A short PhD course (5 lectures, 2 hours each)

Period: 7-16 May 2019
Dipartimento di Matematica
Università di Roma "Tor Vergata"
Room L3

Lecture 1: Weierstrass' theorem for sequences of minimum problems. Definition(s) of Gamma-convergence. Some topological properties of Gamma-convergence. Examples.

Lecture 2: Gamma-convergence and lower semicontinuity. Convergence of local minima. Discretization of the Dirichlet integral. Gamma-convergence in Sobolev spaces. Elliptic homogenization.

Lecture 3:  Homogenization theorems: formulas for the homogenized energy densities. Periodic convex homogenization in dimension one. Remarks on higher dimension. Homogenization of metrics and of Hamiltonian systems.

Lecture 4: Development by Gamma-convergence. Equivalence by Gamma-convergence. Gradient theory of phase transitions as an expansion by Gamma-convergence

Lecture 5: the Mumford-Shah functional and its finite-difference approximation

Abstract
Scope of the course is an introduction to the description of limits of minimum problems using the terminology of Gamma-convergence. Definitions and first examples will be given at the level of first-year calculus. We will then specialize our analysis on some "classical" examples of the Calculus of Variations.

Reference

A. Braides. Gamma-convergence for Beginners. Oxford University Press, Oxford, 2002.