Local Sobolev-Poincare imbedding domains

In this article, we study local Sobolev-Poincaré imbedding domains. The main result reads as below. \begin{enumerate} \item for $1\le p\le n$, 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 $n<p<\text{fz}$, a bounded domain is a local Sobolev-Poincaré imbedding domain of order $p$ if and only if it is an $\alpha$-cigar domain for $\alpha =(p-n)\mathrm{/}(p-1)$. 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}