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


In this book we study problems related to the asymptotic behaviour of energies \(F_{\varepsilon }\) depending on a small parameter \(\varepsilon\) as this parameter tends to zero, facing some issues related to local minimization and variational evolution. For global minimization and quasistatic motion the Γ-limit F of \(F_{\varepsilon }\), if suitably defined, provides a description of the limit behaviour of minimum problems. If the picture of local minima is not maintained by the Γ-limit, the latter can be perturbed in a systematic way so as to have Γ-equivalent energies with the same pattern of local minima. For strict local minimizers of F we can deduce existence (and some times, multiplicity) of local minima for \(F_{\varepsilon }\). Conversely, some Γ-converging sequences can be shown to be have stable critical points that are maintained for F. Dynamical problems can be faced by introducing the notion of minimizing movement along \(F_{\varepsilon }\), depending on a time-scale τ. In general, this minimizing movement depends on the \(\varepsilon\)-τ regime, but can always be compared with the minimizing movement for F. Interesting problems related to this notion are computation of critical \(\varepsilon\)-τ regimes, definition of effective, long-time and backwards motions.


Local Minimizer Euler Scheme Gradient Flow Strict Local Minimizer Quasi Static Evolution 
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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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