Skip to main content
SpringerLink
Log in

On the Large Time Behavior of Solutions of Hamilton–Jacobi Equations Associated with Nonlinear Boundary Conditions

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

In this article, we study the large time behavior of solutions of first-order Hamilton–Jacobi Equations set in a bounded domain with nonlinear Neumann boundary conditions, including the case of dynamical boundary conditions. We establish general convergence results for viscosity solutions of these Cauchy–Neumann problems by using two fairly different methods: the first one relies only on partial differential equations methods, which provides results even when the Hamiltonians are not convex, and the second one is an optimal control/dynamical system approach, named the “weak KAM approach”, which requires the convexity of Hamiltonians and gives formulas for asymptotic solutions based on Aubry–Mather sets.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barles G.: Fully nonlinear Neumann type boundary conditions for second-order elliptic and parabolic equations. J. Differ. Equ. 106(1), 90–106 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barles G.: Solutions de viscosité des équations de Hamilton–Jacobi, Mathématiques & Applications (Berlin), vol. 17. Springer-Verlag, Paris (1994)

    Google Scholar 

  3. Barles G.: Nonlinear Neumann boundary conditions for quasilinear degenerate elliptic equations and applications. J. Differ. Equ. 154, 191–224 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barles G., Lions P.-L.: Fully nonlinear Neumann type boundary conditions for first-order Hamilton–Jacobi equations. Nonlinear Anal. 16(2), 143–153 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. Barles, G., Mitake, H.: A PDE approach to large-time asymptotics for boundary-value problems for nonconvex Hamilton–Jacobi equations. Commun. Partial Differ. Equ. (to appear)

  6. Barles G., Roquejoffre J.-M.: Ergodic type problems and large time behaviour of unbounded solutions of Hamilton–Jacobi equations. Commun. Partial Differ. Equ. 31(7–9), 1209–1225 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Barles G., Souganidis P.E.: On the large time behavior of solutions of Hamilton–Jacobi equations. SIAM J. Math. Anal. 31(4), 925–939 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bernard P., Roquejoffre J.-M.: Convergence to time-periodic solutions in time-periodic Hamilton–Jacobi equations on the circle. Commun. Partial Differ. Equ. 29(3–4), 457–469 (2004)

    MathSciNet  MATH  Google Scholar 

  9. Crandall M.G., Ishii H., Lions P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Da Lio F.: Large time behavior of solutions to parabolic equations with Neumann boundary conditions. J. Math. Anal. Appl. 339(1), 384–398 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Davini A., Siconolfi A.: A generalized dynamical approach to the large time behavior of solutions of Hamilton–Jacobi equations. SIAM J. Math. Anal. 38(2), 478–502 (2006)

    Article  MathSciNet  Google Scholar 

  12. Fathi A.: Sur la convergence du semi-groupe de Lax-Oleinik. C. R. Acad. Sci. Paris Sér. I Math. 327(3), 267–270 (1998)

    MathSciNet  ADS  MATH  Google Scholar 

  13. Fujita Y., Ishii H., Loreti P.: Asymptotic solutions of Hamilton–Jacobi equations in Euclidean n space. Indiana Univ. Math. J. 55(5), 1671–1700 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Giga, Y., Liu, Q., Mitake, H.: Singular Neumann problems and large-time behavior of solutions of non-coercive Hamitonian-Jacobi equations (submitted)

  15. Giga Y., Liu Q., Mitake H.: Large-time behavior of one-dimensional Dirichlet problems of Hamilton–Jacobi equations with non-coercive Hamiltonians. J. Differ. Equ. 252, 1263–1282 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ichihara N., Ishii H.: Asymptotic solutions of Hamilton–Jacobi equations with semi-periodic Hamiltonians. Commun. Partial Differ. Equ. 33(4–6), 784–807 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ichihara N., Ishii H.: The large-time behavior of solutions of Hamilton–Jacobi equations on the real line. Methods Appl. Anal. 15(2), 223–242 (2008)

    MathSciNet  MATH  Google Scholar 

  18. Ichihara N., Ishii H.: Long-time behavior of solutions of Hamilton–Jacobi equations with convex and coercive Hamiltonians. Arch. Ration. Mech. Anal. 194(2), 383–419 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ishii H.: Perron’s method for Hamilton–Jacobi equations. Duke Math. J. 55(2), 369–384 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ishii H.: Fully nonlinear oblique derivative problems for nonlinear second-order elliptic PDEs. Duke Math. J. 62(3), 633–661 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ishii H.: Asymptotic solutions for large time of Hamilton–Jacobi equations in Euclidean n space. Ann. Inst. H. Poincaré Anal. Non Linéaire 25(2), 231–266 (2008)

    Article  ADS  MATH  Google Scholar 

  22. Ishii H.: Weak KAM aspects of convex Hamilton–Jacobi equations with Neumann type boundary conditions. J. Math. Pures Appl. (9) 95(1), 99–135 (2011)

    MathSciNet  MATH  Google Scholar 

  23. Ishii, H.: Long-time asymptotic solutions of convex Hamilton–Jacobi equations with Neumann type bouundary conditions. Calc. Var. Partial Differ. Equ., 2010 (onlinefirst)

  24. Ishii H., Mitake H.: Representation formulas for solutions of Hamilton–Jacobi equations with convex Hamiltonians. Indiana Univ. Math. J. 56(5), 2159–2184 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lions P.-L.: Neumann type boundary conditions for Hamilton–Jacobi equations. Duke Math. J. 52(4), 793–820 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lions, P.-L., Papanicolaou, G., Varadhan, S.R.S.: Homogenization of Hamilton–Jacobi Equations (unpublished work)

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Mitake H.: The large-time behavior of solutions of the Cauchy–Dirichlet problem for Hamilton–Jacobi equations. NoDEA Nonlinear Differ. Equ. Appl. 15(3), 347–362 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Mitake H.: Large time behavior of solutions of Hamilton–Jacobi equations with periodic boundary data. Nonlinear Anal. 71(11), 5392–5405 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Namah G., Roquejoffre J.-M.: Remarks on the long time behaviour of the solutions of Hamilton–Jacobi equations. Commun. Partial Differ. Equ. 24(5–6), 883–893 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  31. Roquejoffre J.-M.: Convergence to steady states or periodic solutions in a class of Hamilton–Jacobi equations. J. Math. Pures Appl. (9) 80(1), 85–104 (2001)

    MathSciNet  MATH  Google Scholar 

  32. Yokoyama E., Giga Y., Rybka P.: A microscopic time scale approximation to the behavior of the local slope on the faceted surface under a nonuniformity in supersaturation. Phys. D 237(22), 2845–2855 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques et Physique Théorique, Fédération Denis Poisson, Université de Tours, Place de Grandmont, 37200, Tours, France

    Guy Barles

  2. Faculty of Education and Integrated Arts and Sciences, Waseda University, Nishi-Waseda, Shinjuku, Tokyo, 169-8050, Japan

    Hitoshi Ishii

  3. Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

    Hitoshi Ishii

  4. Department of Applied Mathematics, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, Japan

    Hiroyoshi Mitake

Authors
  1. Guy Barles
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hitoshi Ishii
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hiroyoshi Mitake
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guy Barles.

Additional information

Communicated by P.L. Lions

This work was partially supported by the ANR (Agence Nationale de la Recherche) through the project “Hamilton–Jacobi et théorie KAM faible” (ANR-07-BLAN-3-187245), by the KAKENHI (Nos. 20340019, 21340032, 21224001, 23340028, 23244015), JSPS and by the Research Fellowship (22-1725) for Young Researchers from JSPS.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Barles, G., Ishii, H. & Mitake, H. On the Large Time Behavior of Solutions of Hamilton–Jacobi Equations Associated with Nonlinear Boundary Conditions. Arch Rational Mech Anal 204, 515–558 (2012). https://doi.org/10.1007/s00205-011-0484-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-011-0484-1

Keywords

Search

Navigation