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