Skip to main content

Global Minimization

Part of the Lecture Notes in Mathematics book series (LNM,volume 2094)

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

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   49.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions
Fig. 2.1

References

  1. Braides, A.: Γ-convergence for Beginners. Oxford University Press, Oxford (2002)

    CrossRef  MATH  Google Scholar 

  2. Braides, A.: A Handbook of Γ-convergence. In: Chipot, M., Quittner, P. (eds.) Handbook of Partial Differential Equations. Stationary Partial Differential Equations, vol. 3, pp. 101–213. Elsevier, Amsterdam (2006)

    CrossRef  Google Scholar 

  3. Braides, A., Defranceschi, A.: Homogenization of Multiple Integrals. Oxford University Press, Oxford (1998)

    MATH  Google Scholar 

  4. Braides, A., Lew, A.J., Ortiz, M.: Effective cohesive behavior of layers of interatomic planes. Arch. Ration. Mech. Anal. 180, 151–182 (2006)

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Braides, A., Truskinovsky, L.: Asymptotic expansions by gamma-convergence. Cont. Mech. Therm. 20, 21–62 (2008)

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. Dal Maso, G.: An Introduction to Γ-convergence. Birkhäuser, Boston (1993)

    CrossRef  Google Scholar 

  7. Modica, L., Mortola, S.: Un esempio di Γ-convergenza. Boll. Un. Mat. It. B 14, 285–299 (1977)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Appendix

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].

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Braides, A. (2014). Global Minimization. In: Local Minimization, Variational Evolution and Γ-Convergence. Lecture Notes in Mathematics, vol 2094. Springer, Cham. https://doi.org/10.1007/978-3-319-01982-6_2

Download citation