# 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.

## Keywords

Minimum Problem Lipschitz Continuity Choice Criterion Recovery Sequence Pointwise Limit## References

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