Inverse anisotropic conductivity from power densities in dimension n ≥ 3

We investigate the problem of reconstructing a fully anisotropic conductivity tensor γ from internal functionals of the form ∇u · γ∇u where u solves ∇ · (γ∇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 ≥ 3, following the previously published case n = 2 [21]. Under knowledge of enough such functionals, and writing γ = τ γ̃ (det γ̃ = 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 γ̃. 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-smooth tensors (0 < α < 1).