Gradient estimates for singular quasilinear elliptic equations with measure data

TL;DR

This paper establishes gradient estimates in Lebesgue spaces for solutions to singular quasilinear elliptic equations with measure data, expanding understanding of regularity in equations with low integrability conditions.

Contribution

It provides new $L^q$-gradient estimates for solutions to singular quasilinear elliptic equations with measure data, addressing cases with low exponent $p$.

Findings

Proved $L^q$-estimates for gradients of solutions.

Extended regularity results to singular cases with measure data.

Applicable to equations with $p$ in (1, 2 - 1/n].

Abstract

In this paper, we prove $L^q$-estimates for gradients of solutions to singular quasilinear elliptic equations with measure data $$-\operatorname{div}(A(x,\nabla u))=\mu,$$ in a bounded domain $\Omega\subset\mathbb{R}^{N}$, where $A(x,\nabla u)\nabla u \asymp |\nabla u|^p$, $p\in (1,2-\frac{1}{n}]$ and $\mu$ is a Radon measure in $\Omega$