Some minimization problems for mean field models with competing forces

TL;DR

This paper reviews recent findings on minimization problems in mean field models with competing attractive and repulsive forces, highlighting phase transitions and existence of minimizers across different regimes.

Contribution

It provides a comprehensive overview of three classes of models, emphasizing the effects of force competition on minimizer existence and phase transitions.

Findings

Existence of minimizers depends on parameter regimes.

Transitions occur between regimes with and without minimizers.

Different models exhibit similar phase transition phenomena.

Abstract

We review recent results on three families of minimization problems, defined on subsets of nonnegative functions with fixed integral. The competition between attractive and repulsive forces leads to transitions between parameter regimes, where minimizers exist and where they do not. The problems considered are generalized liquid drop models, swarming models and generalized Keller-Segel models.