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

Abstract

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 $\partial_t u+H(x,t,Du,D^2u)=0$ in the whole space. We approximate the PDE with a fully nonlinear, possibly degenerate, elliptic operator $\varepsilon F(x,t,D^2u)$. Assuming that $u\in C^\alpha_x$, $u_0\in C^\eta$, $H\in C^\beta_x$ and having power growth $\gamma$ in the gradient entry, we establish a convergence rate of order $\varepsilon^{\min\left\{\frac{\eta}{2},\frac{\beta+\gamma(\alpha-1)}{\beta+\gamma(\alpha-1)+2-\alpha}\right\}}$. 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 H\"older coefficients.