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

Abstract

We investigate the problem of reconstructing a fully anisotropic conductivity tensor $\gamma$ from internal functionals of the form $\nabla u\cdot\gamma\nabla u$ where $u$ solves $\nabla\cdot(\gamma\nabla u) = 0$ over a given bounded domain $X$ with prescribed Dirichlet boundary condition. This work motivated by hybrid medical imaging methods covers the case $n\ge 3$, following the previously published case $n=2$ \cite{Monard2011}. Under knowledge of enough such functionals, and writing $\gamma = \tau \tilde \gamma$ ($\det \tilde\gamma = 1$) with $\tau$ a positive scalar function, we show that all of $\gamma$ can be explicitely and locally reconstructed, with no loss of scales for $\tau$ and loss of one derivative for the anisotropic structure $\tilde\gamma$. The reconstruction algorithms presented require rank maximality conditions that must be satisfied by the functionals or their corresponding solutions, and we discuss different possible ways of ensuring these conditions for $\C^{1,\alpha}$-smooth tensors ($0<\alpha<1$).