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

We give a simplified proof of regularizing effects for first-order Hamilton–Jacobi Equations of the form ut+H(x,t,Du)=0ut+H(x,t,Du)=0 in RN×(0,+∞)RN×(0,+∞) in the case where the idea is to first estimate utut. As a consequence, we have a Lipschitz regularity in space and time for coercive Hamiltonians and, for hypo-elliptic Hamiltonians, we also have an Hölder regularizing effect in space following a result of L.C. Evans and M.R. James.