Advertisement

Min–max formulas and other properties of certain classes of nonconvex effective Hamiltonians

  • Jianliang Qian
  • Hung V. Tran
  • Yifeng Yu
Article

Abstract

This paper is the first attempt to systematically study properties of the effective Hamiltonian \(\overline{H}\) arising in the periodic homogenization of some coercive but nonconvex Hamilton–Jacobi equations. Firstly, we introduce a new and robust decomposition method to obtain min–max formulas for a class of nonconvex \(\overline{H}\). Secondly, we analytically and numerically investigate other related interesting phenomena, such as “quasi-convexification” and breakdown of symmetry, of \(\overline{H}\) from other typical nonconvex Hamiltonians. Finally, in the appendix, we show that our new method and those a priori formulas from the periodic setting can be used to obtain stochastic homogenization for the same class of nonconvex Hamilton–Jacobi equations. Some conjectures and problems are also proposed.

Mathematics Subject Classification

35B10 35B20 35B27 35D40 35F21 

References

  1. 1.
    Achdou, Y., Camilli, F., Capuzzo-Dolcetta, I.: Homogenization of Hamilton–Jacobi equations: numerical methods. Math. Models Methods Appl. Sci. 18(7), 1115–1143 (2008)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Armstrong, S., Cardaliaguet, P.: Stochastic homogenization of quasilinear Hamilton–Jacobi equations and geometric motions. J. Eur. Math. Soc. (to appear)Google Scholar
  3. 3.
    Armstrong, S.N., Souganidis, P.E.: Stochastic homogenization of Hamilton–Jacobi and degenerate Bellman equations in unbounded environments. J. Math. Pures Appl. (9) 97(5), 460–504 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Armstrong, S.N., Souganidis, P.E.: Stochastic homogenization of level-set convex Hamilton–Jacobi equations. Int. Math. Res. Not. 2013, 3420–3449 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Armstrong, S.N., Tran, H.V.: Stochastic homogenization of viscous Hamilton–Jacobi equations and applications. Anal. PDE 7–8, 1969–2007 (2014)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Armstrong, S.N., Tran, H.V., Yu, Y.: Stochastic homogenization of a nonconvex Hamilton–Jacobi equation. Calc. Var. PDE 54(2), 1507–1524 (2015)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Armstrong, S.N., Tran, H.V., Yu, Y.: Stochastic homogenization of nonconvex Hamilton–Jacobi equations in one space dimension. J. Differ. Equ. 261, 2702–2737 (2016)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bangert, V.: Mather sets for twist maps and geodesics on tori. In: Dynamics reported, vol. 1, pp. 1–56Google Scholar
  9. 9.
    Bangert, V.: Geodesic rays, Busemann functions and monotone twist maps. Calc. Var. PDE 2(1), 49–63 (1994)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Barron, E.N., Jensen, R.: Semicontinuous viscosity solutions for Hamilton–Jacobi equations with convex Hamiltonians. Commun. Partial Differ. Equ. 15(12), 1713–1742 (1990)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Cagnetti, F., Gomes, D., Tran, H.V.: Aubry–Mather measures in the non convex setting. SIAM J. Math. Anal. 43, 2601–2629 (2011)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Concordel, M.C.: Periodic homogenization of Hamilton–Jacobi equations: additive eigenvalues and variational formula. Indiana Univ. Math. J. 45(4), 1095–1117 (1996)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Concordel, M.C.: Periodic homogenisation of Hamilton–Jacobi equations. II. Eikonal equations. Proc. R. Soc. Edinb. Sect. A 127(4), 665–689 (1997)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Contreras, G., Iturriaga, R., Paternain, G.P., Paternain, M.: Lagrangian graphs, minimizing measures and Mañé’s critical values. Geom. Funct. Anal. 8, 788–809 (1998)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Davini, A., Kosygina, E.: Homogenization of viscous Hamilton–Jacobi equations: a remark and an application. Calc. Var. 56, 95 (2017)Google Scholar
  16. 16.
    Davini, A., Siconolfi, A.: Exact and approximate correctors for stochastic Hamiltonians: the \(1\)-dimensional case. Math. Ann. 345(4), 749–782 (2009)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    E, W.: Aubry–Mather theory and periodic solutions of the forced Burgers equation. Commun. Pure Appl. Math. 52(7), 811–828 (1999)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Evans, L.C., Gomes, D.: Effective Hamiltonians and averaging for Hamiltonian dynamics. I. Arch. Ration. Mech. Anal. 157(1), 1–33 (2001)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Falcone, M., Rorro, M.: On a variational approximation of the effective Hamiltonian. In: Numerical Mathematics and Advanced Applications, pp. 719–726. Springer, Berlin (2008)Google Scholar
  20. 20.
    Fathi, A.: Weak KAM Theorem in Lagrangian DynamicsGoogle Scholar
  21. 21.
    Feldman, W., Souganidis, P.E.: Homogenization and non-homogenization of certain nonconvex Hamilton–Jacobi equations. J. Math. Pures Appl. (to appear). arXiv:1609.09410 [math.AP]
  22. 22.
    Gao, H.: Random homogenization of coercive Hamilton–Jacobi equations in 1d. Calc. Var. Partial Differ. Equ. 55, 30 (2016)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Gomes, D.A.: A stochastic analogue of Aubry–Mather theory. Nonlinearity 15, 581–603 (2002)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Gomes, D.A., Mitake, H., Tran, H.V.: The selection problem for discounted Hamilton–Jacobi equations: some nonconvex cases. J. Math. Soc. Japan. (to appear). arXiv:1605.07532 [math.AP]
  25. 25.
    Gomes, D.A., Oberman, A.M.: Computing the effective Hamiltonian using a variational formula. SIAM J. Control Optim. 43, 792–812 (2004)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Jing, W., Tran, H.V., Yu, Y.: Inverse problems, non-roundness and flat pieces of the effective burning velocity from an inviscid quadratic Hamilton–Jacobi model Nonlinearity 30, 1853–1875 (2017). arXiv:1602.04728 [math.AP]
  27. 27.
    Kosygina, E., Rezakhanlou, F., Varadhan, S.R.S.: Stochastic homogenization of Hamilton–Jacobi–Bellman equations. Commun. Pure Appl. Math. 59(10), 1489–1521 (2006)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Kosygina, E., Varadhan, S.R.S.: Homogenization of Hamilton–Jacobi–Bellman equations with respect to time-space shifts in a stationary ergodic medium. Commun. Pure Appl. Math. 61(6), 816–847 (2008)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Lions, P.-L., Papanicolaou, G., Varadhan, S.R.S.: Homogenization of Hamilton–Jacobi equations. Unpublished work (1987)Google Scholar
  30. 30.
    Lions, P.-L., Souganidis, P.E.: Homogenization of “viscous” Hamilton–Jacobi equations in stationary ergodic media. Commun. Partial Differ. Equ. 30(1–3), 335–375 (2005)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Lions, P.-L., Souganidis, P.E.: Stochastic homogenization of Hamilton–Jacobi and “viscous” Hamilton–Jacobi equations with convex nonlinearities-revisited. Commun. Math. Sci. 8(2), 627–637 (2010)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Luo, S., Tran, H.V., Yu, Y.: Some inverse problems in periodic homogenization of Hamilton–Jacobi equations. Arch. Ration. Mech. Anal. 221(3), 1585–1617 (2016)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Luo, S., Yu, Y., Zhao, H.: A new approximation for effective Hamiltonians for homogenization of a class of Hamilton–Jacobi equations. Multiscale Model. Simul. 9(2), 711–734 (2011)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Mather, J.N.: Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z. 207(2), 169–207 (1991)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Mañé, R.: Generic properties and problems of minimizing measures of Lagrangian systems. Nonlinearity 9(2), 273–310 (1996)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Nakayasu, A.: Two approaches to minimax formula of the additive eigenvalue for quasiconvex Hamiltonians. arXiv:1412.6735 [math.AP]
  37. 37.
    Oberman, A.M., Takei, R., Vladimirsky, A.: Homogenization of metric Hamilton–Jacobi equations. Multiscale Model. Simul. 8, 269–295 (2009)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Qian, J.-L.: Two Approximations for Effective Hamiltonians Arising from Homogenization of Hamilton–Jacobi Equations, UCLA CAM Report 03-39, University of California, Los Angeles, CA (2003)Google Scholar
  39. 39.
    Rezakhanlou, F., Tarver, J.E.: Homogenization for stochastic Hamilton–Jacobi equations. Arch. Ration. Mech. Anal. 151(4), 277–309 (2000)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Seeger, B.: Homogenization of pathwise Hamilton–Jacobi equations. J. Math. Pures Appl. (to appear). arXiv:1605.00168v3 [math.AP]
  41. 41.
    Souganidis, P.E.: Stochastic homogenization of Hamilton–Jacobi equations and some applications. Asymptot. Anal. 20(1), 1–11 (1999)MathSciNetMATHGoogle Scholar
  42. 42.
    Ziliotto, B.: Stochastic homogenization of nonconvex Hamilton–Jacobi equations: a counterexample. Commun. Pure Appl. Math. (to appear)Google Scholar

Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Department of MathematicsMichigan State UniversityEast LansingUSA
  2. 2.Department of Computational Mathematics, Science and EngineeringMichigan State UniversityEast LansingUSA
  3. 3.Department of MathematicsUniversity of Wisconsin MadisonMadisonUSA
  4. 4.Department of MathematicsUniversity of California, IrvineIrvineUSA

Personalised recommendations