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The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 1: linear coupling

  • Received: 30 September 2019 Accepted: 06 February 2020 Published: 21 August 2020
  • We establish a convergence theorem for the vanishing discount problem for a weakly coupled system of Hamilton-Jacobi equations. The crucial step is the introduction of Mather measures and their relatives for the system, which we call respectively viscosity Mather and Green-Poisson measures. This is done by the convex duality and the duality between the space of continuous functions on a compact set and the space of Borel measures on it. This is part 1 of our study of the vanishing discount problem for systems, which focuses on the linear coupling, while part 2 will be concerned with nonlinear coupling.

    Citation: Hitoshi Ishii. The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 1: linear coupling[J]. Mathematics in Engineering, 2021, 3(4): 1-21. doi: 10.3934/mine.2021032

    Related Papers:

  • We establish a convergence theorem for the vanishing discount problem for a weakly coupled system of Hamilton-Jacobi equations. The crucial step is the introduction of Mather measures and their relatives for the system, which we call respectively viscosity Mather and Green-Poisson measures. This is done by the convex duality and the duality between the space of continuous functions on a compact set and the space of Borel measures on it. This is part 1 of our study of the vanishing discount problem for systems, which focuses on the linear coupling, while part 2 will be concerned with nonlinear coupling.


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    [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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