Interior gradient estimates for quasilinear elliptic equations

Abstract

We study quasilinear elliptic equations of the form $\text{div} \mathbf{A}(x,u,\nabla u) = \text{div}\mathbf{F} $ in bounded domains in $\mathbb{R}^n$, $n\geq 1$. The vector field $\mathbf{A}$ is allowed to be discontinuous in $x$, Lipschitz continuous in $u$ and its growth in the gradient variable is like the $p$-Laplace operator with $1<p<\infty$. We establish interior $W^{1,q}$-estimates for locally bounded weak solutions to the equations for every $q>p$, and we show that similar results also hold true in the setting of {\it Orlicz} spaces. Our regularity estimates extend results which are only known for the case $\mathbf{A}$ is independent of $u$ and they complement the well-known interior $C^{1,\alpha}$- estimates obtained by DiBenedetto \cite{D} and Tolksdorf \cite{T} for general quasilinear elliptic equations.