Morrey space methods in the theory of elliptic difference equations

  • Jens Frehse
Conference paper
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Part of the Lecture Notes in Mathematics book series (LNM, volume 333)
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References

  1. [1]
    BRAMBLE, J.H.; KELLOGG, R.B.; THOMEE, V.: On the rate of convergence of some difference schemes for second order elliptic equations. BIT 8 (1968), 154–173.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    BRAMBLE, J.H.; HUBBARD, B.E., THOMEE, V.: Convergence estimates for essentially positive type discrete ptoblems. Math. Comp. 23 (1969), 695–710.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    CAMPANATO, S.: Proprietà di inclusione per spazi di Morrey. Ricerche di Mat.; 12 (1963), 67–86.MathSciNetMATHGoogle Scholar
  4. [4]
    CIARLET, P.G.: Discrete maximum principle for finite-difference operators. Aequationes Math. 4 (1970), 338–352.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    CIARLET, P.G.; RAVIART, P.-A.: Maximum principle and uniform convergence for the finite element method. To appear.Google Scholar
  6. [6]
    COLLATZ, L.: Bemerkungen zur Fehlerabschätzung für das Differenzenverfahren bei partiellen Differentialgleichungen. Zeitschr. Angew.Math.Mech. 13 (1933), 56–57.CrossRefMATHGoogle Scholar
  7. [7]
    CONTI, F.: Su alcuni spazi funzionale e loro applicazioni ad equazioni differenziale di tipo ellittico. Boll. Un. Mat. Ital. 4 (1969), 554–569.MATHGoogle Scholar
  8. [8]
    FORSYTHE, G.E.; WASOW, W.: Finite difference methods for partial differential equations. New York: Wiley 1960.MATHGoogle Scholar
  9. [9]
    FREHSE, J.: Seminar in Jülich, 1970.Google Scholar
  10. [10]
    FREHSE, J.: Existenz und Konvergenz von Lösungen nichtlinearer elliptischer Differenzengleichungen unter Dirichletrandbedingungen, Math. Z. 109 (1969), 311–343.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    FREHSE, J.: Über die Konvergenz von Differenzen-und anderen Näherungsverfahren bei nichtlinearen Variationsproblemen. Intern. Schriftenreihe z. Numerischen Math.15(1970), 29–44.MathSciNetMATHGoogle Scholar
  12. [12]
    FREHSE, J.: Über pseudomonotone nichtlineare Differentialoperatoren. Anhang zur Habilitationsschrift 1970.Google Scholar
  13. [13]
    GERSCHGORIN, S.: Fehlerabschätzung für das Differenzenverfahren zur Lösung partieller Differentialgleichungen. Zeitschr. Angew. Math. Mech. 10 (1930), 373–382.CrossRefMATHGoogle Scholar
  14. [14]
    GREENSPAN, D.: Introductory numerical analysis of elliptic boundary value problems. New York: Harper and Row, 1965.MATHGoogle Scholar
  15. [15]
    GRIGORIEFF, R.D.: Über die Koerzivität linearer elliptischer Differenzenoperatoren unter allgemeinen Randbedingungen. Frankfurt a.M.: Dissertation 1967.Google Scholar
  16. [16]
    HAINER, K.: Differenzenapproximation elliptischer Differentialgleichungen zweiter Ordnung mit Hilfe des Maximumprinzips und eine allgemeine Konvergenztheorie. Frankfurt a.M.: Dissertation 1968.Google Scholar
  17. [17]
    LERAY, J.; LIONS, J.L.: Quelques résultats de Visik sur les problèmes elliptiques non linéaires par la méthode de Minty-Browder, Bull. Soc. Math. France 93 (1965) 97–107.MathSciNetMATHGoogle Scholar
  18. [18]
    McALLISTER, G.T.: Quasilinear uniformly elliptic partial differential equations and difference equations, J. SIAM Numer. Anal. 3 (1966) 13–33.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    McALLISTER, G.T.: Dirichlet problems for mildly nonlinear elliptic difference equations. Journ. Math. Anal. Appl. 27 (1969), 338–366.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    MORREY, C. B.: Multiple integral problems in the calculus of variations and related topics, Univ. California, Publ.Math., New.Ser., Vol.1(1943), 1–30.MathSciNetMATHGoogle Scholar
  21. [21]
    MORREY, C.B.: Multiple integrals in the calculus of variations, Springer, Berlin-Heidelberg-New York, 1966.MATHGoogle Scholar
  22. [22]
    MOSER, J.: A new proof of de Giorgi's theorem concerning the regularity problem for elliptic differential equations, Comm. Pure and Appl. Math. 13 (1960), 457–468.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    NIRENBERG, L.: On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, Ser, 3, 13 (1959), 115–162.MathSciNetMATHGoogle Scholar
  24. [24]
    NITSCHE, J.; NITSCHE, J.C.C.: Error estimates for the numerical solution of elliptic differential equations. Arch. Rational Mech. Anal. 5 (1960), 293–306.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    PERRON, O.: Eine neue Behandlung der ersten Randwertaufgabe für Δu=o. Math. Zeitschr. 18 (1923), 42–54.MathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    SOBOLEV, S.L.: On estimates for certain sums for functions defined on a grid, Izv. Akad. Nauk. SSSR, Ser.Math. 4 (1940), 5–16. (Russian).Google Scholar
  27. [27]
    STAMPACCHIA, G.: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965), 189–258.MathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    STAMPACCHIA, G.: The spaces Lp,λ, Np,λ and interpolation. Ann. Scuola Norm. Sup. Pisa 19 (1965), 443–462.MathSciNetGoogle Scholar
  29. [29]
    STEPLEMAN, R.: Finite dimensional analogues of variational and quasilinear elliptic Dirichlet problems. Ph. D. Diss. Univ. Maryland 1969.Google Scholar
  30. [30]
    STUMMEL, F.: Elliptische Differenzenoperatoren unter Dirichletrandbedingungen, Math. Zeitschr. 97 (1967), 169–211.MathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    THOMEE, V.; WESTERGREN, B.: Elliptic difference equations and interior regularity, Num. Math. 11 (1968), 196–210.MathSciNetCrossRefMATHGoogle Scholar
  32. [32]
    THOMEE, V.: Some convergence results in elliptic difference equations. Proc. Roy. Soc. Lond. A. 323 (1971), 191–199.MathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    THOMEE, V.: Discrete interior Schauder estimates for elliptic difference operators, SIAM J. Numer. Anal. 5(1968), 626–645.MathSciNetCrossRefMATHGoogle Scholar
  34. [34]
    TÖRNIG, W., MEIS, Th.: Diskretisierung des Dirichletproblems nichtlinearer elliptischer Differentialgleichungen. Erscheint demnächst.Google Scholar
  35. [35]
    VOIGTMANN, K.: Dissertation, Frankfurt a.M. 1971.Google Scholar
  36. [36]
    WIDMAN, K. O.: Hölder continuity of solutions of elliptic systems. To appear in Manuscripta MathGoogle Scholar

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© Springer-Verlag 1973

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  • Jens Frehse

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