Global Minimization

Abstract

The issues related to the behavior of global minimization problems along a sequence of functionals $F_{\varepsilon }$ are by now well understood, and mainly rely on the concept of Γ-limit. In this chapter we review this notion, which will be the starting point of our analysis. Besides the main concepts and the properties of convergence of minimum problems, we present some examples that will be examined in detail from the viewpoint of local minimization, such as elliptic homogenization (which presents local minimizers when lower-order terms are added), the gradient theory of phase transitions (which gives a limit with many local minimizers in one dimension), and linearized fracture mechanics as limit of Lennard-Joned system of atoms. A key issue in these examples is the use of scaling argument in the energy, sometimes formalized as a development by Γ-convergence.

Appendix

For an introduction to Γ-convergence we refer to the ‘elementary’ book [1]. More examples, and an overview of the methods for the computation of Γ-limits can be found in [2]. More detailed information on topological properties of Γ-convergence are found in [6]. The use of higher-order Γ-limits is analyzed in [5].

Homogenization results are described in [3]. The reference to the original Γ-limit of the gradient theory of phase transitions is [7]. A more detailed treatment of Example 2.6 can be found in [4].