Local gradient estimates for degenerate elliptic equations

This paper is focused on the local interior $W^{1,\mathrm{\infty }}$-regularity for weak solutions of degenerate elliptic equations of the form $\text{div}[\mathbf{a}(x,u,\mathrm{\nabla }u)]+b(x,u,\mathrm{\nabla }u)=0$, which include those of $p$-Laplacian type. We derive an explicit estimate of the local $L^{\mathrm{\infty }}$-norm for the solution’s gradient in terms of its local $L^p$-norm. Specifically, we prove \begin{equation*} |\nabla u|{L^\infty(B{\frac{R}{2}}(x_0))}^p \leq \frac{C}{|B_R(x_0)|}\int_{B_R(x_0)}|\nabla u(x)|^p dx. \end{equation*} This estimate paves the way for our forthcoming work in establishing $W^{1,q}$-estimates (for $q>p$) for weak solutions to a much larger class of quasilinear elliptic equations.