Home Mathematics Convergence of solutions of Hamilton–Jacobi equations depending nonlinearly on the unknown function
Article
Licensed
Unlicensed Requires Authentication

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

  • Qinbo Chen ORCID logo EMAIL logo
Published/Copyright: January 20, 2021
Advances in Calculus of Variations
From the journal Advances in Calculus of Variations Volume 16 Issue 1

Abstract

Motivated by the vanishing contact problem, we study in the present paper the convergence of solutions of Hamilton–Jacobi equations depending nonlinearly on the unknown function. Let H ⁢ ( x , p , u ) be a continuous Hamiltonian which is strictly increasing in u, and is convex and coercive in p. For each parameter λ > 0 , we denote by u λ the unique viscosity solution of the Hamilton–Jacobi equation

H ⁢ ( x , D ⁢ u ⁢ ( x ) , λ ⁢ u ⁢ ( x ) ) = c .

Under quite general assumptions, we prove that u λ converges uniformly, as λ tends to zero, to a specific solution of the critical Hamilton–Jacobi equation H ⁢ ( x , D ⁢ u ⁢ ( x ) , 0 ) = c . We also characterize the limit solution in terms of Peierls barrier and Mather measures.

Keywords: Nonlinear discounted systems; Hamilton–Jacobi equations; asymptotic behavior of solutions; weak KAM theory; Aubry–Mather theory
MSC 2010: 35B40; 49L25; 37J50

Communicated by Hitoshi Ishii

Funding source: H2020 European Research Council

Award Identifier / Grant number: 677793

Funding statement: This research is funded by the ERC Project 677793 StableChaoticPlanetM.

A Appendix

Here, we provide an equivalent description for assumption (H2) provided that (H1) and (H3) hold.

Proposition A.1.

If H ∈ C ⁢ ( T * ⁢ M × R ) satisfies the convexity assumption (H1) and the monotonicity assumption (H3), then the following properties are equivalent:

  1. There is a constant 𝐫 𝟎 > 0 such that lim | p | x → ∞ ⁡ H ⁢ ( x , p , - 𝐫 𝟎 ) = + ∞ , uniformly in x ∈ M .

  2. lim | p | x → ∞ ⁡ H ⁢ ( x , p , 0 ) = + ∞ , uniformly in x ∈ M .

Proof.

By the monotonicity assumption (H3), the implication (1)  ⇒  (2) is obvious.

Now, we show that (2) implies (1). To this end, only assumption (H1) will be used. Indeed, by the coercivity of H ⁢ ( x , p , 0 ) and the compactness of M, there is a constant r > 0 such that

H ⁢ ( x , p , 0 ) > max x ∈ M ⁡ H ⁢ ( x , 0 , 0 ) + 2   for all  ⁢ ( x , p ) ∈ ∂ ⁡ B r ,

where ∂ ⁡ B r denotes the compact set { ( x , p ) ∈ T * ⁢ M : x ∈ M , | p | x = r } . By continuity and compactness, there exists a small constant 𝐫 𝟎 > 0 such that

(A.1) H ⁢ ( x , p , - 𝐫 𝟎 ) > max x ∈ M ⁡ H ⁢ ( x , 0 , - 𝐫 𝟎 ) + 1   for all  ⁢ ( x , p ) ∈ ∂ ⁡ B r .

Next, for each ( x , p ) with | p | x > r , we pick a point z x , p := r | p | x ⁢ p on ∂ ⁡ B r . Then the convexity assumption (H1) implies

(A.2) H ⁢ ( x , z x , p , - 𝐫 𝟎 ) ⩽ ( 1 - r | p | x ) ⁢ H ⁢ ( x , 0 , - 𝐫 𝟎 ) + r | p | x ⁢ H ⁢ ( x , p , - 𝐫 𝟎 )   for all  ⁢ | p | x > r .

Since z x , p ∈ ∂ ⁡ B r , we infer from (A.1)–(A.2) that

H ⁢ ( x , p , - 𝐫 𝟎 ) ⩾ H ⁢ ( x , z x , p , - 𝐫 𝟎 ) - H ⁢ ( x , 0 , - 𝐫 𝟎 ) r | p | x + H ⁢ ( x , 0 , - 𝐫 𝟎 ) ⩾ 1 r ⁢ | p | x + H ⁢ ( x , 0 , - 𝐫 𝟎 )

for all | p | x > r . This implies

lim | p | x → ∞ ⁡ H ⁢ ( x , p , - 𝐫 𝟎 ) = + ∞ , uniformly in  x ∈ M . ∎

Acknowledgements

This work was initiated while I was visiting Georgia Tech (December 2018–March 2019), whose warm hospitality is gratefully acknowledged. It was completed while I was working at the University of Padova. I am particularly grateful to Albert Fathi for many stimulating discussions during my stay at Georgia Tech, and for his careful reading of the manuscript and valuable comments. I wish to thank Wei Cheng, Jun Yan and Jianlu Zhang for some helpful discussions. I also would like to thank Gabriella Pinzari for her support.

References

[1] E. S. Al-Aidarous, E. O. Alzahrani, H. Ishii and A. M. M. Younas, A convergence result for the ergodic problem for Hamilton–Jacobi equations with Neumann-type boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), no. 2, 225–242. 10.1017/S0308210515000517Search in Google Scholar

[2] G. Barles, Solutions de viscosité des équations de Hamilton–Jacobi, Math. Appl. (Berlin) 17, Springer, Paris, 1994. Search in Google Scholar

[3] G. Buttazzo, M. Giaquinta and S. Hildebrandt, One-Dimensional Variational Problems. An Introduction, Oxford Lecture Ser. Math. Appl. 15, The Clarendon Press, New York, 1998. 10.1093/oso/9780198504658.001.0001Search in Google Scholar

[4] P. Cannarsa, W. Cheng, L. Jin, K. Wang and J. Yan, Herglotz’ variational principle and Lax–Oleinik evolution, J. Math. Pures Appl. (9) 141 (2020), 99–136. 10.1016/j.matpur.2020.07.002Search in Google Scholar

[5] P. Cannarsa, W. Cheng, K. Wang and J. Yan, Herglotz’ generalized variational principle and contact type Hamilton–Jacobi equations, Trends in Control Theory and Partial Differential Equations, Springer INdAM Ser. 32, Springer, Cham, (2019), 39–67. 10.1007/978-3-030-17949-6_3Search in Google Scholar

[6] P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton–Jacobi Equations, and Optimal Control, Progr. Nonlinear Differential Equations Appl. 58, Birkhäuser, Boston, 2004. 10.1007/b138356Search in Google Scholar

[7] P. Cardaliaguet and A. Porretta, Long time behavior of the master equation in mean field game theory, Anal. PDE 12 (2019), no. 6, 1397–1453. 10.2140/apde.2019.12.1397Search in Google Scholar

[8] Q. Chen, W. Cheng, H. Ishii and K. Zhao, Vanishing contact structure problem and convergence of the viscosity solutions, Comm. Partial Differential Equations 44 (2019), no. 9, 801–836. 10.1080/03605302.2019.1608561Search in Google Scholar

[9] A. Davini, A. Fathi, R. Iturriaga and M. Zavidovique, Convergence of the solutions of the discounted equation: The discrete case, Math. Z. 284 (2016), no. 3–4, 1021–1034. 10.1007/s00209-016-1685-ySearch in Google Scholar

[10] A. Davini, A. Fathi, R. Iturriaga and M. Zavidovique, Convergence of the solutions of the discounted Hamilton–Jacobi equation: Convergence of the discounted solutions, Invent. Math. 206 (2016), no. 1, 29–55. 10.1007/s00222-016-0648-6Search in Google Scholar

[11] A. Davini and L. Wang, On the vanishing discount problem from the negative direction, preprint (2020), https://arxiv.org/abs/2007.12458; to appear in Discrete Contin. Dyn. Syst. 2020. 10.3934/dcds.2020368Search in Google Scholar

[12] A. Davini and M. Zavidovique, Weak KAM theory for nonregular commuting Hamiltonians, Discrete Contin. Dyn. Syst. Ser. B 18 (2013), no. 1, 57–94. 10.3934/dcdsb.2013.18.57Search in Google Scholar

[13] A. Davini and M. Zavidovique, Convergence of the solutions of discounted Hamilton–Jacobi systems, Adv. Calc. Var. (2019), 10.1515/acv-2018-0037. 10.1515/acv-2018-0037Search in Google Scholar

[14] A. Fathi, Weak KAM theorem in Lagrangian dynamics, unpublished book, preliminary version 10 (2008). Search in Google Scholar

[15] A. Fathi, Weak KAM from a PDE point of view: Viscosity solutions of the Hamilton–Jacobi equation and Aubry set, Proc. Roy. Soc. Edinburgh Sect. A 142 (2012), no. 6, 1193–1236. 10.1017/S0308210550000064Search in Google Scholar

[16] A. Fathi and A. Siconolfi, PDE aspects of Aubry–Mather theory for quasiconvex Hamiltonians, Calc. Var. Partial Differential Equations 22 (2005), no. 2, 185–228. 10.1007/s00526-004-0271-zSearch in Google Scholar

[17] D. A. Gomes, Generalized Mather problem and selection principles for viscosity solutions and Mather measures, Adv. Calc. Var. 1 (2008), no. 3, 291–307. 10.1515/ACV.2008.012Search in Google Scholar

[18] D. A. Gomes, H. Mitake and K. Terai, The selection problem for some first-order stationary mean-field games, Networks Heterog. Media 15 (2020), no. 4, 681–710. 10.3934/nhm.2020019Search in Google Scholar

[19] D. A. Gomes, H. Mitake and H. V. Tran, The selection problem for discounted Hamilton–Jacobi equations: Some non-convex cases, J. Math. Soc. Japan 70 (2018), no. 1, 345–364. 10.2969/jmsj/07017534Search in Google Scholar

[20] H. Ishii, Perron’s method for Hamilton–Jacobi equations, Duke Math. J. 55 (1987), no. 2, 369–384. 10.1215/S0012-7094-87-05521-9Search in Google Scholar

[21] H. Ishii, Asymptotic solutions for large time of Hamilton–Jacobi equations in Euclidean n space, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), no. 2, 231–266. 10.1016/j.anihpc.2006.09.002Search in Google Scholar

[22] H. Ishii, The vanishing discount problem for monotone systems of Hamilton–Jacobi equations. Part 1: Linear coupling, preprint (2019), https://arxiv.org/abs/1903.00244. 10.3934/mine.2021032Search in Google Scholar

[23] H. Ishii and L. Jin, The vanishing discount problem for monotone systems of Hamilton–Jacobi equations. Part 2: Nonlinear coupling, Calc. Var. Partial Differential Equations 59 (2020), no. 4, Paper No. 140. 10.1007/s00526-020-01768-8Search in Google Scholar

[24] H. Ishii, H. Mitake and H. V. Tran, The vanishing discount problem and viscosity Mather measures. Part 1: The problem on a torus, J. Math. Pures Appl. (9) 108 (2017), no. 2, 125–149. 10.1016/j.matpur.2016.10.013Search in Google Scholar

[25] H. Ishii, H. Mitake and H. V. Tran, The vanishing discount problem and viscosity Mather measures. Part 2: Boundary value problems, J. Math. Pures Appl. (9) 108 (2017), no. 3, 261–305. 10.1016/j.matpur.2016.11.002Search in Google Scholar

[26] H. Ishii and A. Siconolfi, The vanishing discount problem for Hamilton–Jacobi equations in the Euclidean space, Comm. Partial Differential Equations 45 (2020), no. 6, 525–560. 10.1080/03605302.2019.1710845Search in Google Scholar

[27] R. Iturriaga and H. Sánchez-Morgado, Limit of the infinite horizon discounted Hamilton–Jacobi equation, Discrete Contin. Dyn. Syst. Ser. B 15 (2011), no. 3, 623–635. 10.3934/dcdsb.2011.15.623Search in Google Scholar

[28] P.-L. Lions, G. Papanicolaou and S. R. S. Varadhan, Homogenization of Hamilton–Jacobi equations, unpublished work (1987). Search in Google Scholar

[29] R. Mañé, Generic properties and problems of minimizing measures of Lagrangian systems, Nonlinearity 9 (1996), no. 2, 273–310. 10.1088/0951-7715/9/2/002Search in Google Scholar

[30] S. Marò and A. Sorrentino, Aubry–Mather theory for conformally symplectic systems, Comm. Math. Phys. 354 (2017), no. 2, 775–808. 10.1007/s00220-017-2900-3Search in Google Scholar

[31] J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z. 207 (1991), no. 2, 169–207. 10.1007/BF02571383Search in Google Scholar

[32] J. N. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble) 43 (1993), no. 5, 1349–1386. 10.5802/aif.1377Search in Google Scholar

[33] H. Mitake and H. V. Tran, Selection problems for a discount degenerate viscous Hamilton–Jacobi equation, Adv. Math. 306 (2017), 684–703. 10.1016/j.aim.2016.10.032Search in Google Scholar

[34] X. Su, L. Wang and J. Yan, Weak KAM theory for Hamilton–Jacobi equations depending on unknown functions, Discrete Contin. Dyn. Syst. 36 (2016), no. 11, 6487–6522. 10.3934/dcds.2016080Search in Google Scholar

[35] K. Wang, L. Wang and J. Yan, Aubry–Mather theory for contact Hamiltonian systems, Comm. Math. Phys. 366 (2019), no. 3, 981–1023. 10.1007/s00220-019-03362-2Search in Google Scholar

[36] K. Wang, L. Wang and J. Yan, Variational principle for contact Hamiltonian systems and its applications, J. Math. Pures Appl. (9) 123 (2019), 167–200. 10.1016/j.matpur.2018.08.011Search in Google Scholar

[37] Y. Wang, J. Yan and J. Zhang, Convergence of viscosity solutions of generalized contact Hamilton–Jacobi equations, preprint (2020), https://arxiv.org/abs/2004.12269. 10.1007/s00205-021-01667-ySearch in Google Scholar

[38] M. Zavidovique, Convergence of solutions for some degenerate discounted Hamilton–Jacobi equations, preprint (2020), https://arxiv.org/abs/2006.00779. 10.2140/apde.2022.15.1287Search in Google Scholar

[39] B. Ziliotto, Convergence of the solutions of the discounted Hamilton–Jacobi equation: A counterexample, J. Math. Pures Appl. (9) 128 (2019), 330–338. 10.1016/j.matpur.2019.04.005Search in Google Scholar
Received: 2020-08-28
Revised: 2020-11-25
Accepted: 2020-12-21
Published Online: 2021-01-20
Published in Print: 2023-01-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. The existence of minimizers for an isoperimetric problem with Wasserstein penalty term in unbounded domains
  3. Compactness of 𝑀-uniform domains and optimal thermal insulation problems
  4. Convergence of solutions of Hamilton–Jacobi equations depending nonlinearly on the unknown function
  5. Pansu–Wulff shapes in ℍ1
  6. Integrability of the sub-Riemannian mean curvature at degenerate characteristic points in the Heisenberg group
  7. p-harmonic functions by way of intrinsic mean value properties
  8. The sharp quantitative isocapacitary inequality (the case of p-capacity)
  9. A finer singular limit of a single-well Modica–Mortola functional and its applications to the Kobayashi–Warren–Carter energy
  10. Regularity and monotonicity for solutions to a continuum model of epitaxial growth with nonlocal elastic effects
  11. Remarks on the vanishing discount problem for infinite systems of Hamilton–Jacobi–Bellman equations
  12. Bilateral estimates of solutions to quasilinear elliptic equations with sub-natural growth terms
  13. Rectifiability of entropy defect measures in a micromagnetics model
https://doi.org/10.1515/acv-2020-0089
Keywords for this article
Nonlinear discounted systems; Hamilton–Jacobi equations; asymptotic behavior of solutions; weak KAM theory; Aubry–Mather theory

Articles in the same Issue

  1. Frontmatter
  2. The existence of minimizers for an isoperimetric problem with Wasserstein penalty term in unbounded domains
  3. Compactness of 𝑀-uniform domains and optimal thermal insulation problems
  4. Convergence of solutions of Hamilton–Jacobi equations depending nonlinearly on the unknown function
  5. Pansu–Wulff shapes in ℍ1
  6. Integrability of the sub-Riemannian mean curvature at degenerate characteristic points in the Heisenberg group
  7. p-harmonic functions by way of intrinsic mean value properties
  8. The sharp quantitative isocapacitary inequality (the case of p-capacity)
  9. A finer singular limit of a single-well Modica–Mortola functional and its applications to the Kobayashi–Warren–Carter energy
  10. Regularity and monotonicity for solutions to a continuum model of epitaxial growth with nonlocal elastic effects
  11. Remarks on the vanishing discount problem for infinite systems of Hamilton–Jacobi–Bellman equations
  12. Bilateral estimates of solutions to quasilinear elliptic equations with sub-natural growth terms
  13. Rectifiability of entropy defect measures in a micromagnetics model
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2026 De Gruyter Brill
Downloaded on 3.2.2026 from https://www.degruyterbrill.com/document/doi/10.1515/acv-2020-0089/html
Scroll to top button