Multiscale analysis of a prototypical model for the interaction between microstructure and surface energy

Abstract

The combined effect of fine heterogeneities and small gradient perturbations is analyzed by means of an asymptotic development by $\Gamma$-convergence for a family of energies related to (one-dimensional) phase transformations. We show that multi-scale effects add up to the usual sharp-interface limit, due to the homogenization of microscopic interfaces, internal and external boundary layers, optimal arrangements of microscopic oscillations, etc. Several regimes are analyzed depending on the ``size'' of the heterogeneity ({\em small or large perturbations} of a homogeneous situation) and their relative period as compared with the characteristic length of the phase transitions ({\em slow or fast oscillations}).