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 (small or large perturbations of a homogeneous situation) and their relative period as compared with the characteristic length of the phase transitions (slow or fast oscillations).