Quantitative homogenization for static contact Hamilton-Jacobi equations

Abstract

We characterize possible pairs $(u_{\epsilon },c)\in C({\mathbb{R}}^n\backslash \epsilon {\mathbb{Z}}^n,\mathbb{R})\times \mathbb{R}$ addressing the homogenization problem for Hamilton--Jacobi equations $$H\left(\frac{x}{\epsilon },d{u}_{\epsilon },u_{\epsilon }\right)=c,\left(\mathrm{r}esp.H\left(\frac{x}{\epsilon },d{u}_{\epsilon },u_{\epsilon }\right)=\epsilon \hat{I}”u_{\epsilon }+c\right)$$ 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 \[ \overline H( du,u)=c,\quad ({\mathrm resp.}\quad \overline H(du,u)=Δu+c) \] as $\epsilon \to 0_{+}$ with a uniform rate $\mathcal{O}(\epsilon )$.

25 pages, Lemma 2.10 is revised