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Small-Scale Stability

  • Andrea Braides
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2094)

Abstract

While it is possible to deduce the existence and convergence of local minimizers of \(F_{\varepsilon }\) from the existence of an isolated local minimizer of their Γ-limit F, the knowledge of the existence of local minimizers of \(F_{\varepsilon }\) is not sufficient to deduce the existence of local minimizers for F. In this chapter we examine a notion of stability such that, loosely speaking, a point is stable if it is not possible to reach a lower-energy state from that point without crossing an energy barrier of a specified height. This notion is a quantification of the notion of local minimizer, which instead is “scale-independent” and will allow us to state a convergence theorem for sequences of stable points. Even though a general result is not available, we will show how Γ-converging sequences often give rise to stable convergence.

References

  1. 1.
    Braides, A., Larsen, C.J.: Γ-convergence for stable states and local minimizers. Ann. Scuola Norm. Sup. Pisa 10, 193–206 (2011)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Focardi, M.: Γ-convergence: a tool to investigate physical phenomena across scales. Math. Mod. Meth. Appl. Sci. 35, 1613–1658 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Larsen, C.J.: Epsilon-stable quasi-static brittle fracture evolution. Comm. Pure Appl. Math. 63, 630–654 (2010)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

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

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