Inverse anisotropic conductivity from internal current densities

Abstract

This paper concerns the reconstruction of an anisotropic conductivity tensor $\gamma$ from internal current densities of the form $J=\gamma \nabla u$, where $u$ solves a second-order elliptic equation $\nabla \cdot (\gamma \nabla u)=0$ 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 $\gamma$ can be uniquely reconstructed with a loss of one derivative compared to errors in the measurement of $J$. In the special case where $\gamma$ 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.

27 pages, submitted to Inverse Problems (March 2013)

References in corpus (1)

• Inverse anisotropic conductivity from power densities in dimension $n\ge 3$

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