We study the two-scale homogenization of the diffraction interfacial condition for the diffusion equation relevant to a composite medium which has a periodic structure. The results are applied to the electric field potential within a dielectric composite body when there is a difference in dielectric permittivity between the composite components in the presence of interfacial static charges. The principal result is that the interfacial charge distribution is equivalent to an apparent bulk charge which can be calculated starting from the composite geometry. We perform the corrector analysis and establish that the corrector terms strongly depend on the interfacial charge.

In this paper, we use the adapted periodic unfolding method to study the homogenization and corrector problems for the parabolic problem in a two-component composite with ε-periodic connected inclusions. The condition imposed on the interface is that the jump of the solution is proportional to the conormal derivative via a function of order εγ with γ ≤−1. We give the homogenization results which include those obtained by Jose in [Rev. Roum. Math. Pures Appl. 54 (2009) 189–222]. We also get the corrector results.

A corrector theory for the strong approximation of gradient fields inside periodiccomposites made from two materials with different power law behavior is provided. Eachmaterial component has a distinctly different exponent appearing in the constitutive lawrelating gradient to flux. The correctors are used to develop bounds on the localsingularity strength for gradient fields inside micro-structured media. The bounds aremulti-scale in nature and can be used to measure the amplification of applied macroscopicfields by the microstructure. The results in this paper are developed for materials havingpower law exponents strictly between  −1 and zero.

The aim of this paper is to provide the correctors associated to the homogenization of a parabolic problem describing the heat transfer. The results here complete the earlier study in [Jose, Rev. Roumaine Math. Pures Appl.54 (2009) 189–222] on the asymptotic behaviour of a problem in a domain with two components separated by an ε-periodic interface. The physical model established in [Carslaw and Jaeger, The Clarendon Press, Oxford (1947)] prescribes on the interface the condition that the flux of the temperature is proportional to the jump of the temperature field, by a factor of orderεγ . We suppose that -1 < γ ≤ 1. As far as the energies of the homogenized problems are concerned, we consider the cases -1 < γ < 1 and γ = 1 separately. To obtain the convergence of the energies, it is necessary to impose stronger assumptions on the data. As seen in [Jose, Rev. Roumaine Math. Pures Appl.54 (2009) 189–222] and [Faella and Monsurrò, Topics on Mathematics for Smart Systems, World Sci. Publ., Hackensack, USA (2007) 107–121] (also in [Donato et al., J. Math. Pures Appl.87 (2007) 119–143]), the case γ = 1 is more interesting because of the presence of a memory effect in the homogenized problem.

This article is divided into two chapters. Theclassical problem of homogenization of elliptic operators withperiodically oscillating coefficients is revisited in thefirst chapter. Following a Fourier approach, we discuss someof the basic issues of the subject: main convergence theorem,Bloch approximation, estimates on second order derivatives,correctors for the medium, and so on. The second chapter isdevoted to the discussion of some non-classical behaviour ofvibration problems of periodic structures.