Homogenization of some problems with gradient constraints

A homogenization theorem is established for the problem of minimization of a quadratic integral functional on a set of admissible functions whose gradients are subjected to rapidly changing constraints imposed on a disperse periodic set (regarded as inclusions). At each point of the inclusion, the gradients must belong to a given closed convex set of an arbitrary structure which may vary from point to point within the inclusion. Our approach is based on two-scale convergence and an explicit construction of a Gamma-realizing sequence. This homogenization method can be directly applied to variational problems for vector-valued functions, which is demonstrated on problems of elasticity with convex constraints on the strain tensor at the points of disperse inclusions. We also consider some problems with constraints on periodic sets of zero Lebesgue measure and study homogenization problems for some cases of nondisperse inclusions.