Large time behavior of solutions of Hamilton-Jacobi equations with periodic boundary data

Nonlinear Analysis, Theory, Methods and Applications

Volumn 71, Issue 11, 2009, Pages 5392-5405

  Mitake, Hiroyoshi ^a  

^a WASEDA UNIVERSITY   (Japan)

Author keywords

Aubry set; Cauchy Dirichlet problem; Hamilton Jacobi equations; Large time behavior; Periodic solution

Indexed keywords

AUBRY SET; CAUCHY-DIRICHLET PROBLEM; HAMILTON-JACOBI EQUATIONS; LARGE-TIME BEHAVIOR; PERIODIC SOLUTION;

ASYMPTOTIC ANALYSIS; DIFFERENTIAL EQUATIONS; MECHANICS; VISCOSITY;

SET THEORY;

EID: 68349086917     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.na.2009.04.028     Document Type: Article

References (24)

1. Namah G., and Roquejoffre J.-M. Remarks on the long time behaviour of the solutions of Hamilton-Jacobi equations. Comm. Partial Differential Equations 24 5-6 (1999) 883-893
2. Fathi A. Sur la convergence du semi-groupe de Lax-Oleinik. C. R. Acad. Sci. Paris Sér. I Math. 327 3 (1998) 267-270
3. Barles G., and Souganidis P.E. On the large time behavior of solutions of Hamilton-Jacobi equations. SIAM J. Math. Anal. 31 4 (2000) 925-939
4. Roquejoffre J.-M. Convergence to steady states or periodic solutions in a class of Hamilton-Jacobi equations. J. Math. Pures Appl. (9) 80 1 (2001) 85-104
5. Davini A., and Siconolfi A. A generalized dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations. SIAM J. Math. Anal. 38 2 (2006) 478-502
6. Fujita Y., Ishii H., and Loreti P. Asymptotic solutions of Hamilton-Jacobi equations in Euclidean n space. Indiana Univ. Math. J. 55 5 (2006) 1671-1700
7. Barles G., and Roquejoffre J.-M. Ergodic type problems and large time behaviour of unbounded solutions of Hamilton-Jacobi equations. Comm. Partial Differential Equations 31 7-9 (2006) 1209-1225
8. Ishii H. Asymptotic solutions for large time of Hamilton-Jacobi equations in Euclidean n space. Ann. Inst. H. Poincaré Anal. Non Linéaire 25 2 (2008) 231-266
9. Ichihara N., and Ishii H. Asymptotic solutions of Hamilton-Jacobi equations with semi-periodic Hamiltonians. Comm. Partial Differential Equations 33 4-6 (2008) 784-807
10. Ichihara N., and Ishii H. The large-time behavior of solutions of Hamilton-Jacobi equations on the real line. Methods Appl. Anal. 15 2 (2008) 223-242
11. N. Ichihara, H. Ishii, Long-time behavior of solutions of Hamilton-Jacobi equations with convex and coercive Hamiltonians, Arch. Ration. Mech. Anal. (in press)
12. Bernard P., and Roquejoffre J.-M. Convergence to time-periodic solutions in time-periodic Hamilton-Jacobi equations on the circle. Comm. Partial Differential Equations 29 3-4 (2004) 457-469
13. Bostan M., and Namah G. Time periodic viscosity solutions of Hamilton-Jacobi equations. Commun. Pure Appl. Anal. 6 2 (2007) 389-410
14. Barles G. Asymptotic behavior of viscosity solutions of first Hamilton Jacobi equations. Ricerche Mat. 34 2 (1985) 227-260
15. Mitake H. The large-time behavior of solutions of the Cauchy-Dirichlet problem for Hamilton-Jacobi equations. Nonlinear Differential Equations Appl. 15 3 (2008) 347-362
16. Mitake H. Asymptotic solutions of Hamilton-Jacobi equations with state constraints. Appl. Math. Optim. 58 3 (2008) 393-410
17. Soner H.M. Optimal control with state-space constraint, I. SIAM J. Control Optim. 24 3 (1986) 552-561
18. Ishii H. A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 16 1 (1989) 105-135
19. Barles G., and Perthame B. Discontinuous solutions of deterministic optimal stopping time problems. RAIRO Model. Math. Anal. Numer. 21 4 (1987) 557-579
20. Lions P.-L. Generalized Solutions of Hamilton-Jacobi Equations. Research Notes in Mathematics vol. 69 (1982), Pitman, Boston, Masso. London
21. Bardi M., and Capuzzo-Dolcetta I. Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Systems & Control: Foundations & Applications (1997), Birkhäuser Boston, Inc., Boston, MA
22. Ishii H., and Mitake H. Representation formulas for solutions of Hamilton-Jacobi equations with convex Hamiltonians. Indiana Univ. Math. J. 56 5 (2007) 2159-2184
23. Fathi A., and Siconolfi A. PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians. Calc. Var. Partial Differential Equations 22 2 (2005) 185-228
24. H. Ishii, H. Mitake, Two remarks of periodic solutions of Hamilton-Jacobi equations. Preprint