A relaxation result for non-convex and non-coercive simple integrals

We consider the following classical autonomous variational problem: Minimize {F(u) = \int_a^b f(u(x), u'(x))dx : u ∈ AC([a, b]), u(a) = α, u(b) = β, u([a, b]) ⊆ I} where I is a real interval, α, β ∈ I, and f : I × R → [0, +∞) is possibly neither continuous, nor coercive, nor convex; in particular f(s,·) may be not convex at 0. Assuming the solvability of the relaxed problem, we prove under mild assumptions that the above variational problem has a solution, too.