Rate of convergence for periodic homogenization of convex Hamilton–Jacobi equations in one dimension

Let $u^\varepsilon$ and $u$ be viscosity solutions of the oscillatory Hamilton-Jacobi equation its corresponding effective equation. Given bounded, Lipschitz initial data, we present a simple proof to obtain optimal rate convergence $\mathcal{O}(\varepsilon)$ $u^\varepsilon \rightarrow u$ as $\varepsilon 0^+$ for large class convex Hamiltonians $H(x,y,p)$ in one dimension. This includes from classical mechanics with separable potential. The makes use control theory quantitative version ergodic theorem periodic functions dimension $n = 1$.