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

Abstract

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 ${\bf R}^n$. Sufficient conditions are given in order to obtain an integral representation of the form $F(u,A)=\int_A f(x,u(x))\,dμ+ ν(A)$, where $μ$ and $ν$ are Borel measures and $f$ is convex in the second variable.