The minimum problem for one-dimensional non-semicontinuous functionals

We study the minimum problem for functionals of the form F(u)=∫If(x,u(x),u′(x))dx,where the integrandf:I×Rm×Rm→Ris not convex in the last variable. We providean existence result assuming that the lower convex envelopef=f(x,p,ξ)offsatisfiesa suitable affinity condition on the set on whichf>fand that the map pi→f(x,p,ξ)is monotone with respect to one single componentpiof the vectorp. We show that ourhypotheses are nearly optimal, providing in such a way an almost necessary and sufficientcondition for the existence of minimizers.