In this talk we prove existence and uniqueness of a solution to the problem of minimizing the logarithmic
energy of vector potentials associated to a d-tuple of positive measures supported on closed subsets of the
complex plane. The assumptions we make on the interaction matrix are weaker than the usual ones and
we also let the masses of the measures vary in a compact subset of Rd
+. We characterize the solution in
terms of variational equations. Finally, we review a few examples taken from the recent literature [1,2,3,4]
that are related to our results.