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

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