The Calderón Problem for Conormal Potentials, I: Global Uniqueness and Reconstruction

The goal of this paper is to establish global uniqueness and obtain reconstruction, in dimensions n ≥ 3, for the Calderón problem in the class of potentials conormal to a smooth submanifold H in R. In the case of hypersurfaces, the potentials considered here may have any singularity weaker than that of the delta function δH on the hypersurface H ; in general, these potentials correspond to conductivities which are in C and thus fail to be covered by previously known results. Let Ω ⊂ R be a bounded Lipschitz domain, H ⊂ Ω a smooth submanifold of codimension k, and q ∈ I(H) a real conormal distribution of order μ with μ < 1 − k. Thus, if H = {x : Fj(x) = 0, 1 ≤ j ≤ k} is a local representation of H by means of defining functions with {∇Fj : 1 ≤ j ≤ k} linearly independent on H , then locally q(x) has the Fourier integral representation