Abstract
An electrical potential U on a bordered real surface X in ℝ3 with isotropic conductivity function σ>0 satisfies the equation d(σ d c U)| X =0, where \(d^{c}= i(\bar{ \partial }-\partial )\), \(d=\bar{ \partial }+\partial \) are real operators associated with a complex (conformal) structure on X induced by the Euclidean metric of ℝ3. This paper gives an exact reconstruction of the conductivity function σ on X from the Dirichlet-to-Neumann mapping U| bX →σ d c U| bX . This paper extends to the case of Riemann surfaces the reconstruction schemes of R. Novikov (Funkt. Anal. Prilozh. 22(4):11–22, 1988) and of A. Bukhgeim (J. Inv. Ill-posed Probl. 16:19–34, 2008), given for the case X⊂ℝ2. The paper extends and corrects the statements of Henkin and Michel (J. Geom. Anal. 18:1033–1052, 2008), where the inverse boundary value problem on the Riemann surfaces was first considered.
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Henkin, G.M., Novikov, R.G. On the Reconstruction of Conductivity of a Bordered Two-dimensional Surface in ℝ3 from Electrical Current Measurements on Its Boundary. J Geom Anal 21, 543–587 (2011). https://doi.org/10.1007/s12220-010-9158-8
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DOI: https://doi.org/10.1007/s12220-010-9158-8