On the Main Aspects of the Inverse Conductivity Problem (Journal of Nonlinear Modeling and Analysis, 2024)
We consider a nonlinear inverse problem for an elliptic partial differential equation known as the Calderόn problem or the inverse conductivityproblem. Based on several results, we briefly summarize them to motivatethis research field. We give a general view of the problem by reviewing theavailable results for $C^2$ conductivities. After reducing the original problem tothe inverse problem for a Schrödinger equation, we apply complex geometrical optics solutions to show its uniqueness. After extending the ideas of theuniqueness proof result, we establish a stable dependence between the conductivity and the boundary measurements. By using the Carleman estimate,we discuss the partial data problem, which deals with measurements that aretaken only in a part of the boundary.
Keywords: Calderόn problem; Inverse conductivity problem; Dirichlet-to-Neumann map; Complex geometrical optics solutions; Carleman estimate.