Rate of convergence in periodic homogenization of Hamilton-Jacobi equations: the convex setting

We study the rate of convergence of $u^{ϵ}$, as $ϵ\to 0+$, to $u$ in periodic homogenization of Hamilton-Jacobi equations. Here, $u^{ϵ}$ and $u$ are viscosity solutions to the oscillatory Hamilton-Jacobi equation and its effective equation \begin{equation*} {\rm ©_\epsilon} \qquad \begin{cases} u_t^\epsilon+H\left(\frac{x}{\epsilon},Du^\epsilon\right)=0 \qquad &$\text{in}$ \ $\mathbb{R}$^n \times (0,\infty), u^\epsilon(x,0)=g(x) \qquad &$\text{on}$ \ $\mathbb{R}$^n, \end{cases} \end{equation*} and \begin{equation*} {\rm ©} \qquad \begin{cases} u_t+\overline{H}\left(Du\right)=0 \qquad &$\text{in}$ \ $\mathbb{R}$^n \times (0,\infty), u(x,0)=g(x) \qquad &$\text{on}$
$\mathbb{R}$^n, \end{cases} \end{equation*} respectively. We assume that the Hamiltonian $H=H(y,p)$ is coercive and convex in the $p$ variable and is ${\mathbb{Z}}^n$-periodic in the $y$ variable, and the initial data $g$ is bounded and Lipschitz continuous.