Quantitative homogenization for static contact Hamilton-Jacobi equations

Abstract: We characterize possible pairs $$(u_\varepsilon,c)\in C(\mathbb{R}<sup>n\backslash\varepsilon\mathbb{Z}<sup>n,\mathbb{R})\times\mathbb{R}$$ addressing the homogenization problem for Hamilton--Jacobi equations $$H\left(\frac{x}{\varepsilon}, d u_\varepsilon, u_\varepsilon\right)=c, \quad \left({\mathrm resp.} \quad H\left(\frac{x}{\varepsilon}, d u_\varepsilon, u_\varepsilon\right)=\varepsilonΔu_\varepsilon+c \right)$$ for all $\varepsilon&gt;0$. Under a (not necessarily strict) monotonicity assumption on the Hamiltonian, we proposed certain criteria (based on the structure of Mather measures), under which all possible solutions $u_\varepsilon$ converge to a uniquely identified limit $u\in C(\mathbb{R}<sup>n,\mathbb{R})$ solving the effective equation [ \overline H( du,u)=c,\quad ({\mathrm resp.}\quad \overline H(du,u)=Δu+c) ] as $\varepsilon\rightarrow 0_+$ with a uniform rate $\mathcal{O}(\varepsilon)$.

• The paper introduces a quantitative homogenization framework for static contact Hamilton-Jacobi equations under weak monotonicity.
• It demonstrates O(ε) uniform convergence rates and characterizes extremal solution ordering through novel use of Mather measures.
• The approach accommodates non-uniqueness by defining admissible ergodic constants and extending the analysis to viscous cases.

Quantitative Homogenization for Static Contact Hamilton-Jacobi Equations

Introduction and Context

This work investigates the quantitative homogenization of static contact Hamilton-Jacobi (HJ) equations, specifically considering both first- and second-order forms with contact (i.e., $\theta$-dependent or $u$-dependent) Hamiltonians under weak monotonicity. The study is motivated by the need to address realistic mathematical models arising in fields such as dislocation dynamics and stochastic control, where the periodic or quasi-periodic microstructure is present, and strict monotonicity or uniqueness assumptions may fail.

Central to the paper is the characterization of the limiting behavior (as the microstructure parameter $\varepsilon \to 0$) of viscosity solutions $u_\varepsilon$ to equations

$$H\left(\frac{x}{\varepsilon}, d u_\varepsilon, u_\varepsilon\right) = c \quad\text{and}\quad H\left(\frac{x}{\varepsilon}, d u_\varepsilon, u_\varepsilon\right) = u_\varepsilon + c,$$

as well as the identification of the effective Hamiltonian $\overline{H}$ governing the macroscopic problem.

The key challenge addressed is the potential non-uniqueness of solutions, which precludes classical comparison-based approaches. To manage this, the authors exploit the structure of Mather measures and develop weak comparison principles rooted in Aubry-Mather and weak KAM theory.

Main Results and Methodology

Solvability and Admissible Sets

For a general class of convex and superlinear Hamiltonians $H(x,p,\theta)$ continuous in all variables and only non-negatively monotone in $\theta$, the authors establish:

• The existence of a non-empty, connected set $C$ of admissible ergodic constants $c$ for which the cell problem (static ergodic problem)

$u$0

possesses viscosity solutions for all $u$1.

• For each $u$2, identification of the set $u$3, whose structure determines the uniqueness and stability properties of the homogenization limit.

Quantitative Homogenization and Convergence Rates

A central contribution is quantitative convergence results:

• Under minimal regularity and monotonicity, if $u$4 is a singleton, all families of solutions $u$5 converge uniformly to the unique constant solution $u$6 of the homogenized equation with rate $u$7.
• When $u$8 is not a singleton, the problem may admit multiple local minimizers and solutions differing by constants, and homogenization limits may not be unique.

The results are extended to second-order (viscous) equations of type

$u$9

and the paper provides conditions ensuring uniqueness or identifying maximal/minimal solutions.

The proof strategy circumvents the absence of strict monotonicity and, consequently, the standard comparison principle by:

• Utilizing a novel "weak" comparison principle relying on ordinal Mather measures associated to candidate solutions.
• Employing a variational framework (rather than the classical doubling-of-variables trick), with arguments grounded in dynamical systems and weak KAM theory.
• Establishing Lipschitz regularity and, in the viscous case, smoothness of solutions, which is leveraged for compactness and convergence.

Structure and Ordering of Solutions

One of the novel theoretical findings is a global ordering property: even when the solution set is not a singleton, the family of solutions (with given ergodic constant $\varepsilon \to 0$0) possesses extremal elements, and all other solutions lie between these extremal minimizers and maximizers. This ordering is characterized in terms of the properties of the associated Mather measure sets.

Uniqueness Criteria and Examples

The paper gives explicit criteria—cast in terms of the absence of ordinal Mather measures or positive strict monotonicity of the Hamiltonian with respect to $\varepsilon \to 0$1—for uniqueness of solutions. The authors construct illustrative examples highlighting the possibilities:

• Hamiltonians for which the admissible set $\varepsilon \to 0$2 has empty interior (and hence homogenization fails).
• Hamiltonians where the limiting equation admits non-constant bounded, uniformly continuous (BUC) solutions when $\varepsilon \to 0$3 is not a singleton.

Quantitative and Regularity Results in Related Homogenization Settings

The results are discussed in the context of recent quantitative advances in periodic and multi-scale homogenization, including those achieving optimal convergence rates in various settings (see, e.g., [tran_optimal_2025]). The authors situate their results relative to viscosity solution-based approaches and point out connections to recent advances for equations with nontrivial boundary or multi-scale structures.

Numerical and Contradictory Claims

• Uniform convergence rate: For cases with unique homogenized limits (singleton $\varepsilon \to 0$4), the paper demonstrates uniform convergence of solutions with explicit rate $\varepsilon \to 0$5, improving upon earlier bounds in less general settings.
• Extension beyond uniqueness: In contrast to many classical results relying on strict monotonicity or uniqueness, the authors' framework admits multiple solutions, yet still provides quantitative bounds between extremal solutions.
• Non-necessity of strict monotonicity: The requirements on $\varepsilon \to 0$6 are weakened to non-negativity (rather than positivity), broadening applicability.

Theoretical Implications

The results deepen understanding of homogenization for Hamilton-Jacobi equations with a non-strictly monotone dependence on the solution variable, especially in static and contact cases. The use of variational techniques and ordered Mather measures provides a robust framework for addressing non-uniqueness and noncoercive scenarios. Connections to dynamical systems (weak KAM theory) suggest further unification of methods from optimal control, ergodic theory, and viscosity solutions.

Critical open questions include the detailed structure of the admissible set $\varepsilon \to 0$7 and the relation between the critical Mather set and the ordering/number of effective solutions in more complex settings.

Practical Applications and Future Directions

On the practical side, the theoretical guarantees obtained here have direct consequences for numerical methods and applications in materials science, dynamical systems, and stochastic control with micro-structured or weakly monotone interactions. The sharp convergence rates aid in error control for computational homogenization. The techniques also open avenues for generalizing to weakly coupled PDE systems and to problems with boundary conditions or inhomogeneous media.

Future directions include:

• Quantitative analysis in quasi-periodic, random, or nonconvex Hamiltonian contexts.
• Coupled Hamilton-Jacobi systems and higher-dimensional cell problems.
• Explicit numerical schemes leveraging the extremal solution ordering to control computational errors.

Conclusion

This work establishes a comprehensive quantitative theory for the homogenization of static contact Hamilton-Jacobi equations under weak monotonicity, expanding the mathematical toolbox to treat non-uniqueness and lack of strict comparison. Its results clarify the role of the effective Hamiltonian, Mather measures, and solution ordering in the homogenization limit, and provide a solid foundation for further study of complex micro-structured evolutive and stationary equations.