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On the weak Harnack inequality for quasilinear elliptic equations
L. V. Davydova
Abstract:
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.
UDC:
517.95
MSC: Primary
35R45; Secondary
35J60 Received: 03.10.1983
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