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.

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We consider the problem of identification of a pollution source in a river. The mathematical model is a one-dimensional linear advection-dispersion-reaction equation with the right-hand side spatially supported at a point (the source) and a time-dependent intensity, both unknown. Assuming that the source becomes inactive after the time T * , we prove that it can be identified by recording the evolution of the concentration at two points, one of which is strategic.

We consider the problem of identification of a heat source in a bounded domain Ω. Assuming that the point sources became inactive after the time T * , we prove that they can be identified by measurements of the heat flux on Γ0 × (0, T ), where Γ0 is a part of the boundary of Ω, with non void interior, and T > T * . By a standard trandformation, we derive from these results a method to identify the polluting sources on the surface of a river, a lake . . .