Homogenization and bounds for multiscale problems

In this paper, we consider the homogenization problem for a steady-state heat conduction problem in a heterogeneous medium. The objective is to be able to describe the global behaviour of the heterogeneous medium where its’ constituents are very finely distributed in a periodic manner. First, we modelled the problem using a two-scale elliptic equation where we have 2 variables. The variable x represents the macroscopic scale while \frac{x}{\varepsilon} represents the microscopic scale. Next, we looked at Sobolev spaces which form the basis of weak solutions for the problem in its variational form. From there, we moved on to the definition of the homogenization problem and examined how we could obtain the homogenized matrix and equation in order to solve and find the weak solutions of the original problem. In particular, we looked at the method of asymptotic expansions. Next, we proceeded to derive the Voigt-Reiss’ inequalities using the variational principle which gave us estimates for the homogenized matrix. The search for a better estimate for the homogenized matrix led us to the Hashin-Shtrikman bounds. Ultimately, we were able to gather information about the global behaviour of the heterogeneous medium by considering the effective homogenized medium.Bachelor of Science in Mathematical Science