Remarks on the rate of convergence of the vanishing viscosity process of Hamilton-Jacobi equations

We establish a linear $L^p$ rate of convergence, $1<p<\infty$, with respect to the viscosity $\varepsilon$ for the vanishing viscosity process of semiconcave solutions of Hamilton-Jacobi equations by regularizing the PDE with the half-Laplacian $-\varepsilon(-\Delta)^{1/2}$. Our result reveals a nonlocal phenomenon, since it improves the known estimates obtained via the classical second order vanishing viscosity regularization $\varepsilon\Delta u$. It also highlights a faster rate of convergence than the available $\mathcal{O}(\varepsilon|\log\varepsilon|)$ rate in sup-norm obtained by the doubling of variable technique for this nonlocal approximation. The result is based on integral methods and does not use the maximum principle.