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Quantitative homogenization of Hamilton--Jacobi equations on perforated domains with Dirichlet boundary conditions

Yuxi Han, Son Tu

TL;DR

This work proves an optimal O(ε) convergence rate for the periodic homogenization of convex Hamilton–Jacobi equations on perforated domains with Dirichlet boundary data. The authors develop a Hopf–Lax–type optimal-control representation for the limit solution using an extended running-cost metric, alongside a τ-parameterized cost tilde m whose averaged limit governs the homogenized behavior. A careful analysis of singular paths that do not exhaust the time horizon, together with dynamic programming principles and velocity bounds for minimizing trajectories, yields the quantitative rate and an explicit dependence on domain geometry and boundary data. The results further extend to dilute and defective-domain regimes, establishing analogous rates and highlighting the role of the averaged metric in capturing long-time effective dynamics across complex perforated geometries.

Abstract

We study the periodic homogenization of convex Hamilton-Jacobi equations on perforated domains with Dirichlet boundary conditions. By analyzing the optimal control representation of the solutions and the properties of the metric function associated with the running cost, we establish the optimal convergence rate $\mathcal{O}(\varepsilon)$ for homogenization. A key aspect of our approach is the treatment of the singularity that arises when the optimal path does not fully utilize the available time.

Quantitative homogenization of Hamilton--Jacobi equations on perforated domains with Dirichlet boundary conditions

TL;DR

This work proves an optimal O(ε) convergence rate for the periodic homogenization of convex Hamilton–Jacobi equations on perforated domains with Dirichlet boundary data. The authors develop a Hopf–Lax–type optimal-control representation for the limit solution using an extended running-cost metric, alongside a τ-parameterized cost tilde m whose averaged limit governs the homogenized behavior. A careful analysis of singular paths that do not exhaust the time horizon, together with dynamic programming principles and velocity bounds for minimizing trajectories, yields the quantitative rate and an explicit dependence on domain geometry and boundary data. The results further extend to dilute and defective-domain regimes, establishing analogous rates and highlighting the role of the averaged metric in capturing long-time effective dynamics across complex perforated geometries.

Abstract

We study the periodic homogenization of convex Hamilton-Jacobi equations on perforated domains with Dirichlet boundary conditions. By analyzing the optimal control representation of the solutions and the properties of the metric function associated with the running cost, we establish the optimal convergence rate for homogenization. A key aspect of our approach is the treatment of the singularity that arises when the optimal path does not fully utilize the available time.

Paper Structure

This paper contains 19 sections, 26 theorems, 183 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Relevant Literature
  3. Settings
  4. Simplifications
  5. Main results
  6. Representation of solutions via optimal control
  7. Optimal control formula for $u$
  8. Optimal control for $u^\varepsilon$
  9. Extension of the cost function
  10. The cost function with parameter $\tau$
  11. The value function associated with the average of the extended metric
  12. Proof of Theorem \ref{['thm:main1']}
  13. Discussions: Optimality, Diluted settings, and Domains with defects
  14. Optimality of the rate of convergence
  15. Homogenization in a dilute setting and domains with defects
  16. ...and 4 more sections

Key Result

Theorem 1.1

Assume itm:D1, itm:A1--itm:A6. For $\varepsilon>0$, let $u^\varepsilon$ be the viscosity solution to eqn:PDE_epsilon and let $u$ be the viscosity solution to eqn:PDE_limit, respectively. Then we have where $C>0$ is a constant depends on $n, \partial\Omega, H, \left\|Dg\right\|_{L^\infty\left(\mathbb{R}^n\right)}$ and $\left\|Db\right\|_{L^\infty \left(\mathbb{R}^n\times \partial \Omega\right)}$.

Figures (4)

  • Figure : Case 1
  • Figure : Case 1
  • Figure : Case 2 (a)
  • Figure : Case 2 (b)

Theorems & Definitions (54)

  • Theorem 1.1
  • Proposition 1.2
  • Remark 1
  • Definition 1
  • Remark 2
  • Proposition 2.1
  • Lemma 2.2
  • proof : Proof of Lemma \ref{['lem:DPPHbar']}
  • Proposition 2.3
  • proof : Proof of Proposition \ref{['prop:ocfcont']}
  • ...and 44 more