Mathematische Annalen

, Volume 361, Issue 3–4, pp 647–687 | Cite as

Viscosity solutions of general viscous Hamilton–Jacobi equations

  • Scott N. Armstrong
  • Hung V. TranEmail author


We present comparison principles, Lipschitz estimates and study state constraints problems for degenerate, second-order Hamilton–Jacobi equations.

Mathematics Subject Classification

35D40 35B51 



S. Armstrong thanks the Forschungsinstitut für Mathematik (FIM) of ETH Zürich for support. H. Tran is supported in part by NSF Grant DMS-1361236.


  1. 1.
    Armstrong, S.N., Cardaliaguet, P.: Quantitative stochastic homogenization of viscous Hamilton–Jacobi equations. Comm. Partial Differ. Equ. (to appear)Google Scholar
  2. 2.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Armstrong, S.N., Tran, H.V.: Stochastic homogenization of viscous Hamilton–Jacobi equations and applications. arXiv:1310.1749 [math.AP]
  4. 4.
    Barles, G.: A weak Bernstein method for fully nonlinear elliptic equations. Differ. Integral Equ. 4(2), 241–262 (1991)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Barles, G., Biton, S., Bourgoing, M., Ley, O.: Uniqueness results for quasilinear parabolic equations through viscosity solutions’ methods. Calc. Var. Partial Differ. Equ. 18(2), 159–179 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Barles, G., Da Lio, F.: On the generalized Dirichlet problem for viscous Hamilton–Jacobi equations. J. Math. Pures Appl. (9) 83(1), 53–75 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Barles, G., Perthame, B.: Discontinuous solutions of deterministic optimal stopping time problems. RAIRO Modél. Math. Anal. Numér. 21(4), 557–579 (1987)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Bernstein, S.: Sur la généralisation du problème de Dirichlet. Math. Ann. 69(1), 82–136 (1910)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Da Lio, F., Ley, O.: Uniqueness results for second-order Bellman–Isaacs equations under quadratic growth assumptions and applications. SIAM J. Control Optim. 45(1), 74–106 (2006, electronic)Google Scholar
  11. 11.
    Da Lio, F., Ley, O.: Convex Hamilton–Jacobi equations under superlinear growth conditions on data. Appl. Math. Optim. 63(3), 309–339 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Dolcetta, I.C., Leoni, F., Porretta, A.: Hölder estimates for degenerate elliptic equations with coercive hamiltonians. Trans. Am. Math. Soc. 362(9), 4511–4536 (2010)CrossRefzbMATHGoogle Scholar
  13. 13.
    Fathi, A., Siconolfi, A.: Existence of \(C^1\) critical subsolutions of the Hamilton–Jacobi equation. Invent. Math. 155(2), 363–388 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Kobylanski, M.: Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28(2), 558–602 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Koike, S., Ley, O.: Comparison principle for unbounded viscosity solutions of degenerate elliptic PDEs with gradient superlinear terms. J. Math. Anal. Appl. 381(1), 110–120 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Lasry, J.-M., Lions, P.-L.: Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints. I. The model problem. Math. Ann. 283(4), 583–630 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Lions, P.-L.: Generalized Solutions of Hamilton–Jacobi Equations. Research Notes in Mathematics, vol. 69. Pitman (Advanced Publishing Program). Boston (1982)Google Scholar
  18. 18.
    Lions, P.-L.: Quelques remarques sur les problèmes elliptiques quasilinéaires du second ordre. J. Anal. Math. 45, 234–254 (1985)CrossRefzbMATHGoogle Scholar
  19. 19.
    Mitake, H.: Asymptotic solutions of Hamilton–Jacobi equations with state constraints. Appl. Math. Optim. 58(3), 393–410 (2008)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.CEREMADE (UMR CNRS 7534), Université Paris-DauphineParisFrance
  2. 2.Department of MathematicsThe University of ChicagoChicagoUSA

Personalised recommendations