# American Institute of Mathematical Sciences

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  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 & Continuous Dynamical Systems - A, 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons Inc., New York, 1983.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons Inc., New York, 1983.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
 [1] Olivier Ley, Erwin Topp, Miguel Yangari. Some results for the large time behavior of Hamilton-Jacobi equations with Caputo time derivative. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021007 [2] Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024 [3] Ahmad El Hajj, Hassan Ibrahim, Vivian Rizik. $BV$ solution for a non-linear Hamilton-Jacobi system. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020405 [4] Jose Anderson Cardoso, Patricio Cerda, Denilson Pereira, Pedro Ubilla. Schrödinger Equations with vanishing potentials involving Brezis-Kamin type problems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020392 [5] Bin Wang, Lin Mu. Viscosity robust weak Galerkin finite element methods for Stokes problems. Electronic Research Archive, 2021, 29 (1) : 1881-1895. doi: 10.3934/era.2020096 [6] Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020298 [7] Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020398 [8] Sergey E. Mikhailov, Carlos F. Portillo. Boundary-domain integral equations equivalent to an exterior mixed bvp for the variable-viscosity compressible stokes pdes. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021009 [9] Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380 [10] Tong Yang, Seiji Ukai, Huijiang Zhao. Stationary solutions to the exterior problems for the Boltzmann equation, I. Existence. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 495-520. doi: 10.3934/dcds.2009.23.495 [11] Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272 [12] José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091 [13] Pierre Baras. A generalization of a criterion for the existence of solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 465-504. doi: 10.3934/dcdss.2020439 [14] Ágota P. Horváth. Discrete diffusion semigroups associated with Jacobi-Dunkl and exceptional Jacobi polynomials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021002 [15] Raffaele Folino, Ramón G. Plaza, Marta Strani. Long time dynamics of solutions to $p$-Laplacian diffusion problems with bistable reaction terms. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020403 [16] Jingjing Wang, Zaiyun Peng, Zhi Lin, Daqiong Zhou. On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set. Journal of Industrial & Management Optimization, 2021, 17 (2) : 869-887. doi: 10.3934/jimo.2020002 [17] Yutong Chen, Jiabao Su. Nontrivial solutions for the fractional Laplacian problems without asymptotic limits near both infinity and zero. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021007 [18] Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168 [19] Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137 [20] Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

2019 Impact Factor: 1.338

Article outline