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Linear elliptic difference inequalities with random coefficients. (English) Zbl 0716.39005
Discrete versions of maximum principles, Hölder estimates and Harnack inequalities are obtained for linear elliptic difference operators of positive type. A theorem of T. Motzkin and W. Wasow [J. Math. Phys. 31, 253-259 (1953; Zbl 0050.125)] is then used to obtain new proofs of the corresponding results for differential equations.
Reviewer: L.I.Grimm

39A70 Difference operators
39A12 Discrete version of topics in analysis
39A10 Additive difference equations
Zbl 0050.125
Full Text: DOI
[1] A. D. Aleksandrov, Uniqueness conditions and bounds for the solution of the Dirichlet problem, Vestnik Leningrad. Univ. Ser. Mat. Meh. Astronom. 18 (1963), no. 3, 5 – 29 (Russian, with English summary). · Zbl 0139.05702
[2] I. Ja. Bakel\(^{\prime}\)man, On the theory of quasilinear elliptic equations, Sibirsk. Mat. Ž. 2 (1961), 179 – 186 (Russian).
[3] J. H. Bramble, B. E. Hubbard, and Vidar Thomée, Convergence estimates for essentially positive type discrete Dirichlet problems, Math. Comp. 23 (1969), 695 – 709. · Zbl 0217.21902
[4] J. H. Bramble, R. B. Kellogg, and V. Thomée, On the rate of convergence of some difference schemes for second order elliptic equations, Nordisk Tidskr. Informationsbehandling (BIT) 8 (1968), 154 – 173. · Zbl 0172.20004
[5] A. Brandt, Interior estimates for second-order elliptic differential (or finite-difference) equations via the maximum principle, Israel J. Math. 7 (1969), 95 – 121. · Zbl 0177.37102
[6] K. P. Bube and J. C. Strikwerda, Interior regularity estimates for elliptic systems of difference equations, SIAM J. Numer. Anal. 20 (1983), no. 4, 653 – 670. · Zbl 0561.65070
[7] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. · Zbl 0562.35001
[8] N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 1, 161 – 175, 239 (Russian).
[9] H. J. Kuo and N. S. Trudinger, Discrete methods for fully nonlinear elliptic equations (in preparation). · Zbl 0745.65058
[10] A. B. Merkov, Second-order elliptic equations on graphs, Mat. Sb. (N.S.) 127(169) (1985), no. 4, 502 – 518, 559 – 560 (Russian). · Zbl 0583.35031
[11] T. S. Motzkin and W. Wasow, On the approximation of linear elliptic differential equations by difference equations with positive coefficients, J. Math. Physics 31 (1953), 253 – 259. · Zbl 0050.12501
[12] Aleksey Vasil\(^{\prime}\)yevich Pogorelov, The Minkowski multidimensional problem, V. H. Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons], New York-Toronto-London, 1978. Translated from the Russian by Vladimir Oliker; Introduction by Louis Nirenberg; Scripta Series in Mathematics.
[13] Vidar Thomée, Discrete interior Schauder estimates for elliptic difference operators., SIAM J. Numer. Anal. 5 (1968), 626 – 645. · Zbl 0176.15901
[14] Neil S. Trudinger, Local estimates for subsolutions and supersolutions of general second order elliptic quasilinear equations, Invent. Math. 61 (1980), no. 1, 67 – 79. · Zbl 0453.35028
[15] Neil S. Trudinger, On regularity and existence of viscosity solutions of nonlinear second order, elliptic equations, Partial differential equations and the calculus of variations, Vol. II, Progr. Nonlinear Differential Equations Appl., vol. 2, Birkhäuser Boston, Boston, MA, 1989, pp. 939 – 957. · Zbl 0698.35056
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