Sobolev regularity for the perturbed fractional 1-Laplace equations in the subquadratic case
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911214179713024 |
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| author | Li, Dingding Zhang, Chao |
| author_facet | Li, Dingding Zhang, Chao |
| contents | This work investigates the Sobolev regularity of solutions to perturbed fractional 1-Laplace equations. Under the assumption that weak solutions are locally bounded, we establish that the regularity properties are analogous to those observed in the superquadratic case. By introducing the threshold $\frac{p-1}{p}$, we divide the range of the parameter $s_p$ into two distinct scenarios. Specifically, for any $s_p\in \left(0, \frac{p-1}{p}\right]$ and $q\ge p$, we demonstrate that the solutions possess $W_{\rm loc}^{γ, q}$-regularity for all $γ\in \left(0, \frac{s_p p}{p-1}\right)$ and the $W_{\rm loc}^{1, q}$-regularity for any $s_p\in \left(\frac{p-1}{p}, 1\right)$ and $q\ge p$, respectively. Our analysis relies on the nonlocal finite-difference quotient method combined with a Moser-type iteration scheme, which provides a systematic approach to the regularity theory for such nonlocal and singular problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_14346 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sobolev regularity for the perturbed fractional 1-Laplace equations in the subquadratic case Li, Dingding Zhang, Chao Analysis of PDEs This work investigates the Sobolev regularity of solutions to perturbed fractional 1-Laplace equations. Under the assumption that weak solutions are locally bounded, we establish that the regularity properties are analogous to those observed in the superquadratic case. By introducing the threshold $\frac{p-1}{p}$, we divide the range of the parameter $s_p$ into two distinct scenarios. Specifically, for any $s_p\in \left(0, \frac{p-1}{p}\right]$ and $q\ge p$, we demonstrate that the solutions possess $W_{\rm loc}^{γ, q}$-regularity for all $γ\in \left(0, \frac{s_p p}{p-1}\right)$ and the $W_{\rm loc}^{1, q}$-regularity for any $s_p\in \left(\frac{p-1}{p}, 1\right)$ and $q\ge p$, respectively. Our analysis relies on the nonlocal finite-difference quotient method combined with a Moser-type iteration scheme, which provides a systematic approach to the regularity theory for such nonlocal and singular problems. |
| title | Sobolev regularity for the perturbed fractional 1-Laplace equations in the subquadratic case |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2510.14346 |