Poincaré inequality is stated as follows:



> 
> Let $ \Omega $ be bounded, connected, open subset pf $ \mathbb{R}^n $, with a $ C^1 $ boundary $ \partial \Omega $. Assume that $ 1\leq p<\infty $ and $ u\in W^{1,p}(\Omega) $. Then there exists a constant $ C $, depending only on $ n, p $ and $ \Omega $, such that
> $$
> \left\|u-\frac{1}{|\Omega|}\int\_{\Omega}u\right\|\_{L^p(\Omega)}\leq C\left\|Du\right\|\_{L^p(\Omega)}
> $$
> for each function $ u\in W^{1,p}(\Omega) $.
> 
> 
> 


I have already know the proof from compactness argument. Now I want to prove this inequality by direct computation. First, I consider the condition that $ \Omega $ is convex and prove the inequality. Now I want to deal with the general case by using the extension theorem of Sobolev space. However, I can not deal with the error terms when using the theorem. Can you give me some hints or references?



How to prove Poincaré inequality by using extension theorem?
It seems to me that some useful information you can find in paragraph 1.5 of
"Differentiable Functions on Bad Domains" by V. G. Mazia, S. Pobozchi.


