Global Calderón--Zygmund theory for parabolic $p$-Laplacian system: the case $1<p\le \frac{2n}{n+2}$

Abstract

The aim of this paper is to establish global Calderón--Zygmund theory to parabolic $p$-Laplacian system: $$u_t-\mathrm{div}(|\nabla u{|}^{p-2}\nabla u)=\mathrm{div}(|F{|}^{p-2}F)\text{in}\Omega \times (0,T)\subset {\mathbb{R}}^{n+1},$$ 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 $1<p\le \frac{2n}{n+2}$. We also treat systems with discontinuous coefficients having small BMO (bounded mean oscillation) norm.

We made a minor correction in the proof of Lemma 2.1