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

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.