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

TL;DR

This paper proves boundary regularity results for solutions to nonlinear parabolic systems with nonstandard growth conditions in smooth convex domains, extending understanding of solution behavior near boundaries.

Contribution

It establishes boundary Lipschitz regularity for solutions to parabolic systems with $(p,q)$-growth conditions, a novel result for such nonstandard growth frameworks.

Findings

Local Lipschitz estimates up to the boundary.

Boundary regularity holds for solutions vanishing on the lateral boundary.

Results apply to systems with nonstandard $(p,q)$-growth conditions.

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 $\Omega_T = \Omega \times (0,T)$, where $\Omega \subset \mathbb{R}^n$ ($n \geq 2$) is a bounded, convex $C^2$-domain and $T>0$. The inhomogeneity $f=(f^1,\dots,f^N)$ belongs to $L^{n+2+\sigma}(\Omega_T,\mathbb{R}^N)$ for some $\sigma>0$. 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.