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 ${\mathbf{R}}^n$. 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 $\nu$ are Borel measures and $f$ is convex in the second variable.

51 pages

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