Lipschitz stability for the electrical impedance tomography problem: the complex case

TL;DR

This paper proves Lipschitz stability for the inverse problem of determining a complex-valued conductivity in electrical impedance tomography, assuming the conductivity is piecewise constant with a known number of regions.

Contribution

It establishes Lipschitz stability for complex conductivities in EIT under piecewise constant assumptions, advancing the understanding of stability in the inverse problem.

Findings

Lipschitz stability holds for piecewise constant complex conductivities.

Stability result applies under a priori bounds on the number of unknown regions.

The analysis extends stability results to the complex-valued case in EIT.

Abstract

In this paper we investigate the boundary value problem ${div(\gamma\nabla u)=0 in \Omega, u=f on \partial\Omega$ where $\gamma$ is a complex valued $L^\infty$ coefficient, satisfying a strong ellipticity condition. In Electrical Impedance Tomography, $\gamma$ represents the admittance of a conducting body. An interesting issue is the one of determining $\gamma$ uniquely and in a stable way from the knowledge of the Dirichlet-to-Neumann map $\Lambda_\gamma$. Under the above general assumptions this problem is an open issue. In this paper we prove that, if we assume a priori that $\gamma$ is piecewise constant with a bounded known number of unknown values, then Lipschitz continuity of $\gamma$ from $\Lambda_\gamma$ holds.