Sobolev regularity for the first order Hamilton-Jacobi equation

Pierre Cardaliaguet (1) , Alessio Porretta (2) , Daniela Tonon (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 DIPMAT - Dipartimento di Matematica [Roma II]

We provide Sobolev estimates for solutions of first order Hamilton-Jacobi equations with Hamiltonians which are superlinear in the gradient variable. We also show that the solutions are differentiable almost everywhere. The proof relies on an inverse H\"older inequality. Applications to mean field games are discussed.

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Pierre Cardaliaguet, Alessio Porretta, Daniela Tonon. Sobolev regularity for the first order Hamilton-Jacobi equation. Calculus of Variations and Partial Differential Equations, 2015, 54 (3), pp.3037-3065. ⟨10.1007/s00526-015-0893-3⟩. ⟨hal-01251162⟩