester and Uhlmann) that if ζ · ζ = 0 and |ζ| ≥ 1, then

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

The estimates for G_{ζ}, II.

If we define spaces B_{s}^{∗} by

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

where η is a function which is one on the unit ball and compactly supported in a the ball of radius 2, centered at the origin.

Then, the estimate (1), duality, and interpolation imply.

Theorem 1 If 0 ≤ s ≤ 1, f is compactly supported in a ball of radius R_{0}, then

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

Estimates for the map f → qf. If √γ has s derivatives in L^{p}, and √γ is one near infinity, then q = Δ√γ lies in the Sobolev space W^{s,p} and is compactly supported.

The following theorem is reasonably well-known:

Theorem 2 Suppose that s < n/p, then the bilinear map u, v → uv maps W^{s,p} × W^{s,p} into W^{s,p^{∗}} where 1/p^{∗} = 2/p − s/n.

Transposing to the dual, and looking up the definition of the space B_{.}^{∗} gives:

Corollary 1 If q ∈ W^{−1/2,2n} , then u → qu = m_{q}(u) satisfies

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

The constant depends on the diameter of the support of q.

The estimates for G_{ζ} and m_{q} are enough to imply that the series

∞ ; ∑ ; j=0 (G_{ζ}m_{q})^{j}(1)

converges and gives exponentially growing solutions of Δv −qv = 0 if q is small in W^{s,2n} and compactly supported.

A similar perturbation off of smooth potentials allows us to remove the smallness condition, at least for ζ large.

We can also show that

lim_{|ζ|→∞} ∥ψ∥_{B_{1/2}^{∗}} = 0.

The last steps–technicalities omitted. Suppose we have two conductivities γ_{1} and γ_{2} in W^{3/2,p} (Ω), p > 2n for which the Dirichlet to Neumann maps are equal. Extend γ_{j} to R^{n} and construct the potentials q_{j} as described above.

Fix ξ ∈ R^{n} and R large. Because we are in R^{n}, n ≥ 3, we may choose ζ_{1} and ζ_{2} so that

ζ_{j} · ζ_{j} = 0,  j = 1, 2
ζ_{1} + ζ_{2} = −iξ,  j = 1, 2
|ζ_{j}| = R,  j = 1, 2.

Construct solutions v_{j} of Δv_{j} − q_{j} = 0 of the form v_{j} = e^{x·ζ_{j}}(1 + ψ_{j}).

Because the Dirichlet to Neumann maps are equal, we have

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

Let ζ → ∞ obtain that q̂_{1}(ξ) − q̂_{2}(ξ) = 0. Conclude γ_{1} = γ_{2}.

Some open questions, I. Conjecture 1 The map

γ → Λ_{γ}

is injective on the set of conductivities in W^{1,p}, p > n. This is known in two dimensions.

Some open questions, II. A simpler question is the following: It appears that the natural boundary for a uniqueness theorem for the inverse conductivity problem with conductivities in a Sobolev space is the line where the Sobolev space (almost) embeds into continuous functions. Thus, it is reasonable to conjecture the following:

Conjecture 2 The map

γ → Λ_{γ}

is injective on the set of conductivities in W^{3/2,p}, p > 2n/3.

The proof of this requires better estimates for G_{ζ} than are known.

References.

1. B. Barceló, C. E. Kenig, A. Ruiz, and C. D. Sogge. Weighted Sobolev inequalities and unique continuation for the Laplacian plus lower order terms. Illinois J. Math., 32(2):230– 245, 1988.

2. Russell M. Brown and Gunther A. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Comm. Partial Differential Equations 22 (1997), no. 5-6, 1009–1027.

3. Russell M. Brown, Estimates for the scattering map associated to a two-dimensional first order system, to appear in J. nonlinear science.

4. Adrian I. Nachman. Global uniqueness for a two-dimensional inverse boundary value problem. Ann. of Math. (2), 143:71–96, 1996.

5. Lassi Päivärinta, Alexander Panchenko, and Gunther Uhlmann. Complex geometrical optics solutions for Lipschitz conductivities. Preprint, 2000.