Rate of convergence of the vanishing viscosity method for Hamilton-Jacobi equations with Neumann boundary conditions

Abstract: We study the quantitative small noise limit in the $L<sup>\infty$ norm of certain time-dependent Hamilton-Jacobi equations equipped with Neumann boundary conditions, depending on the regularity of the data and the geometric properties of the domain. We first provide a $\mathcal{O}(\sqrt{\eps})$ rate of convergence for Hamilton-Jacobi equations with locally Lipschitz Hamiltonians posed on convex domains of the Euclidean space. We then enhance this speed of convergence in the case of quadratic Hamiltonians proving one-side rates of order $\mathcal{O}(\eps)$ and $\mathcal{O}(\eps<sup>\beta)$, $\beta\in(1/2,1)$. The results exploit recent $L<sup>1$ contraction estimates for Fokker-Planck equations with bounded velocity fields on unbounded domains used to derive differential Harnack estimates for the corresponding Neumann heat flow.