$\sum $-convergence and reiterated homogenization of nonlinear parabolic operators

We study the reiterated homogenization of nonlinear parabolic differential equations associated with monotone operators. Contrary to what is usually done in the deterministic homogenization theory, we present a new approach based on a deterministic assumption on the coefficients of the operators, which allows us to consider the concrete homogenization problems from a true and natural perspective, taking into account the discontinuities in general. Based on this new approach we obtain very general homogenization results, and we solve several concrete homogenization problems. Our main tool is the theory of homogenization structures, and our homogenization approach falls within the scope of multiscale convergence method.