An application of homogenization theory to a 2-dimensional periodic checkerboard

Homogenization theory studies the relationship between microscopic and macroscopic scales in continuum mechanics and physics. This thesis focuses on partial differential equations with rapidly oscillating coefficients, which model composite materials with fine periodic structures. We introduce the mathematical framework necessary for homogenization: weak convergence in Banach spaces, distributional convergence, and Sobolev spaces. These tools allow us to rigorously define weak solutions and study the limiting behavior of sequences of solutions as the microstructural scale parameter εtends to zero. The central result of this work is the explicit computation of the homogenized matrix for a two-dimensional periodic checkerboard structure composed of two phases with conduc- tivities α and β. Using symmetry arguments and compactness methods, we prove that the homogenized matrix is A^*. = √αβ Id. This example illustrates how homogenization theory “divides and conquers”: the difficult resolution of the original oscillatory problem is replaced by computing the homogenized coefficients and solving a simpler limit equation.