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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.13263 |
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Table of 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}