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

TL;DR

This paper studies how quickly solutions to superquadratic Hamilton--Jacobi equations with state constraints converge as viscosity vanishes, providing explicit rates depending on data regularity.

Contribution

It establishes explicit convergence rates for solutions with different data regularities in the vanishing viscosity limit for superquadratic Hamilton--Jacobi equations.

Findings

Convergence rate of order O(ε^{1/2}) for nonnegative Lipschitz data.

Improved convergence rate of order O(ε^{p/(2(p-1))}) for semiconcave data.

Results apply to equations with superquadratic growth and state constraints.

Abstract

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}(\varepsilon^{1/2}) $ and obtain an improved upper rate of order $ \mathcal{O}\big(\varepsilon^{\frac{p}{2(p-1)}}\big)$ for semiconcave data.