The paper gives quadratic-form reconstruction formulas for the boundary value and normal derivative of the conductivity from the Dirichlet-to-Neumann map, and deduces bulk uniqueness for conductivities in W^{1+(n-5)/(2p)+,p}, n>=5.
The Bilinear Strategy for Calder\'on's Problem
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abstract
Electrical Impedance Imaging would suffer a serious obstruction if for two different conductivities the potential and current measured at the boundary were the same. The Calder\'on's problem is to decide whether the conductivity is indeed uniquely determined by the data at the boundary. In $\mathbb{R}^d$, for $d=5,6$, we show that uniqueness holds when the conductivity is in $W^{1+\frac{d-5}{2p}+, p}(\Omega)$, for $d\le p <\infty$. 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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2019 1verdicts
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Recovery of the Derivative of the Conductivity at the Boundary
The paper gives quadratic-form reconstruction formulas for the boundary value and normal derivative of the conductivity from the Dirichlet-to-Neumann map, and deduces bulk uniqueness for conductivities in W^{1+(n-5)/(2p)+,p}, n>=5.