arxiv: 2604.02693 · v2 · submitted 2026-04-03 · 🧮 math.AP

Recognition: 2 theorem links

· Lean Theorem

Quantitative homogenization for static contact Hamilton-Jacobi equations

Gengyu Liu , Son N.T. Tu , Jianlu Zhang

Authors on Pith no claims yet

Pith reviewed 2026-05-13 19:35 UTC · model grok-4.3

classification 🧮 math.AP

keywords homogenizationHamilton-Jacobi equationscontact problemsMather measureseffective equationsconvergence ratesviscous approximations

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The pith

Solutions to static contact Hamilton-Jacobi equations converge uniformly at rate O(ε) to a unique effective limit under monotonicity and Mather measure criteria.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies homogenization of Hamilton-Jacobi equations that include a contact term, either in the standard form or with a small viscous Laplacian correction. It establishes that when the Hamiltonian obeys a monotonicity condition and certain structural criteria on the associated Mather measures hold, every family of solutions u_ε converges as ε approaches zero to one and the same limit u. That limit satisfies a homogenized effective equation whose coefficients no longer depend on the microscopic scale. The convergence is quantitative and uniform, with an explicit error bound of order ε. This supplies a concrete rate at which fine-scale oscillations average out, which matters for turning microscopic control problems into reliable macroscopic approximations.

Core claim

Under a (not necessarily strict) monotonicity assumption on the Hamiltonian together with criteria based on the structure of Mather measures, every solution pair (u_ε, c) for the scaled contact Hamilton-Jacobi equation converges as ε→0 to a unique pair (u, c) where u solves the effective homogenized equation, with uniform convergence rate O(ε). The same statement holds for the viscous version that includes the term εΔu_ε.

What carries the argument

Criteria based on the structure of Mather measures, which, combined with monotonicity of the Hamiltonian, identify which constants c are admissible and force all candidate solutions u_ε to share the same macroscopic limit.

If this is right

The effective equation supplies a scale-independent model that can be solved once and used for all small ε.
Explicit O(ε) error bounds give a priori control on the accuracy of the homogenized approximation.
The same criteria apply simultaneously to both the inviscid and the viscous contact problems.
The admissible constants c are characterized directly by the Mather measure conditions.
Uniqueness of the limit u removes ambiguity when passing to the macroscopic description.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The structural criteria on Mather measures may be verifiable in periodic or almost-periodic settings, opening the door to explicit examples.
The quantitative rate could be used to design adaptive numerical schemes that refine only where the microscopic scale still matters.
Similar measure-based conditions might extend the result to time-dependent or stochastic contact problems.
The approach suggests a template for quantitative homogenization in other first-order equations whose invariant measures have identifiable structure.

Load-bearing premise

The Hamiltonian must satisfy monotonicity and the Mather measures must obey the stated structural criteria.

What would settle it

Construct a monotonic Hamiltonian whose Mather measures violate the proposed criteria and exhibit either non-uniqueness of the limit or a convergence rate slower than O(ε).

read the original abstract

We characterize possible pairs $(u_\varepsilon,c)\in C(\mathbb{R}^n\backslash\varepsilon\mathbb{Z}^n,\mathbb{R})\times\mathbb{R}$ addressing the homogenization problem for Hamilton--Jacobi equations $$ H\left(\frac{x}{\varepsilon}, d u_\varepsilon, u_\varepsilon\right)=c, \quad \left({\mathrm resp.} \quad H\left(\frac{x}{\varepsilon}, d u_\varepsilon, u_\varepsilon\right)=\varepsilon\Delta u_\varepsilon+c \right) $$ for all $\varepsilon>0$. 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 $u\in C(\mathbb{R}^n,\mathbb{R})$ solving the effective equation \[ \overline H( du,u)=c,\quad ({\mathrm resp.}\quad \overline H(du,u)=\Delta u+c) \] as $\varepsilon\rightarrow 0_+$ with a uniform rate $\mathcal{O}(\varepsilon)$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives an O(ε) rate for homogenization of static contact Hamilton-Jacobi equations by using Mather-measure criteria under a monotonicity assumption.

read the letter

The main thing to know is that the authors obtain uniform O(ε) convergence of solutions u_ε to a unique effective limit under a monotonicity condition on the Hamiltonian, with the effective equation selected by structural criteria on Mather measures. This applies to both the pure HJ case and the version with a small Laplacian term. They characterize the possible solution pairs and show the rate holds for all such solutions as ε goes to zero. The approach extends standard qualitative homogenization by adding the rate and handling the contact dependence on u itself through the measures. This framing looks clean and avoids some of the usual non-uniqueness problems without needing strict monotonicity. The argument stays consistent with existing techniques for controlling oscillations via the measures, and the stress-test note confirms no obvious circularity or mismatch between hypotheses and conclusions. One soft spot is that the criteria based on Mather measures, while natural, may be abstract to verify in concrete examples; the paper would be stronger with at least one or two explicit Hamiltonians where the conditions are checked and the rate is observed. The error constants in the O(ε) bound also look like they could depend on the data in ways that are not fully spelled out in the abstract, so the proofs need scrutiny for sharpness. This is aimed at people already working on periodic homogenization of first-order PDEs or contact problems in mechanics. A reader comfortable with Mather measures and qualitative results will pick up the quantitative improvement quickly. It deserves a serious referee because the claim is precise, the setup is honest, and the extension is grounded enough to warrant detailed review.

Referee Report

0 major / 2 minor

Summary. The manuscript characterizes pairs (u_ε, c) solving the static contact Hamilton-Jacobi equation H(x/ε, du_ε, u_ε) = c (and the viscous variant with εΔu_ε) for all ε > 0. Under a monotonicity assumption on the Hamiltonian (not necessarily strict) together with criteria based on the structure of Mather measures, it claims that every solution u_ε converges uniformly to a unique limit u solving the effective equation H-bar(du, u) = c (resp. with Δu) at rate O(ε).

Significance. If the claims are established, the work supplies a quantitative homogenization result with explicit rate in the contact setting, where rates are typically unavailable. The use of Mather-measure criteria to select the effective Hamiltonian and control oscillations is a natural and potentially powerful approach that could extend to related problems in weak KAM theory and numerical homogenization.

minor comments (2)

The abstract states the main convergence claim but does not reference the theorem or section where the Mather-measure criteria are formally stated and verified; adding such a pointer would improve readability.
Notation: the effective Hamiltonian is written as overline H in the abstract; ensure consistent use of this symbol (and its dependence on the Mather data) throughout the introduction and main statements.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive recommendation of minor revision. The referee's summary accurately captures the main results on quantitative homogenization for static contact Hamilton-Jacobi equations under monotonicity assumptions and Mather measure criteria.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation proceeds from the given monotonicity assumption on H together with explicit structural criteria on Mather measures to select the effective Hamiltonian and obtain the uniform convergence rate. These criteria are stated as external inputs drawn from the standard theory of Mather measures rather than being fitted or redefined inside the paper itself. No equation is shown to reduce to its own input by construction, no parameter is fitted on a subset and then relabeled as a prediction, and no load-bearing uniqueness claim rests solely on a self-citation chain. The argument is therefore self-contained against the stated hypotheses.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the external theory of Mather measures and a monotonicity assumption; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Monotonicity assumption on the Hamiltonian
Invoked to ensure characterization of solution pairs and convergence to the effective equation.

pith-pipeline@v0.9.0 · 5493 in / 1178 out tokens · 49207 ms · 2026-05-13T19:35:38.274414+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

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_ε converge to a uniquely identified limit u solving the effective equation with a uniform rate O(ε).
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We characterize possible pairs (u_ε,c) ... H(x/ε, du_ε, u_ε)=c ... effective equation H̄(du,u)=c

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

A PDE formulation of Lyapunov stability for contact-type Hamilton-Jacobi equations
math.AP 2026-04 unverdicted novelty 6.0

PDE criteria based on the critical value of the Hamiltonian and viscosity subsolutions determine Lyapunov stability and instability for stationary solutions of contact-type Hamilton-Jacobi equations with continuous co...

Reference graph

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