Schauder estimates for parabolic $p$-Laplace systems
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911067704131584 |
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| author | Bögelein, Verena Duzaar, Frank Gianazza, Ugo Liao, Naian Scheven, Christoph |
| author_facet | Bögelein, Verena Duzaar, Frank Gianazza, Ugo Liao, Naian Scheven, Christoph |
| contents | We establish the local Hölder regularity of the spatial gradient of bounded weak solutions $u\colon E_T\to\R^k$ to the non-linear system of parabolic type \begin{equation*}
\partial_tu-\Div\Big( a(x,t)\big(μ^2+|Du|^2\big)^\frac{p-2}2Du\Big)=0
\qquad\mbox{in $E_T$}, \end{equation*} where $p>1$, $μ\in[0,1]$, and the coefficient $a\in L^\infty(E_T)$ is bounded below by a positive constant and is Hölder continuous in the space variable $x$. As an application, we prove Hölder estimates for the gradient of weak solutions to a doubly non-linear parabolic equation in the super-critical fast diffusion regime. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_15722 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Schauder estimates for parabolic $p$-Laplace systems Bögelein, Verena Duzaar, Frank Gianazza, Ugo Liao, Naian Scheven, Christoph Analysis of PDEs 35K65, 35K67, 35B45, 35B65, 35K92, 76S05 We establish the local Hölder regularity of the spatial gradient of bounded weak solutions $u\colon E_T\to\R^k$ to the non-linear system of parabolic type \begin{equation*} \partial_tu-\Div\Big( a(x,t)\big(μ^2+|Du|^2\big)^\frac{p-2}2Du\Big)=0 \qquad\mbox{in $E_T$}, \end{equation*} where $p>1$, $μ\in[0,1]$, and the coefficient $a\in L^\infty(E_T)$ is bounded below by a positive constant and is Hölder continuous in the space variable $x$. As an application, we prove Hölder estimates for the gradient of weak solutions to a doubly non-linear parabolic equation in the super-critical fast diffusion regime. |
| title | Schauder estimates for parabolic $p$-Laplace systems |
| topic | Analysis of PDEs 35K65, 35K67, 35B45, 35B65, 35K92, 76S05 |
| url | https://arxiv.org/abs/2507.15722 |