MATHER measures for space–time periodic nonconvex Hamiltonians

  • Eddaly Guerra-VelascoEmail author
Original Article


In Gomes (Nonlinearity 15(3):581–603, 2002) developed techniques and tools with the purpose of extending the Aubry–Mather theory in a stochastic setting, namely he proved the existence of stochastic Mather measures and their properties. These results were also extended in the time-dependent setting in the doctoral thesis of the Guerra-Velasco (, 2015). However, to construct analogs to the Aubry–Mather measures for nonconvex Hamiltonians, it is necessary to use the adjoint method introduced by Evans (Arch Ration Mech Anal 197:1053–1088, 2010) and Tran (Calc Var Partial Differ Equ 41:301–319, 2011); the construction of the measures is in Cagnetti et al. (SIAM J Math Anal 43(6):2601–262, citelink2011CGT). The main goal of this paper is to construct Mather measures for space–time periodical nonconvex Hamiltonians using the techniques in [10, 21] and [7]. Moreover, we also will prove that there is only one value, such that the viscous Hamilton–Jacobi equation has a smooth periodic solution unique up to an additive constant.


Hamilton–Jacobi Non-convex Periodic Hamiltonians 

Mathematics Subject Classification

37J50 49L25 



The author acknowledges the referees for their valuable suggestions and remarks that substantially improved this article. The author is grateful to Héctor Sánchez Morgado for his suggestions and to Boris Percino Figueroa for his support in the process of this research.


  1. 1.
    Bardi, M., Dolcetta, I.Capuzzo: Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. Birkhausser, Basel (1997)CrossRefzbMATHGoogle Scholar
  2. 2.
    Barles, G.: Solutions de viscosité des équations de Hamilton Jacobi, Mathématiques et Applications 17. Springer, New York (1994)zbMATHGoogle Scholar
  3. 3.
    Bernard, P.: Young measures, superposition, and transport. Indiana Univ. Math. J. 57(1), 247–276 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brandt, A.: Interior Schauder estimates for Parabolic differential- (difference) equations via the maximum principle. Israel J. Math. 7, 254–262 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Barles, G., Souganidis, P.E.: Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations. SIAM J. Math. Anal. 32(6), 1311–1323 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Crandall, M.G., Evans, L.C., Lions, P.-L.: Some properties of viscosity solutions of Hamilton–Jacobi equations. Trans. Am. Math. Soc. 282(2), 487–502 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cagnetti, F., Gomes, D., Tran, H.V.: Aubry-Mather measures in the nonconvex setting. SIAM J. Math. Anal. 43(6), 2601–262 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Contreras, G., Iturriaga, R., Sánchez-Morgado, H.: Weak solutions of the Hamilton–Jacobi equation for time periodic Lagrangians. arXiv:1307.0287
  9. 9.
    Evans, L.C.: Partial differential equations. AMS (2000)Google Scholar
  10. 10.
    Evans, L.C.: Adjoint and compensated compactness methods for Hamilton–Jacobi PDE. Arch. Ration. Mech. Anal. 197, 1053–1088 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Evans, L.C., Gomes, D.: Effective hamiltonians and averaging for hamiltonian dynamics II. Arch. Ration. Mech. Anal. 161, 271–305 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fleming, W., Soner, M.: Controlled Markov Processes and Viscosity Solutions. Springer, New York (1993)zbMATHGoogle Scholar
  13. 13.
    Friedman, A.: Partial Differential Equations of Parabolic Type. Robert E. Krieger Publishing Company, Malabar (1983)Google Scholar
  14. 14.
    Gomes, D.: A stochastic analog of Aubry–Mather theory. Nonlinearity 15(3), 581–603 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gomes, D.: Duality Principles for Fully Nonlinear Elliptic Equations Trends in Partial Differential Equations of Mathematical Physics. Birkhausser, Basel (2005)Google Scholar
  16. 16.
    Gomes, D., Jung, N., Lopes, A.: Minimax probabilities for Aubry–Mather problems. Commun. Contemp. Math. 12(5), 789–813 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Guerra-Velasco, E.: The viscous Hamilton–Jacobi equation for space-time periodic Hamiltonians. (Doctoral Thesis). UNAM Database: (2015)
  18. 18.
    Knerr, B.F.: Parabolic interior schauder estimates by the maximum principle. Arch. Rational Mech. Anal. 75, 51–58 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ladyženskaja, O.A., Solonnikovy V.A., Ural’ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs, AMS, 23 (1968)Google Scholar
  20. 20.
    Mitake, H., Tran, H.V.: Selection problems for a discount degenerate viscous Hamilton–Jacobi equation. Adv. Math. 306, 684–703 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Tran, H.V.: Adjoint methods for static Hamilton–Jacobi equations. Calc. Var. Partial Differ. Equ. 41, 301–319 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Sociedad Matemática Mexicana 2019

Authors and Affiliations

  1. 1.Facultad de Ciencias en Física y MatemáticasCONACYT-Universidad Autónoma de ChiapasMéxicoMexico

Personalised recommendations