{"id":"f5294c4d-a85a-4afd-98dd-00a32898e5dc","arxiv_id":"2507.20472","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Vanishing discount limits for fully nonlinear contact Hamilton-Jacobi equations on R^n converge locally uniformly to the maximal solution selected by a Mather-measure criterion.","lead":"This paper proves that solutions to a broad class of fully nonlinear Hamilton-Jacobi equations on the whole space converge to a distinguished solution as the discount factor vanishes. It supplies a selection principle via Mather-type measures and a localization argument that reduces the whole-space problem to bounded domains.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Prop. 5.4 applies the unbounded-domain comparison principle Theorem A.5 with φ=θuλ or vθ, but uλ is only locally Lipschitz; the global Lipschitz hypothesis needed for (5.6) is not verified.","rationale":"The reader identified the unbounded comparison principle as the weakest assumption; I concur that Theorem A.5 is the load-bearing imported tool. I sharpen the concern: even granting Theorem A.5, the proof of Prop. 5.4 does not verify all its hypotheses, because the comparison functions are built from uλ, which is only locally Lipschitz, and the case labels (P1)/(P2) are swapped in the text. The rest of the paper is coherent and the strategy is plausible; the measure construction and localization arguments are internally consistent once (5.6) is granted. This is not grounds for rejection, but it is a concrete gap that a specialist must close before Theorem 1.1 is fully supported, so conditional acceptance is appropriate.","tokens_in":33631,"tokens_out":24673,"duration_ms":252792,"concrete_test":"Re-derive Prop. 5.4 using the original statement of [17, Theorem 4.1] and check whether φ=θuλ and φ=θuλ+(1-θ)˜v satisfy all its hypotheses. Determine whether Theorem A.5 requires φ to be globally Lipschitz; if it does, either prove a global Lipschitz bound for the Perron solution uλ under (P1),(P2), or construct a global Lipschitz replacement for φ. A negative result would be a Hamiltonian satisfying (H1)-(H3)+(P2) whose maximal solution has unbounded local Lipschitz constants, showing that the comparison step in Prop. 5.4 is unjustified.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central selection result Theorem 1.1 rests on the variational identity (5.6), proved in Prop. 5.4 by comparing the value function W(x,t) from (5.4) with uλ(x)-c(H)t through Theorem A.5. Theorem A.5 requires a globally Lipschitz function φ with H(x,Dφ)≤C and v-φ→∞. In the unbounded cases (P1),(P2), Prop. 5.4 chooses φ=θuλ in the case labelled (ii) and vθ=θuλ+(1-θ)˜v in the case labelled (iii), and uses the inequality H(x,Dφ,λuλ)≤Cθ. However, Prop. 4.3 only yields uλ locally Lipschitz and locally bounded; no global Lipschitz bound for uλ is established. If φ is not globally Lipschitz, Theorem A.5 cannot be invoked on all of R^n×[0,T]. The manuscript's proof also interchanges (P1) and (P2) between cases (ii) and (iii), which indicates this comparison step was not carefully checked. Since both proofs of Theorem 1.1 (Sections 6 and 7.2) rely on (5.6), a failure of this comparison leaves the Mather-measure selection unsupported precisely for noncompact problems with unbounded maximal solutions.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the asymptotic behavior as λ→0+ of the maximal viscosity solution u_λ on R^n of the fully nonlinear contact Hamilton–Jacobi equation H(x,Du,λu)=c(H), under assumptions (H1)–(H3) and one of the structural conditions (P1)–(P3). The main result, Theorem 1.1, asserts that u_λ converges locally uniformly to a solution u_0 of the ergodic equation H(x,Du,0)=c(H), characterized as the supremum of all subsolutions w satisfying an integral inequality against a family M of Mather-type probability measures. A second main result, Theorem 1.2, states a localization property: for each z there exist R_z and λ_z such that u_λ(z) equals the state-constraint solution on B_{R_z}(0). The proof proceeds through a variational representation for u_λ (Section 5), an exponential reweighting with solution-dependent discount indices, construction of discounted measures, and a selection argument (Section 6); an alternative proof via localization and bounded-domain results is given in Section 7.","tokens_in":33897,"tokens_out":9797,"duration_ms":96102,"significance":"If the results are correct, this paper makes a substantial contribution by extending the vanishing-discount selection principle to fully nonlinear contact Hamilton–Jacobi equations on noncompact domains, going beyond the discounted case treated by Ishii and Siconolfi. The introduction of solution-dependent discount indices that yield exact exponential variational formulas is a genuine technical novelty, and the characterization of the limit through Mather-type measures on R^n is a natural and valuable extension of the bounded-domain theory. The paper is clearly organized and the main theorems are precisely stated. However, the manuscript relies heavily on imported results from prior work ([20], [35]) and leaves several key proofs as 'minor modifications'; consequently, the independent verification burden is high, and the proof as written has a load-bearing gap in the comparison argument for the variational formula.","major_comments":[{"comment":"The comparison principle in Theorem A.5 requires an auxiliary function φ that is globally Lipschitz on R^n and satisfies H(x,Dφ)≤C. In cases (ii) and (iii) of Proposition 5.4 the proof chooses φ=θu_λ and φ=v_θ=θu_λ+(1−θ)ṽ, respectively. Proposition 4.3 provides only local Lipschitz regularity and local boundedness of u_λ; no global Lipschitz bound is established under (P1) or (P2). Since the variational identity (5.6) is the foundation for Propositions 5.10–5.11 and for both proofs of Theorem 1.1 (Section 6 and Section 7.2), the central selection result is not supported as written. Please either prove the missing global Lipschitz regularity of u_λ under (P1)/(P2), or replace Theorem A.5 by a comparison principle whose hypotheses are verified for locally Lipschitz φ, and check the resulting growth conditions at infinity.","section":"§5.1, Prop. 5.4; Appendix A.2, Theorem A.5"},{"comment":"The statement lists case (ii) under assumption (P2) and case (iii) under (P1), but the proof of case (ii) uses the estimate H(x,θp,u)≤H(x,p,u)+C_θ, which is condition (1.7) of (P1), while the proof of case (iii) invokes joint convexity of H in (p,u) and the implication (1.8), which is condition (P2). This swap of (P1) and (P2) is not merely a typo, because the two assumptions are not equivalent and the proof currently does not identify which structural condition supports the comparison argument. Please correct either the statement or the proof and confirm the intended hypothesis for each case.","section":"§5.1, Prop. 5.4, cases (ii) and (iii)"},{"comment":"The state-constraint representation (3.5) and the proof of Proposition 3.4 use the global constant c(H), whereas all subsequent formulas in Section 3 (e.g., (3.8), (3.11), (3.18), (3.20)) use the domain-dependent constant c_Ω(H). Since c_Ω(H)≤c(H) and the inequality can be strict for bounded domains not containing the Aubry set, equation (3.5) is false as stated. This inconsistency affects the derivation of the exponential representations and the bounded-domain convergence result Theorem 3.14, which is used in the alternative proof of Theorem 1.1. Please replace c(H) by c_Ω(H) throughout Section 3 and adjust the corresponding proof steps.","section":"Section 3, Prop. 3.4 and Prop. 3.5"}],"minor_comments":[{"comment":"The sentence ending '...complicating localization, i.e., to show that the global solution matches the state-' is truncated and should be completed.","section":"§1.1, p.2"},{"comment":"In assumption (H3), the inequality 'κ_R≤∂uH(x,p,u)≤κ_R' uses the same symbol for the lower and upper bounds; this should be, for example, κ_R≤∂uH(x,p,u)≤\\barκ_R.","section":"§1.2, (H3)"},{"comment":"The word 'miminimizer' appears twice and should be 'minimizer'.","section":"Prop. 3.5(ii) and Prop. 3.8(ii)"},{"comment":"The phrase 'we can ass assume super linearlity' contains a typo and should read 'we can assume superlinearity'.","section":"Proof of Theorem 1.1, §6"},{"comment":"In the proof of Lemma 2.4, 'The fact that x→S_H(y) is a subsolution' should be 'x↦S_H(x,y) is a subsolution'.","section":"Lemma 2.4 proof"},{"comment":"In the display after (6.13), the expression 'e^{λααγ(s)}' uses a stray 'α'; it should be 'e^{λβ^λ_γ(s)}' consistently with (6.11).","section":"Proof of Theorem 1.1, §6"}],"recommendation":"major_revision","confidential_remarks":"The paper contains two independent routes to Theorem 1.1, but both depend on the same variational formula (5.6), so the comparison gap in Proposition 5.4 is the single most important issue to resolve. I also note the heavy reliance on prior results [20] and [35]; if the missing comparison step can be repaired and the (P1)/(P2) confusion clarified, the main theorems are likely to hold. The inconsistency between c(H) and c_Ω(H) in Section 3 should be corrected before the paper is reconsidered."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does something new: it extends vanishing discount selection to fully nonlinear contact Hamilton–Jacobi equations on noncompact domains, with a solution-dependent discount index and a localization theorem that lets the global solution match the state-constraint solution at each point. The Mather-measure characterization and the modified variational formulas are worthwhile contributions, and the paper is mostly well organized.\n\nThe main concern is real. Proposition 5.4 needs to prove equality u(x,t)=uλ(x)-c(H)t by invoking Theorem A.5, an unbounded-domain comparison principle. That theorem requires a globally Lipschitz test function φ with H(x,Dφ)≤C and v-φ→∞. In cases (ii) and (iii) the proof chooses φ=θuλ or vθ=θuλ+(1-θ)ṽ, but uλ is only locally Lipschitz from Prop. 4.3. No global Lipschitz bound is established, so the hypotheses of Theorem A.5 are not verified. Both proofs of Theorem 1.1 rely on the resulting formula (5.6), so this is a load-bearing gap rather than a cosmetic one.\n\nThere is also a clear slip in the same proposition: the statement assigns (P2) to case (ii) and (P1) to case (iii), but the proof uses (P1) in case (ii) and (P2) in case (iii). That is probably just a labeling error, but it suggests the comparison step was not checked as carefully as it should have been. Minor issues: a duplicate κ_R in (H3), \"miminimizer\" repeated, a truncated sentence in Section 1.1, and several central lemmas imported from the authors' prior work [35] and from [20] without full proof. Those are acceptable if the imports are correct, but they make the paper hard to referee quickly.\n\nThe core idea is sound, and the localization theorem (Theorem 1.2) is a nice technical contribution that may well survive the needed repairs. The paper deserves a serious referee: the result is important enough, and the gap, while real, looks fixable by adding a global Lipschitz estimate for uλ under the stated assumptions or by weakening the comparison principle to local Lipschitz functions. I would send it to review, with the explicit instruction to focus on Prop. 5.4 and the applicability of Theorem A.5.","headline":"A genuinely new selection result for fully nonlinear contact HJ equations on the whole space, but the key comparison step in Prop. 5.4 has a real unmet hypothesis that needs fixing before the result is fully supported.","tokens_in":34443,"tokens_out":2233,"would_cite":false,"duration_ms":23970,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35D40","70H20","35J60","37J40","49L25","37K99"],"pacs":[],"model":"deepseek-v4-flash","headline":"The maximal discounted solution of a fully nonlinear Hamilton-Jacobi equation on $\\mathbb{R}^n$ converges locally uniformly, as the discount vanishes, to the largest subsolution of the ergodic equation that passes a Mather-measure…","keywords":["viscosity solution","Hamilton-Jacobi equations","vanishing discount","selection principle","Mather measures","state-constraint problem","noncompact domains","effective Hamiltonian"],"falsifier":"Take a concrete Hamiltonian satisfying (H1)-(H3) and one of (P1)-(P3), compute the Lax-Oleinik value function in (5.4) numerically, and compare it to $u_\\lambda(x)-c(H)t$; if any pair $(x,t)$ shows strict inequality, the representation formula (5.6) fails for an admissible Hamiltonian and the proof of Theorem 1.1 collapses. A cheaper check is to test the localization claim: if for some $z$ the equality $u_\\lambda(z)=\\vartheta_{\\lambda,R}(z)$ fails for all large $R$ and small $\\lambda$, the bounded-domain route to the selection principle is blocked.","tokens_in":33393,"feed_emoji":"🎯","tokens_out":5620,"duration_ms":56273,"temperature":0.7,"pith_summary":"This paper proves a selection principle for the vanishing discount limit of contact Hamilton-Jacobi equations $H(x,Du,\\lambda u)=c(H)$ on the whole space $\\mathbb{R}^n$. As $\\lambda\\to 0^+$, the maximal solution $u_\\lambda$ converges locally uniformly to a specific solution $u_0$ of the ergodic equation $H(x,Du,0)=c(H)$, namely $u_0=\\sup E$ where $E$ collects the subsolutions $w$ satisfying $\\int w\\,\\partial_u L(x,v,0)\\,d\\mu\\ge 0$ for every limiting Mather-type measure $\\mu$. The work matters because on noncompact domains the ergodic equation generally admits many solutions, and this result singles out the one selected by the discounting mechanism. The proof introduces modified exponential discount indices that make global solutions coincide locally with state-constraint solutions, reducing the noncompact problem to known bounded-domain results.","feed_headline":"Vanishing discount selects a unique solution on R^n","feed_subtitle":"The limit solves the ergodic equation and is the largest subsolution passing a Mather-measure test.","key_machinery":"The central object is the modified discount index, defined as a difference quotient of the Lagrangian $L(x,v,\\cdot)$ between the argument $0$ and the value $\\lambda u_\\lambda(x)$ (or between $-\\lambda C_0$ and $\\lambda\\vartheta_\\lambda(x)$ for the state-constraint solution). Using this index, the paper rewrites the variational formula for $u_\\lambda$ and for the state-constraint solution $\\vartheta_{\\lambda,R}$ in exponential form, with weights $e^{\\lambda\\beta_\\gamma(s)}$ and $e^{\\lambda\\alpha_\\gamma(s)}$. A comparison principle for unbounded viscosity solutions converts the control problem into the exact formula (5.6), and the localization theorem (Theorem 1.2) shows $u_\\lambda(z)=\\vartheta_{\\lambda,R}(z)$ for large enough $R$ and small enough $\\lambda$, allowing the bounded-domain vanishing discount result to identify the limit pointwise.","core_discovery":"The paper establishes that, under assumptions (H1)-(H3) plus one of the structural conditions (P1), (P2), or (P3), the maximal viscosity solution $u_\\lambda$ of $H(x,Du,\\lambda u)=c(H)$ in $\\mathbb{R}^n$ converges locally uniformly as $\\lambda\\to 0^+$ to a solution $u_0$ of $H(x,Du,0)=c(H)$, characterized as $u_0=\\sup E$ with $E$ defined by the Mather-measure integral condition $\\int w(x)\\,\\partial_u L(x,v,0)\\,d\\mu(x,v)\\ge 0$ for all $\\mu\\in M$. The set $M$ consists of weak limits of discounted measures built from minimizing curves of an exponentially weighted Lagrangian, and each such measure is holonomic and minimizes $\\int L(x,v,0)\\,d\\mu=-c(H)$. This gives a clear selection rule: the vanishing discount limit is the largest subsolution compatible with all Mather-type measures, extending the known selection mechanism from the discounted case to genuinely nonlinear contact Hamiltonians on noncompact domains.","pith_inferences":["The modified discount-index technique may transfer to second-order or nonlocal contact Hamilton-Jacobi equations, since it only requires the difference-quotient structure of $L$ in the $u$-variable.","The paper leaves open whether $M$ coincides with the full set of holonomic measures minimizing $\\int L(x,v,0)\\,d\\mu$; if it does, the selection condition could be rewritten in a more intrinsic variational form.","The explicit dependence of the selected solution on $\\partial_u L(x,v,0)$ suggests a testable prediction: perturbing the coupling between $u$ and $(x,p)$ while preserving the ergodic constant should change the selected limit in the direction dictated by that derivative.","A quantitative refinement of the localization theorem might yield explicit convergence rates for $u_\\lambda$ on unbounded domains, extending the known bounded-domain rate results."],"forward_implications":["The vanishing discount limit selects a unique solution of the ergodic equation, removing the ambiguity caused by the abundance of solutions on noncompact domains.","The selected solution is the largest subsolution satisfying the Mather-measure integral constraint, so the selection rule is explicit and checkable.","At each point $z$, the global solution coincides with the state-constraint solution on a sufficiently large ball for small discount, so the noncompact selection is locally controlled by compact-domain behavior.","The same conclusion holds under any of the three structural assumption sets (P1), (P2), or (P3), showing that the selection mechanism is robust to different comparison arguments.","The Mather-type measures are holonomic and satisfy $\\int L(x,v,0)\\,d\\mu=-c(H)$, linking the selection principle to the variational structure of the Hamiltonian."],"supporting_citations":[{"why":"Supplies the unbounded-domain comparison principle (Theorem A.5) that converts the Lax-Oleinik value function into the variational formula for $u_\\lambda$.","marker":"[17]"},{"why":"Establishes the vanishing discount problem for the discounted case on $\\mathbb{R}^n$; the present paper extends its localization strategy to nonlinear contact Hamiltonians.","marker":"[20]"},{"why":"Provides the bounded-domain state-constraint vanishing discount result (Theorem 3.14) used in the alternative proof of the main theorem.","marker":"[35]"},{"why":"Gives the original full-sequence convergence in the discounted case via Mather measures, the model for the selection mechanism generalized here.","marker":"[9]"},{"why":"Presents the nonlinear adjoint method for selection problems, an alternative route whose contrast motivates the paper's variational approach.","marker":"[28]"},{"why":"Develops curve-based discounted measures for contact Hamilton-Jacobi equations, the construction adapted here to define the limiting measures $M$.","marker":"[39]"},{"why":"Supplies the state-constraint Lax-Oleinik semigroup and dynamic programming results used to derive the formulas for $\\vartheta_\\lambda$.","marker":"[26]"}],"fun_headline_variants":["Mather measures pick the discount limit on R^n","Nonlinear HJ selects via Mather condition","Vanishing discount: Mather-consistent solution wins","Selection rule for HJ on noncompact domains","Discount limit obeys Mather measure test"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on the unbounded-domain comparison principle of Theorem A.5: without the guarantee that the control value function equals $u_\\lambda(x)-c(H)t$, the variational formula for $u_\\lambda$ and all subsequent Mather-measure arguments have no basis.","fun_headline_variants_meta":{"raw":{"variants":["Mather measures pick the discount limit on R^n","Nonlinear HJ selects via Mather condition","Vanishing discount: Mather-consistent solution wins","Selection rule for HJ on noncompact domains","Discount limit obeys Mather measure test"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000166,"raw_usage":{"total_tokens":1208,"prompt_tokens":851,"completion_tokens":357,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":467,"completion_tokens_details":{"reasoning_tokens":287}},"tokens_in":467,"tokens_out":357,"duration_ms":4442,"temperature":1.0,"reasoning_tokens":287,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:42:45.320363+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a concrete Hamiltonian satisfying (H1)-(H3) and one of (P1)-(P3), compute the Lax-Oleinik value function in (5.4) numerically, and compare it to $u_\\lambda(x)-c(H)t$; if any pair $(x,t)$ shows strict inequality, the representation formula (5.6) fails for an admissible Hamiltonian and the proof of Theorem 1.1 collapses. A cheaper check is to test the localization claim: if for some $z$ the equality $u_\\lambda(z)=\\vartheta_{\\lambda,R}(z)$ fails for all large $R$ and small $\\lambda$, the bounded-domain route to the selection principle is blocked.","supporting_citations":[{"cited_title":"Asymptotic solutions for large time of Hamilton-Jacobi equations in euclidean n space","cited_arxiv_id":null,"evidence_quote":"Supplies the unbounded-domain comparison principle (Theorem A.5) that converts the Lax-Oleinik value function into the variational formula for $u_\\lambda$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the bounded-domain state-constraint vanishing discount result (Theorem 3.14) used in the alternative proof of the main theorem."},{"cited_title":"Convergence of the solutions of the discounted Hamilton–Jacobi equation: Convergence of the discounted solutions","cited_arxiv_id":null,"evidence_quote":"Gives the original full-sequence convergence in the discounted case via Mather measures, the model for the selection mechanism generalized here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Presents the nonlinear adjoint method for selection problems, an alternative route whose contrast motivates the paper's variational approach."},{"cited_title":"Convergence of Viscosity Solutions of Generalized Contact Hamilton–Jacobi Equations","cited_arxiv_id":null,"evidence_quote":"Develops curve-based discounted measures for contact Hamilton-Jacobi equations, the construction adapted here to define the limiting measures $M$."},{"cited_title":"Asymptotic Solutions of Hamilton–Jacobi Equations with State Constraints","cited_arxiv_id":null,"evidence_quote":"Supplies the state-constraint Lax-Oleinik semigroup and dynamic programming results used to derive the formulas for $\\vartheta_\\lambda$."}],"review_version":2}