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Hölder gradient estimates for fully nonlinear elliptic equations. (English) Zbl 0653.35026

The author proves Hölder estimates for the gradient of Lipschitz continuous viscosity solutions of fully nonlinear elliptic equations F(x,u,Du,D \(2u)=0\), thereby generalizing some of his earlier results for classical solutions [see the author, Trans. Am. Math. Soc. 278, 751-769 (1983; Zbl 0518.35036)]. As a special case the results are applicable to Isaac’s equation from stochastic game theory \(F\equiv \inf_{\alpha} \sup_{\beta}(L_{\alpha \beta}u-f_{\alpha \beta})\), where \(L_{\alpha \beta}\) is a family of second order linear operators.
Reviewer: M.Wiegner

MSC:

35J60 Nonlinear elliptic equations
35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs

Citations:

Zbl 0518.35036
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References:

[1] Gilbarg, Elliptic partial differential equations of second order (1983) · Zbl 0562.35001 · doi:10.1007/978-3-642-61798-0
[2] Krylov, Izv. Akad. Nauk SSSR Ser. Mat. 47 pp 75– (1983)
[3] Trudinger, Nat. Univ. Centre for Math. Anal. Research Report 45 (1987)
[4] DOI: 10.1090/S0002-9939-1985-0796449-X · doi:10.1090/S0002-9939-1985-0796449-X
[5] Trudinger, Proc. Centre Math. Anal. Aust. Nat. Univ. 8 pp 65– (1984)
[6] DOI: 10.1090/S0002-9947-1983-0701522-0 · doi:10.1090/S0002-9947-1983-0701522-0
[7] DOI: 10.1215/S0012-7094-87-05521-9 · Zbl 0697.35030 · doi:10.1215/S0012-7094-87-05521-9
[8] DOI: 10.1007/BF01389895 · Zbl 0453.35028 · doi:10.1007/BF01389895
[9] DOI: 10.1080/03605308308820301 · Zbl 0716.49023 · doi:10.1080/03605308308820301
[10] DOI: 10.1007/BF01229801 · Zbl 0616.35028 · doi:10.1007/BF01229801
[11] Krylov, Izv. Akad. Nauk. SSSR Ser. Mat. 40 pp 161– (1980)
[12] DOI: 10.1080/03605308608820422 · Zbl 0589.35036 · doi:10.1080/03605308608820422
[13] Krylov, Dokl. Akad. Nauk UzSSR 274 pp 21– (1983)
[14] Ladyzhenskaya, Linear and quasilinear elliptic equations (1964) · Zbl 0143.33602
[15] Trudinger, Nat. Univ. Centre for Math. Anal. Research Report 46 (1987)
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