Generalized vector potential and Trace Theorem for Lipschitz domains
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916239362752512 |
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| author | Liu, Zhen Wu, Jinbiao |
| author_facet | Liu, Zhen Wu, Jinbiao |
| contents | The vector potential is a fundamental concept widely applied across various fields. This paper presents an existence theorem of a vector potential for divergence-free functions in $W^{m,p}(\mathbb{R}^N,\mathbb{T})$ with general $m,p,N$. Based on this theorem, one can establish the space decomposition theorem for functions in $W^{m,p}_0(\operatorname{curl};Ω,\mathbb{R}^N)$ and the trace theorem for functions in $W^{m,p}(Ω)$ within the Lipschitz domain $Ω\subset \mathbb{R}^N$. The methods of proof employed in this paper are straightforward, natural, and consistent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_05228 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized vector potential and Trace Theorem for Lipschitz domains Liu, Zhen Wu, Jinbiao Mathematical Physics Analysis of PDEs Functional Analysis The vector potential is a fundamental concept widely applied across various fields. This paper presents an existence theorem of a vector potential for divergence-free functions in $W^{m,p}(\mathbb{R}^N,\mathbb{T})$ with general $m,p,N$. Based on this theorem, one can establish the space decomposition theorem for functions in $W^{m,p}_0(\operatorname{curl};Ω,\mathbb{R}^N)$ and the trace theorem for functions in $W^{m,p}(Ω)$ within the Lipschitz domain $Ω\subset \mathbb{R}^N$. The methods of proof employed in this paper are straightforward, natural, and consistent. |
| title | Generalized vector potential and Trace Theorem for Lipschitz domains |
| topic | Mathematical Physics Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2405.05228 |