Boundary regularity for parabolic systems with nonstandard $(p,q)$-growth conditions in smooth convex domains

TL;DR

This work establishes boundary Lipschitz regularity for local weak solutions to parabolic systems with Uhlenbeck-type, nonstandard $(p,q)$-growth in smooth convex domains. By combining a priori gradient estimates for smooth solutions, a reverse Hölder inequality, and a Moser iteration, the authors obtain an explicit local $L^\infty$-gradient bound up to the lateral boundary when $u$ vanishes on the boundary. An approximation scheme regularizes the coefficients and right-hand side to a standard $q$-growth framework, enabling a robust comparison and a limit passage that transfers the gradient bound to the original problem. The results extend interior regularity to the boundary in the nonstandard parabolic $(p,q)$-growth setting and provide quantitative dependence on the data, domain geometry, and growth parameters, thereby broadening the scope of boundary gradient estimates in parabolic systems.

Abstract

We study the boundary regularity of local weak solutions to nonlinear parabolic systems of the form \begin{equation*} \partial_t u^i - \mathrm{div} \big( a(|Du|) Du^i \big)= f^i, \qquad i=1,\dots,N, \end{equation*} in a space-time cylinder $Ω_T = Ω\times (0,T)$, where $Ω\subset \mathbb{R}^n$ ($n \geq 2$) is a bounded, convex $C^2$-domain and $T>0$. The inhomogeneity $f=(f^1,\dots,f^N)$ belongs to $L^{n+2+σ}(Ω_T,\mathbb{R}^N)$ for some $σ>0$. The coefficients $a\colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ are of Uhlenbeck-type and satisfy a nonstandard $(p,q)$-growth condition with \[ 2 \leq p \leq q < p + \frac{4}{n+2}. \] Our main result establishes a local Lipschitz estimate up to the lateral boundary for any local weak solution that vanishes on the lateral boundary of the cylinder.

Theorems & Definitions (27)

• Remark 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Definition 2.1
• Remark 2.2
• Lemma 2.3
• proof