PhD in Pure and Applied Mathematics

The course aims at introducing to the students the concept of variational convergence, which is particularly suited to study limits of integral functionals and of minimization problems.
The minimization of an integral functional is the mathematical tool to search for equilibrium configurations of a physical system. One often encounters the situation in which the limit of a sequence of a family of functionals is sought for, or needs to approximate a given functional (which is possibly hard to study) by some suitable approximations. Γ-convergence provided the right tools to deal with this limit process, also granting, under suitable hypotheses, the convergence of the minima of the approximating functionals to the minimum of the limit one.
After introducing the basic notions of Γ-convergence, the course will cover some classical problems, as well as some research problems.
The exam will consist of a seminar in which the students will present a research paper or their original solution to a problem to be agreed upon with the teacher.

Introduction to Γ-convergence: Γ-convergence by numbers, functions, functionals.
Γ-convergence in abstract spaces. Convergence of minima.
The Modica-Mortola functional.
The Ambrosio-Tortorelli functional.
Relaxation.
Homogenization.