Necessary and Sufficient Conditions for Optimality of Nonconvex, Noncoercive Autonomous Variational Problems with Constraints

We consider the classical autonomous constrained variational problem of minimization of ∫ b a f(v(t), v ′(t))dt in the class Ω:={v ∈ W 1,1(a, b) : v(a) = α, v(b) = β, v′(t) ≥ 0 a.e. in (a, b)}, where f : [α, β] × [0,+∞) → R is a lower semicontinuous, nonnegative integrand, which can be nonsmooth, nonconvex and noncoercive. We prove a necessary and sufficient condition for the optimality of a trajectory v0 ∈ Ω in the form of a DuBois-Reymond inclusion involving the subdifferential of Convex Analysis. Moreover, we also provide a relaxation result and necessary and sufficient conditions for the existence of the minimum expressed in terms of an upper limitation for the assigned mean slope ξ0 = (β−α)/(b−a). Applications to various noncoercive variational problems are also included.