Advertisement

Introduction

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

Abstract

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.

Keywords

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

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

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

Personalised recommendations