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Mathematics in Engineering, 2021, 3(4): 1-21. doi: 10.3934/mine.2021032
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The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 1: linear coupling
Institute for Mathematics and Computer Science, Tsuda University, 2-1-1 Tsuda, Kodaira, Tokyo, 187-8577 Japan
†This contribution is part of the Special Issue: Critical values in nonlinear pdes – Special Issue dedicated to Italo Capuzzo Dolcetta
Guest Editor: Fabiana Leoni
Link: www.aimspress.com/mine/article/5754/special-articles
Received: , Accepted: , Published:
Special Issues: Critical values in nonlinear pdes - Special Issue dedicated to Italo Capuzzo Dolcetta
References
1. Al-Aidarous ES, Alzahrani EO, Ishii H, et al. (2016) A convergence result for the ergodic problem for Hamilton-Jacobi equations with Neumann-type boundary conditions. P Roy Soc Edinb A 146: 225-242.
2. Bardi M, Capuzzo-Dolcetta I (1997) Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Boston: Birkhäuser Boston, Inc.
3. Barles G (1993) Discontinuous viscosity solutions of first-order Hamilton-Jacobi equations: A guided visit. Nonlinear Anal 20: 1123-1134.
4. Barles G (1994) Solutions de viscosité des équations de Hamilton-Jacobi, Paris: Springer-Verlag.
5. Cagnetti F, Gomes D, Tran VH (2013) Adjoint methods for obstacle problems and weakly coupled systems of PDE. ESAIM Contr Optim Ca 19: 754-779.
6. Camilli F, Ley O, Loreti P, et al. (2012) Large time behavior of weakly coupled systems of first-order Hamilton-Jacobi equations. NoDEA Nonlinear Diff 19: 719-749.
7. Chen Q, Cheng W, Ishii H, et al. (2019) Vanishing contact structure problem and convergence of the viscosity solutions. Commun Part Diff Eq 44: 801-836.
8. Crandall MG, Ishii H, Lions PL (1992) User's guide to viscosity solutions of second order partial differential equations. B Am Math Soc 27: 1-67.
9. Crandall MG, Lions PL (1983) Viscosity solutions of Hamilton-Jacobi equations. T Am Math Soc 277: 1-42.
10. Davini A, Fathi A, Iturriaga R, et al. (2016) Convergence of the solutions of the discounted Hamilton-Jacobi equation. Invent Math 206: 29-55.
11. Davini A, Zavidovique M (2014) Aubry sets for weakly coupled systems of Hamilton-Jacobi equations. SIAM J Math Anal 46: 3361-3389.
12. Davini A, Zavidovique M (2019) Convergence of the solutions of discounted Hamilton-Jacobi systems. Adv Calc Var, Available from: https://doi.org/10.1515/acv-2018-0037.
13. Engler H, Lenhart SM (1991) Viscosity solutions for weakly coupled systems of Hamilton-Jacobi equations. P Lond Math Soc 63: 212-240.
14. Evans LC (2004) A survey of partial differential equations methods in weak KAM theory. Commun Pure Appl Math 57: 445-480.
15. Evans LC (2010) Adjoint and compensated compactness methods for Hamilton-Jacobi PDE. Arch Ration Mech Anal 197: 1053-1088.
16. Fathi A (1997) Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens. C R Acad Sci Paris Sér I Math 324: 1043-1046.
17. Fathi A (2008) Weak KAM Theorem in Lagrangian Dynamics, Preliminary Version 10, Available from: https://www.math.u-bordeaux.fr/~pthieull/Recherche/KamFaible/Publications/Fathi2008_01.pdf.
18. Gomes DA (2005) Duality principles for fully nonlinear elliptic equations. In: Trends in Partial Differential Equations of Mathematical Physics, Basel: Birkhäuser, 125-136.
19. Gomes DA, Mitake H, Tran HV (2018) The selection problem for discounted Hamilton-Jacobi equations: Some non-convex cases. J Math Soc JPN 70: 345-364.
20. Ishii H, Jin L (2020) The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. part 2 - Nonlinear coupling. Calc Var 59: 140.
21. Ishii H (1987) Perron's method for Hamilton-Jacobi equations. Duke Math J 55: 369-384.
22. Ishii H, Koike S (1991) Viscosity solutions for monotone systems of second-order elliptic PDEs. Commun Part Diff Eq 16: 1095-1128.
23. Ishii H, Mitake H, Tran HV (2017) The vanishing discount problem and viscosity Mather measures. Part 1: The problem on a torus. J Math Pure Appl 108: 125-149.
24. Ishii H, Mitake H, Tran HV (2017) The vanishing discount problem and viscosity Mather measures. Part 2: Boundary value problems. J Math Pure Appl 108: 261-305.
25. Ishii H, Siconolfi A (2020) The vanishing discount problem for Hamilton-Jacobi equations in the Euclidean space. Commun Part Diff Eq 45: 525-560.
26. Mitake H, Siconolfi A, Tran HV, et al. (2016) A Lagrangian approach to weakly coupled Hamilton-Jacobi systems. SIAM J Math Anal 48: 821-846.
27. Mitake H, Tran HV (2017) Selection problems for a discount degenerate viscous Hamilton-Jacobi equation. Adv Math 306: 684-703.
28. Mitake H, Tran HV (2012) Remarks on the large time behavior of viscosity solutions of quasi-monotone weakly coupled systems of Hamilton-Jacobi equations. Asymptot Anal 77: 43-70.
29. Mitake H, Tran HV (2014) A dynamical approach to the large-time behavior of solutions to weakly coupled systems of Hamilton-Jacobi equations. J Math Pure Appl 101: 76-93.
30. Mitake H, Tran HV (2014) Homogenization of weakly coupled systems of Hamilton-Jacobi equations with fast switching rates. Arch Ration Mech Anal 211: 733-769
31. Lions PL (1982) Generalized Solutions of Hamilton-Jacobi Equations, Boston-London: Pitman.
32. Sion M (1958) On general minimax theorems. Pacific J Math 8: 171-176.
33. Terai K (2019) Uniqueness structure of weakly coupled systems of ergodic problems of Hamilton-Jacobi equations. NoDEA Nonlinear Diff 26: 44.
34. Terkelsen F (1972) Some minimax theorems. Math Scand 31: 405-413.
35. Varga RS (2000) Matrix Iterative Analysis, Berlin: Springer-Verlag.
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