A comparison estimate for singular $p$-Laplace equations and its consequences

TL;DR

This paper establishes a new comparison estimate for singular p-Laplace equations with measure data in the range 1<p<3/2, enabling improved regularity results and potential bounds for solutions and derivatives.

Contribution

It provides the first comparison estimate in the strongly singular case for p-Laplace equations, extending the range of p and enabling new applications in regularity theory.

Findings

Proves comparison estimates for 1<p<3/2 p-Laplace equations with measure data.

Derives pointwise bounds for solutions and derivatives using Wolff's and Riesz's potentials.

Obtains global estimates for bounded domains, applicable to Riccati type equations.

Abstract

Comparison estimates are an important technical device in the study of regularity problems for quasilinear possibly degenerate elliptic and parabolic equations. Such tools have been employed indispensably in many papers of Mingione, Duzaar-Mingione, and Kuusi-Mingione, etc. on certain measure datum problems to obtain pointwise bounds for solutions and their full or fractional derivatives in terms of appropriate linear or nonlinear potentials. However, a comparison estimate for $p$-Laplace type elliptic equations with measure data is still unavailable in the strongly singular case $1< p\leq \frac{3n-2}{2n-1}$, where $n\geq 2$ is the dimension of the ambient space. This issue will be completely resolved in this work by proving a comparison estimate in a slightly larger range $1<p<3/2$. Applications include a `sublinear' Poincar\'e type inequality, pointwise bounds for solutions and their derivatives by Wolff's and Riesz's potentials, respectively. Some global pointwise and weighted estimates are also obtained for bounded domains, which enable us to treat a quasilinear Riccati type equation with possibly sublinear growth in the gradient.