## G. Cardone - A. Corbo Esposito - G. Yosifian - V. V. Zhikov

# Homogenization of some problems with gradient constraints

created on 10 Jan 2002

[

BibTeX]

*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.