Calculus of Variations and Geometric Measure Theory

A. Braides - C. I. Zeppieri

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

created by zeppieri on 28 Sep 2007
modified by braidesa on 21 Apr 2009


Published Paper

Inserted: 28 sep 2007
Last Updated: 21 apr 2009

Journal: Interfaces Free Bound.
Volume: 11
Number: 1
Pages: 61-118
Year: 2009


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

Keywords: Homogenization, phase transitions, $\Gamma$-convergence, $\Gamma$-development