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

TL;DR

The paper proves global higher integrability of the gradient for parabolic systems with $p$-growth in noncylindrical domains, under the condition $\frac{2(n+1)}{n+2} < p < \infty$ and suitable regularity and time-variation controls on the domain. The authors develop a boundary-adapted energy estimate and three-regime intrinsic reverse Hölder inequalities (lateral boundary, initial boundary, and interior) and combine them with a stopping-time argument in intrinsic cylinders to upgrade $|Du|^p$ to $|Du|^{p+\varepsilon}$. Under uniform $p$-fatness of the domain complement and two-sided growth in time, along with compatible boundary data and a source term $F$, they show $Du\in L^{p+\varepsilon}(E)$ for some $\varepsilon>0$, with explicit, data-dependent bounds. This extends global regularity results from cylindrical to noncylindrical domains and even covers the case $p=2$, providing a robust framework for parabolic $p$-Laplace systems in moving domains with optimal boundary-geometric considerations.

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 $p=2$.

Theorems & Definitions (27)

• Theorem 1.1
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Remark 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7
• Theorem 2.8
• Lemma 2.9