Lipschitz Stability for the Electrical Impedance Tomography Problem: The Complex Case

Abstract

In this article we investigate the boundary value problem

where γ is a complex valued L ^∞ coefficient, satisfying a strong ellipticity condition. In electrical impedance tomography, γ represents the admittance of a conducting body. An interesting issue is the one of determining γ uniquely and in a stable way from the knowledge of the Dirichlet-to-Neumann map Λ_γ. Under the above general assumptions this problem is an open issue. In this article we prove that, if we assume a priori that γ is piecewise constant with a bounded known number of unknown values, then Lipschitz continuity of γ from Λ_γ holds.

Keywords:

• Inverse admittivity problem
• Lipschitz continuous dependence

Mathematics Subject Classification:

• 35J47
• 35R30

Acknowledgments

We would like to thank Micol Amar, Daniele Andreucci, Paolo Bisegna and Roberto Gianni for having pointed out the importance of this problem in connection to the study of a model of conduction in biological tissues. We also thank Sergio Vessella for several useful discussions.