Israel Journal of Mathematics

, Volume 36, Issue 3–4, pp 225–247

On solving certain nonlinear partial differential equations by accretive operator methods

  • Lawrence C. Evans
Article

Abstract

We use similar functional analytic methods to solve (a) a fully nonlinear second order elliptic equation, (b) a Hamilton-Jacobi equation, and (c) a functional/partial differential equation from plasma physics. The technique in each case is to approximate by the solutions of simpler problems, and then to pass to limits using a modification of G. Minty’s device to the spaceL.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. Barbu,Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leyden, 1976.MATHGoogle Scholar
  2. 2.
    S. H. Benton, Jr.,The Hamilton Jacobi Equation: A Global Approach, Academic Press, New York, 1977.MATHGoogle Scholar
  3. 3.
    J.-M. Bony,Principe du maximum dans les espaces de Sobolev, C. R. Acad. Sci. Paris265 (1967), 333–336.MATHMathSciNetGoogle Scholar
  4. 4.
    F. Browder,Variational boundary value problems for quasilinear elliptic equations of arbitrary order, Proc. Nat. Acad. Sci. U.S.A.50 (1963), 31–37.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    C. Burch,A semigroup treatment of the Hamilton-Jacobi equation in several space variables, J. Differential Equations23 (1977), 107–124.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    M. G. Crandall,An introduction to evolution governed by accretive operators, Proceedings of the International Symposium on Dynamical Systems, Brown University, Providence, R. I., August 1974.Google Scholar
  7. 7.
    L. C. Evans,A convergence theorem for solutions of nonlinear second order elliptic equations, Indiana Univ. Math. J.27 (1978), 875–887.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    L. C. Evans,Application of nonlinear semigroup theory to certain partial differential equations, inNonlinear Evolution Equations (M. G. Crandall, ed.), Academic Press, New York, 1978.Google Scholar
  9. 9.
    L. C. Evans and A. Friedman,Optimal stochastic switching and the Dirichlet problem for the Bellman equation, Trans. Amer. Math. Soc.253 (1979), 365–389.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    W. H. Fleming,The Cauchy problem for degenerate parabolic equations, J. Math. Mech.13 (1964), 987–1008.MATHMathSciNetGoogle Scholar
  11. 11.
    W. H. Fleming and R. W. Rishel,Deterministic and Stochastic Optimal Control, Springer, New York, 1975.MATHGoogle Scholar
  12. 12.
    A. Friedman,Differential Games, Wiley-Interscience, New York, 1971.MATHGoogle Scholar
  13. 13.
    A. Friedman,The Cauchy problem for first order partial differential equations, Indiana Univ. Math. J.23 (1973), 27–40.MATHCrossRefGoogle Scholar
  14. 14.
    H. Grad, P. N. Hu and D. C. Stevens,Proc. Nat. Acad. Sci. U.S.A. 72 (1975), 3789–3793.CrossRefGoogle Scholar
  15. 15.
    S. N. Kruzkov,First order quasilinear equations in several independent variables, Math. USSR-Sb.10 (1970), 217–243.CrossRefGoogle Scholar
  16. 16.
    J. L. Lions,Quelques méthods de résolution des problèmes aux limites non-linéaires, Dunod, Paris, 1969.Google Scholar
  17. 17.
    P. L. Lions,Résolution des problèmes de Bellman-Dirichlet: une méthode analytique I, to appear.Google Scholar
  18. 18.
    G. J. Minty,Monotone (non-linear) operators in Hilbert space, Duke Math. J.29 (1962), 341–346.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    J. Mossino,Sur certaines inéquations quasi-variationnelles apparaissant en physique, C. R. Acad. Sci. Paris282 (1976), 187–190.MATHMathSciNetGoogle Scholar
  20. 20.
    J. Mossino,Application des inéquations quasi-variationelles a quelques problèmes non linéaires de la physique des plasmas, Israel J. Math.30, (1978), 14–50.MATHMathSciNetGoogle Scholar
  21. 21.
    J. Mossino and J.-P. Zolesio,Solution variationnelle d’un problème non linéaire de la physique des plasmas, C. R. Acad. Sci. Paris285 (1977), 1033–1036.MATHMathSciNetGoogle Scholar
  22. 22.
    K. Sato,On the generators of non-negative contraction semi-groups in Banach lattices, J. Math. Soc. Japan20 (1968), 423–436.MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    E. Sinestrari,Accretive differential operators, Boll. Un. Mat. Ital.13 (1976), 19–31.MATHMathSciNetGoogle Scholar
  24. 24.
    M. B. Tamburro,The evolution operator solution of the Cauchy problem for the Hamilton-Jacobi equation, Israel J. Math.26 (1977), 232–264.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Hebrew University 1980

Authors and Affiliations

  • Lawrence C. Evans
    • 1
  1. 1.Department of MathematicsUniversity of KentuckyLexingtonUSA

Personalised recommendations