Weakly admissible vector equilibrium problems

We establish lower semi-continuity and strict convexity of the energy functionals for a large class of vector equilibrium problems in logarithmic potential theory. This, in particular, implies the existence and uniqueness of a minimizer for such vector equilibrium problems. Our work extends earlier results in that we allow unbounded supports without having strongly confining external fields. To deal with the possible noncompactness of supports, we map the vector equilibrium problem onto the Riemann sphere and our results follow from a study of vector equilibrium problems on compacts in higher dimensions. Our results cover a number of cases that were recently considered in random matrix theory and for which the existence of a minimizer was not clearly established yet.