Γ-convergence: a 50 years long story

Period: Tuesdays and Thursdays from November 5th to December 5th, 2024
Time Schedule: 11h to 13h
November: days 5, 7, 12, 14, 19, 21, 26, 28
December: days 3, 5

Location: FACULTY OF MATHEMATICS (Universitat de Barcelona)

Classrooms:

– Aula S4 – EH: November 5, 7, 12, 19 and 26, December 3.

– Aula B2 – EH: November 14, 21 and 28, December 5.

The Γ-convergence was introduced 50 years ago by Ennio De Giorgi and Tullio Franzoni, and occupies a prominent place in the world of variational convergences due to its applications in mechanics.

It is the commonly recognised notion of convergence for variational problems, and it would be difficult nowadays to think of any other notion of a limit than a Γ-limit, when talking about the asymptotic analysis in a general variational context.

Although other notions of convergences may suit better to specific problems (such as Mosco-convergence, two-scale convergence, G-convergence and H-convergence), it has been well understood that almost all other notions of convergence can be expressed through Γ-convergence.

For these reasons, in recent decades this topic has become of great interest in the mechanics-oriented literature as a powerful application tool for mathematicians and engineers.

In these series of advanced lectures, we devote ourselves both to the theoretical aspects of Γ-convergence, through the famous monograph by Dal Maso, and to the most famous and relevant applications in mechanics, through the monographs by Braides and Braides-Defranceschi. Lecture notes are also provided, aimed to recalling all the results discussed in class.

The main topic of the first part of the course is the Fundamental theorem of Γ-convergence by De Giorgi and Franzoni, which guarantees the good behaviour of sequences of minimizers and minima in the topology of the Γ-convergence. As an application of this result, in the second part of the course we will study problems ranging from the theory of elliptic partial differential equations (G-convergence and H-convergence), to applications in the mathematical theory of composite materials, with particular interest in the theory of periodic homogenization, and in the pioneering result of Modica and Mortola for the mathematical treatment of phase transition problems. For this now classic result, we refer to a detailed survey by Giovanni Alberti.

These lectures are recommended for students in Mathematics and early-career researchers who wish to specialise in the field of analysis. They are also intended for an audience of experienced mathematicians with some background and interest in PDEs and applications in material science.