Lipschitz regularity of fractional $p$-Laplacian
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911115602034688 |
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| author | Biswas, Anup Topp, Erwin |
| author_facet | Biswas, Anup Topp, Erwin |
| contents | In this article, we investigate the Hölder regularity of the fractional $p$-Laplace equation of the form $(-Δ_p)^s u=f$ where $p>1, s\in (0, 1)$ and $f\in L^\infty_{\rm loc}(Ω)$. Specifically, we prove that $u\in C^{0, γ_\circ}_{\rm loc}(Ω)$ for $γ_\circ=\min\{1, \frac{sp}{p-1}\}$, provided that $\frac{sp}{p-1}\neq 1$. In particular, it shows that $u$ is locally Lipschitz for $\frac{sp}{p-1}>1$. Moreover, we show that for $\frac{sp}{p-1}=1$, the solution is locally Lipschitz, provided that $f$ is locally Hölder continuous. Additionally, we discuss further regularity results for the fractional double-phase problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_09457 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lipschitz regularity of fractional $p$-Laplacian Biswas, Anup Topp, Erwin Analysis of PDEs 35B65, 35J70, 35R09 In this article, we investigate the Hölder regularity of the fractional $p$-Laplace equation of the form $(-Δ_p)^s u=f$ where $p>1, s\in (0, 1)$ and $f\in L^\infty_{\rm loc}(Ω)$. Specifically, we prove that $u\in C^{0, γ_\circ}_{\rm loc}(Ω)$ for $γ_\circ=\min\{1, \frac{sp}{p-1}\}$, provided that $\frac{sp}{p-1}\neq 1$. In particular, it shows that $u$ is locally Lipschitz for $\frac{sp}{p-1}>1$. Moreover, we show that for $\frac{sp}{p-1}=1$, the solution is locally Lipschitz, provided that $f$ is locally Hölder continuous. Additionally, we discuss further regularity results for the fractional double-phase problems. |
| title | Lipschitz regularity of fractional $p$-Laplacian |
| topic | Analysis of PDEs 35B65, 35J70, 35R09 |
| url | https://arxiv.org/abs/2504.09457 |