Global Calder\'{o}n--Zygmund theory for parabolic $p$-Laplacian system: the case $1<p\leq \frac{2n}{n+2}$

TL;DR

This paper extends the global Calderón-Zygmund theory to the parabolic p-Laplacian system for the case 1<p≤2n/(n+2), including systems with discontinuous coefficients with small BMO norm.

Contribution

It establishes the Calderón-Zygmund estimate for the parabolic p-Laplacian system in the previously unresolved case 1<p≤2n/(n+2), including systems with discontinuous coefficients.

Findings

Proves gradient estimates for p-Laplacian systems with p in the critical range.

Extends Calderón-Zygmund theory to systems with small BMO coefficients.

Provides a unified approach covering all p>1.

Abstract

The aim of this paper is to establish global Calder\'{o}n--Zygmund theory to parabolic $p$-Laplacian system: $$ u_t -\operatorname{div}(|\nabla u|^{p-2}\nabla u) = \operatorname{div} (|F|^{p-2}F)~\text{in}~\Omega\times (0,T)\subset \mathbb{R}^{n+1}, $$ proving that $$F\in L^q\Rightarrow \nabla u\in L^q,$$ for any $q>\max\{p,\frac{n(2-p)}{2}\}$ and $p>1$. Acerbi and Mingione \cite{Acerbi07} proved this estimate in the case $p>\frac{2n}{n+2}$. In this article we settle the case $1<p\leq \frac{2n}{n+2}$. We also treat systems with discontinuous coefficients having small BMO (bounded mean oscillation) norm.