Quantitative homogenization for static contact Hamilton-Jacobi equations

We characterize possible pairs $(u_{\epsilon },c)\in C({\mathbb{R}}^n\mathrm{\backslash }\epsilon {\mathbb{Z}}^n,\mathbb{R})\times \mathbb{R}$ addressing the homogenization problem for Hamilton–Jacobi equations $$H(\frac{x}{\epsilon },d{u}_{\epsilon },u_{\epsilon })=c,(\mathrm{r}esp\mathrm{.}H(\frac{x}{\epsilon },d{u}_{\epsilon },u_{\epsilon })=\epsilon \mathrm{\Delta }{u}_{\epsilon }+c)$$ for all $\epsilon >0$. Under a (not necessarily strict) monotonicity assumption on the Hamiltonian, we proposed certain criteria (based on the structure of Mather measures), under which all possible solutions $u_{\epsilon }$ converge to a uniquely identified limit $u\in C({\mathbb{R}}^n,\mathbb{R})$ solving the effective equation $$\bar{H}(du,u)=c,(\mathrm{r}esp\mathrm{.}\bar{H}(du,u)=\mathrm{\Delta }u+c)$$ as $\epsilon \to 0_{+}$ with a uniform rate $\mathcal{O}(\epsilon )$.