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 $\int_a^bf(v(t),v'(t)) dt$ in the class $\Omega:=\{v \in W^{1,1}(a,b):$ $v(a)=\alpha, v(b)= \beta, v'(t)\ge 0$   a.e. in $(a,b) \}$, where $f:[\alpha, \beta]\times [0,+\infty) \to \mathbb{R}$ 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 $v_0\in \Omega$ 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 $\xi_0=(\beta-\alpha)/(b-a)$. Applications to various noncoercive variational problems are also included.