Geodesic
On the weak Harnack inequality for quasilinear elliptic equations
Sbornik. Mathematics, Tome 53 (1986) no. 2, pp. 335-349 Cet article a éte moissonné depuis la source Math-Net.Ru

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A generalization of Harnack's inequality is given for solutions of the differential inequality \begin{equation} |Lu|\leqslant K_1|\nabla u|^{1+\alpha}+K_2, \end{equation} in which $L$ is a uniformly elliptic operator with measurable and bounded coefficients, $K_1$ and $K_2$ are fixed positive constants, and $\alpha$, $0<\alpha<1$, is some number. It is shown that there exist $\alpha_0$, $0<\alpha_0<1$, depending on the ellipticity constant and the dimension of the space, and $M_0>1$, depending on the ellipticity constant, the dimension of the space and the numbers $K_1$, $K_2$ and $\alpha$, such that for solutions $u$ of inequality (1) with $\alpha<\alpha_0$ which are positive in the ball of radius $R$ with center at the origin, and such that $u(0)=M>M_0$, Harnack's inequality holds if $R$ is commensurate with $M^{-\alpha/(1-\alpha_0)}$ with the constant in Harnack's inequality depending only on the dimension of the space and the ellipticity constant. Bibliography: 9 titles. 
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     title = {On the weak {Harnack} inequality for quasilinear elliptic equations},
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L. V. Davydova. On the weak Harnack inequality for quasilinear elliptic equations. Sbornik. Mathematics, Tome 53 (1986) no. 2, pp. 335-349. http://geodesic.mathdoc.fr/item/SM_1986_53_2_a2/

[1] Aleksandrov A. D., “Mazhorirovanie reshenii lineinykh uravnenii vtorogo poryadka”, Vestn. LGU. Ser. matem., mekh., astron., 1966, no. 1, vyp. 1, 5–25 | Zbl

[2] Landis E. M., Uravneniya vtorogo poryadka ellipticheskogo i parabolicheskogo tipov, Nauka, M., 1971 | MR

[3] Krylov N. V., Safonov M. V., “Otsenka veroyatnosti popadaniya diffuzionnogo protsessa v mnozhestve polozhitelnoi mery”, DAN SSSR, 245:1 (1979), 18–20 | MR

[4] Krylov N. V., Safonov M. V., “Nekotoroe svoistvo reshenii parabolicheskikh uravnenii s izmerimymi koeffitsientami”, Izv. AN SSSR. Ser. matem., 44:1 (1980), 161–175 | MR | Zbl

[5] Safonov M. V., “Neravenstvo Kharnaka dlya ellipticheskikh uravnenii i gelderovost ikh reshenii”, Zap. nauchn. sem. Leningr. otd. MIAN, 96 (1980), 272–287 | MR | Zbl

[6] Trudinger N. S., “Local estimates for subsolutions and supersolutions of general second order elliptic quasilinear equations”, Invent. Math., 61:1 (1980), 67–79 | DOI | MR | Zbl

[7] Ladyzhenskaya O. A., Ural'ceva N. N., Estimates of Holder constants for bounded solutions of second order quasilinear parabolic equations of nondivergent form, Preprint, E-ll-81, LOMI, Leningrad, 1981

[8] Landis E. M., “O slabom neravenstve Kharnaka dlya pochti lineinykh ellipticheskikh uravnenii”, Tr. Tbil. un-ta. Matem., mekh., astron., 232–233:13–14 (1982), 159–182 | MR

[9] Novruzov A. A., “Ob odnom podkhode k issledovaniyu kachestvennykh svoistv reshenii nedivergentnykh ellipticheskikh uravnenii vtorogo poryadka”, Matem. sb., 122(164) (1983), 360–387 | MR | Zbl