On the vanishing viscosity limit of Hamilton-Jacobi equations with nearly optimal discount

In this paper, we establish the convergence of solutions to the viscous Hamilton-Jacobi equation (with a Tonelli Hamiltonian): $$\lambda u+H(x,du)=\epsilon (\lambda )\mathrm{\Delta }u,\lambda >0$$ as $\lambda \to 0_{+}$, once the modulus $\epsilon (\lambda )$ satisfies ${\mathrm{}}_{\lambda \to 0_{+}}\epsilon (\lambda )\mathrm{/}\lambda =0$. Such an exponent of $\epsilon (\lambda )$ is nearly optimal in the convergence.