On uniqueness in nonlocal diffuse optical tomography

We investigate the inverse problem of recovering the diffusion and absorption coefficients (σ,q) in the nonlocal diffuse optical tomography equation (-div( σ ∇))ˢ u+q u =0 from the (partial) Dirichlet-to-Neumann map. The purpose of this article is twofold: (i) Firstly, we show that the diffusion coefficient σ and absorption coefficient q can be recovered simultaneously. (ii) Secondly, we prove that the absorption coefficient q can be determined provided σ is known in a neighborhood of the boundary ∂Ω. The key ingredients to prove these uniqueness results are the Caffarelli--Silvestre type extension technique and a novel Runge approximation related to solution spaces of two different partial differential equations, which is based on the geometric form of the Hahn--Banach theorem. The results in this work hold for any dimension n≥ 3.