# Extension operator for Lipschitz domain in Sobolev spaces

Suppose we have a Lipschitz domain $\Omega \subset \mathbb{R}^2$. Let $u$ be a function in the Sobolev space $W^{1\,,\,p}(\Omega)$. Since $\Omega$ is Lipschitz, there is an extension operator $P: W^{1\,,\,p}(\Omega) \to W^{1\,,\,p}(\mathbb{R}^2)$ such that $$P \, u \, |_{\Omega}=u \tag{1}$$

Q$1$. Can someone give me a reference for $(1)\,$?

Q$2$. How to extend $u$ in this case?

Let us say that a domain $\Omega \subset \mathbb{R}^n$ is an extension domain if there exists a bounded linear operator $\mathcal{E} \colon W^{1,p}(\Omega) \to W^{1,p}(\mathbb{R}^n)$ such that $\mathcal{E}u(x) = u(x)$ for $x \in \Omega$.
In general, the construction of the extension is done via a local reduction (change of variables that reduced the boundary of $\Omega$ to a particular shape) and a patching by means of partitions of unity. You can find several examples in standard books about Sobolev spaces.