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): \[ λu +H(x, du)=\varepsilon(λ)Δu,\quad λ>0 \] as $λ\rightarrow 0_+$, once the modulus $\varepsilon(λ)$ satisfies $\varlimsup_{λ\rightarrow 0_+}\varepsilon(λ)/λ=0$. Such an exponent of $\varepsilon(λ)$ is nearly optimal in the convergence.