${\mathbf{C}}^{\mathbf{1},\mathbf{1}}$-smoothness of constrained solutions in the calculus of variations with application to mean field games

We derive necessary optimality conditions for minimizers of regular functionals in the calculus of variations under smooth state constraints. In the literature, this classical problem is widely investigated. The novelty of our result lies in the fact that the presence of state constraints enters the Euler-Lagrange equations as a local feedback, which allows to derive the $C^{1,1}$-smoothness of solutions. As an application, we discuss a constrained Mean Field Games problem, for which our optimality conditions allow to construct Lipschitz relaxed solutions, thus improving the existence result in [Cannarsa, P., Capuani, R., \textsl{Existence and uniqueness for Mean Field Games with state constraints}, arXiv:1711.01063].