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

TL;DR

This paper studies the rate at which solutions to state-constraint Hamilton-Jacobi equations with vanishing viscosity converge, providing precise rates under various boundary and data conditions using advanced analytical techniques.

Contribution

It establishes new convergence rate results for the vanishing viscosity process in state-constraint Hamilton-Jacobi equations, including improved rates for specific data types.

Findings

Convergence rate is (\u221avarepsilon) in the interior for nonnegative Lipschitz data.

One-sided convergence rate can be improved to () for compactly supported data.

For smooth data with zero boundary values, the rate is (^{1/(p-rac{1}{2})}).

Abstract

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.