In this article we highlight how the Fonseca and Muller blow-up technique is particularly well suited for homogenization problems. As examples we give a simple proof of the non-linear homogenization theorem for integral functionals and we prove a homogenization theorem for sets of finite perimeter

We study by-convergence the discrete-to-continuum limit of the Blume-Emery-Griffiths model describing the phase transition of a binary mixture in presence of a third surfactant phase. In the case of low surfactant concentration we study the dependence of the surface tension on the density of the surfactant and we describe the microstructure of the ground states. We then consider more general (n-dimensional) energies modeling phase transitions in presence of different species of surfactants and, in the spirit of homogenization theory, we provide an integral representation result for theirlimit. As an application we study the ground states of these systems for prescribed volume fractions of the phases.

We give a Γ-convergence result for vector-valued nonlinear energies defined on periodically perforated domains. We consider integrands with n-growth where n is the space dimension, showing that there exists a critical scale for the perforations such that the Γ-limit is non-trivial. We prove that the limit extra-term is given by a formula of homogenization type, which simplifies in the case of n-homogeneous energy densities.

We analyze the overall behavior of discrete systems in the presence of defects (modeled by truncated quadratic potentials for some of the interactions) by exhibiting approximate free-discontinuity continuous energies. We give bounds on the limit surface energy densities, and we prove that these bounds are sharp in the classes of energy densities depending only on the normal to the discontinuity set, or concave and depending only on the jump across the interface. As a preliminary result we give a continuous descriptions of the ‘discrete Neumann sieve’

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