A short PhD course (5 lectures, 2 hours each)
Program
Period: 716 May 2019 Dipartimento di Matematica Università di Roma "Tor Vergata" Room L3 Schedule:
Lecture 1: Weierstrass'
theorem for sequences of minimum problems.
Definition(s) of Gammaconvergence. Some topological
properties of Gammaconvergence. Examples.
Lecture 2: Gammaconvergence and lower semicontinuity. Convergence of local minima. Discretization of the Dirichlet integral. Gammaconvergence 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 Gammaconvergence. Equivalence by Gammaconvergence. Gradient theory of phase transitions as an expansion by Gammaconvergence Lecture 5: the MumfordShah functional and its finitedifference approximation Abstract
Scope of the course is an introduction to the description of limits of minimum problems using the terminology of Gammaconvergence. Definitions and first examples will be given at the level of firstyear calculus. We will then specialize our analysis on some "classical" examples of the Calculus of Variations. Reference A. Braides. Gammaconvergence for Beginners. Oxford University Press, Oxford, 2002. 
