Remarks on the vanishing viscosity process of state-constraint Hamilton-Jacobi equations

We investigate the convergence rate in the vanishing viscosity process of the solutions to the subquadratic state-constraint Hamilton-Jacobi equations. We give two different proofs of the fact that, for nonnegative Lipschitz data that vanish on the boundary, the rate of convergence is $\mathcal{O}(\sqrt{\varepsilon})$ in the interior. Moreover, the one-sided rate can be improved to $\mathcal{O}(\varepsilon)$ for nonnegative compactly supported data and $\mathcal{O}(\varepsilon^{1/(p-\frac{1}{2})})$ (where $1<p\leq 2$ is the exponent of the gradient term) for nonnegative data $f\in \mathrm{C}^2(\overline{\Omega})$ such that $f = 0$ and $Df = 0$ on the boundary. Our approach relies on deep understanding of the blow-up behavior near the boundary and semiconcavity of the solutions.