Search Advanced

Mathematics in Engineering

2021, Volume 3, Issue 4: 1-21. doi: 10.3934/mine.2021032
Previous Article Next Article
Research article Special Issues

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
  • 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.


    加载中


    [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.
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Mathematics in Engineering
1.4 2.2

Metrics

Article views(3601) PDF downloads(588) Cited by(2)

Preview PDF
Download XML
Export Citation
Article outline

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog