Concentration-compactness method in calculus of variations

(Reading seminar Winter 2024-2025)

Phan Thành Nam, Moodle (ID: 37144, pass: Compactness).

General Information

Description: The concentration-compactness method is a powerful technique to deal with the lack of compactness in various function spaces. It is very helpful to obtain the existence of optimizers, as well as the compactness of minimizing sequences, for a large class of variational problems, including for example Sobolev embeddings and nonlinear Schrödinger equations. In the seminar, we will discuss in particular the missing mass method of Lieb and the profile decomposition method of Lions and their applications to nonlinear problems in mathematical physics.

Audience : Bachelor and Master students of Mathematics.

Credits: 3 ECTS.

Language: English.

Time and place: Friday 14:15-16:00 (B251).

References:

• E. H. Lieb, Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities, Annals of Math., 1983.
• P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Part 1, Ann. de L'IHP, 1984.
• P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Part 2, Ann. de L'IHP, 1984.
• P.L. Lions, The concentration-compactness principle in the Calculus of Variations. The limit case, Part 1, Revista Mat. Iberoamericana, 1985.
• P.L. Lions, The concentration-compactness principle in the Calculus of Variations. The limit case, Part 2, Revista Mat. Iberoamericana, 1985.
• P. Gérard. Description du défaut de compacité de l'injection de Sobolev. ESAIM COCV 1998.
• R. Killip and M. Visan, Nonlinear Schrödinger Equations at Critical Regularity. Clay Mathematics Proceedings. 2013
• T. Tao, Concentration compactness and the profile decomposition. Blog November 5, 2008.
• M. Lewin, Describing lack of compactness in Sobolev spaces, from Variational Methods in Quantum Mechanics. Lecture Notes 2010.
• J. Sabin. Compactness methods in Lieb’s work. Review paper 2022.
• C. Dietze and P. T. Nam. Minimizing sequences of Sobolev inequalities revisited. Preprint 2024.

Schedule:

25.10.2024. Introduction and distribution of the reading material.

15.11. Phan Thành Nam: Rearrangement inequalities and existence of sub-critical Sobolev optimizers.

22.11. David Scholz: Existence of critical Sobolev optimizers via rearrangement inequalities.

29.11. Riccardo Panza: Lion's L^1-concentration compactness lemma and existence of Thomas-Fermi type minimizers.

13.12. Boyan Angelov Kugiyski: Lion's H^1-concentration compactness lemma and existence of sub-critical Sobolev optimizers.

10.1.2025 Boyan Angelov Kugiyski: Lion's H^1-concentration compactness lemma and existence of sub-critical Sobolev optimizers (part 2).

17.1. Phan Thành Nam: Lion's proof of the existence of critical Sobolev optimizers and compactness of minimizing sequences.

24.1. David Scholz: Critical Sobolev inequality: compactness via a refined inequality.

31.1. Further approaches and concluding remarks.