Quantitative homogenization of convex Hamilton-Jacobi equations with $u/\varepsilon$-periodic Hamiltonians

TL;DR

This paper develops a precise rate of convergence for homogenizing convex Hamilton-Jacobi equations with $u/ ext{}\varepsilon$-periodic Hamiltonians, and proves regularity results under certain growth conditions.

Contribution

It introduces a novel approach using the fundamental solution and variational principles to achieve optimal convergence rates and regularity in homogenization.

Findings

Established optimal convergence rate for homogenization.

Proved global Hölder regularity for solutions and correctors.

Applied results to dislocation dynamics models.

Abstract

Here, we study quantitative homogenization of first-order convex Hamilton-Jacobi equations with $(u/\varepsilon)$-periodic Hamiltonians which typically appear in dislocation dynamics. Firstly, we establish the optimal convergence rate by using the inherent fundamental solution and the implicit variational principle of Hamilton dynamics with their Hamiltonian depending on the unknown. Secondly, under additional growth assumptions on the Hamiltonian, we establish global H\"older regularity for both the solutions and the correctors, serving as a notable application of our quantitative homogenization theory.