Scale-integration and scale-disintegration in nonlinear homogenization

This work is devoted to scale transformations of stationary nonlinear problems. A class of coarse-scale problems is first derived by integrating a family of two-scale minimization problems (scale-integration), in presence of appropriate orthogonality conditions. The equivalence between the two formulations is established by showing that conversely any solution of the coarse-scale problem can be represented as the fine-scale average of a solution of the two-scale problem (scale-disintegration). This procedure may be applied to the homogenization of several quasilinear problems, and is related to De Giorgi’s notion of Γ-convergence. As an example the homogenization of a simple nonlinear model of magnetostatics is illustrated: a two-scale minimization problem is first derived via Nguetseng’s notion of two-scale convergence, and afterwards the equivalence with a coarse-scale problem is proved.