Boundary regularity for parabolic systems with nonstandard $(p,q)$-growth conditions in smooth convex domains
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| Format: | Preprint |
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2026
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| _version_ | 1866914275363127296 |
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| author | Strunk, Michael |
| author_facet | Strunk, Michael |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_16546 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Boundary regularity for parabolic systems with nonstandard $(p,q)$-growth conditions in smooth convex domains Strunk, Michael Analysis of PDEs 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. |
| title | Boundary regularity for parabolic systems with nonstandard $(p,q)$-growth conditions in smooth convex domains |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2601.16546 |