Abstract
We prove uniqueness in the inverse conductivity problem for uniformly elliptic conductivities in \({W^{s,p}(\Omega)}\), where \({\Omega \subset \mathbb{R}^{n}}\) is Lipschitz, \({3\leq n \leq 6}\), and s and p are such that \({ W^{s,p}(\Omega)\not \subset W^{1,\infty}(\Omega)}\). In particular, we obtain uniqueness for conductivities in \({W^{1,n}(\Omega)}\) (n = 3, 4). This improves on the result of the author and Tataru, who assumed that the conductivity is Lipschitz.
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Communicated by P. Deift
This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE 1106400. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.
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Haberman, B. Uniqueness in Calderón’s Problem for Conductivities with Unbounded Gradient. Commun. Math. Phys. 340, 639–659 (2015). https://doi.org/10.1007/s00220-015-2460-3
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DOI: https://doi.org/10.1007/s00220-015-2460-3