The paper is concerned with relaxation problems for a functional of
the type

u®

ó õ

W

g(x,Ñu)dx,

where W is a bounded smooth subset of R^{N} and
g is a Carathéodory function, when the admissible
functions u are forced to satisfy a pointwise gradient constraint of
the type

Ñu(x) Î C(x) fora.e. x Î W,

C(x) being, for every x Î W, a bounded convex subset of
R^{N}, in general varying with x not necessarily in a
smooth way.
In this case some new problems appear. First of all, one must expect
that, because of the above pointwise gradient
constraint condition and of the nonsmooth dependence of C on
x, the relaxation process depends heavily on the
smoothness properties of the admissible functions. So, we need to
consider both the relaxed functionals below