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

We investigate the problem of reconstructing a fully anisotropic conductivity tensor $γ$ from internal functionals of the form $\nabla u\cdotγ\nabla u$ where $u$ solves $\nabla\cdot(γ\nabla u) = 0$ 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 $γ= τ\tilde γ$ ($\det \tildeγ= 1$) with $τ$ a positive scalar function, we show that all of $γ$ can be explicitely and locally reconstructed, with no loss of scales for $τ$ and loss of one derivative for the anisotropic structure $\tildeγ$. 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 $\C^{1,α}$-smooth tensors ($0<α<1$).