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On solving certain nonlinear partial differential equations by accretive operator methods

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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 .

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References

  1. V. Barbu,Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leyden, 1976.

    MATH  Google Scholar 

  2. S. H. Benton, Jr.,The Hamilton Jacobi Equation: A Global Approach, Academic Press, New York, 1977.

    MATH  Google Scholar 

  3. J.-M. Bony,Principe du maximum dans les espaces de Sobolev, C. R. Acad. Sci. Paris265 (1967), 333–336.

    MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  5. C. Burch,A semigroup treatment of the Hamilton-Jacobi equation in several space variables, J. Differential Equations23 (1977), 107–124.

    Article  MATH  MathSciNet  Google Scholar 

  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.

  7. L. C. Evans,A convergence theorem for solutions of nonlinear second order elliptic equations, Indiana Univ. Math. J.27 (1978), 875–887.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  10. W. H. Fleming,The Cauchy problem for degenerate parabolic equations, J. Math. Mech.13 (1964), 987–1008.

    MATH  MathSciNet  Google Scholar 

  11. W. H. Fleming and R. W. Rishel,Deterministic and Stochastic Optimal Control, Springer, New York, 1975.

    MATH  Google Scholar 

  12. A. Friedman,Differential Games, Wiley-Interscience, New York, 1971.

    MATH  Google Scholar 

  13. A. Friedman,The Cauchy problem for first order partial differential equations, Indiana Univ. Math. J.23 (1973), 27–40.

    Article  MATH  Google Scholar 

  14. H. Grad, P. N. Hu and D. C. Stevens,Proc. Nat. Acad. Sci. U.S.A. 72 (1975), 3789–3793.

    Article  Google Scholar 

  15. S. N. Kruzkov,First order quasilinear equations in several independent variables, Math. USSR-Sb.10 (1970), 217–243.

    Article  Google Scholar 

  16. J. L. Lions,Quelques méthods de résolution des problèmes aux limites non-linéaires, Dunod, Paris, 1969.

    Google Scholar 

  17. P. L. Lions,Résolution des problèmes de Bellman-Dirichlet: une méthode analytique I, to appear.

  18. G. J. Minty,Monotone (non-linear) operators in Hilbert space, Duke Math. J.29 (1962), 341–346.

    Article  MATH  MathSciNet  Google Scholar 

  19. J. Mossino,Sur certaines inéquations quasi-variationnelles apparaissant en physique, C. R. Acad. Sci. Paris282 (1976), 187–190.

    MATH  MathSciNet  Google Scholar 

  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.

    MATH  MathSciNet  Google Scholar 

  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.

    MATH  MathSciNet  Google Scholar 

  22. K. Sato,On the generators of non-negative contraction semi-groups in Banach lattices, J. Math. Soc. Japan20 (1968), 423–436.

    Article  MATH  MathSciNet  Google Scholar 

  23. E. Sinestrari,Accretive differential operators, Boll. Un. Mat. Ital.13 (1976), 19–31.

    MATH  MathSciNet  Google Scholar 

  24. M. B. Tamburro,The evolution operator solution of the Cauchy problem for the Hamilton-Jacobi equation, Israel J. Math.26 (1977), 232–264.

    Article  MATH  MathSciNet  Google Scholar 

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Alfred P. Sloan fellow 1979–1981.

Supported in part by NSF grant MCS 77-01952.

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Evans, L.C. On solving certain nonlinear partial differential equations by accretive operator methods. Israel J. Math. 36, 225–247 (1980). https://doi.org/10.1007/BF02762047

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  • DOI: https://doi.org/10.1007/BF02762047

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