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


W 
g(x,u)dx,
where W is a bounded smooth subset of RN and g is a Carathéodory function, when the admissible functions u are forced to satisfy a pointwise gradient constraint of the type
u(x) C(x) for a.e. x W,
C(x) being, for every x W, a bounded convex subset of RN, 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

GPC1(W)
 
(u)= inf



liminf
h+ 



\ 
Omegag(x,uh)dx :

{uh} PC1(W), uh(x) C(x) fora