A partial data result for less regular conductivities in admissible geometries

We consider the Calder\'on problem with partial data in certain admissible geometries, that is, on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. We show that measuring the Dirichlet-to-Neumann map on roughly half of the boundary determines a conductivity that has essentially 3/2 derivatives. As a corollary, we strengthen a partial data result due to Kenig, Sj\"ostrand, and Uhlmann.


Introduction
In 1980 A. P. Calderón published a short paper [3] where he posed the following question: is it possible to determine the electrical conductivity of a medium by making voltage and current measurements on the boundary? This pioneering contribution motivated many developments in inverse problems, in particular the construction of 'complex geometrical optics' solutions of partial differential equations to solve several inverse problems. The precise mathematical formulation of the problem is the following.
Let Ω ⊆ R n , n ≥ 2, be a bounded domain with smooth boundary. The electrical conductivity of Ω is modeled by a bounded positive function γ. In the absence of sources or sinks, given a boundary potential f ∈ H 1/2 (∂Ω) the induced potential u ∈ H 1 (Ω) solves the Dirichlet problem div(γ∇u) = 0 in Ω u = f on ∂Ω.
The Dirichlet-to-Neumann map, or voltage-to-current map, is given by Λ γ (f ) = (γ∂ ν u) ∂Ω where ν denotes the unit outer normal to ∂Ω. The Calderón problem is to determine γ from measurements of Λ γ . Substantial progress has been made on the Calderón problem since Calderón's paper. See [14] for an exposition of many of the main results. Uhlmann and Sylvester proved in [13] that for n ≥ 3, knowledge of Λ γ on the whole boundary uniquely determines γ ∈ C 2 (Ω) uniquely. Since then, there has been considerable work on reducing the assumption that γ ∈ C 2 (Ω). In [7] Haberman and Tataru improved this assumption to γ ∈ C 1 (Ω). Recently in [8] Haberman improved this assumption to γ with unbounded gradient in low dimensions, and in [4] Caro and Rogers proved uniqueness for Lipschitz γ in higher dimensions. For n = 2, Astala and Päivärinta proved in [1] that knowledge of Λ γ on the whole boundary uniquely determines γ ∈ L ∞ (Ω).
In the case that Λ γ is measured only on part of the boundary, it was first shown by Bukhgeim and Uhlmann in [2] that for n ≥ 3, knowledge of Λ γ on roughly half of the boundary determines γ ∈ C 2 (Ω) uniquely. The assumptions made in [2] on the structure of the subset where Λ γ is measured were greatly improved by Kenig et al. in [11], but their assumption on the regularity of the conductivity was the same. The regularity assumption made on the conductivity in [2] was improved by Knudsen in [12] to γ ∈ W 3/2+,2n (Ω). In [15] Zhang used ideas from [7] to reduce this further to γ ∈ C 1 (Ω). To the author's knowledge, there has been no improvement made to the regularity assumption in [11]. By considering a more general setting for the Calderón problem, we will improve the regularity assumption in the result from [11].
In this paper, we replace Ω with a compact n-dimensional Riemannian manifold (M, g). In particular, let γ be a bounded positive function on M . Let ν denote the unit outer normal to ∂M with respect to g, ∇ g denote the gradient operator on M , div g denote the divergence operator on M , and |g| = det g. In local coordinates (x i ) where g = (g ij ), for smooth functions u and vector fields X = X i ∂ i .
The inverse problem is to determine γ from the knowledge of Λ g,γ . In order to state our main result, we need a definition. Here a compact manifold (M 0 , g 0 ) with boundary is simple if for any ω ∈ M 0 , the exponential map exp ω with its maximal domain of definition is a diffeomorphism onto M 0 and if ∂M 0 is strictly convex (that is, the second fundamental form of ∂M 0 ֒→ M 0 is positive definite). Examples of admissible manifolds include compact submanifolds of Euclidean space, the sphere minus a point, and hyperbolic space.
The following theorem is the main result of this paper.
Theorem 1.2. Let (M, g) be an admissible manifold. Given ǫ > 0, define This theorem can be seen as a generalization of [12]. As a corollary, we obtain a strengthening of Theorem 1.2 from [11].
Let Ω ⊂ R n be a bounded domain with smooth boundary, and assume x 0 is not in the convex hull of Ω. Given ǫ > 0, define where ϕ(x) = log |x − x 0 |. Let γ 1 , γ 2 ∈ W 3/2+η,2n (Ω) for some η > 0. Suppose further that In the proof of Theorem 1.2, we will combine ideas from [12], [5], [6], [9], and [10]. We outline the method of proof. Let γ ∈ C 1 (M ) and let u be a solution to (1.1). Then u satisfies The first step is to construct a suitable family of complex geometrical optics (CGO) solutions to (1.2) using ideas from [12] and [10]. The second step is to use an Allessandri type integral identity to relate information on the boundary to the attenuated geodesic ray transform of the conductivities. A uniqueness result for the attenuated ray transform on simple manifolds allows us to conclude that the two conductivities agree on M .
The outline of this paper is as follows. In Section 2, we derive Carleman estimates for first and zeroth order perturbations of the Laplacian ∆ g . In Section 3, we constuct CGO solutions to (1.2). Finally, in Section 4 we prove Theorem 1.2 and and Theorem 1.3.
Acknowledgments: This work was completed during the author's doctoral studies. The author would like to thank his advisor, Carlos Kenig, for suggesting the problem and for his invaluable patience and guidance.

Carleman Estimate
Throughout the remainder of the paper, (M, g) is an admissible manifold with g = e ⊕ g 0 and ϕ(x) = x 1 . In particular, it suffices to prove Theorem 1.2 in the case c = 1. This reduction follows easily from the relations div cg (γ∇ cg u) = c − n 2 div g (γ∇ g u) , where the subscripts indicate that the normal derivatives are taken with respect to cg and g respectively. We also denote the inner product and norm (on tangent spaces) given by g by ·, · g and | · | g respectively.
Then there exists a constant τ 0 > 0 such that for |τ | ≥ τ 0 , we have the estimate where C, C ′ , C ′′ > 0 depends only on τ 0 and M .
Proof. The proof is similar to the proof of Proposition 2.1 in [12]. For v ∈ H 2 (M ) we write The Poincaré inequality with boundary term implies that for arbitrary δ > 0. Combining the above estimates and choosing δ > 0 sufficiently small implies that for |τ | ≥ τ 0 . Finally, a short calculation using local coordinates on M 0 shows that

Combining (2.2) with (2.3) and (2.4) yields (2.1).
An immediate corollary of the previous proposition is a Carleman estimate for the operator −∆ g + A, ∇ g g + q when the coefficient A is sufficiently small and q is bounded.
Suppose v ∈ H 2 (M ) and q, A ∈ L ∞ (M ). Let C be the constant in (2.1), and let τ 0 be given in the previous proposition. Then there exists A 0 > 0 andτ 0 > 0 satisfying

CGO Solutions
The main method for constructing CGO solutions to (1.1) and (1.2) is to first conjugate in the equation is an admissible manifold, then constructing CGO solutions to (1.1) and (1.2) is therefore reduced to constructing CGO solutions to a zeroth order perturbation of the Laplacian ∆ g = ∂ 2 1 + ∆ g 0 . However, the first step requires more smoothness that we assume on γ. Following [12], we proceed by introducing a smooth approximation of γ and conjugate the equation with the approximation.
Let γ ∈ W 3/2+η,2n (M ) be positive. We first extend γ to a function outside M such that γ − 1 ∈ W 3/2+η,2n 0 (R × M int 0 ). Let φ = log γ and A = ∇ g φ. We recall the following approximation result for functions W 3/2+η,2n 0 We remark that Lemma 3.1 from [12] is stated for M int 0 = R n . By a standard partition of unity argument, these estimates carry over to the current setting.
The CGO solutions to the equation (1.1) we will construct are of the form where ψ, v τ ∈ C ∞ (M ) will be chosen andr is a remainder term that tends to zero in a suitable norm as |τ | → ∞ (along a sequence). For the construction, fix a slightly larger simple manifoldM 0 such that M 0 ⊆M int 0 . We denote the determinant of the metric g 0 by |g 0 |. The main result of the section is the following.
. Write x = (x 1 , r, θ) where (r, θ) are polar normal coordinates with center ω. For |τ | sufficiently large outside a countable set, there exists a solution u ∈ H 1 (M ) of div g (γ∇ g u) = 0 in M of the form For the proof, we will need the following conjugation relations which are straightforward calculations: Note that by (3.2) and (3.4) we have that (3.10) as |τ | → ∞.
If we take as our ansatz v τ = e −iτ ψ a for some a ∈ C ∞ (M ), then we have that u solves (1.1) if and only if ψ a). It was established in [10] that −∆ g,τ has a bounded right inverse in appropriate spaces with norm decaying like |τ | −1 for τ outside a discrete set. In particular, the following proposition holds.

This operator satisfies
Proceeding perturbatively, we obtain an inverse for the operator −∆ g,τ + A − A τ , ∇ g g +q τ .

This operator satisfies
Proof. Given f ∈ L 2 (M ), we wish to solve We take as our ansatz v = G 0,τ g. Then v solves (3.12) if and only if g solves the integral equation Hence for |τ | sufficiently large and outside a discrete set we have that K L 2 (M )→L 2 (M ) < 1. Thus, we may solve (3.13) for g via a Neumann series and obtain v. The norm estimates for v follow from the corresponding norm estimates for G 0,τ .
The proof of Proposition 3.2 now follows easily from the previous preparations.

Uniqueness for Partial Data
We first state a result from [5].

Consider the integrals
where (r, θ) are polar normal coordinates in (M 0 , g 0 ) centered at some ω ∈ ∂M 0 . Here ·, · M int 0 denotes the dual pairing on M int 0 between F and C ∞ functions. If |λ| is sufficiently small, and if these integrals vanish for all ω ∈ ∂M 0 and all b ∈ C ∞ (S n−2 ), then F = 0. Finally, we have the following boundary integral identity from [12].
The above result is stated in [12] for M ⊆ R n , but their argument is based on the divergence theorem and easily carries over to the current setting.
For τ > 0 sufficiently large, let be our CGO solutions to div g (γ j ∇ g u j ) = 0 in M , j = 1, 2 which satisfy (3.8). In the notation from Section 3, we chose Lemma 4.3. Suppose the assumptions of Theorem 1.2 hold, and defineũ 1 , u 2 , and u 1 as above. Then Proof. The proof is similar to the proof from [12]. The fact that By the trace theorem and (3.8), we have for 0 < δ < η To estimate (4.3) further we introduce the function Since φ 1,τ for large τ is uniformly bounded from below by a positive constant andũ 1 = u 2 on ∂M , it follows that To prove the claim, observe γ 1 | ∂M = γ 2 | ∂M and ∂ ν γ 1 | ∂M = ∂ ν γ 2 | ∂M , (3.5) and (3.6) imply that By the trace theorem, for 0 < δ < η we have that The bound for ∂M e −2τ x 1 |δu| 2 dS g is proved similarly. This proves Claim 1.
The geodesic ray transform with constant attenuation −λ, acts on C ∞ functions on M 0 by Suppose the hypotheses of Lemma 4.1 are satisfied, and F = f ∈ C ∞ 0 (M int 0 ). Suppose (r, θ) are polar normal coordinates in (M 0 , g 0 ) centered at some ω ∈ ∂M 0 . Since M 0 is simple, (r, θ) are global coordinates on M 0 . Then the hypotheses of Lemma (4.1) read for any ω ∈ ∂M 0 and any θ 0 ∈ S n−2 . Hence the attenuated ray transform T λ f of f is identically zero for small λ. By the following injectivity result (see [6], Theorem 7.1), it follows that f = 0.
We can reduce the case when F is a distribution that is compactly supported in M int 0 to the previous case F ∈ C ∞ 0 (M int 0 ) by using duality and the ellipticity of the operator T * λ T λ . We will need a few facts about T λ and T * λ . We write for h ∈ C ∞ (∂ + (SM 0 )), and where µ(x, ξ) = − ξ, ν(x) and dN is the Riemannian volume form on a manifold N . From Section 5 of [5], we have the following facts.
Lemma A.2 (Santaló Formula). If F : SM 0 → R is continuous then Lemma A.4. T * λ T λ is a self-adjoint elliptic pseudodifferential operator of order −1 in M int 0 . We now turn to the proof of Lemma 4.1.
Since T * λ T λ is self-adjoint, the previous line implies that T * λ T λ F = 0. By ellipticity, there exists a pseudodifferential operator of order 1 in M int 0 denoted by B and a smoothing operator R : Thus, T * λ T λ F = 0 implies that F = −RF ∈ C ∞ (M int 0 ). We can now use the argument for smooth F from before to conclude the proof that F = 0.