Convergence rates for the vanishing viscosity approximation of fully nonlinear, non-convex, second-order Hamilton-Jacobi equations

We obtain new quantitative estimates of the vanishing viscosity approximation for time-dependent, degenerate, Hamilton-Jacobi equations that are neither concave nor convex in the gradient and Hessian entries of the form ₜ u+H (x, t, Du, D²u) =0 in the whole space. We approximate the PDE with a fully nonlinear, possibly degenerate, elliptic operator F (x, t, D²u). Assuming that u C^αₓ, u₀ C^η, H C^βₓ and having power growth γ in the gradient entry, we establish a convergence rate of order ^\{η{2, β+γ (α-1) β+γ (α-1) +2-α\}}. Our novel approach exploits the regularizing properties of sup/inf-convolutions for viscosity solutions and the comparison principle. We also obtain explicit constants and do not assume differentiability properties neither on solutions nor on H. The same method provides new convergence rates for the vanishing viscosity approximation of the stationary counterpart of the equation and for transport equations with Hölder coefficients.