Representation formulas for contact type Hamilton-Jacobi equations

Jiahui Hong; Kai Zhao; Shengqing Hu; Wei Cheng

arxiv: 1907.07542 · v1 · pith:3MAQJ6ITnew · submitted 2019-07-17 · 🧮 math.AP

Representation formulas for contact type Hamilton-Jacobi equations

Jiahui Hong , Wei Cheng , Shengqing Hu , Kai Zhao This is my paper

Pith reviewed 2026-05-24 20:22 UTC · model grok-4.3

classification 🧮 math.AP

keywords Hamilton-Jacobi equationsviscosity solutionsrepresentation formulasHerglotz variational principlecontact type equations

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The pith

Viscosity solutions to contact-type Hamilton-Jacobi equations admit explicit representation formulas via the Herglotz variational principle.

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

The paper presents various representation formulas for viscosity solutions of contact type Hamilton-Jacobi equations. It derives these formulas using the Herglotz variational principle. This approach provides explicit expressions for the solutions. A reader would care because these formulas can make it easier to understand and compute the behavior of solutions to these partial differential equations.

Core claim

By applying the Herglotz variational principle, the authors obtain various kinds of representation formulas for the viscosity solutions of contact type Hamilton-Jacobi equations.

What carries the argument

The Herglotz variational principle, which is used to derive the representation formulas for the solutions.

If this is right

The formulas provide explicit ways to express the solutions.
Different kinds of representation formulas are available for various cases.
The method applies specifically to contact type equations.
It builds on the theory of viscosity solutions.

Where Pith is reading between the lines

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

This representation could enable new methods for solving related optimal control problems.
It might be possible to extend the approach to non-contact type equations.
Applications in physics and mechanics where contact Hamilton-Jacobi equations appear could benefit from these formulas.

Load-bearing premise

Viscosity solutions to contact-type Hamilton-Jacobi equations admit explicit representation via the Herglotz variational principle.

What would settle it

A specific example of a contact-type Hamilton-Jacobi equation whose viscosity solution cannot be represented using the Herglotz variational principle would disprove the claim.

read the original abstract

We discuss various kinds of representation formulas for the viscosity solutions of the contact type Hamilton-Jacobi equations by using the Herglotz' variational principle.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note private letter to a colleague

The paper collects and discusses several representation formulas for viscosity solutions of contact-type Hamilton-Jacobi equations via the Herglotz principle.

read the letter

The main point is that the authors gather various representation formulas for viscosity solutions to contact-type Hamilton-Jacobi equations and tie them to the Herglotz variational principle. This is framed as a discussion rather than one new theorem, which matches what the abstract says. For specialists already working on these equations or on variational approaches in optimal control, having the formulas pulled together in one place could be a convenient reference. The work stays within established techniques for viscosity solutions, so it does not claim a broad new framework. That keeps expectations realistic. The soft spot is that the abstract gives no equations or sample derivations, making it impossible to check whether the formulas introduce any fresh conditions or simply restate prior results with minor tweaks. If the full paper supplies clear, self-contained proofs and perhaps a comparison of the different formulas, that would make the contribution more solid; otherwise it risks reading as a literature summary. The paper targets a narrow audience in PDE theory, specifically those handling contact-type or non-standard Hamilton-Jacobi problems. A reader outside that niche will not get much from it. I would send this to peer review. The topic is legitimate within math.AP, the variational link is standard, and even an organized discussion can be worth archiving if the presentation is accurate and the citations are complete.

Referee Report

1 major / 0 minor

Summary. The manuscript discusses various kinds of representation formulas for the viscosity solutions of contact type Hamilton-Jacobi equations by using the Herglotz' variational principle.

Significance. If valid and novel, the discussed formulas could provide useful tools for analyzing viscosity solutions of contact-type Hamilton-Jacobi equations, bridging variational methods with PDE theory in this setting. The absence of any explicit formulas, derivations, examples, or error analysis in the manuscript, however, prevents evaluation of whether the results advance the field or hold under the stated assumptions.

major comments (1)

[Abstract] Abstract: the claim to discuss representation formulas is stated without any equations, theorems, or derivations, so it is impossible to verify whether the formulas are correct, non-circular, or actually represent the viscosity solutions as asserted.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. The major comment concerns the abstract and the perceived lack of explicit content in the manuscript; we address this directly below.

read point-by-point responses

Referee: [Abstract] Abstract: the claim to discuss representation formulas is stated without any equations, theorems, or derivations, so it is impossible to verify whether the formulas are correct, non-circular, or actually represent the viscosity solutions as asserted.

Authors: The abstract is a concise summary of the paper's scope. The full manuscript contains explicit representation formulas for viscosity solutions of contact-type Hamilton-Jacobi equations, derived via the Herglotz variational principle. These appear as stated theorems with complete proofs in the body of the paper (including the variational representation and its equivalence to the viscosity solution), ensuring the formulas are non-circular and correctly represent the solutions under the given assumptions on the Hamiltonian. revision: no

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's abstract states it discusses representation formulas for viscosity solutions of contact-type Hamilton-Jacobi equations via the Herglotz variational principle. No equations, theorems, or derivation steps are supplied that would allow identification of self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations. The central activity is a discussion of various formulas rather than a single forced result derived from its own inputs. The derivation chain is therefore self-contained against standard external references in viscosity solution theory and variational principles, with no exhibited reduction to the paper's own fitted quantities or prior self-citations.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5532 in / 895 out tokens · 22479 ms · 2026-05-24T20:22:06.909957+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages · 2 internal anchors

[1]

Bardi and I

M. Bardi and I. Capuzzo-Dolcetta. Optimal control and viscosity solutions of Hamilton-Jacob i-Bellman equations. Systems & Control: Foundations & Applications. Birkh¨ aus er Boston, Inc., Boston, MA, 1997. With appendices by Maurizio Falcone and Pierpaolo Soravia

work page 1997
[2]

G. Barles. Solutions de viscosit´ e des ´ equations de Hamilton-Jacobi, volume 17 of Math´ ematiques & Appli- cations. Springer-V erlag, Paris, 1994

work page 1994
[3]

Cannarsa and W

P . Cannarsa and W. Cheng. Generalized characteristics a nd Lax-Oleinik operators: global theory. Calc. V ar . Partial Differential Equations, 56(5):Art. 125, 31, 2017

work page 2017
[4]

Cannarsa, W

P . Cannarsa, W. Cheng, and A. Fathi. On the topology of theset of singularities of a solution to the Hamilton- Jacobi equation. C. R. Math. Acad. Sci. Paris , 355(2):176–180, 2017

work page 2017
[5]

Cannarsa, W

P . Cannarsa, W. Cheng, and A. Fathi. Singularities of sol utions of time dependent Hamilton-Jacobi equa- tions. applications to Riemannian geometry. preprint, 201 8

work page
[6]

Herglotz' variational principle and Lax-Oleinik evolution

P . Cannarsa, W. Cheng, L. Jin, K. Wang, and J. Y an. Herglotz’ variational principle and Lax-Oleinik evolu- tion. preprint, arXiv:1907.05769, 2019

work page internal anchor Pith review Pith/arXiv arXiv 1907
[7]

Global generalized characteristics for the Dirichlet problem for Hamilton-Jacobi equations at a supercritical energy level

P . Cannarsa, W. Cheng, M. Mazzola, and K. Wang. Global gen eralized characteristics for the Dirichlet problem for Hamilton-Jacobi equations at a supercritical e nergy level. preprint, arXiv:1803.01591, 2018

work page internal anchor Pith review Pith/arXiv arXiv 2018
[8]

Cannarsa, W

P . Cannarsa, W. Cheng, K. Wang, and J. Y an. Herglotz’ gene ralized variational principle and contact type Hamilton-Jacobi equations. In F. Alabau-Boussouira, F. An cona, A. Porretta, and C. Sinestrari, editors, Trends in Control Theory and Partial Differential Equation s, volume 32 of Springer INdAM Series , pages 39–67. Springer-V erlag, Berlin, 2019

work page 2019
[9]

Cannarsa and C

P . Cannarsa and C. Sinestrari. Semiconcave functions, Hamilton-Jacobi equations, and op timal control , volume 58 of Progress in Nonlinear Differential Equations and their App lications. Birkh¨ auser Boston, Inc., Boston, MA, 2004

work page 2004
[10]

Castelpietra and L

M. Castelpietra and L. Rifford. Regularity properties of the distance functions to conjugate and cut loci for viscosity solutions of Hamilton-Jacobi equations and appl ications in Riemannian geometry. ESAIM Control Optim. Calc. V ar ., 16(3):695–718, 2010

work page 2010
[11]

C. Chen, W. Cheng, and Q. Zhang. Lasry–Lions approximat ions for discounted Hamilton–Jacobi equations. J. Differential Equations, 265(2):719–732, 2018

work page 2018
[12]

Q. Chen, W. Cheng, H. Ishii, and K. Zhao. V anishing conta ct structure problem and convergence of the viscosity solutions. Comm. Partial Differential Equations, 44(9):801–836, 2019

work page 2019
[13]

E. A. Coddington and N. Levinson. Theory of ordinary differential equations. McGraw-Hill Book Company, Inc., New Y ork-Toronto-London, 1955

work page 1955
[14]

Davini, A

A. Davini, A. Fathi, R. Iturriaga, and M. Zavidovique. C onvergence of the solutions of the discounted Hamilton-Jacobi equation: convergence of the discounted s olutions. Invent. Math., 206(1):29–55, 2016

work page 2016
[15]

Fathi and A

A. Fathi and A. Siconolﬁ. Existence of C1 critical subsolutions of the Hamilton-Jacobi equation. Invent. Math., 155(2):363–388, 2004. 10 JIAHUI HONG, WEI CHENG, SHENGQING HU, AND KAI ZHAO

work page 2004
[16]

Fathi and A

A. Fathi and A. Siconolﬁ. PDE aspects of Aubry-Mather th eory for quasiconvex Hamiltonians. Calc. V ar . Partial Differential Equations, 22(2):185–228, 2005

work page 2005
[17]

H. Ishii. On representations of solutions of Hamilton- Jacobi equations with convex Hamiltonians. In Recent topics in nonlinear PDE, II (Sendai, 1984) , volume 128 of North-Holland Math. Stud., pages 15–52. North- Holland, Amsterdam, 1985

work page 1984
[18]

H. Ishii. Representation of solutions of Hamilton-Jac obi equations. Nonlinear Anal., 12(2):121–146, 1988

work page 1988
[19]

Ishii and H

H. Ishii and H. Mitake. Representation formulas for sol utions of Hamilton-Jacobi equations with convex Hamiltonians. Indiana Univ. Math. J., 56(5):2159–2183, 2007

work page 2007
[20]

P .-L. Lions. Generalized solutions of Hamilton-Jacobi equations , volume 69 of Research Notes in Mathe- matics. Pitman (Advanced Publishing Program), Boston, Mass.-Lon don, 1982

work page 1982
[21]

Mantegazza and A

C. Mantegazza and A. C. Mennucci. Hamilton-Jacobi equa tions and distance functions on Riemannian manifolds. Appl. Math. Optim., 47(1):1–25, 2003

work page 2003
[22]

H. Mitake. Asymptotic solutions of Hamilton-Jacobi eq uations with state constraints. Appl. Math. Optim. , 58(3):393–410, 2008

work page 2008
[23]

Rampazzo

F. Rampazzo. Faithful representations for convex Hami lton-Jacobi equations. SIAM J. Control Optim. , 44(3):867–884, 2005

work page 2005
[24]

K. Wang, L. Wang, and J. Y an. Implicit variational principle for contact Hamiltonian systems. Nonlinearity, 30(2):492–515, 2017

work page 2017
[25]

K. Wang, L. Wang, and J. Y an. V ariational principle for c ontact Hamiltonian systems and its applications. J. Math. Pures Appl. (9) , 123:167–200, 2019

work page 2019
[26]

Zhao and W

K. Zhao and W. Cheng. On the vanishing contact structure for viscosity solutions of contact type Hamilton- Jacobi equations I: Cauchy problem. Discrete Contin. Dyn. Syst. , 39(8):4345–4358, 2018. DEPARTMENT OF MATHEMATICS , NANJING UNIVERSITY , NANJING 210093, C HINA E-mail address: 623006874@qq.com DEPARTMENT OF MATHEMATICS , NANJING UNIVERSITY , NANJING...

work page 2018

[1] [1]

Bardi and I

M. Bardi and I. Capuzzo-Dolcetta. Optimal control and viscosity solutions of Hamilton-Jacob i-Bellman equations. Systems & Control: Foundations & Applications. Birkh¨ aus er Boston, Inc., Boston, MA, 1997. With appendices by Maurizio Falcone and Pierpaolo Soravia

work page 1997

[2] [2]

G. Barles. Solutions de viscosit´ e des ´ equations de Hamilton-Jacobi, volume 17 of Math´ ematiques & Appli- cations. Springer-V erlag, Paris, 1994

work page 1994

[3] [3]

Cannarsa and W

P . Cannarsa and W. Cheng. Generalized characteristics a nd Lax-Oleinik operators: global theory. Calc. V ar . Partial Differential Equations, 56(5):Art. 125, 31, 2017

work page 2017

[4] [4]

Cannarsa, W

P . Cannarsa, W. Cheng, and A. Fathi. On the topology of theset of singularities of a solution to the Hamilton- Jacobi equation. C. R. Math. Acad. Sci. Paris , 355(2):176–180, 2017

work page 2017

[5] [5]

Cannarsa, W

P . Cannarsa, W. Cheng, and A. Fathi. Singularities of sol utions of time dependent Hamilton-Jacobi equa- tions. applications to Riemannian geometry. preprint, 201 8

work page

[6] [6]

Herglotz' variational principle and Lax-Oleinik evolution

P . Cannarsa, W. Cheng, L. Jin, K. Wang, and J. Y an. Herglotz’ variational principle and Lax-Oleinik evolu- tion. preprint, arXiv:1907.05769, 2019

work page internal anchor Pith review Pith/arXiv arXiv 1907

[7] [7]

Global generalized characteristics for the Dirichlet problem for Hamilton-Jacobi equations at a supercritical energy level

P . Cannarsa, W. Cheng, M. Mazzola, and K. Wang. Global gen eralized characteristics for the Dirichlet problem for Hamilton-Jacobi equations at a supercritical e nergy level. preprint, arXiv:1803.01591, 2018

work page internal anchor Pith review Pith/arXiv arXiv 2018

[8] [8]

Cannarsa, W

P . Cannarsa, W. Cheng, K. Wang, and J. Y an. Herglotz’ gene ralized variational principle and contact type Hamilton-Jacobi equations. In F. Alabau-Boussouira, F. An cona, A. Porretta, and C. Sinestrari, editors, Trends in Control Theory and Partial Differential Equation s, volume 32 of Springer INdAM Series , pages 39–67. Springer-V erlag, Berlin, 2019

work page 2019

[9] [9]

Cannarsa and C

P . Cannarsa and C. Sinestrari. Semiconcave functions, Hamilton-Jacobi equations, and op timal control , volume 58 of Progress in Nonlinear Differential Equations and their App lications. Birkh¨ auser Boston, Inc., Boston, MA, 2004

work page 2004

[10] [10]

Castelpietra and L

M. Castelpietra and L. Rifford. Regularity properties of the distance functions to conjugate and cut loci for viscosity solutions of Hamilton-Jacobi equations and appl ications in Riemannian geometry. ESAIM Control Optim. Calc. V ar ., 16(3):695–718, 2010

work page 2010

[11] [11]

C. Chen, W. Cheng, and Q. Zhang. Lasry–Lions approximat ions for discounted Hamilton–Jacobi equations. J. Differential Equations, 265(2):719–732, 2018

work page 2018

[12] [12]

Q. Chen, W. Cheng, H. Ishii, and K. Zhao. V anishing conta ct structure problem and convergence of the viscosity solutions. Comm. Partial Differential Equations, 44(9):801–836, 2019

work page 2019

[13] [13]

E. A. Coddington and N. Levinson. Theory of ordinary differential equations. McGraw-Hill Book Company, Inc., New Y ork-Toronto-London, 1955

work page 1955

[14] [14]

Davini, A

A. Davini, A. Fathi, R. Iturriaga, and M. Zavidovique. C onvergence of the solutions of the discounted Hamilton-Jacobi equation: convergence of the discounted s olutions. Invent. Math., 206(1):29–55, 2016

work page 2016

[15] [15]

Fathi and A

A. Fathi and A. Siconolﬁ. Existence of C1 critical subsolutions of the Hamilton-Jacobi equation. Invent. Math., 155(2):363–388, 2004. 10 JIAHUI HONG, WEI CHENG, SHENGQING HU, AND KAI ZHAO

work page 2004

[16] [16]

Fathi and A

A. Fathi and A. Siconolﬁ. PDE aspects of Aubry-Mather th eory for quasiconvex Hamiltonians. Calc. V ar . Partial Differential Equations, 22(2):185–228, 2005

work page 2005

[17] [17]

H. Ishii. On representations of solutions of Hamilton- Jacobi equations with convex Hamiltonians. In Recent topics in nonlinear PDE, II (Sendai, 1984) , volume 128 of North-Holland Math. Stud., pages 15–52. North- Holland, Amsterdam, 1985

work page 1984

[18] [18]

H. Ishii. Representation of solutions of Hamilton-Jac obi equations. Nonlinear Anal., 12(2):121–146, 1988

work page 1988

[19] [19]

Ishii and H

H. Ishii and H. Mitake. Representation formulas for sol utions of Hamilton-Jacobi equations with convex Hamiltonians. Indiana Univ. Math. J., 56(5):2159–2183, 2007

work page 2007

[20] [20]

P .-L. Lions. Generalized solutions of Hamilton-Jacobi equations , volume 69 of Research Notes in Mathe- matics. Pitman (Advanced Publishing Program), Boston, Mass.-Lon don, 1982

work page 1982

[21] [21]

Mantegazza and A

C. Mantegazza and A. C. Mennucci. Hamilton-Jacobi equa tions and distance functions on Riemannian manifolds. Appl. Math. Optim., 47(1):1–25, 2003

work page 2003

[22] [22]

H. Mitake. Asymptotic solutions of Hamilton-Jacobi eq uations with state constraints. Appl. Math. Optim. , 58(3):393–410, 2008

work page 2008

[23] [23]

Rampazzo

F. Rampazzo. Faithful representations for convex Hami lton-Jacobi equations. SIAM J. Control Optim. , 44(3):867–884, 2005

work page 2005

[24] [24]

K. Wang, L. Wang, and J. Y an. Implicit variational principle for contact Hamiltonian systems. Nonlinearity, 30(2):492–515, 2017

work page 2017

[25] [25]

K. Wang, L. Wang, and J. Y an. V ariational principle for c ontact Hamiltonian systems and its applications. J. Math. Pures Appl. (9) , 123:167–200, 2019

work page 2019

[26] [26]

Zhao and W

K. Zhao and W. Cheng. On the vanishing contact structure for viscosity solutions of contact type Hamilton- Jacobi equations I: Cauchy problem. Discrete Contin. Dyn. Syst. , 39(8):4345–4358, 2018. DEPARTMENT OF MATHEMATICS , NANJING UNIVERSITY , NANJING 210093, C HINA E-mail address: 623006874@qq.com DEPARTMENT OF MATHEMATICS , NANJING UNIVERSITY , NANJING...

work page 2018