  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
$\R_+^d$. The solution is characterized in terms of variational inequalities.
Finally, we review a few examples taken from the recent literature that are
related to our results.
