Note on Calder\'on's inverse problem for measurable conductivities

The unique determination of a measurable conductivity from the Dirichlet-to-Neumann map of the equation $\mathrm{div} (\sigma \nabla u) = 0$ is the subject of this note. A new strategy, based on Clifford algebras and a higher dimensional analogue of the Beltrami equation, is here proposed. This represents a possible first step for a proof of uniqueness for the Calder\'on problem in three and higher dimensions in the $L^\infty$ case.

This inverse problem have triggered an impressive amount of pure and applied research in the last decades. Global uniqueness, meaning the injectivity of the map σ → Λ σ , under some smoothness assumptions on σ, has been first shown in [18] in three and higher dimensions, and in [17] in two dimensions.
The latter result have been greatly improved in [3], where uniqueness was obtained for measurable conductivities in two dimensions (later generalized in [2]). In higher dimensions this problem is still open. The best results so far show that the lowest regularity required to guarantee uniqueness is Sobolev W 1,n in dimension n = 3, 4 [15] and Lipschitz in higher dimensions [10].
It is unclear if global uniqueness in three and higher dimensions for measurable conductivites holds true. No counterexamples have been found but it has been conjectured [8] that the lowest regularity possible is W 1,n , in dimension n ≥ 3 (because of related results on unique continuation). It is rather clear, though, that the techniques used until now have reached some sort of limit and a new framework must be introduced in order to tackle the problem.
The present note suggests a new strategy to study this problem. The main idea is to extend the two-dimensional approach of Astala-Päivärinta [3] to higher dimensions. It seemed that the most natural framework to do so is via Clifford algebras, which in the three dimensional case is the algebra of quaternions. The main result obtained in this note is to rewrite the conductivity equation, in the three dimensional case, as a higher dimensional analogue of the Beltrami equation, also known as Clifford-Betrami equation: where F is a Clifford algebra valued function and D is a so-called Cauchy-Riemann operator. Incidentally, the Beltrami coefficient µ coincides with the one from [3].
The next natural step is to construct so-called complex geometrical optics (CGO) solutions (also known as exponentially growing or Faddeev-type solutions [13]) for this equation and study their properties. In this way one could obtain either a higher dimensional analogue of the∂ equation in some parameter space, a linear (or nonlinear) transform of µ from high frequency asymptotics, or other indirect information on the unknown conductivity. Here we only propose a possible definition of CGO solutions, leaving their construction and analysis to future work.
The proposed CGO solutions are characterized by an asymptotic behaviour defined by a new family of exponential functions inspired by [16]. These exponential functions are not only harmonic, but also monogenic, i.e. they belong to the kernel of D. To show the usefulness of these functions, a new proof of uniqueness for the linearized Calderón problem at a constant conductivity is given.
Note that quaternionic analytic techniques have been used in connection with the inverse conductivity problem also in the works [6,5,4,11,12].
The structure of this note is the following. In Section 2 we present the main ideas. Some basic notations of Clifford analysis are introduced, as well as the reduction of the conductivity equation to a Clifford-Beltrami equation and the new uniqueness proof for the linearized problem. We propose a definition of CGO solutions in Section 3 and give few concluding remarks in Section 4.

The Clifford-Beltrami equation
Following the same argument as in [3,Section 2], it is possible to reduce the problem to the case where Ω is a smooth domain, for instance the unit ball in R n . Let us briefly review it.
The map Λ σ can be defined on general domains by identifying H 1/2 (∂Ω) = Now let B ⊂ R n be the unit ball, Ω ⊂ B a simply connected domain and σ 1 , σ 2 two L ∞ conductivities defined in Ω such that Λ σ 1 = Λ σ 2 . Extend σ 1 and σ 2 as the constant 1 outside Ω and letσ 1 ,σ 2 be the new conductivities on B.
Note that in B \ Ω we haveũ 1 =ũ 2 andσ 1 =σ 2 . This immediately yields From now on we assume that Ω = B and extend σ ≡ 1 outside Ω. We will rewrite the conductivity equation using some notation from differential geometry. Let d be the exterior derivative on differential forms and ⋆ the Hodge star operator. Then, the conductivity equation (1.1) can be written as We will now extend the notion of σ-harmonic conjugate, introduced in [3] on the plane, to higher dimensions.
Let Ω ⊂ R n , n ≥ 3 be the unit ball, σ ∈ L ∞ (Ω) bounded from below and u ∈ H 1 (Ω) a solution of the conductivity equation (2.1). Then there exists a n − 2 form ω, unique up to dφ, for a n − 3 form φ, such that Proof. The proof follows from Poincaré lemma and the properties of the Hodge star operator. Nevertheless it is useful to point out the main relationship between u and the form ω: The form ω will be called σ-harmonic conjugate of u and it is (locally) given by n(n−1) 2 functions. We will now restrict ourselves to the case n = 3. In this case there exists three functions u 1 , u 2 , u 3 such that identity (2.3) can be written as The triplet (u 1 , u 2 , u 3 ) is defined up to ∇φ for some function φ, which will be precised later.
The system (2.4) is a 3D analogue of the one obtained in [3] on the plane. In that case this was equivalent to a Beltrami equation and thanks to the Ahlfors-Vekua theory of quasiconformal maps it was possible to construct CGO solutions for L ∞ conductivities.
It does not seem clear how to construct CGO solutions directly for (2.4). We will instead use the framework of Clifford analysis and Dirac operators [7] to write the system in a more convenient form.
We consider R (2) , the real universal Clifford algebra over R 2 . It is generated as an algebra over R by the elements {e 0 , e 1 , e 2 }, where e 1 , e 2 is a basis of R 2 with e i e j + e j e i = −2δ ij , for i, j = 1, 2, and e 0 = 1 is the identity and commutes with the basis elements. The dimension of R (2) is 4 and it can be identified with H, the algebra of quaternions. We denote e 3 = e 1 e 2 for the sake of simplicity. An element of R (2) can be written as where a j , j = 0, . . . , 3 are real. We define the conjugateĀ of an element A as For A, B ∈ R (2) we write AB for the resulting Clifford product. The product AB defines a Clifford valued inner product on R (2) . We have AB =BĀ and A = A. For A ∈ R (2) , Sc(A) denotes the scalar part of A, that is the coefficient of the element e 0 . The scalar part of a Clifford inner product, Sc(ĀB), is the usual inner product in R 4 when A and B are identified as vectors. We will write it A, B . With this inner product the space R (2) is an Hilbert space and the resulting norm is the usual Euclidean norm A = ( j A 2 j ) 1/2 . A Clifford valued function f : R 3 → R (2) can be written as f = f 0 e 0 + f 1 e 1 + f 2 e 2 + f 3 e 3 , where f j are real valued.
The Banach spaces C α , L p , W 1,p of R (2) -valued functions are defined by requiring that each component f j belong to such spaces. On L 2 (Ω) we introduce the R (2) -valued inner product We define the following Cauchy-Riemann operators, with (x 0 , x 1 , x 2 ) coordinates of R 3 , The operator ∂ = e 1 This last condition can always be achieved since we can add to (u 0 , u 1 , u 2 ) the gradient of a function φ such that ∆φ = −div(u 0 , u 1 , u 2 ). In other words, equation (2.7) is equivalent to the system (2.8) curl(u 0 , u 1 , u 2 ) = σ∇u, div(u 0 , u 1 , u 2 ) = 0.
More precisely, the following identities hold:
A generalization of Alessandrini's identity can be now readily proven.
Proof. By Green's formulas one readily obtain the classical Alessandrini's identity for the Calderón problem [1]: Let now U j the vector field such that curl(U j ) = σ j ∇u j . Then Using identities (2.9), (2.10), and the definition of the scalar product ·, · in R (2) , we can write the quantity under the integral sign as the scalar part of a Clifford product as follows: thanks to the Clifford-Beltrami equation satisfied by F 1 , F 2 .
We now consider the complex Clifford algebra C (2) , generated over C with the same basis elements of R (2) . Note that the Clifford conjugation is always defined as in (2.6), so that it does not extend to the complex conjugation on the coefficients (which is never used in this note). Following [16], we define the following exponential function with values in C (2) : and i is the imaginary unit. It is a holomorphic function of ζ ∈ C 2 for each x ∈ R 3 and satisfies This yields D (l) E 1 = D (r) E 1 = 0, that is E 1 is left and right monogenic.
Note that we also have Here we have denoted |ζ| C a square root of |ζ| 2 C , the holomorphic extension of the Euclidean norm ξ 2 , for ξ ∈ R 2 , defined as |ζ| C E 1 (x, ζ), for |ζ| C = 0, which is left and right monogenic and can be written as Now consider the following combination of the two: Using the identity e z = cos(z) + i sin(z), for z ∈ C, one readily obtains: This function is left and right monogenic and its scalar part coincides with the harmonic exponential of the classical CGO solutions, i.e. e ix·ζ , for ζ ∈ C 3 , ζ · ζ = 0.

CGO solutions of the Clifford-Beltrami equation
We want to construct a special family of solutions to equation (2.7), often referred as complex geometrical optics (CGO) solutions, with prescribed asymptotic behavior. Such behavior has usually been determined by an exponential function with some special properties. In this case we use the function E introduced in Section 2.
We first need to establish a Leibniz formula for the operator D. This was already obtained in [14, Theorem 1.3.2] for a slightly different Cauchy-Riemann operator. Lemma 3.1 (Leibniz's formula). Let f = 3 k=0 f k e k , g = 3 l=0 g l e l be two Clifford valued functions. Then where where the sums are taken over all indices j = 0, 1, 2 and k, l = 0, 1, 2, 3. Now the last term can be rewritten as The proof follows by combining the two identities. We seek solutions to equation (2.7) of the form with M (x, ζ) → 1 as |x| → +∞, for ζ ∈ C 2 , |ζ| C = 0, and M a C (2) -valued function. Plugging (3.2) into equation (2.7), using Leibniz's formula (3.1) and the fact that DE =DĒ = 0, we find that M = 3 j=0 M j e j satisfies: where we have denoted ζ = ζ 1 e 1 + ζ 2 e 2 . We also used the fact that

Concluding remarks
A new strategy to study the Calderón problem in three and higher dimensions has been proposed in this note. The approach is inspired by the two dimensional uniqueness result by Astala-Päivärinta [3] and it is based on Clifford analytic techniques. For the three dimensional case, the conductivity equation has been reduced to a Clifford-Beltrami equation and two family of CGO solutions have been proposed. A careful study of such solutions and their properties still needs to be done. These new tools might shed light on the uniqueness question for the inverse conductivity problem for measurable conductivities in three and higher dimensions.