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Fully nonlinear, uniformly elliptic equations under natural structure conditions. (English) Zbl 0518.35036


MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35B45 A priori estimates in context of PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
49J55 Existence of optimal solutions to problems involving randomness
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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[1] Lawrence C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math. 35 (1982), no. 3, 333 – 363. · Zbl 0469.35022 · doi:10.1002/cpa.3160350303
[2] Lawrence C. Evans, Classical solutions of the Hamilton-Jacobi-Bellman equation for uniformly elliptic operators, Trans. Amer. Math. Soc. 275 (1983), no. 1, 245 – 255.
[3] Lawrence C. Evans and Avner Friedman, Optimal stochastic switching and the Dirichlet problem for the Bellman equation, Trans. Amer. Math. Soc. 253 (1979), 365 – 389. · Zbl 0425.35046
[4] Lawrence C. Evans and Pierre-Louis Lions, Résolution des équations de Hamilton-Jacobi-Bellman pour des opérateurs uniformément elliptiques, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 22, A1049 – A1052 (French, with English summary). · Zbl 0461.49017
[5] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin-New York, 1977. Grundlehren der Mathematischen Wissenschaften, Vol. 224. · Zbl 0361.35003
[6] A. V. Ivanov, A priori estimates for the solutions of nonlinear elliptic second order equations, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 59 (1976), 31 – 59, 255 (Russian, with English summary). Boundary value problems of mathematical physics and related questions in the theory of functions, 9. · Zbl 0338.35010
[7] 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).
[8] O. A. Ladyženskaja and N. N. Ural\(^{\prime}\)ceva, Quasilinear elliptic equations and variational problems in several independent variables, Uspehi Mat. Nauk 16 (1961), no. 1 (97), 19 – 90 (Russian).
[9] Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. · Zbl 0164.13002
[10] -, Hölder estimates for solutions of second order quasilinear elliptic equations in general form, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 96 (1980), 161-168.
[11] P.-L. Lions, Résolution analytique des problèmes de Bellman-Dirichlet, Acta Math. 146 (1981), no. 3-4, 151 – 166 (French). · Zbl 0467.49016 · doi:10.1007/BF02392461
[12] 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
[13] 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 · doi:10.1007/BF01389895
[14] -, Elliptic equations in non-divergence form, Proc. Miniconf. Partial Differential Equations, Canberra, 1981, pp. 1-16.
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