*Book*

**Inserted:** 13 feb 2013

**Last Updated:** 23 jul 2013

**Journal:** Lecture Notes in Mathematics, Springer

**Volume:** 2094

**Year:** 2013

**Notes:**

Note that the book version is slightly different from the preprint version. In particular the numbering of chapters and theorems is different. If you quote some result please use the numbering of the book.

**Abstract:**

Local minimization, variational evolution and Gamma-convergence

In these lecture notes we examine some issues related to 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 local minima, or their effect on the related gradient flows. After a first part where we review the most relevant examples of multi-scale energies, we focus on: 1) some examples when an asymptotic 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) definition of effective evolution problems for energies with many local minima, obtained by a time-discrete scheme (minimizing movements along a sequence of energies).

Contents:

1 Global minimization

2 Parameterized motion driven by global minimization

3 Local minimization as a selection criterion

4 Convergence of local minimizers

5 Small-scale stability

6 Minimizing movements

7 Minimizing movements along a sequence of functionals

8 Geometric minimizing movements

9 Different time scales

10 Stability theorems

Lecture notes of a PhD Course (Sapienza Roma 2012, Pavia 2012-2013)

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