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.