Small-Scale Stability

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


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.


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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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