Local gradient estimates for degenerate elliptic equations

TL;DR

This paper establishes explicit local gradient bounds for solutions to degenerate elliptic equations, including p-Laplacian types, facilitating future regularity results for broader classes of quasilinear elliptic equations.

Contribution

It provides a new explicit estimate of the local $L^ abla$-norm of solutions' gradients in terms of their $L^p$-norm, advancing regularity theory for degenerate elliptic equations.

Findings

Derived an explicit local $L^ abla$-norm estimate for solutions' gradients.

Established a bound linking the $L^ abla$-norm to the $L^p$-norm.

Facilitated future work on $W^{1,q}$-estimates for quasilinear elliptic equations.

Abstract

This paper is focused on the local interior $W^{1,\infty}$-regularity for weak solutions of degenerate elliptic equations of the form $\text{div}[\mathbf{a}(x,u, \nabla u)] +b(x, u, \nabla u) =0$, which include those of $p$-Laplacian type. We derive an explicit estimate of the local $L^\infty$-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.