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

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Pith's one-line read This paper proves a time-uniform optimal $O(\varepsilon)$ convergence rate for convex Hamilton–Jacobi equations whose Hamiltonian depends periodically on $u/\varepsilon$, via an implicit variational principle and a curve-surgery argument…

desk verdict Likely correct optimal O(ε) rate for u/ε-periodic convex HJ, with two fixable issues (deferred rate lemma and a sign typo in Lemma 3.6) before I'd sign off. read the letter →

arxiv 2507.00663 v1 pith:UTKYIRKM submitted 2025-07-01 math.AP

Hiroyoshi Mitake , Panrui Ni , Hung V. Tran This is my paper · ORCID
classification math.AP MSC 35B2735B4049L25
keywords homogenizationHamilton-JacobiequationsconvexHamiltoniansu/epsilon-periodicdislocationdynamicsoptimalconvergenceratesfundamentalsolutionHölderregularity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The authors prove that solutions $u^\varepsilon$ of the convex Hamilton–Jacobi equation $u^\varepsilon_t + H(x/\varepsilon, u^\varepsilon/\varepsilon, Du^\varepsilon)=0$ converge uniformly in space and in time to a limit $u$ with error at most $C\varepsilon$, where $C$ depends only on the Hamiltonian and the Lipschitz norm of the initial datum, and that the linear rate is optimal. This is the first time-uniform optimal rate for homogenization problems in which the Hamiltonian oscillates periodically in the unknown itself, a structure that arises in dislocation dynamics. The proof constructs a fundamental solution through an implicit variational principle, establishes its subadditivity and superadditivity by a curve-surgery argument in the joint $(x,u)$-space, and then derives the effective Lagrangian, effective Hamiltonian, and bounded continuous correctors as consequences. Under extra polynomial growth, the same theory yields global Hölder estimates for the approximate solutions and the correctors, with explicit exponents and no dependence on $\varepsilon$.

What carries the argument

The load-bearing object is the fundamental solution $m(t,x,y,c)$, the unique continuous function satisfying the implicit variational principle $m(t,x,y,c)=c+\inf_\gamma \int_0^t L(\gamma(s), m(s,\gamma(s),y,c), \dot\gamma(s))\,ds$ for the Lagrangian $L$ conjugate to $H$. This is a fixed-point version of the Herglotz variational principle: the cost of a curve depends on the running value of the solution itself, which is why the unknown-dependent $u/\varepsilon$ can be handled. Two structural estimates on $m$ carry the proof: a subadditivity bound and a superadditivity bound, obtained by a cutting lemma for periodic metrics that slices optimal curves in $(x,u)$-space, shifts the slices by the period lattice, and patches them back with controlled cost. A subadditivity limit then converts these bounds into the sharp $O(\varepsilon)$ gap between the rescaled fundamental solution and the effective metric.

What would settle it

Compute for $H(y,r,p)=|p|^2/2+\cos(2\pi r)$ and $\varphi\equiv 0$ the quantity $\sup_{\varepsilon\in(0,1),\,t>0} |u^\varepsilon(0,t)-u(0,t)|/\varepsilon$; Theorem 1.1 says it is bounded by the constant $C$, while a divergent value would falsify the claimed optimal time-uniform rate.

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Extended reading notes

Core claim

The central assertion is Theorem 1.1: under assumptions (H1)–(H5), $\|u^\varepsilon - u\|_{L^\infty(\mathbb{R}^n\times[0,\infty))} \le C\varepsilon$ for all $\varepsilon\in(0,1)$, with $C$ depending only on $H$ and $\|D\varphi\|_{L^\infty}$, and the exponent one cannot be improved because the case where $H$ is independent of $r$ already saturates it. The proof identifies the limit through an effective metric $\bar m(t,x,y,c)=c+t\bar L((x-y)/t)$, so $u(x,t)=\inf_y\{\varphi(y)+t\bar L((x-y)/t)\}$, and $u$ solves $u_t+\bar H(Du)=0$ with $\bar H$ the convex conjugate of $\bar L$. The machinery also produces bounded continuous correctors for the cell problem $v_\tau+H(y,p\cdot y+v-\bar H(p)\tau, p+D_y v)=\bar H(p)$, and, under additional growth hypotheses, global Hölder estimates with explicit exponents for both the approximate solutions and the correctors.

Load-bearing premise

The argument depends on the Hamiltonian being superlinear in the momentum variable $p$ (assumption (H3)); with mere coercivity the available Lipschitz bounds for $u^\varepsilon$ grow like $e^{Kt/\varepsilon}$, so the time-uniform $O(\varepsilon)$ rate is not obtained.

Editorial extensions

If this is right

  • Because the error bound does not depend on the terminal time $T$, the approximation $u^\varepsilon \approx u$ is equally accurate on arbitrarily long time horizons, in contrast to the earlier bound $O(e^T \varepsilon^{1/3})$.
  • The limit is characterized without solving a cell problem first: the effective metric gives $u$ in Lax–Oleinik form and the effective Hamiltonian by Legendre transform.
  • Bounded continuous correctors exist for the cell problem (1.4), improving on semi-continuous sub/supercorrectors; under (H6) the correctors are globally Hölder continuous with explicit exponents.
  • For $r$-independent Hamiltonians, Theorem 1.1 recovers the known optimal rate for classical convex periodic homogenization, so the new result is a genuine extension.
  • The dislocation-motivated example $H(y,r,p)=|p|^2-F(r)$ yields an $O(\varepsilon)$ bound for periodic ODEs $\dot y^\varepsilon=F(y^\varepsilon/\varepsilon)$ converging to an affine limit, and the same rate for one-dimensional oscillatory transport equations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The implicit variational principle should extend to weakly coupled systems or time-dependent periodic coefficients, where the same fixed-point-plus-surgery scheme could give uniform rates despite the absence of a single-cell effective Hamiltonian.
  • A natural next step, already hinted at in Section 4.4, is to develop a contact analogue of Aubry–Mather theory: minimizing $\int L(x,u,v)\,d\mu$ over dual-flow invariant measures with prescribed rotation vector would refine $\bar L$ and connect the $O(\varepsilon)$ rate to a Mather-type $\beta$-function.
  • A testable hypothesis is whether mere coercivity of $H$ in $p$ is enough: if a merely coercive Hamiltonian such as $H(y,r,p)=|p|-\cos(2\pi r)$ also produced bounded $\sup_{\varepsilon,t}|u^\varepsilon-u|/\varepsilon$, then the superlinearity condition (H3) would be an artifact of the method rather than a necessary condition.
  • The uniform-in-time error could be combined with standard numerical schemes to certify long-horizon approximations of the effective solution; the one-dimensional transport coefficient of Proposition 4.6 is a convenient benchmark for such a test.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper studies the Cauchy problem u_t^ε + H(x/ε, u^ε/ε, Du^ε)=0 with H periodic in the first two arguments and convex and superlinear in p. The authors construct a fundamental solution m via an implicit variational principle (Theorem 2.1) and then establish approximate subadditivity and superadditivity properties for m in Section 3. On this basis they claim a time-uniform O(ε) convergence rate for u^ε (Theorem 1.1), with optimality inherited from the r-independent case studied by Tran–Yu. They then derive an effective Lagrangian and Hamiltonian, prove existence of bounded continuous correctors (Theorem 1.2 and Section 4), and, under an additional growth hypothesis (H6), establish global Hölder estimates for u^ε and for the correctors (Theorems 1.3 and 1.4). Appendix B provides a derivation of the model from dislocation dynamics.

Significance. If the proof of Corollary 3.7 can be completed, Theorem 1.1 would be a substantial advance: it would extend the optimal O(ε) rate of [43] to Hamiltonians with the u/ε dependence and would answer a question raised in [44]. The implicit variational principle, the derivation of bounded correctors from the quantitative rate, and the global Hölder estimates are valuable contributions. The paper is largely self-contained, with the fixed-point construction of the fundamental solution given in detail and the curve-surgery adaptation to the u-dependent case being genuinely nontrivial. However, the main rate claim currently rests on an unproved corollary, and there is a sign inconsistency in the proof of the superadditivity lemma. The significance is therefore conditional on a complete proof of Corollary 3.7.

major comments (3)
  1. [§3, Corollary 3.7] Corollary 3.7 is the engine of Theorem 1.1, yet it is asserted without proof: the text says it is a straightforward consequence of Lemmas 3.2 and 3.6 and refers to [17, Theorem 4.2]. This is not immediate. Lemmas 3.2 and 3.6 give approximate subadditivity and superadditivity with additive errors that do not visibly depend on the scale; a direct Fekete-type argument with such bounded errors yields convergence of the scaled quantities but not an O(ε) rate. To obtain O(ε) one must carefully track how the errors behave under rescaling by 1/ε and how they accumulate over O(1/ε) blocks. Moreover, [17] treats state-constraint Hamilton–Jacobi equations, not the present u/ε-periodic setting, so the cited theorem cannot be applied without a detailed transfer argument. Since Corollary 3.7 is also used in Lemma 4.1 to prove continuity of the effective metric, this gap affects the effective Hamiltonian and corrector results as well. Please include a complete proof of Corollary 3.7.
  2. [§3, Lemma 3.6] The Burago cutting step in the proof of Lemma 3.6 states that the selected pieces satisfy Σ_i (γ(b_i)-γ(a_i), b_i-a_i, w(b_i)-w(a_i)) = (y, t, (w(2t)-2c)/2). Since the minimizer γ of m(2t,0,2y,2c) runs from γ(0)=2y to γ(2t)=0, the total displacement over [0,2t] is -2y, and the selected vector should sum to -y if the concatenated shifted curve is to run from y to 0. The subsequent construction indeed sets η(0)=y and η(t)=0. As written, the displayed sum gives endpoint 2y, not 0, so the sign is inconsistent. If this is a typo, it must be corrected and the periodic-shift estimates checked against the corrected sign; if it is not a typo, the superadditivity estimate is not established.
  3. [§5, Theorem 1.3 and Lemma 5.1] Theorem 1.3 and Lemma 5.1 assume φ ∈ BUC(Rn) but state that the constants depend on ∥φ∥_{W^{1,∞}(Rn)}. This is inconsistent as written: a general BUC function need not have a weak derivative in L∞. Furthermore, the proof of Lemma 5.1 invokes Theorem 1.1, which requires φ ∈ Lip(Rn), and the minimizer estimates in the proof are Lipschitz-based. The hypotheses should be corrected, for example by assuming φ ∈ BUC(Rn) ∩ W^{1,∞}(Rn), or by proving the result for genuinely bounded uniformly continuous initial data. This is load-bearing for Theorem 1.3; Theorem 1.4 is not affected because it assumes w0 ∈ BUC(Rn) ∩ Lip(Rn).
minor comments (3)
  1. [Section 2, definition of C(x,y;b,a)] In the line defining C(x,y;b,a), the condition 'a ≤ b' is written with a,b ∈ R^n; it should read a,b ∈ R.
  2. [Lemma 5.2] In the first line of the proof, 'We first show u(x,t) - u(x,t−δ) ≤ C3δ' should refer to uε(x,t) - uε(x,t−δ); the subsequent argument uses minimizers of uε, so this is a typo.
  3. [Throughout] There are several typographical errors, including 'Legenedre' in Section 4.1, 'indepedent' in the proof of Proposition 4.4, and 'uϵ' in the proof of Proposition 3.1. These should be corrected.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction: the effective Hamiltonian and correctors are obtained from the limiting fundamental solution, with Theorem 1.1 used forward rather than assumed.

full rationale

The paper's central derivation defines m via the implicit variational principle (Theorem 2.1), defines m_epsilon by rescaling, proves existence of the limit m (Corollary 3.4), and then proves the O(epsilon) rate by comparing m_epsilon with m through subadditivity and superadditivity (Lemmas 3.2 and 3.6). The rate result Theorem 1.1 is not obtained by assuming the limit equation or by fitting a parameter; m is genuinely defined before the rate is proved. The effective Lagrangian L is then read off from m in Lemma 4.1, and the effective Hamiltonian is its Legendre transform; this is a definition, not an input. Theorem 1.2 uses Theorem 1.1 to construct bounded correctors, which is a forward application of a result already proved. The only point requiring scrutiny is Corollary 3.7, which is asserted as 'a straightforward result of Lemmas 3.2, 3.6' with a pointer to [17, Theorem 4.2] for a similar proof. That citation involves two of the present authors (Mitake and Tran), and Corollary 3.7 is load-bearing for the optimal rate, but the cited theorem concerns a different state-constraint problem and is not the same statement as the present result. Thus this is a proof-deferral or correctness gap rather than a circular reduction: no equation of the present paper is shown to be equivalent to its own input, and no fitted quantity is renamed as a prediction. The optimality argument uses the r-independent special case [43] as an external benchmark. Accordingly, the paper exhibits no significant circularity, and the score reflects only the minor self-citation in the deferred proof of Corollary 3.7.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central claim rests on the stated structural hypotheses (H1)-(H5), plus standard results from convex analysis and optimal control. No parameters are fitted to data. The main object, the fundamental solution m, is constructed in the paper rather than assumed.

assumptions (8)
  • domain assumption (H1) Z^{n+1}-periodicity of H in (y,r)
    Standing structural assumption defining the u/epsilon-periodic class; used throughout.
  • domain assumption (H2) H is K-Lipschitz in r uniformly in (y,p)
    Guarantees the implicit variational problem has a unique solution (Theorem 2.1) via the contraction estimate (2.5).
  • domain assumption (H3) H is superlinear in p
    Controls speed of minimizers and prevents time-dependent blow-up; explicitly essential for the optimal rate.
  • domain assumption (H4) p -> H(y,r,p) is convex
    Needed for the Lagrangian/variational representation and for the effective Lagrangian to be convex.
  • domain assumption (H5) initial data phi is Lipschitz
    Used for comparison, representation formula, and rate estimates.
  • domain assumption (H6) growth bounds q1,q2 for Holder regularity
    Additional assumptions for Theorems 1.3-1.4; not used for the main rate.
  • standard math Burago's cutting lemma
    Used in Lemma 3.6 to select disjoint intervals for the superadditivity curve surgery; cited from [4, Lemma 2].
  • standard math Classical Lipschitz continuity of the standard metric function m0 (Davini's theorem)
    Used in the proof of Theorem 2.1, Step 1; cited from [13, Theorem 3.1].

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Pith. "Pith review of Quantitative homogenization of convex Hamilton-Jacobi equations with $u/\varepsilon$-periodic Hamiltonians." pith.science (2026). https://pith.science/paper/UTKYIRKM

@misc{pith2026250700663,
  author       = {Pith},
  title        = {Pith review of: Quantitative homogenization of convex Hamilton-Jacobi equations with $u/\varepsilon$-periodic Hamiltonians},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UTKYIRKM}},
  note         = {Machine review of arXiv:2507.00663}
}
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.

Figures

Figures reproduced from arXiv: 2507.00663 by the authors.

Figure 1
Figure 1. Image of η. Let ξ2 = ξ2,η be the solution of ( ˙ξ2(s) = L(η(s), ξ2(s), η˙(s)) for a.e. s ∈ (0,(σ + l)t), ξ2(0) = (σ + l)c. We take n1 ∈ Z such that lc + n1 − ξ2(σt) ∈ [0, 1). Note that ˜ξ2(s) := ξ2(σt + s) satisfies ˜˙ ξ2(s) = ˙ξ2(σt + s) = L(η(σt + s), ξ2(σt + s), η˙(σt + s)) = L(γl(s), ˜ξ2(s), γ˙l(s)) for s ∈ (0, lt) with the initial condition ˜ξ2(0) = ξ2(σt). Let ˜ξ1(s) := ξ l 1 (s)+n1. Noting that ˜ξ1(0) = lc + … view at source ↗
Figure 2
Figure 2. Graphs of ξ σ 1 , ξ σ 1 + n2, ξ σ 1 + n3, ξ l 1 + n1 and ξ2. We take a straight line α : [0, σt] → R n connecting σy and 0 with speed |α˙ | ≤ M0. Then, ξ σ 1 (σt) − σc = m(σt, 0, σy, σc) − σc ≤ Z σt 0 L(α(s), m(s, α(s), y, c), α˙(s)) ds ≤  max y∈Rn, |v|≤M0 L(y, 0, v) + K  σt, since L(x, r, v) is periodic in r. If (3.3) does not hold, then ξ σ 1 (⌊σt⌋) − σc = Z ⌊σt⌋ 0 L(γσ(s), ξσ 1 (s), γ˙σ(s)) ds ≥ 4⌊σt⌋Cd, which … view at source ↗
Figure 3
Figure 3. Image of η. Now, we consider the solution v(s) of ( v˙(s) = L(η(s), v(s), η˙(s)) for s ∈ [0, t], v(0) = c. Without loss of generality, we assume id > 1. We choose suitable n1 ∈ Z such that w(a1) + n1 − v  1 8(k ′ + 1)  ∈ [0, 1). Define ˜v(s) = v(s + 1 8(k ′+1) ), we have    v˜˙(s) = L(˜γ(s), v˜(s), γ˜˙(s)) for s ∈ [0, t1], v˜(0) = v  1 8(k ′+1) [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Graphs of v, w, and periodic shifts of parts of the graph of w. Summing up (3.12)–(3.16) and using Lemma 2.6, we get m(t, 0, y, c) − c ≤ v(t) − c ≤ X k i=1 (w(bi) − w(ai)) + C = w(2t) − 2c 2 + C = 1 2 m(2t, 0, 2y, 2c) − c + C for some C > 0. □ Corollary 3.7. Let mε and…

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