Advertisement

Global Minimization

  • Andrea Braides
Chapter
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.

References

  1. 1.
    Braides, A.: Γ-convergence for Beginners. Oxford University Press, Oxford (2002)CrossRefzbMATHGoogle Scholar
  2. 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)CrossRefGoogle Scholar
  3. 3.
    Braides, A., Defranceschi, A.: Homogenization of Multiple Integrals. Oxford University Press, Oxford (1998)zbMATHGoogle Scholar
  4. 4.
    Braides, A., Lew, A.J., Ortiz, M.: Effective cohesive behavior of layers of interatomic planes. Arch. Ration. Mech. Anal. 180, 151–182 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Braides, A., Truskinovsky, L.: Asymptotic expansions by gamma-convergence. Cont. Mech. Therm. 20, 21–62 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dal Maso, G.: An Introduction to Γ-convergence. Birkhäuser, Boston (1993)CrossRefGoogle Scholar
  7. 7.
    Modica, L., Mortola, S.: Un esempio di Γ-convergenza. Boll. Un. Mat. It. B 14, 285–299 (1977)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Andrea Braides
    • 1
  1. 1.Dipartimento di MatematicaUniversità di Roma Tor VergataRomaItaly

Personalised recommendations