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On the Main Aspects of the Inverse Conductivity Problem
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KeyWord:Calder{\'o}n problem; Inverse conductivity problem; Dirichlet-to-Neumann map; Complex geometrical optics solutions; Carleman estimate
Author NameAffiliation
Manal Aoudj School of Mathematics and Statistics, Central China Normal University, Wuhan, Hubei 430079, China 
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      We consider a nonlinear inverse problem for an elliptic partial differential equation known as the Calder{\'o}n problem or the inverse conductivity problem. Based on several results, we briefly summarize them to motivate this research field. We give a general view of the problem by reviewing the available results for $C^2$ conductivities. After reducing the original problem to the inverse problem for a Schr\"odinger equation, we apply complex geometrical optics solutions to show its uniqueness. After extending the ideas of the uniqueness 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 are taken only in a part of the boundary.