An inverse source problem in potential analysis

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Abstract
This paper discusses some aspects of an inverse source problem for elliptic equations, with observations on the boundary of the domain. The main application aimed at is the problem of identifying electrostatic dipoles in the human head where the boundary data are collected via electrodes placed on a part of the head. An uniqueness result is established for dipolar sources. Through solving a finite number of Cauchy problems, one arrives at an inverse problem in the homogeneous case. Assuming the number of dipoles bounded by a known integer M, we have established an algorithm which allows us to identify the number, the locations and moments of the dipoles by algebraic considerations. Other types of sources are also considered.