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

TL;DR

This paper presents a simplified proof demonstrating regularizing effects for solutions of first-order Hamilton-Jacobi equations, establishing Lipschitz regularity for coercive Hamiltonians and H"older regularity for hypo-elliptic cases.

Contribution

It provides a streamlined proof technique for regularizing effects, extending known results to broader classes of Hamilton-Jacobi equations.

Findings

Lipschitz regularity in space and time for coercive Hamiltonians

H"older regularizing effect in space for hypo-elliptic Hamiltonians

Simplified proof approach for regularizing effects

Abstract

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.