Inverse conductivity problem on Riemann surfaces

TL;DR

This paper extends the inverse conductivity problem to Riemann surfaces, proposing a reconstruction scheme for conductivity functions using advanced kernels, thereby broadening the applicability of electrical impedance tomography.

Contribution

It generalizes Novikov's reconstruction method from simply connected domains to Riemann surfaces using new kernels for the dbar operator.

Findings

Reconstruction scheme successfully extended to Riemann surfaces.

New kernels for dbar enable effective conductivity reconstruction.

Method applicable to affine algebraic Riemann surfaces.

Abstract

An electrical potential U on a bordered Riemann surface X with conductivity function sigma>0 satisfies equation d(sigma d^cU)=0. The problem of effective reconstruction of sigma is studied. We extend to the case of Riemann surfaces the reconstruction scheme given, firstly, by R.Novikov (1988) for simply connected X. We apply for this new kernels for dbar on affine algebraic Riemann surfaces constructed in Henkin, arXiv:0804.3761