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

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.