Pith. sign in

REVIEW 3 major objections 6 minor 42 references

Vanishing discount limits for first-order fully nonlinear Hamilton-Jacobi equations on noncompact domains

T0 review · 3 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2507.20472 v1 pith:RE7RKGND submitted 2025-07-28 math.AP

Son N.T. Tu , Jianlu Zhang This is my paper · ORCID
classification math.AP MSC 35D4070H2035J6037J4049L2537K99
keywords viscositysolutionHamilton-JacobiequationsvanishingdiscountselectionprincipleMathermeasuresstate-constraintproblemnoncompactdomainseffectiveHamiltonian
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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

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.

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 (3)
  1. [§5.1, Prop. 5.4; Appendix A.2, Theorem A.5] 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.
  2. [§5.1, Prop. 5.4, cases (ii) and (iii)] 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.
  3. [Section 3, Prop. 3.4 and Prop. 3.5] 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.
minor comments (6)
  1. [§1.1, p.2] The sentence ending '...complicating localization, i.e., to show that the global solution matches the state-' is truncated and should be completed.
  2. [§1.2, (H3)] 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.
  3. [Prop. 3.5(ii) and Prop. 3.8(ii)] The word 'miminimizer' appears twice and should be 'minimizer'.
  4. [Proof of Theorem 1.1, §6] The phrase 'we can ass assume super linearlity' contains a typo and should read 'we can assume superlinearity'.
  5. [Lemma 2.4 proof] 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'.
  6. [Proof of Theorem 1.1, §6] 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).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the selection characterization and localization theorem are derived from the variational representation and external comparison results, not assumed.

full rationale

The central claim (Theorem 1.1) is derived in Section 6 from the variational representation (5.6), the exponential formula (5.18), and the measure construction of Definition 6.2. The set E is defined using Mather-type measures M built from minimizing curves of u_lambda, which is a construction from the approximating solutions, not a repackaging of the claimed limit; the proof that any limit u_0 lies in E and dominates every w in E is carried out by explicit inequalities (6.8)-(6.13). The localization theorem (Theorem 1.2) is proven independently in Section 7.1 via the modified discount indices K_lambda and k_lambda, and it is not assumed. The alternative proof of Theorem 1.1 in Section 7.2 invokes the bounded-domain result Theorem 3.14, whose proof is omitted and credited to [35] by the same authors, but this is not load-bearing because Section 6 provides an independent proof of the same theorem. The comparison principle Theorem A.5 is adapted from the external reference [17], and the omitted proofs of Lemma 3.13 and Theorem 3.14 are bounded-domain background results. No parameter is fitted and then renamed as a prediction, and no step reduces by construction to its own input. Technical concerns about global Lipschitz hypotheses in Proposition 5.4 are correctness risks, not circularity.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central results rest on the structural assumptions (H1)-(H3) plus one of (P1)-(P3), along with imported comparison principles and the bounded-domain theorem from the authors' prior work. No empirical parameters are fitted; all constants are existential bounds proven in the text. The Mather-type measure set M is constructed internally and has no external falsifiable handle beyond the paper's own theorems, so it is not listed as an invented entity.

free parameters (2)
  • C0 = not specified (exists by Proposition 4.3)
    Lower bound for u_lambda, introduced in Definition 7.1 to construct modified discount indices K_lambda and k_lambda satisfying K_lambda >= k_lambda. This monotonicity is the crux of the localization proof.
  • lambda_z, R_z, M_z = existential, depend on z
    Constants in Proposition 5.11 and Theorem 1.2 bounding the support of minimizing curves. Their existence is proved, not numerically fitted.
assumptions (6)
  • domain assumption (H1): H is continuous, nondecreasing in u, convex in p, coercive locally in (x,u).
    Structural assumption on H used throughout to define viscosity solutions, Legendre transform, and the ergodic constant.
  • domain assumption (H2): There exists epsilon > 0 such that limsup_{|x|->infinity} max_{|p|<=epsilon} H(x,p,0) < max_x min_p H(x,p,0).
    Ensures the critical value c(H) is finite and the Aubry set is compact, following [20].
  • domain assumption (H3): H is differentiable in u with bounded derivative satisfying (1.6).
    Provides the modulus needed for convergence of the difference quotients of L in u and for the discount index limits.
  • domain assumption One of (P1), (P2), (P3) to guarantee a comparison principle for unbounded solutions.
    These alternative conditions ensure the variational formula (5.6) for u_lambda via Theorem A.5. Without one of them, the representation formulas may fail.
  • standard math Comparison principle for unbounded HJ equations (Theorem A.5, adapted from [17, Theorem 4.1]).
    Imported from the literature; central to deriving the variational representation of the maximal solution.
  • domain assumption Theorem 3.14 from [35]: vanishing discount convergence for state-constraint problems on bounded domains.
    Used in the alternative proof of Theorem 1.1 (Section 7.2) after localization. It is a prior result of the same authors.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Vanishing discount limits for first-order fully nonlinear Hamilton-Jacobi equations on noncompact domains." pith.science (2026). https://pith.science/paper/RE7RKGND

@misc{pith2026250720472,
  author       = {Pith},
  title        = {Pith review of: Vanishing discount limits for first-order fully nonlinear Hamilton-Jacobi equations on noncompact domains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RE7RKGND}},
  note         = {Machine review of arXiv:2507.20472}
}
abstract

We study the asymptotic behavior of solutions to the fully nonlinear Hamilton-Jacobi equation $H(x, Du, \lambda u) = 0$ in $\mathbb{R}^n$ as $\lambda \to 0^+$. Under the assumption that the Aubry set is localized, we employ a variational approach to derive limiting Mather-type measures and formulate a selection principle. Central to our analysis is a modified variational formula that bridges global and local state-constraint solutions, thereby extending localization techniques to the nonlinear framework.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

42 extracted references · 37 canonical work pages

  1. [35]

    TU, S. N. T., AND ZHANG , J. Generalized convergence of solutions for nonlinear Hamilton–Jacobi equations with state-constraint. Journal of Differential Equations 406 (Oct. 2024), 87–125

    work page 2024

  2. [20]

    The vanishing discount problem for Hamilton–Jacobi equations in the Euclidean space

    ISHII , H., AND SICONOLFI , A. The vanishing discount problem for Hamilton–Jacobi equations in the Euclidean space. Communications in Partial Differential Equations 0 , 0 (Jan. 2020), 1–36. Publisher: Taylor & Francis _eprint: https://doi.org/10.1080/03605302.2019.1710845

    arXiv 2020

  3. [1]

    S., A LZAHRANI , E

    AL-AIDAROUS , E. S., A LZAHRANI , E. O., I SHII , H., AND YOUNAS , A. M. M. A convergence result for the ergodic problem for Hamilton–Jacobi equations with Neumann-type boundary conditions. Royal Society of Edinburgh - Proceedings A (Mar. 2016), 1–18. Publisher: Cambridge University Press

    work page 2016

  4. [2]

    BOZORGNIA , F., K WON , D., AND TU, S. N. T. The regularity with respect to domains of the additive eigenvalues of superquadratic Hamilton–Jacobi equation. Journal of Differential Equations 402 (Sept. 2024), 518–553

    work page 2024

  5. [3]

    CAGNETTI , F., G OMES , D., AND TRAN , H. V. Aubry–Mather Measures in the Nonconvex Setting. SIAM Journal on Mathematical Analysis 43, 6 (Nov. 2011), 2601–2629. Publisher: Society for Industrial and Applied Mathematics

    work page 2011

  6. [4]

    On the vanishing discount approximation for compactly supported perturbations of periodic Hamiltonians: the 1d case

    CAPUZZO DOLCETTA , I., , AND DAVINI , A. On the vanishing discount approximation for compactly supported perturbations of periodic Hamiltonians: the 1d case. Communications in Partial Differential Equations 48, 4 (Apr. 2023), 576–622. Publisher: Taylor & Francis _eprint: https://doi.org/10.1080/03605302.2023.2183409

    arXiv 2023

  7. [5]

    Hamilton-Jacobi Equations with State Constraints

    CAPUZZO -D OLCETTA , I., AND LIONS , P.-L. Hamilton-Jacobi Equations with State Constraints. Transactions of the American Mathematical Society 318, 2 (1990), 643–683. Publisher: American Mathematical Society

    work page 1990

  8. [7]

    Convergence of solutions of Hamilton–Jacobi equations depending nonlinearly on the unknown function

    CHEN , Q. Convergence of solutions of Hamilton–Jacobi equations depending nonlinearly on the unknown function. Advances in Calculus of Variations 16, 1 (Jan. 2023), 45–68. Publisher: De Gruyter Section: Advances in Calculus of Variations

    work page 2023

Show all 42 references
  1. [8]

    Convergence of the solutions of the nonlinear discounted Hamilton– Jacobi equation: The central role of Mather measures

    CHEN , Q., F ATHI , A., Z AVIDOVIQUE , M., AND ZHANG , J. Convergence of the solutions of the nonlinear discounted Hamilton– Jacobi equation: The central role of Mather measures. Journal de Mathématiques Pures et Appliquées 181 (Jan. 2024), 22–57

    2024

  2. [9]

    Convergence of the solutions of the discounted Hamilton–Jacobi equation: Convergence of the discounted solutions

    DAVINI , A., F ATHI , A., I TURRIAGA , R., AND ZAVIDOVIQUE , M. Convergence of the solutions of the discounted Hamilton–Jacobi equation: Convergence of the discounted solutions. Inventiones mathematicae 206, 1 (2016), 29–55

    2016

  3. [10]

    The vanishing discount problem for nonlocal Hamilton-Jacobi equations, Apr

    DAVINI , A., AND ISHII , H. The vanishing discount problem for nonlocal Hamilton-Jacobi equations, Apr. 2025. arXiv:2504.11789 [math]

    2025 arXiv

  4. [11]

    Convergence/divergence phenomena in the vanishing discount limit of Hamilton-Jacobi equations, Nov

    DAVINI , A., N I, P., YAN, J., AND ZAVIDOVIQUE , M. Convergence/divergence phenomena in the vanishing discount limit of Hamilton-Jacobi equations, Nov. 2024. arXiv:2411.13780 [math]

    2024 arXiv

  5. [12]

    On the vanishing discount problem from the negative direction

    DAVINI , A., AND WANG , L. On the vanishing discount problem from the negative direction. Discrete and Continuous Dynamical Systems 41, 5 (May 2021), 2377–2389. Publisher: Discrete and Continuous Dynamical Systems

    2021

  6. [13]

    Weak kam theorem on non compact manifolds

    FATHI , A., AND MADERNA , E. Weak kam theorem on non compact manifolds. Nonlinear Differential Equations and Applications NoDEA 14, 1 (Oct. 2007), 1–27. 26

    2007

  7. [14]

    PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians

    FATHI , A., AND SICONOLFI , A. PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians. Calculus of Variations and Partial Differential Equations 22, 2 (Feb. 2005), 185–228

    2005

  8. [15]

    H., AND SONER , H

    FLEMING , W. H., AND SONER , H. M. Controlled Markov processes and viscosity solutions , 2nd ed ed. No. 25 in Applications of mathematics. Springer, New York, 2006

    2006

  9. [16]

    GOMES , D. A. Duality Principles for Fully Nonlinear Elliptic Equations. In Trends in Partial Differential Equations of Mathematical Physics. Birkhauser Basel, 2005, pp.125–136. ISSN: 2374-0280

    2005

  10. [17]

    Asymptotic solutions for large time of Hamilton-Jacobi equations in euclidean n space

    ISHII , H. Asymptotic solutions for large time of Hamilton-Jacobi equations in euclidean n space. Annales de l’I.H.P . Analyse non linéaire 25, 2 (2008), 231–266

    2008

  11. [18]

    ISHII , H., M ITAKE , H., AND TRAN , H. V. The vanishing discount problem and viscosity Mather measures. Part 1: The problem on a torus. Journal de Mathématiques Pures et Appliquées 108, 2 (2017), 125–149

    2017

  12. [19]

    ISHII , H., M ITAKE , H., AND TRAN , H. V. The vanishing discount problem and viscosity Mather measures. Part 2: Boundary value problems. Journal de Mathématiques Pures et Appliquées 108, 3 (2017), 261 – 305

    2017

  13. [21]

    Nonlinear Semigroup Approach to the Hamilton-Jacobi Equation– a Toy Model.Minimax Theory and its Applications 8, 1 (Aug

    JIN, L., Y AN, J., AND ZHAO , K. Nonlinear Semigroup Approach to the Hamilton-Jacobi Equation– a Toy Model.Minimax Theory and its Applications 8, 1 (Aug. 2023), 61–84. Number: 1

    2023

  14. [22]

    JING , W., M ITAKE , H., AND TRAN , H. V. Generalized ergodic problems: Existence and uniqueness structures of solutions. Journal of Differential Equations 268, 6 (Mar. 2020), 2886–2909

    2020

  15. [23]

    V., AND TU, S

    KIM, Y., T RAN , H. V., AND TU, S. N. State-Constraint Static Hamilton–Jacobi Equations in Nested Domains. SIAM Journal on Mathematical Analysis 52, 5 (Jan. 2020), 4161–4184. Publisher: Society for Industrial and Applied Mathematics

    2020

  16. [24]

    Generalized Solutions of Hamilton–Jacobi Equations

    LIONS , P.-L. Generalized Solutions of Hamilton–Jacobi Equations. Chapman & Hall/CRC research notes in mathematics series. Pitman, 1982

    1982

  17. [25]

    LIONS , P.-L., P APANICOLAOU , G., AND VARADHAN , S. R. Homogenization of Hamilton-Jacobi equations. Unpublished preprint (1986)

    1986

  18. [26]

    Asymptotic Solutions of Hamilton–Jacobi Equations with State Constraints

    MITAKE , H. Asymptotic Solutions of Hamilton–Jacobi Equations with State Constraints. Applied Mathematics and Optimization 58, 3 (Dec. 2008), 393–410

    2008

  19. [27]

    Weak KAM theory for discounted Hamilton–Jacobi equations and its application.Calculus of Variations and Partial Differential Equations 57, 3 (Apr

    MITAKE , H., AND SOGA , K. Weak KAM theory for discounted Hamilton–Jacobi equations and its application.Calculus of Variations and Partial Differential Equations 57, 3 (Apr. 2018), 78

    2018

  20. [28]

    MITAKE , H., AND TRAN , H. V. Selection problems for a discount degenerate viscous Hamilton–Jacobi equation. Advances in Mathematics 306 (Jan. 2017), 684–703

    2017

  21. [29]

    V., AND YU, Y

    MITAKE , H., T RAN , H. V., AND YU, Y. Rate of Convergence in Periodic Homogenization of Hamilton–Jacobi Equations: The Convex Setting. Archive for Rational Mechanics and Analysis 233, 2 (Aug. 2019), 901–934

    2019

  22. [30]

    Multiple asymptotic behaviors of solutions in the generalized vanishing discount problem

    NI, P. Multiple asymptotic behaviors of solutions in the generalized vanishing discount problem. Proceedings of the American Mathematical Society 151, 12 (Dec. 2023), 5239–5250

    2023

  23. [31]

    Optimal Control with State-Space Constraint I

    S ONER , H. Optimal Control with State-Space Constraint I. SIAM Journal on Control and Optimization 24, 3 (1986), 552–561

    1986

  24. [32]

    T RAN , H. V. Hamilton-Jacobi Equations: Theory and Applications. American Mathematical Society, 2021

    2021

  25. [33]

    TRAN , H. V. Representation formulas and large time behavior for solutions to some nonconvex Hamilton-Jacobi equations, May

  26. [34]

    TU, S. N. T. Vanishing discount problem and the additive eigenvalues on changing domains. Journal of Differential Equations 317 (Apr. 2022), 32–69

    2022

  27. [36]

    Aubry–Mather Theory for Contact Hamiltonian Systems

    WANG , K., W ANG , L., AND YAN, J. Aubry–Mather Theory for Contact Hamiltonian Systems. Communications in Mathematical Physics 366, 3 (Mar. 2019), 981–1023

    2019

  28. [37]

    Variational principle for contact Hamiltonian systems and its applications

    WANG , K., W ANG , L., AND YAN, J. Variational principle for contact Hamiltonian systems and its applications. Journal de Mathématiques Pures et Appliquées 123 (Mar. 2019), 167–200

    2019

  29. [38]

    A variational principle for contact Hamiltonian systems

    WANG , Y.-N., AND YAN, J. A variational principle for contact Hamiltonian systems. Journal of Differential Equations 267, 7 (Sept. 2019), 4047–4088

    2019

  30. [39]

    Convergence of Viscosity Solutions of Generalized Contact Hamilton–Jacobi Equations

    WANG , Y.-N., Y AN, J., AND ZHANG , J. Convergence of Viscosity Solutions of Generalized Contact Hamilton–Jacobi Equations. Archive for Rational Mechanics and Analysis 241, 2 (Aug. 2021), 885–902

    2021

  31. [40]

    On the Negative Limit of Viscosity Solutions for Discounted Hamilton–Jacobi Equations

    WANG , Y.-N., Y AN, J., AND ZHANG , J. On the Negative Limit of Viscosity Solutions for Discounted Hamilton–Jacobi Equations. Journal of Dynamics and Differential Equations 36, 2 (June 2024), 1347–1365

    2024

  32. [41]

    Convergence of solutions for some degenerate discounted Hamilton–Jacobi equations

    ZAVIDOVIQUE , M. Convergence of solutions for some degenerate discounted Hamilton–Jacobi equations. Analysis & PDE 15, 5 (Sept. 2022), 1287–1311. Publisher: Mathematical Sciences Publishers

    2022

  33. [42]

    Limit of solutions for semilinear Hamilton–Jacobi equations with degenerate viscosity

    ZHANG , J. Limit of solutions for semilinear Hamilton–Jacobi equations with degenerate viscosity. Advances in Calculus of Variations 17, 4 (Oct. 2024), 1185–1200. Publisher: De Gruyter Section: Advances in Calculus of Variations. †DEPARTMENT OF MATHEMATICS , B AYLOR UNIVERSITY...

    2024

  34. [2025]

    arXiv:2505.01377 [math]

    arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.