Universal potential estimates for $1<p\leq 2-\frac{1}{n}$

TL;DR

This paper extends universal potential estimates to the singular range of p for quasilinear equations with measure data, broadening the understanding of solutions to these nonlinear PDEs.

Contribution

It generalizes the universal potential estimates to the case 1<p≤2−1/n for quasilinear equations with measure data, including the singular p-Laplacian case.

Findings

Extended potential estimates to the singular p-range.

Applicable to equations with measure data.

Provides new regularity insights for solutions.

Abstract

We extend the so-called universal potential estimates of the Kuusi-Mingione type (J.Funct. Anal. 2012) to the singular case $1<p\leq 2-1/n$ for the quasilinear equation with measure data \begin{equation*} -\operatorname{div}(A(x,\nabla u))=\mu \end{equation*} in a bounded open subset $\Omega$ of $\mathbb{R}^n$, $n\geq 2$, with a finite signed measure $\mu$ in $\Omega$. The operator $\operatorname{div}(A(x,\nabla u))$ is modeled after the $p$-Laplacian $\Delta_p u:= {\rm div}\, (|\nabla u|^{p-2}\nabla u)$, where the nonlinearity $A(x, \xi)$ ($x, \xi \in \mathbb{R}^n$) is assumed to satisfy natural growth and monotonicity conditions of order $p$, as well as certain additional regularity conditions in the $x$-variable.