Regularity theory for sub-critical $p$-parabolic systems with measurable coefficients
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866909988814848000 |
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| author | Bögelein, Verena Duzaar, Frank Gianazza, Ugo Liao, Naian |
| author_facet | Bögelein, Verena Duzaar, Frank Gianazza, Ugo Liao, Naian |
| contents | A quantitative regularity theory is developed for weak solutions to the parabolic system $$ \partial_t u-\mathrm{div}\,{\boldsymbol{\mathsf A}}(x,t,Du)=0 \quad\text{in }E_T\subset \mathbb{R}^N\times\mathbb{R}, $$ which features the $p$-Laplacian with measurable coefficients. We focus on the sub-critical range $1<p\le \tfrac{2N}{N+2}$ and obtain two main results. \emph{Local boundedness:} starting from an $L^{\boldsymbol{\mathsf r}}$-control of $u$ with ${\boldsymbol{\mathsf r}}>\frac{N(2-p)}{p}$, we derive sharp, scale-invariant $L^\infty$-estimates. \emph{Higher integrability of the gradient:} $|Du|$ self-improves from $L^p_{\mathrm{loc}}$ to $L^{p(1+\varepsilon)}_{\mathrm{loc}}$ for some $\varepsilon>0$ depending only on the data. The same results still hold given proper source terms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_08466 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Regularity theory for sub-critical $p$-parabolic systems with measurable coefficients Bögelein, Verena Duzaar, Frank Gianazza, Ugo Liao, Naian Analysis of PDEs 35B65, 35K67, 35K40, 35K55 A quantitative regularity theory is developed for weak solutions to the parabolic system $$ \partial_t u-\mathrm{div}\,{\boldsymbol{\mathsf A}}(x,t,Du)=0 \quad\text{in }E_T\subset \mathbb{R}^N\times\mathbb{R}, $$ which features the $p$-Laplacian with measurable coefficients. We focus on the sub-critical range $1<p\le \tfrac{2N}{N+2}$ and obtain two main results. \emph{Local boundedness:} starting from an $L^{\boldsymbol{\mathsf r}}$-control of $u$ with ${\boldsymbol{\mathsf r}}>\frac{N(2-p)}{p}$, we derive sharp, scale-invariant $L^\infty$-estimates. \emph{Higher integrability of the gradient:} $|Du|$ self-improves from $L^p_{\mathrm{loc}}$ to $L^{p(1+\varepsilon)}_{\mathrm{loc}}$ for some $\varepsilon>0$ depending only on the data. The same results still hold given proper source terms. |
| title | Regularity theory for sub-critical $p$-parabolic systems with measurable coefficients |
| topic | Analysis of PDEs 35B65, 35K67, 35K40, 35K55 |
| url | https://arxiv.org/abs/2601.08466 |