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arXiv · 2109.02595

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

Ke Chen·Quoc-Hung Nguyen·Na Zhao
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Abstract

The aim of this paper is to establish global Calderón--Zygmund theory to parabolic $p$-Laplacian system: $$ u_t -\operatorname{div}(|\nabla u|^{p-2}\nabla u) = \operatorname{div} (|F|^{p-2}F)~\text{in}~Ω\times (0,T)\subset \mathbb{R}^{n+1}, $$ 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.

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Ke Chen, Quoc-Hung Nguyen, Na Zhao. 2021-09-13. Global Calderón--Zygmund theory for parabolic $p$-Laplacian system: the case $1<p\leq \frac{2n}{n+2}$. https://arxiv.org/abs/2109.02595

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