# American Institute of Mathematical Sciences

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

## 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

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

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##### 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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