Mathematics of Complexity and Dynamical Systems

2011 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Non-linear Partial Differential Equations, Viscosity Solution Method in

  • Shigeaki Koike
Reference work entry

Article Outline


Definition of the Subject



Comparison Principle

Existence Results

Boundary Value Problems

Asymptotic Analysis

Other Notions

Future Directions


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Primary Literature

  1. 1.
    Alvarez O, Hoch P, Le Bouar Y, Monneau R (2006) Dislocation dynamics: short time existence and uniqueness of the solution. Arch Ration Mech Anal 85:371–414Google Scholar
  2. 2.
    Aronsson G, Crandall MG, Juutinen P (2004) A tour of the theory of absolutely minimizing functions. Bull Amer Math Soc 41:439–505MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bardi M, Capuzzo Dolcetta I (1997) Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. Birkhäuser, BostonMATHCrossRefGoogle Scholar
  4. 4.
    Bardi M, Mannuci P (2006) On the Dirichlet problem for non‐totally degenerate fully nonlinear elliptic equations. Comm Pure Appl Anal 5:709–731MATHCrossRefGoogle Scholar
  5. 5.
    Bardi M, Crandall MG, Evans LC, Soner HM, Souganidis PE (1997) Viscosity Solutions and Applications. Springer, BerlinGoogle Scholar
  6. 6.
    Barles G (1993) Discontinuous viscosity solutions of first-order Hamilton–Jacobi equations: A guided visit. Nonlinear Anal 20:1123–1134MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Barles G (1994) Solutions de Viscosité des Équations de Hamilton–Jacobi. Springer, BerlinMATHGoogle Scholar
  8. 8.
    Barles G, Perthame B (1987) Discontinuous solutions of deterministic optimal stopping‐time problems. Model Math Anal Num 21:557–579MathSciNetMATHGoogle Scholar
  9. 9.
    Barles G, Souganidis PE (1998) A new approach to front propagation problems: theory and applications. Arch Ration Mech Anal 141:237–296MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Cabré X, Caffarelli LA (2003) Interior C 2,α regularity theory for a class of nonconvex fully nonlinear elliptic equations. J Math Pures Appl 83:573–612Google Scholar
  11. 11.
    Caffarelli LA (1989) Interior a priori estimates for solutions of fully non‐linear equations. Ann Math 130:180–213MathSciNetCrossRefGoogle Scholar
  12. 12.
    Caffarelli LP, Cabré X (1995) Fully Nonlinear Elliptic Equations. Amer Math Soc, ProvidenceGoogle Scholar
  13. 13.
    Caffarelli LA, Crandall MG, Kocan M, Świȩch A (1996) On viscosity solutions of fully nonlinear equations with measurable ingredients. Comm Pure Appl Math 49:365–397Google Scholar
  14. 14.
    Caffarelli LA, Souganidis PE, Wang L (2005) Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media. Comm Pure Appl Math 58:319–361MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Crandall MG, Lions P-L (1981) Condition d'unicité pour les solutions generalisées des équations de Hamilton–Jacobi du premier ordre. CR Acad Sci Paris Sér I Math 292:183–186MathSciNetMATHGoogle Scholar
  16. 16.
    Crandall MG, Lions P-L (1983) Viscosity solutions of Hamilton–Jacobi equations. Tran Amer Math Soc 277:1–42MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Crandall MG, Evans LC, Lions P-L (1984) Some properties of viscosity solutions of Hamilton–Jacobi equations. Trans Amer Math Soc 282:487–435MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Crandall MG, Ishii H, Lions PL (1992) User's guide to viscosity solutions of second order partial differential equations. Bull Amer Math Soc 27:1–67MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Evans LC (1992) Periodic homogenization of certain fully nonlinear partial differential equations. Proc Roy Soc Edinb 120:245–265MATHCrossRefGoogle Scholar
  20. 20.
    Fleming WH, Soner HM (1993) Controlled Markov Processes and Viscosity Solutions. Springer, BerlinMATHGoogle Scholar
  21. 21.
    Friedman A, Souganidis PE (1986) Blow-up of solutions of Hamilton–Jacobi equations. Comm Partial Differ Equ 11:397–443MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Giga Y (2002) Viscosity solutions with shocks. Comm Pure Appl Math 55:431–480MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Giga Y (2006) Surface Evolutions Equations: A Level Set Approach. Birkäuser, BaselGoogle Scholar
  24. 24.
    Gilbarg D, Trudinger NS (1983) Elliptic Partial Differential Equations of Second Order. Springer, New YorkMATHCrossRefGoogle Scholar
  25. 25.
    Gozzi F, Sritharan SS, Świȩch A (2005) Bellman equations associated to optimal control of stochastic Navier–Stokes equations. Comm Pure Appl Math 58:671–700Google Scholar
  26. 26.
    Gutiérrez CE (2001) The Monge–Ampère Equation. Birkhäuser, BostonGoogle Scholar
  27. 27.
    Ishii H (1987) Perron's method for Hamilton–Jacobi equations. Duke Math J 55:369–384MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Ishii H, Koike S (1993/94) Viscosity solutions of functional‐differential equations. Adv Math Sci Appl 3:191–218Google Scholar
  29. 29.
    Jensen R (1988) The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations. Arch Rat Mech Anal 101:1–27MATHCrossRefGoogle Scholar
  30. 30.
    Jensen R (1993) Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient. Arch Ration Mech Anal 123:51–74MATHCrossRefGoogle Scholar
  31. 31.
    Karatzas I, Shreve SE (1998) Methods of Mathematical Finance. Springer, New YorkMATHGoogle Scholar
  32. 32.
    Koike S (1995) On the state constraint problem for differential games. Indiana Univ Math J 44:467–487MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Koike S, Święch A (2004) Maximum principle for fully nonlinear equations via the iterated comparison function method. Math Ann 339:461–484Google Scholar
  34. 34.
    Lions P-L (1983) Bifurcation and optimal stochastic control. Nonlinear Anal 7:177–207MathSciNetCrossRefGoogle Scholar
  35. 35.
    Lions PL, Souganidis PE (1998) Fully nonlinear stochastic partial differential equations. CR Acad Sci Paris 326:1085–1092; 326:753–741MathSciNetMATHGoogle Scholar
  36. 36.
    Lions PL, Souganidis PE (2000) Uniqueness of weak solutions of fully nonlinear stochastic partial differential equations. CR Acad Sci Paris 331:783–790MathSciNetMATHGoogle Scholar
  37. 37.
    Lions PL, Souganidis PE (2003) Correctors for the homogenization of Hamilton–Jacobi equations in the stationary ergodic setting. Comm Pur Appl Math 56:1501–1524MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Manfredi JJ (2002) A version of the Hopf–Lax formula in the Heisenberg group. Comm Partial Differ Equ 27:1139–1159MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Nadirashvili N, Vlăduţ S (2007) Nonclassical solutions of fully nonlinear elliptic equations. Geom Funct Anal 17:1283–1296Google Scholar
  40. 40.
    Nadirashvili N, Vlăduţ S (2008) Singular viscosity solutions to fully nonlinear elliptic equations. J Math Pures Appl 89(9):107–113Google Scholar
  41. 41.
    Savin O (2005) C 1 regularity for infinity harmonic functions in two dimensions. Arch Ration Mech Anal 176:351–361MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Soner MH (1986) Optimal control with state-space constraint I. SIAM J Control Optim 24:552–562MathSciNetMATHCrossRefGoogle Scholar

Books and Reviews

  1. 43.
    Bellman R (1957) Dynamic Programming. Princeton University Press, PrincetonMATHGoogle Scholar
  2. 44.
    Evans LC, Gangbo W (1999) Differential Equations Methods for the Monge–Kantrovich Mass Transfer Problem. Amer Math Soc, ProvidenceGoogle Scholar
  3. 45.
    Fleming WH, Rishel R (1975) Deterministic and Stochastic Optimal Control. Springer, New YorkMATHCrossRefGoogle Scholar
  4. 46.
    Koike S (2004) A Beginner's Guide to the Theory of Viscosity Solutions. Math Soc Japan, Tokyo. Corrections:
  5. 47.
    Lions P-L (1982) Generalized Solutions of Hamilton–Jacobi Equations. Pitman, BostonMATHGoogle Scholar
  6. 48.
    Maugeri A, Palagachev DK, Softova LG (2000) Elliptic and Parabolic Equations with Discontinuous Coefficients. Wiley, BerlinMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Shigeaki Koike
    • 1
  1. 1.Department of MathematicsSaitama UniversitySaitamaJapan