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Estimates for Solutions of Fully Nonlinear Discrete Schemes

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Book cover Trends in Partial Differential Equations of Mathematical Physics

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 61))

Abstract

We describe some estimates for solutions of nonlinear discrete schemes, which are analogues of fundamental estimates of Krylov and Safonov for linear elliptic partial differential equations and the resultant Schauder estimates for nonlinear elliptic equations of Evans, Krylov and Safonov.

The first author was supported by the Taiwan National Science Council; the second author was supported by an Australian Research Council Grant.

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References

  1. L.A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, Amer. Math. Soc., 1995.

    Google Scholar 

  2. L.C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math., 35 (1982), 333–363.

    Article  MATH  MathSciNet  Google Scholar 

  3. D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, 2nd edition, Springer Verlag, 1983.

    Google Scholar 

  4. D. Holtby, Higher-order estimates for fully nonlinear difference equations I, Proc. Edinb. Math. Soc. 43 (2000), 485–510.

    Article  MATH  MathSciNet  Google Scholar 

  5. _____, Higher-order estimates for fully nonlinear difference equations II, Proc. Edinb. Math. Soc. 44 (2001), 87–102.

    Article  MATH  MathSciNet  Google Scholar 

  6. J.S. Hwang, Rational approximation to orthogonal bases and of the solutions of elliptic equaitons, Numer. Math. 81 (1999), 561–575.

    Article  MATH  MathSciNet  Google Scholar 

  7. M. Kocan, Approximation of viscosity solutions of elliptic partial differential equations on minimal grids Numer. Math. 72 (1995), 73–92.

    Article  MATH  MathSciNet  Google Scholar 

  8. N.V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations, English translation: Math. USSR Izv., 20 (1983), 459–492.

    Article  Google Scholar 

  9. N.V. Krylov and M. Safonov, A property of the solutions of parabolic equations with measurable coeffcients, Izv. Akad. Nauk SSSR Ser. mat., 44 (1980), 161–175 (in Russian); English translation in Math. USSR Izv. 16 (1981), 151–164.

    MATH  MathSciNet  Google Scholar 

  10. T. Kunkle, Lagrange interpolation on a lattice: bounding derivatives by divided differences, J. Approx. Theory, 71 (1992), 94–103.

    Article  MATH  MathSciNet  Google Scholar 

  11. H.J. Kuo and N.S. Trudinger, Linear elliptic difference inequalities with random coefficients, Math. Comp. 55 (1990), 37–53.

    Article  MATH  MathSciNet  Google Scholar 

  12. _____, Discrete methods for fully nonlinear elliptic equations, SIAM J. Numer. Anal. 29 (1992), 123–135.

    Article  MATH  MathSciNet  Google Scholar 

  13. _____, Maximum principles for difference operators, Topics in Partial Differential Equations & Applications: Collected papers in Honor of Carlo Pucci, Lecture Notes in Pure and Applied Mathematics Series/177, 209–219, Marcel Dekker, Inc.

    Google Scholar 

  14. _____, Positive difference operators on general meshes, Duke Math. J. 83 (1996), 415–433.

    Article  MATH  MathSciNet  Google Scholar 

  15. _____, Evolving monotone difference operators on general space-time meshes, Duke Mathematical Journal, Vol. 91, (1998), 587–607.

    Article  MATH  MathSciNet  Google Scholar 

  16. _____, A note on the discrete Aleksandrov-Bakelman maximum principle, Taiwanese Journal of Mathematics, 4 (2000), 55–67.

    MATH  MathSciNet  Google Scholar 

  17. _____, Schauder estimates for fully nonlinear elliptic difference operators, Proc. Roy. Soc. Edin. 132A, 1395–1406 (2002).

    MathSciNet  Google Scholar 

  18. T.S. Motzkin and W. Wasow, On the approximation of linear elliptic differential equations by difference equations with positive coefficients, J. Math. Phys. 31 (1953), 253–259.

    MATH  MathSciNet  Google Scholar 

  19. M.V. Safonov, On the classical solution of nonlinear elliptic equations of second order, Izv. Akad. Nauk SSSR, Ser. Mat. 52 (1988), 1272–1287. (Engish transl.: Math. USSR Izvestiya) 33 (1989), 597–612.

    Google Scholar 

  20. M.V. Safonov, Nonlinear elliptic equations of second order, Lecture Notes Univ. di Firenze, 1991.

    Google Scholar 

  21. V. Thomée, Discrete interior Schauder estimates for elliptic difference operators, SIAM J. Numer. Anal. 5 (1968), 626–645.

    Article  MATH  MathSciNet  Google Scholar 

  22. N.S. Trudinger, Local estimates for subsolutions and supersolutions of general second order elliptic quasilinear equations, Inventiones Mathematicae, 61 (1980), 67–79.

    Article  MATH  MathSciNet  Google Scholar 

  23. N.S. Trudinger, Regularity of solutions of fully nonlinear elliptic equations, Boll. UMI, 3 (1984), 421–430.

    MATH  MathSciNet  Google Scholar 

  24. N.S. Trudinger, Lectures on nonlinear elliptic equations of second order, Lectures in Mathematical Sciences, 9, Univ. Tokyo, 1995.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Applied Mathematics, National Chung-Hsing University, Taichung, 402, Taiwan

    Hung-Ju Kuo

  2. Centre for Mathematics and its Applications, Australian National University, Canberra, ACT, 0200, Australia

    Neil S. Trudinger

Authors
  1. Hung-Ju Kuo
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  2. Neil S. Trudinger
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Editor information

Editors and Affiliations

  1. Universidade de Lisboa/CMAF, Av. Prof. Gama Pinto, 2, 1649-003, Lisboa, Portugal

    José Francisco Rodrigues

  2. Steklov Institute of Mathematics, Russian Academy of Science, 27, Fontanka, St. Petersburg, 191023, Russia

    Gregory Seregin

  3. Departamento de Matemática, Universidade de Coimbra/FCT, Apartado 3008, 3001-454, Coimbra, Portugal

    José Miguel Urbano

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© 2005 Birkhäuser Verlag Basel/Switzerland

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Kuo, HJ., Trudinger, N.S. (2005). Estimates for Solutions of Fully Nonlinear Discrete Schemes. In: Rodrigues, J.F., Seregin, G., Urbano, J.M. (eds) Trends in Partial Differential Equations of Mathematical Physics. Progress in Nonlinear Differential Equations and Their Applications, vol 61. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7317-2_20

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