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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.19834 |
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| _version_ | 1866915288107188224 |
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| author | Kangasniemi, Ilmari |
| author_facet | Kangasniemi, Ilmari |
| contents | Given a bounded domain $Ω\subset \mathbb{R}^n$, a result by Bourgain, Brezis, and Mironescu characterizes when a function $f \in L^p(Ω)$ is in the Sobolev space $W^{1,p}(Ω)$ based on the limiting behavior of its Besov seminorms. We prove a direct analogue of this result which characterizes when a differential $k$-form $ω\in L^p(\wedge^k T^* Ω)$ has a weak exterior derivative $dω\in L^p(\wedge^{k+1} T^* Ω)$, where the analogue of the Besov seminorm that our result uses is based on integration over simplices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_19834 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Bourgain-Brezis-Mironescu -type characterization for Sobolev differential forms Kangasniemi, Ilmari Analysis of PDEs Differential Geometry Functional Analysis 46E35 (Primary) 53C65 (Secondary) Given a bounded domain $Ω\subset \mathbb{R}^n$, a result by Bourgain, Brezis, and Mironescu characterizes when a function $f \in L^p(Ω)$ is in the Sobolev space $W^{1,p}(Ω)$ based on the limiting behavior of its Besov seminorms. We prove a direct analogue of this result which characterizes when a differential $k$-form $ω\in L^p(\wedge^k T^* Ω)$ has a weak exterior derivative $dω\in L^p(\wedge^{k+1} T^* Ω)$, where the analogue of the Besov seminorm that our result uses is based on integration over simplices. |
| title | A Bourgain-Brezis-Mironescu -type characterization for Sobolev differential forms |
| topic | Analysis of PDEs Differential Geometry Functional Analysis 46E35 (Primary) 53C65 (Secondary) |
| url | https://arxiv.org/abs/2406.19834 |