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

In this paper we investigate the boundary value problem ${div(\gamma\nabla u)=0 in \Omega, u=f on \partial\Omega$where$\gamma$isacomplexvalued$L^\infty$ 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 ${\mathrm{\Lambda }}_{\gamma }$. 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 ${\mathrm{\Lambda }}_{\gamma }$ holds.