Optimal convergence rate for homogenization of convex Hamilton-Jacobi equations in the periodic spatial-temporal environment

TL;DR

This paper establishes the optimal convergence rate of O(ε) for homogenization of convex Hamilton-Jacobi equations with periodic and time-dependent Hamiltonians, advancing understanding of their asymptotic behavior.

Contribution

It proves the optimal convergence rate for homogenization of convex Hamilton-Jacobi equations with periodic, time-dependent Hamiltonians, extending previous results.

Findings

Optimal convergence rate is O(ε) for the homogenization problem.

Homogenization results apply to time-dependent, periodic Hamiltonians.

Theoretical proof aligns with prior conjectures on convergence rates.

Abstract

We study the optimal convergence rate for homogenization problem of convex Hamilton-Jacobi equations when the Hamitonian is periodic with respect to spatial and time variables, and notably time-dependent. We prove a result similar to that of [8], which means the optimal convergence rate is also $O(\varepsilon)$.