2019/09/26 by Genqian Liu, Liu, Genqian
Engineering · Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #Electrical and Bioimpedance Tomography #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Microwave Imaging and Scattering Analysis #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1909.12803
openalex publication_date 2019/09/26 · openalex created_date 2019/10/03 · openalex updated_date 2026/07/28
For a compact, connected, oriented Riemannian 3-manifold (M, g) with smooth boundary ∂ M, we explicitly give a local representation and a full symbol expression for the electromagnetic Dirichlet-to-Neumann map by factorizing Maxwell's equations and using an isometric transform. We prove that one can reconstruct a compact, connected, real-analytic Riemannian 3-manifold M with boundary from the set of tangential electric fields and tangential magnetic fields, given on a non-empty open subset Γ of the boundary, of all electric and magnetic fields with tangential electric data supported in Γ. We note that for this result we need no assumption on the topology of the manifold other than compactness and connectedness, nor do we need a priori knowledge of all of ∂ M. In addition, as a by-product of the explicit symbol expression of Λg,Γ, we show that for a given smooth Riemannian metric g, the electromagnetic Dirichlet-to-Neumann map Λg,Γ uniquely determines all order tangential and normal derivatives of electromagnetic parameters μ and σ on Γ. Therefore, μ and σ are completely determined in M by Λg,Γ if these two parameter functions and metric g are all real analytic in M up to Γ.