A near-optimal rate of periodic homogenization for convex Hamilton-Jacobi equations

TL;DR

This paper establishes a near-optimal rate of homogenization for convex Hamilton-Jacobi equations with periodic Hamiltonians, using a novel combination of control theory and percolation results, applicable across all dimensions.

Contribution

It introduces a new homogenization rate for convex Hamilton-Jacobi equations by integrating optimal control representation and a theorem from first-passage percolation, improving understanding of convergence speed.

Findings

Homogenization rate within a log-factor of optimal

Applicable to all spatial dimensions

Combines control theory with percolation results

Abstract

We consider a Hamilton-Jacobi equation where the Hamiltonian is periodic in space and coercive and convex in momentum. Combining the representation formula from optimal control theory and a theorem of Alexander, originally proved in the context of first-passage percolation, we find a rate of homogenization which is within a log-factor of optimal and holds in all dimensions.