On the homogenization of second order differential equations

We discuss the homogenization process of second order differential equations involving highly oscillating coefficients in the time and space variables. It generate memory or nonlocal effect. For initial value problems, the memory kernels are described by Volterra integral equations; and for boundary value problems, they are characterized by Fredholm integral equations. When the equation is translation (in time or in space) invariant, the memory or nonlocal kernel can be represented explicitly in terms of the Young's measure.