-
Previous Article
Genetics of iterative roots for PM functions
- DCDS Home
- This Issue
-
Next Article
Well-posedness of renormalized solutions for a stochastic $ p $-Laplace equation with $ L^1 $-initial data
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A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, Preliminary version 10, Lyon, unpublished, June 15, 2008. |
[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. |
[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. |
[1] |
Kai Zhao, Wei Cheng. On the vanishing contact structure for viscosity solutions of contact type Hamilton-Jacobi equations I: Cauchy problem. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4345-4358. doi: 10.3934/dcds.2019176 |
[2] |
Mihai Bostan, Gawtum Namah. Time periodic viscosity solutions of Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2007, 6 (2) : 389-410. doi: 10.3934/cpaa.2007.6.389 |
[3] |
Olga Bernardi, Franco Cardin. Minimax and viscosity solutions of Hamilton-Jacobi equations in the convex case. Communications on Pure and Applied Analysis, 2006, 5 (4) : 793-812. doi: 10.3934/cpaa.2006.5.793 |
[4] |
Kaizhi Wang, Jun Yan. Lipschitz dependence of viscosity solutions of Hamilton-Jacobi equations with respect to the parameter. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1649-1659. doi: 10.3934/dcds.2016.36.1649 |
[5] |
Eddaly Guerra, Héctor Sánchez-Morgado. Vanishing viscosity limits for space-time periodic Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2014, 13 (1) : 331-346. doi: 10.3934/cpaa.2014.13.331 |
[6] |
Thomas Strömberg. A system of the Hamilton--Jacobi and the continuity equations in the vanishing viscosity limit. Communications on Pure and Applied Analysis, 2011, 10 (2) : 479-506. doi: 10.3934/cpaa.2011.10.479 |
[7] |
Olga Bernardi, Franco Cardin. On $C^0$-variational solutions for Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 385-406. doi: 10.3934/dcds.2011.31.385 |
[8] |
Gawtum Namah, Mohammed Sbihi. A notion of extremal solutions for time periodic Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 647-664. doi: 10.3934/dcdsb.2010.13.647 |
[9] |
Gui-Qiang Chen, Bo Su. Discontinuous solutions for Hamilton-Jacobi equations: Uniqueness and regularity. Discrete and Continuous Dynamical Systems, 2003, 9 (1) : 167-192. doi: 10.3934/dcds.2003.9.167 |
[10] |
David McCaffrey. A representational formula for variational solutions to Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1205-1215. doi: 10.3934/cpaa.2012.11.1205 |
[11] |
Nalini Anantharaman, Renato Iturriaga, Pablo Padilla, Héctor Sánchez-Morgado. Physical solutions of the Hamilton-Jacobi equation. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 513-528. doi: 10.3934/dcdsb.2005.5.513 |
[12] |
Claudio Marchi. On the convergence of singular perturbations of Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1363-1377. doi: 10.3934/cpaa.2010.9.1363 |
[13] |
Isabeau Birindelli, J. Wigniolle. Homogenization of Hamilton-Jacobi equations in the Heisenberg group. Communications on Pure and Applied Analysis, 2003, 2 (4) : 461-479. doi: 10.3934/cpaa.2003.2.461 |
[14] |
Gonzalo Dávila. Comparison principles for nonlocal Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022061 |
[15] |
Olga Bernardi, Franco Cardin, Antonio Siconolfi. Cauchy problems for stationary Hamilton-Jacobi equations under mild regularity assumptions. Journal of Geometric Mechanics, 2009, 1 (3) : 271-294. doi: 10.3934/jgm.2009.1.271 |
[16] |
Thi Tuyen Nguyen. Large time behavior of solutions of local and nonlocal nondegenerate Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator. Communications on Pure and Applied Analysis, 2019, 18 (3) : 999-1021. doi: 10.3934/cpaa.2019049 |
[17] |
Xia Li. Long-time asymptotic solutions of convex hamilton-jacobi equations depending on unknown functions. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5151-5162. doi: 10.3934/dcds.2017223 |
[18] |
Laura Caravenna, Annalisa Cesaroni, Hung Vinh Tran. Preface: Recent developments related to conservation laws and Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems - S, 2018, 11 (5) : i-iii. doi: 10.3934/dcdss.201805i |
[19] |
Fabio Camilli, Paola Loreti, Naoki Yamada. Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1291-1302. doi: 10.3934/cpaa.2009.8.1291 |
[20] |
Yasuhiro Fujita, Katsushi Ohmori. Inequalities and the Aubry-Mather theory of Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2009, 8 (2) : 683-688. doi: 10.3934/cpaa.2009.8.683 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]