On the regularizing effect for unbounded solutions of first-order Hamilton-Jacobi equations

Guy Barles (1, 2) , Emmanuel Chasseigne (2, 1)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 FDP - Fédération de recherche Denis Poisson

We give a simplified proof of regularizing effects for first-order Hamilton-Jacobi Equations of the form $u_t+H(x,t,Du)=0$ in $\R^N\times(0,+\infty)$ in the case where the idea is to first estimate $u_t$. As a consequence, we have a Lipschitz regularity in space and time for coercive Hamiltonians and, for hypo-elliptic Hamiltonians, we also have an H\"older regularizing effect in space following a result of L. C. Evans and M. R. James.

Mots clés: First-order Hamilton-Jacobi Equations; viscosity solutions; regularizing effects

Domaines: Equations aux dérivées partielles [math.AP]

Guy Barles, Emmanuel Chasseigne. On the regularizing effect for unbounded solutions of first-order Hamilton-Jacobi equations. Journal of Differential Equations, 2016, 260 (9), pp.7020-7031. ⟨hal-01214425⟩