Diagonal non-semicontinuous variational problems

We study the minimum problem for non sequentially weakly lower semicontinuos functionals of the form F(u) = integral(I) f(x, u(x), u' (x)) dx, defined on Sobolev spaces, where the integrand f : I x R-m x R-m -> R is assumed to be non convex in the last variable. Denoting by (f) over bar the lower convex envelope of f with respect to the last variable, we prove the existence of minimum points of F assuming that the application p (sic) (f) over bar(., p, .) is separately monotone with respect to each component p(i) of the vector p and that the Hessian matrix of the application xi (sic) (f) over bar(., ., xi) is diagonal. In the special case of functionals of sum type represented by integrands of the form f(x, p, xi) = g(x, xi) + h(x, p), we assume that the separate monotonicity of the map p (sic) h(., p) holds true in a neighbourhood of the (unique) minimizer of the relaxed functional and not necessarily on its whole domain.