Monotonicity properties of minimizers and relaxation for autonomous variational problems

We consider the following classical autonomous variational problem minimize F (v) =\int_a^b f (v(x), v (x)) dx : v ∈ AC([a, b]), v(a) = α, v(b) = β , where the Lagrangian f is possibly neither continuous, nor convex, nor coercive. We prove a monotonicity property of the minimizers stating that they satisfy the maximum principle or the minimum one. By virtue of such a property, applying recent results concerning constrained variational prob- lems, we derive a relaxation theorem, the DuBois-Reymond necessary condition and some existence or non-existence criteria.