Introduction

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.