C\(^{1,1}\)-smoothness of constrained solutions in the calculus of variations with application to mean field games (Q2305117)

The authors derive necessary optimality conditions for minimizers of regular functionals in the calculus of variations under smooth state constraints. The novelty of their 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. They apply their findings by constructing Lipschitz relaxed solutions to a constrained Mean Field Games problem.