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

Abstract

In this paper we investigate the boundary value problem ${div(γ\nabla u)=0 in Ω, u=f on \partialΩ$ 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.

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