Gradient estimates for the fractional $p$-Poisson equation
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909531567554560 |
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| author | Bögelein, Verena Duzaar, Frank Liao, Naian Moring, Kristian |
| author_facet | Bögelein, Verena Duzaar, Frank Liao, Naian Moring, Kristian |
| contents | We consider local weak solutions to the fractional $p$-Poisson equation of order $s$, i.e. $\left( - Δ_p\right)^s u = f$. In the range $p>1$ and $s\in \big(\frac{p-1}{p},1\big)$ we prove Calderón & Zygmund type estimates at the gradient level. More precisely, we show for any $q>1$ that \begin{equation*}
f\in L^{\frac{qp}{p-1}}_{\rm loc}
\quad\Longrightarrow\quad
\nabla u\in L^{qp}_{\rm loc}. \end{equation*} The qualitative result is accompanied by a local quantitative estimate. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_05903 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gradient estimates for the fractional $p$-Poisson equation Bögelein, Verena Duzaar, Frank Liao, Naian Moring, Kristian Analysis of PDEs 35B65, 35J60, 35R09, 47G20 We consider local weak solutions to the fractional $p$-Poisson equation of order $s$, i.e. $\left( - Δ_p\right)^s u = f$. In the range $p>1$ and $s\in \big(\frac{p-1}{p},1\big)$ we prove Calderón & Zygmund type estimates at the gradient level. More precisely, we show for any $q>1$ that \begin{equation*} f\in L^{\frac{qp}{p-1}}_{\rm loc} \quad\Longrightarrow\quad \nabla u\in L^{qp}_{\rm loc}. \end{equation*} The qualitative result is accompanied by a local quantitative estimate. |
| title | Gradient estimates for the fractional $p$-Poisson equation |
| topic | Analysis of PDEs 35B65, 35J60, 35R09, 47G20 |
| url | https://arxiv.org/abs/2503.05903 |