Pointwise gradient estimates for a class of singular quasilinear equation with measure data

TL;DR

This paper derives local and global pointwise gradient estimates for solutions to a class of singular quasilinear elliptic equations with measure data, extending previous results to more singular cases.

Contribution

It provides new gradient estimates for solutions to singular quasilinear equations with measure data, broadening the applicability of existing theories.

Findings

Established gradient bounds for solutions in singular regimes

Extended previous results to the case where (3n-2)/(2n-1)<p≤2−1/n

Applicable to nonsmooth domains with measure data

Abstract

Local and global pointwise gradient estimates are obtained for solutions to the quasilinear elliptic equation with measure data $-\operatorname{div}(A(x,\nabla u))=\mu$ in a bounded and possibly nonsmooth domain $\Omega$ in $\mathbb{R}^n$. Here $\operatorname{div}(A(x,\nabla u))$ is modeled after the $p$-Laplacian. Our results extend earlier known results to the singular case in which $\frac{3n-2}{2n-1}<p\leq 2-\frac{1}{n}$.