Local Sobolev-Poincare imbedding domains

In this article, we study local Sobolev-Poincar\'e 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-Poincar\'e imbedding domain of order $p$; conversely a local Sobolev-Poincar\'e 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-Poincar\'e imbedding domain of order $p$ if and only if it is an $\alpha$-cigar domain for $\alpha=(p-n)/(p-1)$. Hence, a domain is a local Sobolev-Poincar\'e imbedding domain of oder $p$ if and only if it is a (global) Sobolev-Poincar\'e imbedding domain. \end{enumerate}