Lorentz-Morrey global bounds for singular quasilinear elliptic equations with measure data

TL;DR

This paper establishes global gradient bounds in Lorentz-Morrey spaces for solutions to singular quasilinear elliptic equations with measure data, extending previous local estimates to the boundary and broader parameter ranges.

Contribution

It extends local gradient estimates to global bounds in Lorentz-Morrey spaces for singular elliptic equations with measure data, under minimal domain regularity conditions.

Findings

Global Lorentz-Morrey bounds for gradient solutions

Extension of local estimates to boundary results

Applicable to singular p-range with measure data

Abstract

The aim of this paper is to present the global estimate for gradient of renormalized solutions to the following quasilinear elliptic problem: \begin{align*} \begin{cases} -div(A(x,\nabla u)) &= \mu \quad \text{in} \ \ \Omega, \\ u &=0 \quad \text{on} \ \ \partial \Omega, \end{cases} \end{align*} in Lorentz-Morrey spaces, where $\Omega \subset \mathbb{R}^n$ ($n \ge 2$), $\mu$ is a finite Radon measure, $A$ is a monotone Carath\'eodory vector valued function defined on $W^{1,p}_0(\Omega)$ and the $p$-capacity uniform thickness condition is imposed on the complement of our domain $\Omega$. It is remarkable that the local gradient estimates has been proved firstly by G. Mingione in \cite{Mi3} at least for the case $2 \le p \le n$, where the idea for extending such result to global ones was also proposed in the same paper. Later, the global Lorentz-Morrey and Morrey regularities were obtained by N.C.Phuc in \cite{55Ph1} for regular case $p>2 - \frac{1}{n}$. Here in this study, we particularly restrict ourselves to the singular case $\frac{3n-2}{2n-1}<p\le 2-\frac{1}{n}$. The results are central to generalize our technique of good-$\lambda$ type bounds in previous work \cite{MP2018}, where the local gradient estimates of solution to this type of equation was obtained in the Lorentz spaces. Moreover, the proofs of most results in this paper are formulated globally up to the boundary results.