Global higher integrability for systems with $p$-growth structure in noncylindrical domains

TL;DR

This paper proves global higher integrability of the gradient for systems with p-growth in noncylindrical domains, extending known results to a broader class of PDEs including the linear case p=2.

Contribution

It establishes the first global higher integrability results for such systems in noncylindrical domains, including the linear case p=2.

Findings

Proves global higher integrability of Du for p>2(n+1)/(n+2).

Results apply to domains with suitable regularity and controlled growth.

Extends known results to the linear case p=2.

Abstract

We consider the Cauchy-Dirichlet problem to systems with $p$-growth structure with $1 < p < \infty$, whose prototype is \begin{equation*} \partial_t u- \operatorname{div} \big( |Du|^{p-2} Du \big) = \operatorname{div} \left( |F|^{p-2} F \right), \end{equation*} in a bounded noncylindrical domain $E \subset \mathbb{R}^{n+1}$. For $p> \frac{2(n+1)}{n+2}$ and domains $E$ that satisfy suitable regularity assumptions and do not grow or shrink too fast, we prove global higher integrability of $Du$. The result is already new in the case $p=2$.