On the rate of convergence in superquadratic Hamilton--Jacobi equations with state constraints

In this paper, we investigate the convergence rate in the vanishing viscosity limit for solutions to superquadratic Hamilton–Jacobi equations with state constraints. For every $p>2$, we establish the rate of convergence for nonnegative Lipschitz data vanishing on the boundary to be of order $\mathcal{O}({\epsilon }^{1\mathrm{/}2})$ and obtain an improved upper rate of order $\mathcal{O}\left({\epsilon }^{\frac{p}{2(p-1)}}\right)$ for semiconcave data.