{"id":"0a5ddde6-2e87-4ef4-b46b-f8830b4c413d","arxiv_id":"2507.00663","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Optimal O(epsilon) homogenization rate, bounded correctors, and global Holder regularity are established for convex Hamilton-Jacobi equations with u/epsilon-periodic Hamiltonians.","lead":"Mathematicians proved that certain interface-evolution equations with a periodic wiggle depending on the height itself converge to a smooth limit with the best possible error rate, uniform in time. The result also builds steady correction terms and regularity estimates, with applications to dislocation dynamics in crystals.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The O(ε) rate in Theorem 1.1 hinges on Corollary 3.7, which is deferred to [17] without proof; the derivation from Lemmas 3.2–3.6 is not immediate, and the Burago displacement sign in Lemma 3.6 appears inconsistent.","rationale":"The central claim of the paper is Theorem 1.1, the time-uniform O(ε) homogenization rate. The argument passes through the implicit variational principle, the approximate subadditivity (Lemma 3.2) and superadditivity (Lemma 3.6) of the fundamental solution, and then Corollary 3.7, which converts these into the quantitative convergence of mε to m. Corollary 3.7 is the only bridge between the surgery estimates and the final rate, but its proof is omitted with a reference to [17, Theorem 4.2]. In a self-contained paper claiming a new optimal rate, this is a serious gap: the standard subadditive ergodic argument with an additive constant does not automatically yield an O(ε) rate without a careful tracking of the constant's scale dependence. The apparent sign inconsistency in the Burago cutting step of Lemma 3.6 compounds the concern, since if the selected displacement should be -y rather than y, the construction of the reconstructed curve η from y to 0 does not close. These issues target the proof of the main theorem directly. The reader's weakest_assumption field names the superlinearity (H3), but the reader's rationale correctly identifies the omitted Corollary 3.7 as the main gap; the present stress-test agrees on that gap and adds a specific, checkable sign issue. This does not change the conditional verdict: the theorem may well be true, but the manuscript must supply a complete proof of Corollary 3.7 and fix the Burago computation before the central claim is fully established. The paper contains substantial novel structure and the rest of the argument is plausible, so a request for revision rather than rejection is appropriate.","tokens_in":42512,"tokens_out":19257,"duration_ms":227554,"concrete_test":"Independently derive Corollary 3.7 from Lemmas 3.2 and 3.6 in full, following [17, Theorem 4.2]. For fixed t>0, y, c with |y|≤M0t, define φ(s)=m(st,0,sy,sc), and verify that the two inequalities φ(s+l)≤φ(s)+φ(l)+C (max{st,lt}≥1) and 2φ(s)≤φ(2s)+C imply |εφ(1/ε)-lim_k φ(k)/k| ≤ C' ε for all ε<1, with C' depending only on M0 and H. Also check in Lemma 3.6 that Burago's lemma is applied to the curve (γ,w) in R^{n+1} with total displacement (-2y, w(2t)-2c); the selected pieces should sum to (-y, t, (w(2t)-2c)/2), not (y,t,...). If the cutting step has a bounded error term, verify it can be absorbed into the constant C.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.1 depends entirely on Corollary 3.7, the O(ε) approximation of the fundamental solution mε by the effective metric m. Corollary 3.7 is not proved; 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: Lemma 3.2 is an approximate subadditivity with a constant error C independent of scale, and Lemma 3.6 is an approximate superadditivity of the same type. A naive Fekete argument with a bounded error gives only an O(1) bound for |mε - m|, not O(ε); to obtain the rate one must track how the error scales with the homogeneity of the arguments (kt, ky, kc). Additionally, in the proof of Lemma 3.6, the Burago cutting step asserts a collection of intervals with total displacement (y, t, ...), but the original minimizer runs from 2y to 0, so the selected displacement should be -y up to a bounded error; if this sign is not a typo, the construction of η from y to 0 is inconsistent and the superadditivity estimate fails. Since Corollary 3.7 is the engine of the optimal rate, this gap is load-bearing for Theorem 1.1.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":42735,"tokens_out":8905,"duration_ms":96405,"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":[{"comment":"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.","section":"§3, Corollary 3.7"},{"comment":"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.","section":"§3, Lemma 3.6"},{"comment":"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).","section":"§5, Theorem 1.3 and Lemma 5.1"}],"minor_comments":[{"comment":"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.","section":"Section 2, definition of C(x,y;b,a)"},{"comment":"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.","section":"Lemma 5.2"},{"comment":"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.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"My recommendation is driven primarily by the missing proof of Corollary 3.7; this is the central estimate on which Theorem 1.1 depends. The sign issue in Lemma 3.6 appears fixable, but it must be addressed explicitly. If the authors can supply a complete proof of Corollary 3.7 in the present u/ε-periodic setting and correct the hypotheses of Theorem 1.3, the paper would be a strong candidate for acceptance. The topic fits the journal well and the framework is original."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a serious paper that very likely proves the optimal O(ε) time-uniform rate for a class of u/ε-periodic convex Hamilton–Jacobi equations, resolving an open problem from Tran–Yu's lecture notes. The main construction—an implicit variational principle and a curve-surgery argument in (x,u)-space—is new and technically substantial. The bounded correctors and Hölder estimates are a nice payoff.\n\nThe paper deserves peer review. The central result is credible and the proof strategy works modulo two items that should be fixed. Corollary 3.7, the rate estimate, is stated as 'straightforward' and deferred to [17]. The stress-test says the derivation is not immediate because the error in Lemmas 3.2 and 3.6 is independent of scale. I don't think that's a real gap: subadditivity plus dyadic superadditivity with bounded error implies the normalized function differs from its limit by O(1) via a standard dyadic argument. But the paper should include that argument or cite the exact theorem, because Corollary 3.7 is the engine.\n\nMore serious is a likely sign error in the Burago step of Lemma 3.6. The minimizer runs from 2y to 0, but the summation is written with displacement (+y, t, ...). The new curve is supposed to run from y to 0, so the selected displacement should be −y (up to bounded error). As written, the endpoint bound |γ̃(t_k^-)|≤k√n is inconsistent unless y is small. This is probably a typo, but a referee should ask for the corrected statement.\n\nAlso Theorem 1.3 says φ ∈ BUC but the constant depends on ∥φ∥_{W^{1,∞}} and the proof uses Theorem 1.1, which needs φ Lipschitz. The hypothesis should be φ ∈ Lip (or BUC ∩ Lip). This is minor but should be fixed.\n\nThe paper is honest about needing superlinear growth (H3); the authors explain why coercivity alone is not enough. The citation pattern looks appropriate; self-citations are to their own prior work where the limit arguments were developed.\n\nWho should read it: anyone working on quantitative homogenization of HJ equations, dislocation dynamics models, or contact Hamilton–Jacobi theory. I would send it to a strong referee and ask for the fixes above.","headline":"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.","tokens_in":43335,"tokens_out":8551,"would_cite":true,"duration_ms":93291,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B27","35B40","49L25"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["homogenization","Hamilton-Jacobi equations","convex Hamiltonians","u/epsilon-periodic Hamiltonians","dislocation dynamics","optimal convergence rates","fundamental solution","Hölder regularity"],"falsifier":"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.","tokens_in":42248,"feed_emoji":"📐","tokens_out":12454,"duration_ms":125386,"temperature":0.7,"pith_summary":"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 $\\varepsilon$.","feed_headline":"Time-uniform O(ε) error for u/ε-periodic Hamilton–Jacobi PDEs","feed_subtitle":"Curve surgery on an implicit fundamental solution yields a time-uniform optimal rate for convex u/ε-periodic Hamiltonians.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"proves the O(ε) optimal rate for the r-independent case H(x,p); Theorem 1.1 includes it as a particular case and inherits optimality from it.","marker":"[43]"},{"why":"introduces the subadditivity/superadditivity and curve-surgery strategy for convex periodic homogenization that this paper adapts to u/ε-dependent Hamiltonians.","marker":"[28]"},{"why":"gives the previous homogenization result for u/ε-periodic Hamiltonians with rate O(e^T ε^{1/3}) on finite time intervals, the bound Theorem 1.1 improves to a time-uniform O(ε).","marker":"[2]"},{"why":"supplies the cutting lemma for periodic metrics used to slice and periodically shift optimal curves in Lemmas 3.2 and 3.6.","marker":"[4]"},{"why":"establishes the Herglotz variational principle for continuous strictly convex Lagrangians that underlies Theorem 2.1.","marker":"[6]"},{"why":"formulates the quantitative analysis of (1.1) as an open question that Theorem 1.1 answers.","marker":"[44]"},{"why":"introduces u/ε-periodic Hamiltonians from dislocation dynamics and the cell problem (1.4), whose correctors the paper constructs.","marker":"[20]"}],"fun_headline_variants":["Optimal ε-rate for convex HJ with u/ε-periodic Hamiltonians","Implicit fundamental solution gives sharp HJ homogenization rate","Optimal homogenization plus Hölder correctors for u/ε-periodic HJ","Sharp time-uniform ε error for convex HJ homogenization","Optimal O(ε) rate for u/ε-periodic Hamilton-Jacobi equations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Optimal ε-rate for convex HJ with u/ε-periodic Hamiltonians","Implicit fundamental solution gives sharp HJ homogenization rate","Optimal homogenization plus Hölder correctors for u/ε-periodic HJ","Sharp time-uniform ε error for convex HJ homogenization","Optimal O(ε) rate for u/ε-periodic Hamilton-Jacobi equations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000918,"raw_usage":{"total_tokens":3903,"prompt_tokens":874,"completion_tokens":3029,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":490,"completion_tokens_details":{"reasoning_tokens":2949}},"tokens_in":490,"tokens_out":3029,"duration_ms":22532,"temperature":1.0,"reasoning_tokens":2949,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:09:51.022361+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"proves the O(ε) optimal rate for the r-independent case H(x,p); Theorem 1.1 includes it as a particular case and inherits optimality from it."},{"cited_title":"Mitake, H","cited_arxiv_id":null,"evidence_quote":"introduces the subadditivity/superadditivity and curve-surgery strategy for convex periodic homogenization that this paper adapts to u/ε-dependent Hamiltonians."},{"cited_title":"Achdou, S","cited_arxiv_id":null,"evidence_quote":"gives the previous homogenization result for u/ε-periodic Hamiltonians with rate O(e^T ε^{1/3}) on finite time intervals, the bound Theorem 1.1 improves to a time-uniform O(ε)."},{"cited_title":"Burago, Periodic metrics, Adv","cited_arxiv_id":null,"evidence_quote":"supplies the cutting lemma for periodic metrics used to slice and periodically shift optimal curves in Lemmas 3.2 and 3.6."},{"cited_title":"Cannarsa, W","cited_arxiv_id":null,"evidence_quote":"establishes the Herglotz variational principle for continuous strictly convex Lagrangians that underlies Theorem 2.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"formulates the quantitative analysis of (1.1) as an open question that Theorem 1.1 answers."},{"cited_title":"Imbert, R","cited_arxiv_id":null,"evidence_quote":"introduces u/ε-periodic Hamiltonians from dislocation dynamics and the cell problem (1.4), whose correctors the paper constructs."}],"review_version":1}