Dottorato di Ricerca in Matematica - UniversitÓ di Pavia



Ph D course: Local minimization, variational evolution and Gamma-convergence
(prof. Andrea Braides)


Description of the course
We examine some issues on the description of sequences of energies with local minima. It is known that "coarse grained" theories (e.g. those obtained by Gamma-convergence) in general do not describe those local minima or their effect on the related gradient flows. We  focus then on:
1) some examples when a description of local minima of complex systems is possible thanks to a simpler energy which captures the main features of the systems;
2) elaboration of criteria that ensure the passage to the limit for "stable solutions" or local minimizers;
3) evolution problems for energies with many local minima obtained by a time-discrete scheme (minimizing movements along a sequence of energies).


Lecture Notes
Introduction (version of Nov 8 2012)
Chapter 1. Global minimization
(version of Nov 8 2012)
Chapter 2. Parameterized motion driven by global minimization (version of Nov 30 2012)
Chapter 3. Local minimization as a selection criterion (version of Nov 30 2012)
Chapter 4. Convergence of local minimizers
(version of Dec 6 2012)
Chapter 5. Stability (version of Jan 11 2013)
Chapter 6. Minimizing movements (version of Jan 11 2013)
Chapter 7. Minimizing movements along a sequence of functionals (version of Jan 19 2013)
Chapter 8. Geometric minimizing movements (version of Jan 21 2013)
Chapter 9. Different time scales (version of Jan 31 2013)
Chapter 10. Stability theorems (version of Feb 1 2013)

The notes of the course have been posted as soon as possible. This means that they are full of mistakes and misprints; please let me know as soon as you find them.

A slightly different (and corrected) version of the notes is contained in the book
A. Braides.
Local minimization, variational evolution and Gamma-convergence. Lecture Notes in Mathematics. Vol. 2094. Springer, Berlin, 2013.
If you need to quote results from the notes, please quote that book, following the numbering therein.

Schedule

November 2012
Wed 7, 14:30-16:30
Thu 8, 10:00-12:30
Wed 14, 15:30-17:30
Thu 15, 10:00-13:00
Wed 28,
15:30-17:30
Thu 29, 10:00-13:00

January 2013
Wed 9, 10:00-12:00
Thu 10, 10:00-13:00

Thu 17, 10:30-13:00
Fri 18, 9:30-12:00
Wed 30, 10:30-13:00
Thu 31, 10:30-13:00


Exams: March 14