Integral representation for a class of $C^1$-convex functionals

In view of the applications to the asymptotic analysis of a family of obstacle problems, we consider a class of convex local functionals $F(u,A)$, defined for all functions $u$ in a suitable vector valued Sobolev space and for all open sets $A$ in ${\mathbf{R}}^n$. Sufficient conditions are given in order to obtain an integral representation of the form $F(u,A)=\int_A f(x,u(x)),d\mu + \nu(A)$,where$\mu$and$\nu$areBorelmeasuresand$f$ is convex in the second variable.