Lower gradient estimates for viscosity solutions to first-order Hamilton--Jacobi equations depending on the unknown function

TL;DR

This paper establishes lower gradient bounds for viscosity solutions of first-order Hamilton--Jacobi equations with convex Hamiltonians depending on the unknown, using two innovative methods involving inf-convolution and Lie equations.

Contribution

It introduces new techniques for deriving lower gradient estimates for viscosity solutions with Hamiltonians depending on the solution itself.

Findings

Lower bounds for gradients of viscosity solutions are derived.

Two methods are used: inf-convolution analysis and Lie equation examination.

Results improve understanding of gradient propagation in Hamilton--Jacobi equations.

Abstract

In this paper, we derive the lower bounds for the gradients of viscosity solutions to the Hamilton--Jacobi equation, where the convex Hamiltonian depends on the unknown function. We obtain gradient estimates using two different methods. First, we utilize the equivalence between viscosity solutions and Barron--Jensen solutions to study the properties of the inf-convolution. Second, we examine the Lie equation to understand how initial gradients propagate along its solutions.