Inverse Conductivity Problem on Riemann Surfaces

An electrical potential U on a bordered Riemann surface X with conductivity function σ>0 satisfies equation d(σdcU)=0. The problem of effective reconstruction of σ from electrical currents measurements (Dirichlet-to-Neumann mapping) on the boundary: U|bX↦σdcU|bX is studied. We extend to the case of Riemann surfaces the reconstruction scheme given, firstly, by R. Novikov (Funkc. Anal. Ego Priloz. 22:11–22, 2008) for simply connected X. We apply for this new kernels for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\bar{ \partial }$\end{document} on the affine algebraic Riemann surfaces constructed in Henkin (arXiv:0804.3761, 2008).