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C1;1-smoothness of constrained solutions in the calculus of variations with application to mean field games

1 Dipartimento di Matematica, Università di Roma “Tor Vergata”, 00133 Roma, Italy
2 Dipartimento di Matematica, Università di Roma “Tor Vergata”, 00133 Roma, Italy, & Université Paris-Dauphine, PSL Research University, CNRS UMR 7534, CEREMADE, 75016 Paris, France
3 Université Paris-Dauphine, PSL Research University, CNRS UMR 7534, CEREMADE, 75016 Paris, France

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 C1;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 an existence result due to the first two authors.
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