ΣΣ-convergence of nonlinear parabolic operators

We discuss the nonstochastic homogenization of nonlinear parabolic differential operators in an abstract setting framed to bridge the gap between periodic and stochastic homogenization theories. Instead of the classical periodicity hypothesis, we have here an abstract assumption covering a great variety of concrete behaviours in both space and time variables, such as the periodicity, the almost periodicity, the convergence at infinity, and others. Our basic approach is the ΣΣ-convergence method generalizing the well-known two-scale convergence technique. Fundamental homogenization theorems are proved and several concrete examples are worked out, which reveal that homogenization beyond the periodic setting may involve difficulties that are beyond imagination.