We consider a fractal refinement of the Carleson problem for the Schrödinger equation, that is to identify the minimal regularity needed by the solutions to converge pointwise to their initial data almost everywhere with respect to the α-Hausdorff measure (α-a.e.). We extend to the fractal setting (α < n) a recent counterexample of Bourgain [5], which is sharp in the Lebesque measure setting (α = n). In doing so we recover the necessary condition from [23] for pointwise convergence α-a.e. and we extend it to the range n/2 < α ≤ (3n + 1)/4.

We describe a method to reconstruct the conductivity and its normal derivative at the boundary from the knowledge of the potential and current measured at the boundary. The method of reconstruction works for isotropic conductivities with low regularity. This boundary determination for rough conductivities implies the uniqueness of the conductivity in the whole domain Ω when it lies in W 1 n¡5 2p ,p pΩq, for dimensions n ¥ 5 and for n ¤ p 8.

Electrical Impedance Imaging would suffer a serious obstruction if two different conductivities yielded the same measurements of potential and current at the boundary. The Calderón's problem is to decide whether the conductivity is indeed uniquely determined by the data at the boundary. In R d , for d ě 5, we show that uniqueness holds when the conductivity is in W 1`d´5 2p`, p pΩq, for d ď p ă 8. This improves on recent results of Haberman, and of Ham, Kwon and Lee. The main novelty of the proof is an extension of Tao's Bilinear Theorem.

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