*Preprint*

**Inserted:** 10 jan 2002

**Year:** 2001

**Abstract:**

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 disperse periodic 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 homogenization problems with constraints on a set of zero Lebesgue measure. For scalar problems with gradient constraints on nondisperse inclusions we prove a homogenization theorem in the case of the gradients belonging to a periodic family of uniformly bounded convex sets containing a neighborhood of the origin.