On the vanishing discount problem from the negative direction

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May 2021, 41(5): 2377-2389. doi: 10.3934/dcds.2020368

On the vanishing discount problem from the negative direction

Andrea Davini ^1, and Lin Wang ^2,

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

Received January 2020 Revised July 2020 Published May 2021 Early access November 2020

It has been proved in [7] that the unique viscosity solution of

$ \begin{equation} \lambda u_\lambda+H(x,d_x u_\lambda) = c(H)\qquad\hbox{in $M$}, \;\;\;\;\;\;\;\;\;(*)\end{equation} $

uniformly converges, for

$ \lambda\rightarrow 0^+ $

, to a specific solution

$ u_0 $

of the critical equation

$ H(x,d_x u) = c(H)\qquad\hbox{in $M$}, $

where

$ M $

is a closed and connected Riemannian manifold and

$ c(H) $

is the critical value. In this note, we consider the same problem for

$ \lambda\rightarrow 0^- $

. In this case, viscosity solutions of equation (*) are not unique, in general, so we focus on the asymptotics of the minimal solution

$ u_\lambda^- $

of (*). Under the assumption that constant functions are subsolutions of the critical equation, we prove that the

$ u_\lambda^- $

also converges to

$ u_0 $

as

$ \lambda\rightarrow 0^- $

. Furthermore, we exhibit an example of

$ H $

for which equation (*) admits a unique solution for

$ \lambda<0 $

as well.

Keywords: Hamilton-Jacobi equations, vanishing discount problems, viscosity solutions.

Mathematics Subject Classification: Primary: 37J51, 35F21; Secondary: 35D40.

Citation: Andrea Davini, Lin Wang. On the vanishing discount problem from the negative direction. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2377-2389. doi: 10.3934/dcds.2020368

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.

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