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On the vanishing discount problem from the negative direction
1. | Dip. di Matematica, Sapienza Università di Roma, P.le Aldo Moro 2, 00185 Roma, Italy |
2. | Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China |
$ \begin{equation} \lambda u_\lambda+H(x,d_x u_\lambda) = c(H)\qquad\hbox{in $M$}, \;\;\;\;\;\;\;\;\;(*)\end{equation} $ |
$ \lambda\rightarrow 0^+ $ |
$ u_0 $ |
$ H(x,d_x u) = c(H)\qquad\hbox{in $M$}, $ |
$ M $ |
$ c(H) $ |
$ \lambda\rightarrow 0^- $ |
$ u_\lambda^- $ |
$ u_\lambda^- $ |
$ u_0 $ |
$ \lambda\rightarrow 0^- $ |
$ H $ |
$ \lambda<0 $ |
References:
[1] |
G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi, vol. 17 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Paris, 1994. |
[2] |
P. Bernard,
Existence of $C^{1, 1}$ critical sub-solutions of the Hamilton-Jacobi equation on compact manifolds, Ann. Sci. École Norm. Sup., 40 (2007), 445-452.
doi: 10.1016/j.ansens.2007.01.004. |
[3] |
P. Bernard,
Smooth critical sub-solutions of the Hamilton-Jacobi equation, Math. Res. Lett., 14 (2007), 503-511.
doi: 10.4310/MRL.2007.v14.n3.a14. |
[4] |
P. Cannarsa and H. M. Soner,
Generalized one-sided estimates for solutions of Hamilton-Jacobi equations and applications, Nonlinear Anal., 13 (1989), 305-323.
doi: 10.1016/0362-546X(89)90056-4. |
[5] |
F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons Inc., New York, 1983. |
[6] |
G. Contreras, R. Iturriaga, G. P. Paternain and M. Paternain,
Lagrangian graphs, minimizing measures and Mañé's critical values, Geom. Funct. Anal., 8 (1998), 788-809.
doi: 10.1007/s000390050074. |
[7] |
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), 29-55.
doi: 10.1007/s00222-016-0648-6. |
[8] |
A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, Preliminary version 10, Lyon, unpublished, June 15, 2008. Google Scholar |
[9] |
A. Fathi and A. Siconolfi,
Existence of $C^1$ critical subsolutions of the Hamilton-Jacobi equation, Invent. Math., 155 (2004), 363-388.
doi: 10.1007/s00222-003-0323-6. |
[10] |
N. Kryloff and N. Bogoliuboff,
La théorie générale de la mesure et son application à l'étude des systèmes dynamiques de la mécanique non linéaire, Ann. of Math., 38 (1937), 65-113.
doi: 10.2307/1968511. |
[11] |
S. Marò and A. Sorrentino,
Aubry-Mather theory for conformally symplectic systems, Comm. Math. Phys., 354 (2017), 775-808.
doi: 10.1007/s00220-017-2900-3. |
[12] |
A. Siconolfi, Hamilton-Jacobi equations and weak KAM theory, in Mathematics of Complexity and Dynamical Systems, Vols. 1–3, Springer, New York, (2012), 683–703.
doi: 10.1007/978-1-4614-1806-1_42. |
[13] |
K. Wang, L. Wang and J. Yan,
Implicit variational principle for contact Hamiltonian systems, Nonlinearity, 30 (2017), 492-515.
doi: 10.1088/1361-6544/30/2/492. |
[14] |
K. Wang, L. Wang and J. Yan,
Aubry-Mather theory for contact Hamiltonian systems, Comm. Math. Phys., 366 (2019), 981-1023.
doi: 10.1007/s00220-019-03362-2. |
[15] |
K. Wang, L. Wang and J. Yan,
Variational principle for contact Hamiltonian systems and its applications, J. Math. Pures Appl., 123 (2019), 167-200.
doi: 10.1016/j.matpur.2018.08.011. |
show all references
References:
[1] |
G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi, vol. 17 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Paris, 1994. |
[2] |
P. Bernard,
Existence of $C^{1, 1}$ critical sub-solutions of the Hamilton-Jacobi equation on compact manifolds, Ann. Sci. École Norm. Sup., 40 (2007), 445-452.
doi: 10.1016/j.ansens.2007.01.004. |
[3] |
P. Bernard,
Smooth critical sub-solutions of the Hamilton-Jacobi equation, Math. Res. Lett., 14 (2007), 503-511.
doi: 10.4310/MRL.2007.v14.n3.a14. |
[4] |
P. Cannarsa and H. M. Soner,
Generalized one-sided estimates for solutions of Hamilton-Jacobi equations and applications, Nonlinear Anal., 13 (1989), 305-323.
doi: 10.1016/0362-546X(89)90056-4. |
[5] |
F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons Inc., New York, 1983. |
[6] |
G. Contreras, R. Iturriaga, G. P. Paternain and M. Paternain,
Lagrangian graphs, minimizing measures and Mañé's critical values, Geom. Funct. Anal., 8 (1998), 788-809.
doi: 10.1007/s000390050074. |
[7] |
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), 29-55.
doi: 10.1007/s00222-016-0648-6. |
[8] |
A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, Preliminary version 10, Lyon, unpublished, June 15, 2008. Google Scholar |
[9] |
A. Fathi and A. Siconolfi,
Existence of $C^1$ critical subsolutions of the Hamilton-Jacobi equation, Invent. Math., 155 (2004), 363-388.
doi: 10.1007/s00222-003-0323-6. |
[10] |
N. Kryloff and N. Bogoliuboff,
La théorie générale de la mesure et son application à l'étude des systèmes dynamiques de la mécanique non linéaire, Ann. of Math., 38 (1937), 65-113.
doi: 10.2307/1968511. |
[11] |
S. Marò and A. Sorrentino,
Aubry-Mather theory for conformally symplectic systems, Comm. Math. Phys., 354 (2017), 775-808.
doi: 10.1007/s00220-017-2900-3. |
[12] |
A. Siconolfi, Hamilton-Jacobi equations and weak KAM theory, in Mathematics of Complexity and Dynamical Systems, Vols. 1–3, Springer, New York, (2012), 683–703.
doi: 10.1007/978-1-4614-1806-1_42. |
[13] |
K. Wang, L. Wang and J. Yan,
Implicit variational principle for contact Hamiltonian systems, Nonlinearity, 30 (2017), 492-515.
doi: 10.1088/1361-6544/30/2/492. |
[14] |
K. Wang, L. Wang and J. Yan,
Aubry-Mather theory for contact Hamiltonian systems, Comm. Math. Phys., 366 (2019), 981-1023.
doi: 10.1007/s00220-019-03362-2. |
[15] |
K. Wang, L. Wang and J. Yan,
Variational principle for contact Hamiltonian systems and its applications, J. Math. Pures Appl., 123 (2019), 167-200.
doi: 10.1016/j.matpur.2018.08.011. |
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