Weakly Admissible Vector Equilibrium Problems

TL;DR

This paper proves the existence and uniqueness of solutions for a broad class of vector equilibrium problems in logarithmic potential theory, even with unbounded supports, by mapping to the Riemann sphere.

Contribution

It extends previous results by allowing unbounded supports and noncompactness, providing a unified approach to establish minimizer existence and uniqueness.

Findings

Established lower semi-continuity and strict convexity of energy functionals.

Proved existence and uniqueness of minimizers for vector equilibrium problems.

Extended applicability to cases in random matrix theory.

Abstract

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.