Good-$\lambda$ and Muckenhoupt-Wheeden type bounds in quasilinear measure datum problems, with applications

TL;DR

This paper establishes weighted good-$\lambda$ inequalities and Muckenhoupt-Wheeden bounds for gradients of solutions to quasilinear elliptic equations with measure data, including the singular case, leading to new existence and removable singularity results.

Contribution

It introduces novel global bounds for quasilinear elliptic equations with measure data, especially addressing the singular $p$-Laplacian case, and applies these to solve Riccati type equations.

Findings

Established weighted inequalities for gradients of solutions.

Derived sharp existence criteria for solutions.

Provided bounds on removable singular sets.

Abstract

Weighted good-$\lambda$ type inequalities and Muckenhoupt-Wheeden type bounds are obtained for gradients of solutions to a class of quasilinear elliptic equations with measure data. Such results are obtained globally over sufficiently flat domains in $\mathbb{R}^n$ in the sense of Reifenberg. The principal operator here is modeled after the $p$-Laplacian, where for the first time singular case $\frac{3n-2}{2n-1}<p\leq 2-\frac{1}{n}$ is considered. Those bounds lead to useful compactness criteria for solution sets of quasilinear elliptic equations with measure data. As an application, sharp existence results and sharp bounds on the size of removable singular sets are deduced for a quasilinear Riccati type equation having a gradient source term with linear or super-linear power growth.