Local Sobolev-Poincare imbedding domains
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916260563910656 |
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| author | Liang, Tian Zhu, Zheng |
| author_facet | Liang, Tian Zhu, Zheng |
| contents | In this article, we study local Sobolev-Poincaré imbedding domains. The main result reads as below. \begin{enumerate} \item for $1\leq p\leq n$, a bounded uniform domain is also a local Sobolev-Poincaré imbedding domain of order $p$; conversely a local Sobolev-Poincaré imbedding domain of order $p$ is locally linearly connected $(LLC)$. A uniform domain is $(LLC)$. Conversely, with some very weak connecting assumption, a $(LLC)$ domain is uniform. \item for $n<p<\fz$, a bounded domain is a local Sobolev-Poincaré imbedding domain of order $p$ if and only if it is an $α$-cigar domain for $α=(p-n)/(p-1)$. Hence, a domain is a local Sobolev-Poincaré imbedding domain of oder $p$ if and only if it is a (global) Sobolev-Poincaré imbedding domain. \end{enumerate} |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_13263 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local Sobolev-Poincare imbedding domains Liang, Tian Zhu, Zheng Functional Analysis In this article, we study local Sobolev-Poincaré imbedding domains. The main result reads as below. \begin{enumerate} \item for $1\leq p\leq n$, a bounded uniform domain is also a local Sobolev-Poincaré imbedding domain of order $p$; conversely a local Sobolev-Poincaré imbedding domain of order $p$ is locally linearly connected $(LLC)$. A uniform domain is $(LLC)$. Conversely, with some very weak connecting assumption, a $(LLC)$ domain is uniform. \item for $n<p<\fz$, a bounded domain is a local Sobolev-Poincaré imbedding domain of order $p$ if and only if it is an $α$-cigar domain for $α=(p-n)/(p-1)$. Hence, a domain is a local Sobolev-Poincaré imbedding domain of oder $p$ if and only if it is a (global) Sobolev-Poincaré imbedding domain. \end{enumerate} |
| title | Local Sobolev-Poincare imbedding domains |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2401.13263 |