Hölder estimates in space-time for viscosity solutions of Hamilton-Jacobi equations

Piermarco Cannarsa (1) , Pierre Cardaliaguet (2)

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

It is well-known that solutions to the basic problem in the calculus of variations may fail to be Lipschitz continuous when the Lagrangian depends on t. Similarly, for viscosity solutions to time-dependent Hamilton-Jacobi equations one cannot expect Lipschitz bounds to hold uniformly with respect to the regularity of coefficients. This phenomenon raises the question whether such solutions satisfy uniform estimates in some weaker norm. We will show that this is the case for a suitable Hölder norm, obtaining uniform estimates in (x,t) for solutions to first and second order Hamilton-Jacobi equations. Our results apply to degenerate parabolic equations and require superlinear growth at infinity, in the gradient variables, of the Hamiltonian. Proofs are based on comparison arguments and representation formulas for viscosity solutions, as well as weak reverse Hölder inequalities.

Mots clés: Hamilton-Jacobi equations; viscosity solutions; Hölder continuity; degenerate parabolic equations; reverse Hölder inequalities

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

Piermarco Cannarsa, Pierre Cardaliaguet. Hölder estimates in space-time for viscosity solutions of Hamilton-Jacobi equations. Communications on Pure and Applied Mathematics, 2010, 63 (5), pp.559-696. ⟨10.1002/cpa.20315⟩. ⟨hal-00371681⟩