Some minimization problems for mean field models with competing forces

Creators: Frank, Rupert L.

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.

Additional Information

© 2021 by the author. This paper may be reproduced, in its entirety, for non-commercial purposes. The author would like to thank the organizers of the 8th European Congress of Mathematics for the organization of the meeting and for the invitation to speak. Since the topic of his invited talk was recently and rather exhaustively reviewed in [25], this contribution is based on a talk in a minisymposium at the congress, organized by L. Pick, to whom the author is very grateful. The results reviewed here were obtained in collaboration with many researchers and it is a pleasure thank, in particular, José Carrillo, Matías Delgadino, Jean Dolbeault, Franca Hoffmann, Rowan Killip, Mathieu Lewin, Elliott Lieb and Phan Thành Nam for many stimulating discussions. Partial support through U.S. National Science Foundation grants DMS-1363432 and DMS-1954995 and through the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through Germany's Excellence Strategy EXC-2111-390814868 is acknowledged.

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