# How to prove Poincaré inequality by using extension theorem?

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?

• Hey! Did you figure out how to use the extension theorem? I am wondering the same question. Jan 23 at 14:24