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 ${\mathbb{R}}^n$, $n\ge 1$. 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.

References in corpus (1)

• Local gradient estimates for degenerate elliptic equations