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 $L^{\infty }$ 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 $\mathcal{O}(\sqrt{\epsilon })$ 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 $\mathcal{O}(\epsilon )$ and $\mathcal{O}({\epsilon }^{\beta })$, $\beta \in (1/2,1)$. The results exploit recent $L^1$ 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.