Leveraging on practical applications such as composite materials with periodic heterogeneity, the homogenization (upscaling, averaging, etc.) is a typical framework to deal with problems in a class of complicated perforated media, and enables us to obtain good overall approximations.
In fact, we usually characterize the properties of the aforementioned materials by means of a local scale, i.e. taking each component into account. In reality, a global scale is much more attractive to gain considerable attention of many researchers. In principle, the global behavior by homogenization is driven by replacing the heterogeneity the so-called homogenized material whose behavior must be as close as possible to the composite’s behavior.
Another reason comes from numerics. For example, when one takes into consideration a superconducting multifilamentary composite (SMC) with several thousands of fibers whose diameter is very small, a direct numerical computation is impossible. As one could readily expect, one needs to have an extremely fine discretization mesh in hands, and obviously, it requires a long and long time elapsed for computation. Concerning more complex materials, let us also mention the honeycomb structures, the reinforced structures and their generalized versions.
In order to have a lucid mathematical explanation, a one-dimensional equation with oscillating coefficients is presented in this note as the first glance in the study of homogenization.
The problem here is governed by
in , and associated with with zero Dirichlet boundary conditions, i.e. on , and is a given function.
Included here, the oscillating coefficient is supposed to be periodic with period and generated by a bounded positive function satisfying
The coefficient is thus assigned by
For each , the considered problem admits a unique solution. Moreover, an important thing shows that by the Poincaré inequality, we obtain the a priori estimate of over -norm. In addition, if we define the stress function
then the a priori estimate also holds.
We are led to the fact that there exists a subsequence, ignored the sub-index, such that in , in , in in the weak or weak-* sense.
By the average convergence for periodic functions, is exactly the mean value of , i.e.
Moreover, we also obtain in , and go through by the Rellich-Kondrachov theorem the fact that the convergence is strong in .
Notice that is the product of two weakly converging quantities. In general, this does not imply that the limit does not coincide with the product of the limits and , saying that does not hold. In fact, we show that
which leads to the equation
In conclusion, we considered here the periodic case with a bounded sequence (not necessarily periodic) satisfying weak-* in . The solution of the considered problem is shown to be weakly convergent to in . And the function is proved to be a unique solution of the problem given by
in and along with the zero Dirichlet boundary conditions.