Inverse anisotropic conductivity from internal current densities

This paper concerns the reconstruction of an anisotropic conductivity tensor $γ$ from internal current densities of the form $J = γ\nabla u$, where $u$ solves a second-order elliptic equation $\nabla\cdot(γ\nabla u) = 0$ on a bounded domain $X$ with prescribed boundary conditions. A minimum number of such functionals equal to $n + 2$, where $n$ is the spatial dimension, is sufficient to guarantee a local reconstruction. We show that $γ$ can be uniquely reconstructed with a loss of one derivative compared to errors in the measurement of $J$. In the special case where $γ$ is scalar, it can be reconstructed with no loss of derivatives. We provide a precise statement of what components may be reconstructed with a loss of zero or one derivatives.

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