  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 $\mathcal{O}(\varepsilon^{1/p})$ (where $1<p<2$ is the
exponent of the gradient term) for nonnegative data $f\in
\mathrm{C}^2(\bar{\Omega})$ 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.
